text:
Simply add the `manifold-exceptions` dependency to your project. Now checked exceptions behave like unchecked
exceptions!  No more compiler errors, no more unintentional exception swallowing, no more boilerplate
try/catch/wrap/rethrow nonsense. This is how most modern JVM languages behave, now you have the option to make Java do
the same.

## Table of Contents
* [No More Catch-n-Wrap](#no-more-catch-n-wrap)
* [Lambdas](#lambdas)
* [IDE Support](#ide-support)
* [Setup](#setup)
* [Javadoc](#javadoc)
* [License](#license)
* [Versioning](#versioning)
* [Author](#author)

## No More Catch-n-Wrap
The vast majority of checked exceptions go unhandled, instead they are caught, wrapped in unchecked exceptions, and
rethrown like this:
```java
URL url;
try {
  url = new URL("http://manifold.systems");
}
catch(MalformedURLException e) {
  throw new RuntimeException(e);
}
process(url);
```
This code alone explains why the designers of modern languages such as Scala, Kotlin, and others chose not to follow
Java's example.  The `exceptions` plugin option provides you with the same choice.  With it enabled you can write the
same code like this:
```java
process(new URL("http://manifold.systems"));
```
Sells itself.

## Lambdas
Perhaps the most offensive checked exception use-cases involve lambdas:
```java
List<String> strings = ...;
List<URL> urls = strings.stream()
  .map(URL::new) // Unhandled exception error: MalformedURLException
  .collect(Collectors.toList());
```
The checked exception prevents concise lambda usage here.  With Manifold, however, you are free to write code as you
like:
```java
List<String> strings = ...;
List<URL> urls = strings.stream()
  .map(URL::new) // Mmm, life is good
  .collect(Collectors.toList());
```

# IDE Support

Manifold is fully supported in [IntelliJ IDEA](https://www.jetbrains.com/idea/download) and [Android Studio](https://developer.android.com/studio).

## Install

Get the [Manifold plugin](https://plugins.jetbrains.com/plugin/10057-manifold) directly from within the IDE via:

Settings ➜ Plugins ➜ Marketplace ➜ search: `Manifold`

## Sample Project

Experiment with the [Manifold Sample Project](https://github.com/manifold-systems/manifold-sample-project) via:

File ➜ New ➜ Project from Version Control ➜ Git

Enter: https://github.com/manifold-systems/manifold-sample-project.git

Use the [plugin](https://plugins.jetbrains.com/plugin/10057-manifold) to really boost your productivity. With the plugin
installed IntelliJ is aware of all things Manifold, including checked exception handling e.g., the editor won't complain
about checked exceptions.

# Setup

## Building this project

The `manifold-exceptions` project is defined with Maven.  To build it install Maven and run the following command.

```
mvn compile
```

## Using this project

The `manifold-exceptions` dependency works with all build tooling, including Maven and Gradle. It also works with Java
versions 8 - 19.

## Binaries

If you are *not* using Maven or Gradle, you can download the latest binaries [here](http://manifold.systems/docs.html#download).

## Gradle

>Note, if you are targeting **Android**, please see the [Android](http://manifold.systems/android.html) docs.

Here is a sample `build.gradle` script. Change `targetCompatibility` and `sourceCompatibility` to your desired Java
version (8 - 19), the script takes care of the rest.
```groovy
plugins {
    id 'java'
}

group 'systems.manifold'
version '1.0-SNAPSHOT'

targetCompatibility = 11
sourceCompatibility = 11

repositories {
    jcenter()
    maven { url 'https://oss.sonatype.org/content/repositories/snapshots/' }
}

dependencies {
    compileOnly 'systems.manifold:manifold-rt:2022.1.27'
    testImplementation 'junit:junit:4.12'
    // Add manifold to -processorpath for javac
    annotationProcessor group: 'systems.manifold', name: 'manifold-exceptions', version: '2022.1.27'
    testAnnotationProcessor group: 'systems.manifold', name: 'manifold-exceptions', version: '2022.1.27'
}

if (JavaVersion.current() != JavaVersion.VERSION_1_8 &&
    sourceSets.main.allJava.files.any {it.name == "module-info.java"}) {
    tasks.withType(JavaCompile) {
        // if you DO define a module-info.java file:
        options.compilerArgs += ['-Xplugin:Manifold', '--module-path', it.classpath.asPath]
    }
} else {
    tasks.withType(JavaCompile) {
        // If you DO NOT define a module-info.java file:
        options.compilerArgs += ['-Xplugin:Manifold']
    }
}
```
Use with accompanying `settings.gradle` file:
```groovy
rootProject.name = 'MyProject'
```

## Maven

```xml
<?xml version="1.0" encoding="UTF-8"?>
<project xmlns="http://maven.apache.org/POM/4.0.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://maven.apache.org/POM/4.0.0 http://maven.apache.org/maven-v4_0_0.xsd">
    <modelVersion>4.0.0</modelVersion>

    <groupId>com.example</groupId>
    <artifactId>my-exceptions-app</artifactId>
    <version>0.1-SNAPSHOT</version>

    <name>My Exceptions App</name>

    <properties>
        <!-- set latest manifold version here -->
        <manifold.version>2022.1.27</manifold.version>
    </properties>

    <dependencies>
        <dependency>
            <!-- Necessary only during compile-time to resolve generated source-level annotations -->
            <groupId>systems.manifold</groupId>
            <artifactId>manifold-rt</artifactId>
            <version>${manifold.version}</version>
            <scope>provided</scope> <!-- dependency is only applied during compile-time for manifold-exceptions -->
        </dependency>
    </dependencies>

    <!--Add the -Xplugin:Manifold argument for the javac compiler-->
    <build>
        <plugins>
            <plugin>
                <groupId>org.apache.maven.plugins</groupId>
                <artifactId>maven-compiler-plugin</artifactId>
                <version>3.8.0</version>
                <configuration>
                    <source>11</source>
                    <target>11</target>
                    <encoding>UTF-8</encoding>
                    <compilerArgs>
                        <!-- Configure manifold plugin-->
                        <arg>-Xplugin:Manifold</arg>
                    </compilerArgs>
                    <!-- Add the processor path for the plugin -->
                    <annotationProcessorPaths>
                        <path>
                            <groupId>systems.manifold</groupId>
                            <artifactId>manifold-exceptions</artifactId>
                            <version>${manifold.version}</version>
                        </path>
                    </annotationProcessorPaths>
                </configuration>
            </plugin>
        </plugins>
    </build>
</project>
```

# Javadoc

`manifold-exceptions`:

[![javadoc](https://javadoc.io/badge2/systems.manifold/manifold-exceptions/2022.1.27/javadoc.svg)](https://javadoc.io/doc/systems.manifold/manifold-exceptions/2022.1.27)

# License

Open source Manifold is free and licensed under the [Apache 2.0](http://www.apache.org/licenses/LICENSE-2.0) license.

# Versioning

For the versions available, see the [tags on this repository](https://github.com/manifold-systems/manifold/tags).

# Author

* [Scott McKinney](mailto:scott@manifold.systems)


The below content is from social media taken from external sites and is not representative of the views of businessagent.com or Upgrade Pack.
DocuChain's 4Trust Platform connects the physical and virtual worlds in a secure, permissioned doman. It allows users to define, archive and authenticate business performance processes, related data and participating people in real time in Linux Foundation HyperLedger blockchains. For DocuChain, artifacts and annual licence fees, for every chain of custody, could provide the opportunity for recurring business to business revenue. DocuChain had previous discussions with 5 global organisations and aims to meet the investment and operational necessities of markets including aviation, manufacturing, luxury goods, financial services and government documents.


PME Bright White Candy buttons are easy to melt and re-melt! They have great taste, smooth texture and are microwaveable.
They are used for moulding, dipping and drizzling and covering. Use to cover cakepops, cookies etc.
Ingredients: Sugar* Vegetable oils (palm kernels,Hydrogenated palm kernel), Whole MILK powder, whey powder (MILK), skimmed MILK powder, Emulsifier (SOYA* lecithin), salt, Flavour. * GENETICALLY MODIFIED.
May contain amounts of peanuts due to common processing equipment.


Walsall will finish as group winners if they avoid defeat against a Leicester Under-21s side who - like the Saddlers - are already through to the knock-out stages of the Checkatrade Trophy. Jon Whitney's men sit top of Northern Group H with six points from a possible six, after 5-2 and 2-1 victories over Grimsby and Sheffield United respectively. The young Foxes, meanwhile, lie a point behind the Saddlers in second place. They drew their first match 0-0 at Bramall Lane but won the subsequent penalty shoot-out 5-4 to gain an extra point, before beating Grimsby 1-0 to seal their place in the last 32. The team that finishes top of the group will be guaranteed a home tie in the next round. Grimsby and Sheff Utd – who meet tomorrow night in the final group fixture – are both out.
Kairo Mitchell scored the only goal as Leicester Under-21s beat ten-man Grimsby to qualify for the knock-out stages of the Checkatrade Trophy. Mariners centre-back Shaun Pearson played in goal for 74 minutes following the dismissal of goalkeeper James McKeown for deliberate handball. The young Foxes' winner came on 63 minutes, Mitchell latching onto Raul Uche's pass and bundling the ball home.
Walsall booked their place in the second round of the Checkatade Trophy with victory at Sheffield United. Scott Laird's 16th-minute tap-in put the Saddlers ahead before Amadou Bakayoko fired home a second on 66 minutes. Paul Coutts pulled one back for United late on, but Jon Whitney's side held on to make it back-to-back wins in Northern Group H.
The Foxes' Under-21s are managed by Steve Beaglehole, whose official title is Senior Professional Development Coach. He spent two years in charge of his hometown team Doncaster in the early 1990s, and has also held coaching positions at Middlesbrough, Nottingham Forest and Leeds.
The powerful frontman has scored Leicester Under-21s' only goal in the competition to date – a scruffy 63rd-minute winner at Grimsby back in October. Mitchell enjoyed a productive 2015/16 with the Foxes' Under-18s and has made several appearances in Premier League 2, as well as the Hong Kong Soccer Sevens.
Manager Jon Whitney is expected to give a number of his younger players a chance to impress. Flo Cuvelier, Maz Kouhyar, Reece Flanagan and Rory Oliver remain on the sidelines.
The West Yorkshire-based official has taken charge of 13 matches this season, including Leicester U21s' 1-0 win at Grimsby on matchday two of the Checkatrade Trophy. He has brandished 49 yellow cards and three red, while the last Walsall game he officiated was our 3-2 win at Bury back in January.


With our world facing new and old systemic challenges, there is a need for coherent, cooperative, and healing ways to imagine, generate, and shape new possibilities. "The Hummingbird Cricket Hour" draws on ancestral knowledge and mythological imaginations, mostly from the first peoples of the continents of this hemisphere. Hosts Luis and Trini Rodriguez explore ideas, propositions, and organizing principles for a world trying to be born. Starting from the archetypal concept that the medicine is already inside of us, Luis and Trini will have discussions with each other and invited guests about the current state of culture and society, as well as what gives life, meaning, and vitality to the beauty, truth, and systemic change needed today.
With Patreon support, "The Hummingbird Cricket Hour" creates podcasts that inspire conversation across a global community. The intention is to move ideas and imaginations, thoughts and hearts, and align understandings to the immense possibilities everywhere for a peaceful, just, clean & green, abundant, and transparent economically moral world.
Trini is a daughter of Mexican migrants who raised her in working class Pacoima, CA. She has been a teacher, writer, editor, court interpreter,and community arts leader who grew up in Los Angeles, lived in Chicago for 17 years, then returned to L.A. with her husband Luis and two sons. She then co-founded and directed Tia Chucha's Centro Cultural & Bookstore for 17 years in the Northeast San Fernando Valley area of the city. Twenty years ago she was spiritually adopted by Anthony and Delores Lee, Dine elders on the Navajo reservation in Arizona and has been taught in Native American and Mexican spiritual practices. She is a waterpourer at the Hummingbird Sweat Lodge and women's healing circle in the community of Sylmar.
Luis is a writer as well as a community and urban peace activist, mentor, healer, youth and arts activist. Luis has been involved in Native American and Mexican spiritual practices for close to 25 years among the Mexica, Lakota, Dine, Mayan, and Quechua people in Pine Ridge, the Navajo Nation, Mexico, Guatemala, and Peru. He is also a bestselling author of the memoir "Always Running, La Vida Loca, Gang Days in L.A." and has over 15 other books in poetry, children's literature, fiction, and nonfiction. He is co-founder of Tia Chucha's Centro Cultural & Bookstore, and has been founding editor of the independent cross-cultural press, Tia Chucha Press, for close to 30 years. From 2014-2016, he served as Los Angeles's Poet Laureate.
Together Luis and Trini have had a positive impact in the various communities they have lived in. Their contributions have been sought across the country and other parts of the world, most recently as a couple in Germany, Honduras, and Spain.
We thank you for supporting the Hummingbird Cricket Hour. You will receive access to the patron-only feed, in which you may ask questions, voice concerns, and suggest future topics and guests!
We thank you for supporting the Hummingbird Cricket Hour. In addition to the lower tier rewards, you will receive behind-the-scenes content, including exclusive episodes!
We thank you for supporting the Hummingbird Cricket Hour. In addition to the lower tier rewards, you will receive a single autographed book by Luis J. Rodriguez with our deepest appreciation for your contribution and support!
Upgrade the audio quality of the podcast with better microphones and sound-proofing!


Busbud shows you the best bus ticket fares and bus schedules so you can easily plan and book a trip by bus from Phoenix to Toronto.
Bus service from Phoenix to Toronto will be provided by the most trusted bus companies.
Whether it is cheap bus tickets or more luxurious buses going to Toronto from Phoenix, we help you find what you're looking for based on how much of a budget you have.
Bus service from Phoenix to Toronto is covered by Greyhound.
Buses leaving from Phoenix to Toronto will depart from Downtown Greyhound Station, Tufesa Station - N 27th Ave, Glendale Ave Greyhound Station, PHX Airport Sky Train, Desert Ridge Mellow Mushroom or North Phoenix.
A bus ride to Toronto will drop you off at Scarborough Town Centre, Toronto Coach Terminal, Yorkdale Bus Terminal, University Ave and Wellington Street, Front St & York St, Bay St, Black Creek Village, Don Valley Parkway or Pearson YYZ Airport.
If you're abroad, study some local language. Not only could it get you out of a pickle during your bus trip from Phoenix to Toronto but it'll also be endearing to native speakers who will surely appreciate your effort.
Much more than a mere vehicle taking you from Phoenix to Toronto, riding the bus will expose you to some of most stunning panoramas while enroute to your final bus stop.
Buses are energy-efficient. Carrying a passenger over 100 kms by coach only takes 0.6-0.9 liters of gas. Compare that to the 2.6 liters required by high-speed train, 6.6 liters by airplane and 7.6 liters by gas-powered car, and it's clear that the bus is a more environmentally-conscious option for your bus transportation from Phoenix to Toronto.
Make a music playlist for your bus ride from Phoenix to Toronto, and enjoy the scenery to tunes from your own personal soundtrack.
Will you be returning to Phoenix by bus?
If you are looking for the return bus journey, check out buses from Toronto to Phoenix.


text:
Guest post: A smile and legacy to bridge the political divide
A tribute to Hunter Goodman
Dan Sytman, with introduction by Rob McKenna	April 19, 2019
Hunter Goodman, Secretary of the State Senate from 2013-2017, served as Government Affairs Director for the Attorney General's Office from 2005-2013. Hunter was an outstanding public servant and exceptional person.
Dan Sytman, who served on the AGO's communications team, has written a fantastic tribute to Hunter, with the assistance of Chris Johnson and Maureen Scharber from our policy and communications teams. My thanks to Washington State Wire for sharing Dan's tribute here.
— Rob McKenna, former Washington State Attorney General
Amidst the political strife afflicting the United States, hope is found in the story of a man who knew how to cure conflict with friendship, a smile, and a sincere desire to bring people together. Unfortunately, the barrage of daily headlines about partisan conflict have left untold the many stories of those who toil behind the scenes in state politics, in good faith and with great personal character. Among them is the story of my friend, Hunter Goodman, the former Secretary of the Washington State Senate. Hunter passed away on April 17, leaving a great many friends and colleagues from all political sides to ponder the profound impact he had on each of us, and what life will be like without him.
We are in a moment of political mistrust that poisons our ability to work together on common problems. Yet there are so many lessons we can glean from Hunter's life and work. By sharing his story, I hope we can each do our part to advance his remarkable legacy of public service.
I arrived at the Washington state capitol in the spring of 2008, as a senior communications manager for Washington State Attorney General Rob McKenna. Along with a box of family photos, I brought several preconceived notions about what I was walking into. My expectation was that McKenna faced tough odds as a Republican serving in the environment in which Democrats controlled both chambers of the Legislature and the Governor's mansion. In such a challenging climate, it was not unrealistic to imagine little would get done. Thankfully, nothing was further from the truth.
McKenna worked, with terrific success, in close collaboration with all state government entities. Among the centerpieces of his leadership was expanding the office's role in not just carrying out its duties in providing legal counsel to state agencies and representing state government in court, but in modifying our laws to better serve the public. In order to do that, McKenna appointed Hunter as his Director of Government Affairs, and tasked him with uniting lawmakers to pass laws to protect consumers, victims of human trafficking and domestic violence, and more. I was incredibly fortunate to work with Hunter, a K-12 classmate from Mercer Island, Washington. I was awestruck on a regular basis – yet never surprised – by the positive influence he wielded in state government.
By the time McKenna's two, four-year terms were over, the Washington State Attorney General's Office had written and persuaded elected officials to pass a whopping 45 bills, all with strong support from both parties, and in many cases nearly unanimous support.
Left to right: Chris Johnson, AGO Policy and Gov't Affairs Specialist Rebecca Podszus, and Hunter, in 2008.
Hunter worked with a small staff that included his long-time friend and Eagle Scout, Chris Johnson, who served as Policy Director. Together, they helped Attorney General McKenna obtain funding to expand the Attorney General's Consumer Protection Division that shields consumers – with an emphasis on seniors and those in lower-income groups – from unfair and deceptive business practices. With identity theft on the rise, they passed a law — which was later replicated around the country — to require companies to pay for free credit monitoring when customers are the victims of data breaches. They passed a groundbreaking bill to enact tough penalties for those who install spyware on consumers' computers. And since the Attorney General is also the state's chief law enforcement officer, McKenna, Hunter, and team worked with legislators to reform the sexually violent predator statute to better shield the public from dangerous criminals about to be released from prison.
On a national level, again working through Hunter, Johnson and others, McKenna helped Washington State become a leader in the fight against human trafficking. These are just a few examples of what a Republican Attorney General, who charged his government affairs director to work with a Legislature and Governor's Office controlled by the opposing party, was able to achieve.
Act two of Hunter's contributions to the state of Washington was perhaps more dramatic and inspiring, as it took place as he was battling cancer. In January 2013, as we looked on with pride, Hunter was appointed by unanimous vote to be the Secretary of the Senate. He was re-appointed, again by unanimous vote, in 2016. The Secretary of the Senate serves as that chamber's CEO, responsible for everything required to ensure it functions properly. But Hunter did more than that. As a seasoned policy expert, he helped state senators from both parties work together to enact important reforms. In 2015, the Legislature invested over $1 billion in additional funds for public education, without a tax increase. In the same year, the Senate led the charge to cut college tuition, bucking a long national trend of annual increases. A national Associated Press story reported that "A decision this week to cut tuition for Washington state's public universities by 15 to 20 percent over the next two years is a rare move that national experts believe could influence other states as they come out from under the recession." The Senate and House also passed in 2015, and the Governor signed, a $16 billion transportation package that was years in the making.
Gov. Jay Inslee, Hunter, Grayson and Sarah Goodman.
It's no coincidence that Hunter was present during this time of intense cooperation between the two political parties, and a simple question logically flows from his work: how did he help pull it off, and how can we emulate his approach to address important issues outside of the state of Washington? One key is to hear from the lawmakers who worked with Hunter. I did that in 2015, in the process of nominating him for an award. Their responses were, until now, never published.
"Hunter Goodman is absolutely, without question, one of the best human beings I have had the pleasure of working with over a long career of working with exceptional people," said King Council Council member Jeanne Kohl-Welles, a Democrat who represents one of Seattle's most liberal districts. "His engaging personality, his earnest interest in others, and his deep commitment to truth and fairness define him and make him stand out like a blooming rose in a field of daisies."
"Hunter Goodman has been an outstanding public servant that is respected by people from both parties," said Sen. Andy Hill, a Republican who, before his death in 2016, represented communities in and around Redmond, Washington. "He has the ability to forge relationships with just about anyone, usually via a smile and a joke."
In a remarkable moment last weekend, State Senators honored Hunter's contributions.
"He would insert himself in a conflict in order to defuse the conflict and prevent it from going further," Sen. Ann Rivers remarked on the Senate floor, revealing one of the secrets of Hunter's success. "He took on the weight of the issues on his broad shoulders."
Having heard from lawmakers and having had the honor to work with Hunter for five years, here are the lessons I draw from his approach to working across party lines:
1. First connect on a human level: Hunter was helped in this regard because he was the kind of person who simply likes people. For Hunter, it was easy to establish relationships with lawmakers and their staff members because he was genuinely interested in them. We should all take a cue from his example. Each person, regardless of political views, has a family, friends, hobbies, likes and dislikes. Learn about the person first, and you are likely to uncover things you admire and have in common. Politics come second, or maybe even third, after you talk about your children, an ailing parent, a planned vacation or favorite sports team. I would argue that these are things that define us, not our political views, which are often inherited or a function of where we live.
2. Listen, compromise, and share the credit: Hunter spent a lot of time listening to lawmakers. He understood that each has constituents to serve, and just like any of us, each worries about doing a good job. Hunter appreciated that, even visiting the lawmakers in their home districts prior to the legislative session while serving as the AG's Government Affairs Director (an annual pilgrimage Hunter's colleagues dubbed "Hunter-Palooza"). He took what he learned and adjusted his approach based on what lawmakers felt they needed to get done. That's how he spurred forward momentum and shared accomplishments built on unity and compromise. To the naked eye, such compromise could have been perceived as going against one of Hunter's core values. After all, you couldn't know Hunter without knowing that he attended school, and played football, at the University of Southern California. The battle-cry of Hunter's Trojans, a phrase he inserted into most conversations, is "fight on." But the truth is Hunter never came to work looking for a fight. Instead he sought, and achieved, mutually beneficial outcomes. For example, when we announced our legislative agenda at the Attorney General's Office, we invited dozens of lawmakers to participate in media engagements. On the urging of Hunter and others, Communications Director Janelle Guthrie informed me that we were to carefully ensure that all sponsors of our bills were listed in our press materials. We offered each of them quotes in those materials, which we provided to reporters as a kind of appendix.Many of us forget, especially in national politics, that nothing is lost, and there is everything to gain, from sharing the credit.
3. Character is the key: In politics, like in other elements of life, negotiations are best conducted within sphere of good faith. Elected officials, and their staff members always knew Hunter was a person of character – someone they could trust. While they didn't explore his personal background, they likely wouldn't have been surprised to learn that Hunter was Chair of the board for Mercer Island Youth and Family Services, which provides services to lower-income youth and seniors, when he was just 15 years-old, and through the end of high school. He served on the Board of Directors of the Mercer Island Boys and Girls Club during the same approximate time period. And he was the Mercer Island Club's Youth of the Year during his senior year of high school. That distinction goes to a student who goes above and beyond in lending a hand to younger kids in need of guidance, and in serving the larger community. Hunter's concern for others remained a key element of his life. While struggling through his own debilitating cancer treatments, yet still carrying out his duties in the Senate, he worked with the Legislature to fund a new blood-donation-mobile for Bloodworks Northwest, which supplies lifesaving fluids to 90 northwest hospitals. He also helped secure nearly $1.5 million in state funding for Cancer Pathways, a nonprofit that provides services for those struggling with cancer and their families. Those services include Camp Sparkle, a week-long summer camp for children impacted by cancer.
In part because of the elements of personal character they shared, Hunter's long association with Rob McKenna was a natural partnership. Like Hunter, Rob's personal dedication to nonprofits, including but not limited to those committed to education and domestic violence survivors, stretches back to his teens. And like Hunter, his everyday conduct demonstrates deep compassion. I once stood by as Rob, after a 16-hour day on the road, and with his dinner getting cold at the kitchen table and a legitimately impatient spouse waiting, conducted some final business for the day with our driver. He stood in front of a mirror helping the young intern, who had lost his father at a young age, learn to how to tie a tie. The lesson I draw from Hunter's service, and from his longtime association with Rob, the Eagle Scout, is that personal character is essential to good faith negotiations on behalf of constituents. I believe we need to re-tune our priorities to seek more leaders who exhibit it. When judging a candidate, remember that personal character is exhibited by a clear record good conduct, often reflected in personal interactions and volunteer service over a number of years. This kind of conduct should be given a higher priority than party affiliation, because it's the foundation for good-faith negotiation and compromise, which our nation desperately needs if we are to escape our present state mistrust and conflict.
From left, Luke Esser, Hunter Goodman, Rob McKenna, and Chris Johnson, who started working together in the mid-1990s. Photo credit: Luke Esser.
We live in a divided country. Let's bridge the chasm by heeding Hunter's advice, never spoken but always modeled. If we are to "fight on," perhaps the real fight is with our own worst impulses, such as leaping to assume the worst intentions of our political counterparts. Hunter provided a different path, and we'd be wise to follow.
In Hunter's honor, please donate to Cancer Pathways, and give blood and plasma to Bloodworks Northwest.
Hunter, with Grayson Goodman, in the State Senate.
All photography in this piece, of the State Senate, is credited to the Washington State Republican Senate Caucus.
This tribute was posted with the permission of Dan Sytman, who originally published the piece here.
Cottage food product labeling requirements might change
Madeline Shannon	January 17, 2020
Advocates, opponents turn out for sex education bill
State Treasurer releases new study on the state's debt
Office of State Treasurer Duane A. Davidson	January 16, 2020
Senate committee considers sexual offender registration waiver
Gov. delivers State of the State
2020 session starts amid rallies on farming, environment

redpajama_set_name: RedPajamaCommonCrawl


text:
Summary of the Kingdom Animalia
By Jasey Kelly
By far the most diverse kingdom in the scientific classification system, Animalia contains all types of creatures from the slimy earthworm to humans. The classification system arranges organisms in the following order: kingdom, phylum, class, order, family, genus and species. As the classification goes down from kingdom, the creatures are more closely related; for example, species within the same genus are more closely related to each other than to those of other genera.
Animalia, also known as Metazoa, is the most diverse kingdom with an estimated nine to ten million different species inhabiting the earth. Some of the differences between Animalia and the other kingdoms is that most animals produce sexually; have an interior digestive system and that they're multi-cellular unlike the uni-cellular protozoans.
The vast differences in the animal kingdom also make it unique among the other classification kingdoms. Animals range from the microscopic creatures consisting of only a few cells to the largest living animal, the blue whale, which weighs over 20 tons on average. The most diverse and largest class, Insecta, contains around 1 million named species and an estimate of several times that number that remain undiscovered.
Most members of Animalia reproduce sexually by means of sperm and eggs. While the process can vary between different types of animals, most share the same development sequence of an embryo developing after fertilization. Embryo development is either oviparous, in which the embryo develops externally; ovoviparous, in which the embryo develops internally but the mother does not contribute nutrition; and viviparous, in which the embryo develops internally and the mother contributes nutrients to the embryo. Some insects store sperm for later use; some types of animals fertilize the eggs after they've left the female's body.
Heterotroph and Digestion
Animals are heterotrophs, which means an organism that relies on other organisms for food. In short, animals eat other organisms to stay alive. Most of the creatures in the kingdom Animalia ingest food through a mouth and digest it inside their bodies. For example, food goes in the body through the mouth and then begins being digested through the stomach and the digestive tract.
University of Michigan Animal Diversity Web: Animalia
Sidwell Friends School: Kingdom Animalia
Wayne's Word: Major Phyla of Animals
With a professional background in gardening, landscapes, pests and natural ecosystems, Jasey Kelly has been sharing her knowledge through writing since 2009 and has served as an expert writer in these fields. Kelly's background also includes childcare, and animal rescue and care.
How Do Armadillos Reproduce?
Relatives of the Platypus
Is a Starfish a Mollusk?
What Are the Main Characteristics of Nematoda?
This is How Long It Takes Most Mammals To Poop
Characteristics of Rotifera
Life Cycle of a Nematode
List of Facts About Kingdom Animalia
Information on the Reproductive System of a Koala Bear
The Classification of Ladybugs
What Mammals Have the Biggest Brains?
What Kind of Animal Is a Honey Badger?
What Is Kingdom Animalia?
Why Are Arthropods Important to Humans?
What is the Basic Body Plan of an Annelid?
Do Snakes Have Wombs?
How Do Tapeworms Obtain Food Without a Digestive System?


text:
require 'ngin_http'

class PlayerStat < NginHttp::Resource

	field :player_name
	field :goals

	 def self.get_stats
  	build_request("http://localhost:3000/stats/2")
  end

end

redpajama_set_name: RedPajamaGithub


text:
\section{ }
\subsection{ }
\subsubsection{ }

See kbsjrnl.ins for documentation on using Bibtex,
also on making an alphabetical reference section using Bibtex.

\section{Introduction}

Consider the problems ``What are appropriate distortion measures for
images of handwritten characters, or images of faces, or
representations of speech signals?'' Simple measures such as squared
Euclidean distance, while widely used in vector quantisation
applications\cite{Graytext}, do not correlate well with our own
subjective notion of distance in these problems.  For example, an
image of an ``A'' and a slightly larger image of an ``A'' look
subjectively very similar to a human observer, although their squared
Euclidean separation (measured on a pixel by pixel basis) is very
large. The same can be said for two images of my face viewed from
slightly different angles.  Finding distortion measures that more
accurately correlate with our subjective experience is of great
practical utility in vector-quantisation and machine learning.
Quantisation with a poor distortion measure will cause the encoder to
make inefficient use of the available codebook vectors.  Also,
learning using nearest neighbour techniques or by generating a
piecewise constant approximation to the target function will be more
effective if an appropriate measure of the distortion between input
vectors is available.

From a purely philosophical perspective there is no {\em a priori}
natural distortion measure for a particular space $X$ of images or
signals. To generate a distortion measure some extra structure must be
added to the problem.  In this paper it is argued that the required
extra structure is given in the form of an {\em environment} of
functions ${\cal F}$ on $X$.  For example, consider each possible face
classifier as a function on the space of images $X$. The classifier
for ``jon'', $f_{\text{jon}}$, behaves as follows: $f_{\text{jon}}(x)=1$
if $x$ is an image of my face and $f_{\text{jon}}(x) = 0$ if $x$ is
not an image of my face. Note that $f_{\text{jon}}$ is {\em constant}
across images of the same face: it gives constant value ``1'' to
images of my face and constant value ``0'' to images of anyone else's
face. Similarly, there exists classifiers in the environment that
correspond to ``mary'' ($f_{\text{mary}}$), ``joe'' ($f_{\text{joe}}$)
and so on. All these classifiers possess the same property: they are
constant across images of similar looking faces. Thus, information
about the appropriate distortion measure to use for faces is somehow
stored in the environment of face classifiers. Similar considerations
suggest that it is the class of character classifiers that generate
the natural distortion measure for characters, it is the class of
spoken words that generate the natural distortion measure for speech
signals, it is the class of smooth functions that generate the natural
distortion measure for regression problems, and so on. A more formal
justification for this assertion will be given in section \ref{metric}
where an explicit formula is presented for the distortion measure
generated by a function class.  Such a distortion measure is termed
the {\em Canonical Distortion Measure} or CDM for the function
class. Loosely speaking, the canonical distortion between two inputs
$x$ and $x'$ is defined to be the difference between $f(x)$ and
$f(x')$, averaged over all functions $f$ in the environment.

In section \ref{proof} an optimality property of the CDM is proved,
namely that it generates optimal quantization points and Voronoi
regions for generating piecewise constant approximations of the
functions in the environment. In section \ref{examples} the CDM is
explicitly calculated for several simple environments. In particular
it is shown that the squared Euclidean distance function is the CDM
for a {\em linear environment}. This leads to the interesting
observation that the squared Euclidean distance is optimal for
approximating linear classes (and in fact {\em only} optimal for
approximating linear classes).

In section \ref{qex} the optimal quantization points for a quadratic
environment are calculated, and the relationship between the spacing
of the points and the behaviour of the functions in the environment is
discussed.

The relationship between the CDM and other approaches to learning
distance metrics is discussed in section \ref{relation} (see also
section \ref{rel} below), where we will
see that the CDM provides a unifying conceptual framework for many
seemingly disparate threads of research.

In section \ref{learn} it is shown how the CDM may be {\em learnt} by
sampling both from the input space {\em and} the environment, and an
(albeit toy) experiment is reported in which the CDM is learnt for a
``robot arm'' environment. The resulting CDM is then used to
facilitate the learning of piecewise constant approximations to the
functions in the environment. The same functions are also learnt
without the help of the CDM and the results are compared. At least on
this toy problem, learning with the CDM gives far better
generalisation from small training sets than learning without the
CDM.

\subsection{Related work}
\label{rel}
Other authors have investigated the possibility for using specially
tailored distance functions in both machine learning and vector
quantization contexts. The authors of \cite{Simard} used a measure of
distance that takes into account invariance with respect to affine
transformations and thickness transformations of handwritten
characters. They achieved a notable improvement in performance using
this measure in a nearest neighbour classifier.  Independently but
simultaneously with some of the work in a previous incarnation of this
paper \cite{thesis}, the authors of \cite{Thrun} proposed an
``invariance measure'' on images that has a close relationship to the
CDM defined here, under certain restrictions on the functions in the
environment. They also presented techniques for using the invariance
measure to facilitate the learning of novel tasks. In work
that is close to the spirit of the present paper, the authors of
\cite{Gray} considered vector quantization in a Bayes classifier
environment.  They modified the usual squared Euclidean reconstruction
error (\eqref{recerr} in section \ref{vq} below, with $d =
\|\cdot\|^2$) by adding a Bayes risk term, and presented results
showing improved combined quantization and classification performance
in several real-world problems.  The relationship between all these
approaches and the CDM is discussed further in section \ref{relation}.

As an historical note, the idea of using a set of functions to
generate a pseudo-metric was used at least twenty years ago as a
technical tool in the theory of topology \cite{Negropontis}, although
that application has little to do with the present paper.

\section{The distortion measure and vector quantization}
\subsection{Vector quantization}
\label{vq}
As any real digital communication channel has only finite capacity,
transmitting continuous data (e.g speech signals or images) requires
first that such data be transformed into a discrete representation.
Typically, given a probability space $(X,\Sigma,P)$ ($\Sigma$ is a
$\sigma$-algebra of subsets of $X$ and $P$ is the probability
distribution on $X$), one chooses a {\em quantization} or codebook
$\{x_1,\dots,x_k\}\subset X$, and instead of transmitting $x\in X$,
the index of the ``nearest'' quantization point
\begin{equation}
\label{qd}
q_d(x) := {\mbox{argmin}}_{x_i} d(x_i,x)
\end{equation}
is transmitted, where $d$ is a distortion measure (not necessarily a
metric) on $X$.  The quantization points ${\vec x} = \{x_1,\dots,x_k\}$ are
chosen so that the expected distortion between $x$ and its
quantization $q_d(x)$ is minimal, i.e ${\vec x}$ are chosen to minimize the
{\em reconstruction error}
\begin{equation}
\label{recerr}
E_d({\vec x}) =}%%{{\,\,\,\,\overset{\text{def}}{=}\,\,\,\,} \int_X d(x,q_d(x))\, dP(x).
\end{equation}
A common approach to minimizing \eqref{recerr} is Lloyd's
algorithm which iteratively improves a set of quantization points
based on a centroidal update (see \cite{lloyd,CT,Graytext}).

Some examples of distortion measures are the {\em Hamming metric},
$$
d(x,x') = 1 - \delta(x,x'),
$$
where $\delta$ is the Kronecker delta function, and the {\em squared
Euclidean distortion measure} for vector-valued $X$,
$$
d(x,x') = \|x-x'\|^2.
$$
The use of these distortion measures has more to do with their
convenient mathematical properties than their applicability to any
particular problem domain. For example, suppose $X$ is a space of
images and it is images of characters that are being transmitted over
the channel. An image of the character ``A'' and another translated
image of the same character would be considered ``close'' in this
context, although the squared Euclidean distance between the two
images would be large, quite likely larger than the distance between
an image of ``A'' and an image of ``B'' in the same location. Thus the
squared Euclidean distortion measure does not capture well the idea of
two images being ``close''. Another example is that of speech
coding---there is a large squared Euclidean distance between a speech
signal and a small translation of it in time, although both sound very
similar to a human observer. A vector quantizer constructed using an
inappropriate distortion measure will require a far larger set of
quantization points to achieve satisfactory performance in
environments where we are primarily interested in the {\em subjective}
quality of the reconstruction. And in almost all applications of
vector quantization it is the subjective quality of the reconstruction
that is the most important: eventually the quality of the
reconstructed speech signal or image is measured by how close it
appears to be to the original for some observer.

In the next section we will see how the problem of choosing an
appropriate distortion measure may be solved, at least {\em in
principle}, by using the idea of the {\em environment} of a
quantization process.

\subsection{The Environment}
\label{metric}
What makes the translated image of an ``A'' close to the original,
while an untranslated image of a ``B'' quite distant? And what makes
two speech signals nearly identical even though they are miles apart
from a Euclidean perspective? As discussed in the introduction, it is
because there is an {\em environment} of functions (the character
classifiers) that vary little across images of the same
character, and similarly there is an environment of ``speech
classifiers'' that vary little across utterances of the same
sentences or words.

Note that the set of {\em possible} character classifiers
is much larger than the set of particular classifiers for
the English alphabet (English language).  The particular form of the
letters we use is arbitrary (as evidenced by the existence of vastly
different alphabets such as Hebrew or Kanji), the only thing that is
required of a character is that different examples of it are
recognisably the same object (to us). Thus the number of different
character classifiers is potentially astronomical. A similar
conclusion holds for speech: the set of spoken word classifiers is
nearly infinite if one includes all possible words from all possible
languages.

\subsection{Formal definition of the Canonical Distortion Measure (CDM)}
\label{formal}
To formalize the idea that it is an environment of functions that
determines the distortion measure on the input space, define the {\em
environment} of any probability space $(X,\Sigma,P)$ to be a pair
$({\cal F},Q)$ where ${\cal F}$ is a set of functions mapping $X$ into a space
$(Y,\sigma)$, where $\sigma\colon Y\times Y\to {\bb R}$ ($\sigma$ may be a
metric), and $Q$ is a probability measure on ${\cal F}$\footnote{For $Q$ to
be well defined there needs to be a $\sigma$-algebra on ${\cal F}$. We leave
that algebra unspecified in what follows, and simply assume that an
appropriate one exists.}. An environment so defined induces the
following natural distortion measure on $X$:
\begin{equation}
\label{rho}
\rho(x,x') =}%%{{\,\,\,\,\overset{\text{def}}{=}\,\,\,\,} \int_{\cal F} \sigma(f(x),f(x'))\, dQ(f),
\end{equation}
for all $x,x'\in X$. In words, $\rho(x,x')$ is the average distance
between $f(x)$ and $f(x')$ where $f$ is a function chosen at random
from the environment.

Note that if $\sigma$ is a metric on $Y$ then $\rho$ is a
pseudo-metric on $X$ (recall that a pseudo-metric is the same as a
metric except that $\rho(x,x') = 0$ does not necessarily imply
$x=x'$).  From now on $\rho$ will be referred to as the {\em Canonical
Distortion Measure} or {\em CDM} for the environment $({\cal F},Q)$.

In relation to the character transmission problem, ${\cal F}$ would consist
of all character like classifiers, $Y$ would be the set $[0,1]$ and we
could take $\sigma(y,y') = (y-y')^2$ (here we are assuming that the
functions $f \in {\cal F}$ are {\em probabilistic classifiers}, that is
$f(x)$ represents the probability that $x$ belongs to the category
represented by $f$). Given our limited capacity to
learn characters, we could take the environmental probability measure
$Q$ to have support on at most 10,000 distinct elements of ${\cal F}$ (so
${\cal F}$ must contain more than just the 26 English letters).  Then if $x$
and $x'$ are two images of the same character, $f(x) \approx f(x')$
for all $f$ in the support of $Q$ and so by \eqref{rho}, $\rho(x,x')
\approx 0$, as required. If $x$ and $x'$ are images of different
characters, the classifiers $f_x$ and $f_{x'}$ corresponding to $x$
and $x'$ would have $f_x(x) \approx 1, f_x(x') \approx 0$ and
$f_{x'}(x) \approx 0, f_{x'}(x') \approx 1$. Classifiers for
characters that are subjectively similar to $x$ ($x'$) will behave in
a similar way to $f_x$ ($f_{x'}$), except that they will not assign
such a high value to positive examples as $f_x$ and $f_{x'}$ do. All
other classifiers $f$ will have $f(x) \approx f(x') \approx
0$. Substituting this into \eqref{rho} shows $\rho(x,x')$ will be
larger if $x$ and $x'$ are images of different characters than if they
are images of the same characters.

Note that $\rho$ depends only upon the environment $({\cal F},Q)$ and not
upon $X$ or its probability measure $P$.  Thus problems with the same
$X$ but different environments (for example character classification
and face recognition---different environments for the space of images)
will generate different canonical distortion measures.  In the next
section we will show that $\rho$ is the optimal distortion measure to
use if piecewise constant approximation of the functions in the
environment is the aim. Thus the fact that different environments
generate different $\rho$'s shows that the optimal similarity measure
between data points is highly dependent on what one is planning to do
with the data.

The definition of $\rho$ does not restrict us to considering only
classifier environments, any kind of functions will do. For example,
the environment might consist of all bounded linear functions on some
compact subset of $X={\bb R}^n$, all threshold functions or all polynomials
up to a certain degree.  Noisy environments can also be modelled
within this framework by making the functions $f\in{\cal F}$ {\em
distribution} valued. For example, if the problem is to learn noisy
concepts then these can be represented as a distribution $P$ on $X$
combined with a conditional distribution on
$\{0,1\}$. Different concepts would have different conditional
distributions, which we can represent as mappings $f\colon X\to Y$
where now $Y$ is the set of all {\em distributions} on $\{0,1\}$.  The
distance measure $\sigma$ is now a distance measure on probability
distributions, such as Kullback-Liebler divergence or Hellinger distance.
The CDM between two inputs $x$ and $x'$ would
then equal the average distance between the {\em distributions} $f(x)$
and $f(x')$.

\section{Examples calculatable in closed form}
\label{examples}
\subsection{A linear environment}
\label{linear}
Suppose that $X={\bb R}^n$ and ${\cal F}$ consists of all linear maps from $X$
into ${\bb R}$. ${\cal F}$ is the vector space dual of $X$ and so is itself
isomorphic to ${\bb R}^n$. With this in mind, take the measure $Q$ on ${\cal F}$
to be Lebesgue measure on ${\bb R}^n$, but restrict $Q$'s support to the
cube $[-\alpha,\alpha]^n$ for some $\alpha > 0$, and renormalise so
that $Q([-\alpha,\alpha]^n) = 1$.  Let $\sigma(y,y') = (y-y')^2$ for
all $y,y'\in{\bb R}$.  $\rho$ can then be reduced as follows:
\begin{eqnarray*}
\rho(x,x') &=& \int_{\cal F}\sigma(f(x),f(x'))\, dQ(f)\\ &=&
\frac1{(2\alpha)^n} \int\limits_{a\in [-\alpha,\alpha]^n} \(a\cdot x -
a\cdot x'\)^2\, d^na \\ &=& \frac1{(2\alpha)^n}
\int_{-\alpha}^\alpha\dots\int_{-\alpha}^\alpha \(\sum_{i=1}^n a_i x_i
- \sum_{i=1}^n a_i x'_i\)^2\, da_1\dots da_n \\* &=&
\frac{4\alpha^2}{3}\|x-x'\|^2.
\end{eqnarray*}
Thus a linear environment induces the squared Euclidean distance on
$X$. The reverse conclusion is also true, {\em i.e.\ } if one assumes that
$\rho(x,x') = K\|x - x'\|^2$ for all $x,x'$ then ${\cal F}$ must be a linear
function class (almost everywhere). So based on the optimality
result of the next section, using the squared Euclidean distortion is
optimal if one wishes to approximate linear functions on the input
space, but is not optimal for any other environments.  As it is very
rare that one is interested in applying linear functions to images, or
speech signals (for example face classifiers are not linear maps on
image space, nor are word classifiers linear maps on speech signals),
the use of squared Euclidean distortion in these environments is not
the best thing to do.

Note that a uniform distribution over any {\em symmetric} region of
weight space will yield a CDM that is proportional to Euclidean
distance, while a general distribution will yield a CDM that is a
quadratic form $\rho(x,x') = \langle x|A|x'\rangle$ where $A$ is an
$n\times n$ matrix.

\subsection{A thresholded linear environment}
Take the same example as above but this time threshold the output of
each $f\in{\cal F}$ with the Heaviside step function, and take $Q$ to have
support only on the unit ball in ${\bb R}^n$, rather than the cube,
$[-\alpha,\alpha]^n$ (this is done to make the calculations simple).
After some algebra we find
$$
\rho(x,x') = \frac{\theta}{\pi},
$$
where $\theta$ is the angle between $x$ and $x'$. Thus in an
environment consisting of linear classifiers ({\em i.e.\ } thresholded linear
functions) whose coefficients are distributed uniformly in the unit
ball, the natural distortion measure on the input space is the {\em
angle} between two input vectors.

\subsection{A quadratic environment}
\label{ex3}
Let $X=Y=[-1,1]$, $\sigma(y,y')=|y-y'|$ for all $y,y'\in Y$ and let
${\cal F}=\{f\colon x\mapsto a x^2\}$ with $a$ uniformly distributed in the
range $[-1,1]$. With this environment,
\begin{eqnarray*}
\rho(x,y) &=& \int_{-1}^1 |a x^2 - a y^2|\, da \\* &=& |x-y| |x+y|.
\end{eqnarray*}
Note that $\rho(x,y)=0$ if $x=y$ and if $x=-y$, so that $x$ and $-x$
are zero distance apart under $\rho$.  This reflects the fact that
$f(x)=f(-x)$ for all $f\in{\cal F}$. Notice also that $\rho(x,y)$ is the
ordinary Euclidean distance between $x$ and $y$, {\em scaled} by
$|x+y|$. Thus two points with fixed Euclidean distance become further
and further apart under $\rho$ as they are moved away from the
origin. This reflects the fact that the quadratic functions in ${\cal F}$
have larger variation in their range around large values of $x$ than
they do around small values of $x$. This can also be seen by
calculating the ${\varepsilon}$-ball around a point $x$ under $\rho$ (i.e the
set of points $x'\in X$ such that $\rho(x,x') \leq {\varepsilon}$). To first
order in ${\varepsilon}/{x}$ this is
$$
\[-x-\frac{{\varepsilon}}{2 x}, -x + \frac{{\varepsilon}}{2 x}\] \bigcup
\[x-\frac{{\varepsilon}}{2 x}, x + \frac{{\varepsilon}}{2 x}\].
$$
Note that the Euclidean diameter of the ${\varepsilon}$-ball around $x$
decreases inversely linearly with $x$'s---Euclidean again---distance
from the origin.

\section{The optimality of the Canonical Distortion Measure}
\label{proof}
In this section it is shown that the CDM is the optimal distortion
measure to use if the goal is to find piecewise constant
approximations to the functions in the environment.

Piecewise constant approximations to $f\in{\cal F}$ are generated by
specifying a quantization ${\vec x}=\{x_1,\dots,x_k\}$ ($x_i\in X$) of $X$ and
a partition ${\vec X}=\{X_1,\dots,X_k\}$ ($X_i\subseteq X, X_i\cap X_j =
\phi, X=\cup X_i$) of $X$ that is {\em faithful} to $\{x_1,\dots,x_k\}$
in the sense that $x_i\in X_i$ for $1\leq i\leq k$.  The piecewise
constant approximation ${\hat f}$ to any function $f$ is then defined by
${\hat f}(x) = f(x_i)$ for all $x\in X_i$, $1\leq i\leq k$.

If information about the function $f$ is being transmitted from one
person to another using the quantization ${\vec x}$ and the partition ${\vec X}$
then ${\hat f}$ is the function that will be constructed by the person on
the receiving end.

The most natural way to measure the deviation between $f$ and ${\hat f}$
in this context is with the pseudo-metric $d_P$,
$$
d_P(f,{\hat f}) := \int_X\sigma(f(x),{\hat f}(x))\,dP(x).
$$
$d_P(f,{\hat f})$ is the expected difference between $f(x)$ and
${\hat f}(x)$ on a sample $x$ drawn at random from $X$ according to
$P$. The {\em reconstruction error of ${\cal F}$} with respect to the pair
${\vec x}=\{x_1,\dots,x_k\}$ and ${\vec X}=\{X_1,\dots,X_k\}$ is defined to be
the expected deviation between $f$ and its approximation ${\hat f}$,
measured according to the distribution $Q$ on ${\cal F}$:
\begin{equation}
\label{define1}
E_{\cal F}({\vec x},{\vec X}):= \int_{\cal F} d_P(f,{\hat f})\, dQ(f).
\end{equation}
The quantization ${\vec x}$ and partition ${\vec X}$ should be chosen so as to
minimize $E_{\cal F}({\vec x},{\vec X})$.

Given any quantization ${\vec x}=\{x_1,\dots, x_k\}$ and distortion measure
$\rho$, define the partition ${\vec X}^\rho = \{X^\rho_1,\dots, X^\rho_k\}$
by
$$
X^\rho_i := \{x\in X\colon \rho(x,x_i) \leq \rho(x,x_j),\,\text{for
all}\, j\neq i\}
$$ (break any ties by choosing the partition with the
smallest index).  Call this the partition of $X$ {\em induced} by
$\rho$ and ${\vec x}$ (it is the Voronoi partition). Define
\begin{equation}
\label{define}
E_{\cal F}^\rho({\vec x}) := E_{\cal F}({\vec x},{\vec X}^\rho).
\end{equation}

\begin{lem}
\label{twoerrs}
\begin{equation}
\label{equals}
E_{\cal F}^\rho({\vec x}) = E_\rho({\vec x}).
\end{equation}
\end{lem}
\sloppy{
{\bf Proof.}  Let ${\vec x}=\{x_1,\dots,x_k\}$ be any quantization of $X$
and let ${\vec X}^\rho =\{X^\rho_1,\dots,X^\rho_k\}$ be the corresponding
partition induced by $\rho$. Denote the approximation of $f\in{\cal F}$ with
respect to this partition by ${\hat f}_\rho$. Then,
\begin{eqnarray*}
E_{\cal F}^\rho({\vec x}) &=& \int_{\cal F} d_P(f,{\hat f}_\rho)\,dQ(f) \\ &=&
     \int_{\cal F}\int_X \sigma(f(x),{\hat f}_\rho(x))\,dP(x)\,dQ(f) \\ &=&
     \int_X\int_{\cal F} \sigma(f(x),f(q_\rho(x)))\,dQ(f)\,dP(x) \\ &=&
     \int_X \rho(x,q_\rho(x))\, dP(x) \\* &=& E_\rho({\vec x})
\end{eqnarray*}
$\qed$

\begin{thm}
\label{optimal}
The reconstruction error $E_{\cal F}$ of ${\cal F}$ is minimal with respect to a
quantization ${\vec x} = \{x_1,\dots,x_k\}$ minimizing the reconstruction
error $E_\rho({\vec x})$ of $X$, and the partition ${\vec X}^\rho$ induced by
the CDM $\rho$ and ${\vec x}$.
\end{thm}
{\bf Proof.}  Let ${\vec x}=\{x_1,\dots,x_k\}$ be any quantization of $X$
and let ${\vec X}^\rho =\{X^\rho_1,\dots,X^\rho_k\}$ be the corresponding
partition induced by $\rho$. Denote the approximation of $f\in{\cal F}$ with
respect to this partition by ${\hat f}_\rho$. Let ${\vec X}=\{X_1,\dots,X_k\}$
be any other partition of $X$ that is faithful to ${\vec x}$ and let
${\hat f}$ denote the approximation of $f$ with respect to this second
partition. Define $X_{ij} = X_i \cap X^\rho_j, 1\leq i \leq k, 1\leq
j\leq k$. Note that the $X_{ij}$'s are also a partition of $X$. The
reconstruction error of ${\cal F}$ with respect to the partition ${\vec X}$
satisfies:
\begin{eqnarray*}
E_{\cal F}({\vec x},{\vec X}) &=& \int_{\cal F} d_P(f,{\hat f})\, dQ(f) \\
&=& \int_{\cal F}\int_X
     \sigma(f(x),{\hat f}(x))\,dP(x)\,dQ(f) \\ &=&
     \sum_{i,j=1}^k\int_{X_{ij}}\int_{\cal F}\sigma(f(x),f(x_i))\,dQ(f)\,dP(x)
     \\ &=&
 \sum_{i,j=1}^k\int_{X_{ij}}\rho(x,x_i)\,dP(x) \\ &\geq&
     \sum_{i,j=1}^k\int_{X_{ij}}\rho(x,x_j)\,dP(x) \\ &=& \int_{\cal F}
     d_P(f,{\hat f}_\rho)\, dQ(f) \\* &=& E_{\cal F}^\rho({\vec x}).
\end{eqnarray*}
The theorem now follows from lemma \ref{twoerrs}.  $\Box$

Theorem \ref{optimal} states that as far as generating piecewise
constant approximations to the functions in the environment is
concerned, there is {\em no better} partition of the input space than
that induced by the CDM and its optimal quantization set.
}
\subsection{Quadratic environment revisited}
\label{qex}
For the quadratic environment of example \ref{ex3}, the optimal
quantization for $k=6$ is shown in figure \ref{fhat} along with $f$
and ${\hat f}_\rho$ for $f(x)=x^2$.  Note how the optimal quantization
reduces the deviation between $f$ and its approximation ${\hat f}_\rho$
by spacing the points closer together for larger values of $x$.

To calculate the optimal quantization in this case, first note that by
the symmetry of the environment, the quantization points $\{x_1,\dots,
x_6\}$ can all be assumed to be positive, and without loss of
generality suppose that $x_1\leq x_2 \leq\dots\leq x_6$. Direct
calculation of the reconstruction error
$E_\rho({\vec x})$ \eqref{recerr} shows that
$$
x_i^2 = \frac14\(x_{i-1}^2 + x_{i+1}^2\) +
\frac1{4\sqrt{2}}\sqrt{x_{i-1}^4 + 6 x_{i-1}^2 x_{i+1}^2 + x_{i+1}^4}.
$$
A similar procedure can be used to show that $x_1$ and $x_6$ must
satisfy,
\begin{eqnarray*}
x_1 &=& \frac{x_2}{\sqrt{7}},\\* x_6 &=& \frac{4+\sqrt{2+7 x_5^2}}{7}.
\end{eqnarray*}
Optimal quantization points can be found by solving these equations
numerically.
\begin{figure}
\begin{center}
\leavevmode\epsfxsize=3in\epsfbox{quad.eps}
\caption{Optimal quantization points and the corresponding
approximation to $f(x)=x^2$ for the quadratic environment of example
\ref{ex3}. The six quantization points are (to three significant
figures) $x_1=0.142$, $x_2=0.377$, $x_3=0.545$, $x_4=0.690$,
$x_5=0.821$, $x_6=0.942$.
\label{fhat}}
\end{center}
\end{figure}

\section{Relationship between the CDM and other distance measures}
\label{relation}
\subsection{Transformation invariant distance}
The authors of \cite{Simard} introduced a technique for comparing
handwritten characters called {\em Transformation Distance}. They
observed that images of characters are invariant under transformations
such as rotation, dilation, shift, line thickening and so on. Denoting
the set of all such transformations by $G$, and assuming that $G$ can be
parameterised by $k$ real parameters, they noted that for each
character $x$, the set
$$
{\cal M}(x) := \{gx\colon g\in G\}
$$
forms a $k$-dimensional manifold in the input space $X$. They defined
the distance bewteen two images $x$ and $x'$ to be:
\begin{equation}
\label{trans}
D(x,x') := \inf_{y\in {\cal M}(X),y'\in {\cal M}(x')} \|y - y'\|,
\end{equation}
that is, $D(x,x')$ is the smallest Euclidean distance between any
transformed image of $x$ and $x'$ (and is called the {\em
transformation distance between $x$ and $x'$}). In order to simplify
the computation, in \cite{Simard} $D(x,x')$ was approximated by
a linearised version. However we will concentrate on the exact expression
\eqref{trans}.

Relating the transformation distance to the CDM, note that invariance
of characters under the action of $G$ is equivalent to assuming that
all character classifiers in the environment ${\cal F}$ are invariant under
the action of $G$. In other words, for all $f\in{\cal F}$, $f(gx) = f(x)$
for all $g\in G$. Then clearly, if $x'\in {\cal M}(x)$, $\rho(x,x') = D(x,x') =
0$, so the CDM gives the same answer as the transformation distance in
this case. However, if $x'\notin {\cal M}(x)$ then it is unlikely that
$D(x,x')$ and $\rho(x,x')$ will be the same, because $D(x,x')$
(somewhat arbitrarily) measures the Euclidean distance between the
manifolds ${\cal M}(x)$ and ${\cal M}(x')$, whereas $\rho(x,x')$ performs an
average over the functions in the environment to compute the distance.

The transformation distance and the CDM share the important property
that the distance between any pair of points in ${\cal M}(x)$ and ${\cal M}(x')$
respectively is always the same. This is trivial by the construction
of $D$; to see it for the CDM let $x,x'\in{\cal M}(x)$ and $y,y'\in M(y)$.
Then $\rho(x,x') = \rho(y,y') = 0$ and combining this with the
triangle inequality gives:
\begin{eqnarray*}
\rho(x,y) &=& \rho(x,y) + \rho(x,x') + \rho(y,y') \\
  	&\geq& \rho(x',y) + \rho(y,y') \\*
	&\geq& \rho(x',y').
\end{eqnarray*}
Running the same argument with $x$ and $x'$ and $y$ and $y'$
interchanged
shows that $\rho(x,y) = \rho(x',y')$ always.

\subsection{The CDM and Edelman's Chorus of Prototypes}
In \cite{Edelman1}, Edelman introduced a concept of representation he
called the {\em Chorus of Prototypes}. The idea is to train a set of
real-valued classifiers, $f_1,\dots,f_N$, for a domain of {\em
prototype} objects (so that $f_i(x)\in [0,1]$ is interpreted as the
probability that $x$ is an example of object $i$). All objects (not
just the prototypes) are then represented by the vector of activations
they induce at the output of the prototype classifiers. This vector of
activations is a {\em Low Dimensional Representation} (LDR) of the input
space. It is argued that the Euclidean distance between two
representation vectors corresponds to the distal similarity of the
objects.

Given that both the CDM and the Chorus of Prototypes represent
similarity by making use of an environment of classifiers, it is
natural to look for a connection between the two. There is a
connection if one assumes that all functions in the environment can be
implemented as linear maps composed with a fixed low dimensional
representation. So let $h = (h_1,\dots,h_k) \colon {\bb R}^d\to {\bb R}^k$ ($d
\gg k$) be the fixed LDR, and then each $f\in {\cal F}$ is of the form $\sum
w_i h_i$ where the $w_i$ depend on $f$. Suppose that $k$ is
minimal. Note that the $f$ take values outside the range $[0,1]$ and
so cannot be interpreted as probabilities. However we can still
interpret the output of $f$ as a ``degree of classification''---large
positive values being high confidence and large negative values low
confidence. In this case the environmental distribution $Q$ is a
distribution over the weights $w$. If $Q$ is uniform over a symmetric
region of weight space then by section \ref{linear}, the CDM between
two inputs $x$ and $x'$ will be proportional to the Euclidean distance
between their transformed representations $h(x)$ and $h(x')$.

Now choose any $f_1,\dots, f_k$ such that their respective weight vectors
$w^1,\dots,w^k$ are linearly independent (such a set of functions can
always be found because we assumed $k$ is minimal). $(f_1,\dots,f_k)$
will be the chorus of prototypes representation. Set
\begin{equation}
W =
\begin{array}{ccc}
w_1^1 & \dots & w_1^k \\
\vdots & \ddots & \vdots \\
w^1_k & \dots & w^k_k
\end{array}
\end{equation}
and note that $W$ is nonsingular. For any input $x$, $(f_1(x),
\dots,f_k(x))$ (the representation of $x$ by its similarity to the
prototypes $i=1,\dots, k$) is equal to $W h(x)$.  Let $f = w\cdot h$
be any classifier in the environment. Set $w':= w W^{-1}$. Then $f =
w' W h = \sum w'_i f_i(x)$. Thus any classifier in the environment is
representable as a linear combination of prototype classifiers, as
required for the chorus of prototypes idea.

\subsection{Similarity measure of Thrun and Mitchell}
The authors of \cite{Thrun} defined an ``invariance function''
$\sigma\colon X\times X\to \{0,1\}$ for a finite environment ${\cal F}$ with
the property that if there exists $f\in{\cal F}$ such that $f(x) = 1$ then
$f'(x) = 0$ for all other $f'\in {\cal F}$:
$$
\sigma(x,x') := \left\{ \begin{array}{ll}
			1& \mbox{if $\exists f\in{\cal F}$ with $f(x)= f(x') = 1$} \\
			0& \mbox{if $\exists f\in{\cal F}$ with $f(x) \neq f(x')$} \\
			\mbox{undefined}& \mbox{otherwise}
			\end{array} \right.
$$
If we assume the environmental distribution $Q$ is uniform on ${\cal F}$,
a quick calculation shows that for all $x,x'$ for which
$\sigma(x,x')$ is defined,
$$
\sigma(x,x') = 1 - \frac{|{\cal F}|}{2} \rho(x,x').
$$
Thrun and Mitchell also showed how $\sigma$ can be used to facilitate
learning of novel tasks within their {\em lifelong learning framework}.

\section{Learning the CDM with neural networks}
\label{learn}

For most environments encountered in practise (e.g speech recognition
or image recognition), $\rho$ will be unknown. In this section it is
shown how $\rho$ may be estimated or {\em learnt} using feed-forward
neural networks.  An experiment is presented in which the CDM $\rho$
is learnt for a toy ``robot arm'' environment. The learnt CDM is then used to
generate optimal Voronoi regions for the input
space, and these are compared with the Voronoi regions of the true CDM
(which can be calculated exactly for this environment).
Piecewise-constant approximations to the functions in the
environment are then learnt with respect to the Voronoi partition, and
the results are compared with direct learning using feedforward
nets. We conclude that learning piecewise-constant approximations
using the CDM gives far better generalisation performance than the
direct learning approach, at least on this toy problem. It should be
emphasised that this problem is designed primarily to illustrate the
ideas behind learning the CDM, and not as a practical test of the
theory on real-world problems. The latter awaits more sophisticated
experimentation.

\subsection{Sampling the environment}

To generate training sets for learning the CDM,
both the distribution $Q$ over the environment ${\cal F}$ and the distribution
$P$ over the input space $X$ must be sampled.
So let $\{f_1,\dots,f_M\}$ be $M$ {i.i.d.\ } samples from ${\cal F}$ according to $Q$
and let $\{x_1,\dots,x_N\}$ be $N$ {i.i.d.\ } samples from $X$ according to $P$.
For any pair $x_i,x_j$ an estimate of $\rho(x_i,x_j)$ is given by
\begin{equation}
\label{rhat}
{\hat \rho}(x_i, x_j) = \frac1M\sum_{i=1}^M \sigma(f_i(x_i), f_i(x_j)).
\end{equation}
This generates $N^2$ training {\em triples},
$$\{(x_1,x_1,{\hat \rho}(x_1,x_1)), (x_1,x_2,{\hat \rho}(x_1,x_3)),\dots, (x_N, x_N,
{\hat \rho}(x_N, x_N))\},$$
which can be used as data to train a neural network. That is,
the neural network would have two sets of inputs---one set for $x_i$ and one
set for $x_j$---and a real-valued output $\rho^*(x_i,x_j)$
representing the network's estimate of ${\hat \rho}(x_i,x_j)$. The mean-squared
error of the network on the training set is then
\begin{equation}
\label{emp}
E = \frac1{N^2}\sum_{i=1}^N\sum_{j=1}^N
	\[\rho^*(x_i,x_j) - {\hat \rho}(x_i,x_j)\]^2.
\end{equation}
$E$ is an estimate of the true distance between the network's
$\rho^*$,  and the true CDM $\rho$, where this is defined by:
\begin{equation}
\label{true}
d(\rho,\rho^*) := \int_{X^2}\[\rho(x,x') - \rho^*(x,x')\]^2\,dP(x)\,dP(x').
\end{equation}

Note that the process of sampling from ${\cal F}$ to generate
$f_1,\dots,f_M$ is a form of multi-task learning (see {\em e.g.\ }
\cite{colt95,caruana1,Thrun,pratt}) and that such sampling is a
necessary condition for the empirical estimate of the CDM, $\rho^*$,
to converge to the true CDM $\rho$.

\subsection{``Robot arm'' experiment}
An artificial environment was created to test the effectiveness of
training neural networks to learn the CDM.  The environment was chosen
to consist of a set of two link ``robot arm'' problems. That is, each
function in the environment corresponded to a robot arm with two links
of length $r_1$ and $r_2$ (see Figure \ref{robot}. Note that the term
``robot'' is used fairly loosely here: the example doesn't have much to do
with robotics). The function
corresponding to $r_1,r_2$ is the map $f_{r_1,r_2}
\colon [-\pi,\pi]^2 \to [0,(r_1+r_2)^2]$
that computes the square of the
distance of the end of the arm from the origin, given the
angles of the links $\theta_1,\theta_2$.
Thus, $f_{r_1,r_2}(\theta_1,\theta_2) = r_1^2 + r_2^2 + 2r_1r_2
\cos(\theta_1 - \theta_2)$.
The link lengths $r_1$ and $r_2$ were chosen uniformly in the range $[0,1]$,
so that ${\cal F} = \{f_{r_1,r_2}\colon (r_1,r_2) \in [0,1]^2\}$. The goal
was to train a neural network to correctly implement the CDM
$\rho((\theta_1,\theta_2),
(\theta'_1,\theta'_2))$. Note that in this case $\rho$ can be calculated
in closed form:
\begin{equation}
\label{rhoex}
\rho((\theta_1,\theta_2),(\theta'_1,\theta'_2)) =
\frac49\[\cos(\theta_1-\theta_2) - \cos(\theta'_1-\theta'_2)\]^2.
\end{equation}

\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=3.0in\epsfbox{robot.eps}
\caption{\label{robot} The robot arm environment is generated by all two
link robot arms with link lengths $0\leq r_1 \leq 1$ and $0\leq r_2 \leq 1$.
For each pair of angles $(\theta_1,\theta_2)$,
$f_{r_1,r_2}(\theta_1,\theta_2)$ is the square of the distance of the end
of the robot arm from the origin.}
\end{center}
\end{figure}

The network architecture used was a single hidden layer neural network with
$\tanh$ activation function on the hidden layer nodes and a linear
output node. After some experimentation twenty hidden nodes was found to
be sufficient.
The network had four inputs, one each for $\theta_1,\theta_2,\theta'_1$ and
$\theta'_2$. The knowledge that $\rho$ is symmetric was built into
the network by taking the output of the network for inputs $x_i$ and
$x_j$ to be
\begin{equation}
\label{sym}
\frac{\rho^*(x_i,x_j) + \rho^*(x_j,x_i)}{2}
\end{equation}
rather than just the ``raw'' network output $\rho^*(x_i,x_j)$.
Note that
\eqref{sym} is automatically symmetric in $x_i$ and $x_j$. With this choice
of the estimate of $\rho$, the error measure  \eqref{emp} becomes
\begin{equation}
\label{Ed}
E = \frac{2}{N(N+1)} \sum_{i=1}^N\sum_{j=i}^N
\[\frac{\rho^*(x_i,x_j) + \rho^*(x_j,x_i)}2 - {\hat \rho}(x_i,x_j)\]^2.
\end{equation}
Back-propagation was used to compute the gradient of \eqref{Ed}.

Training sets were generated by first sampling $M$ times from the environment
to generate $f_1,\dots,f_M$, which in this case meant generating M pairs
$(r_1,r_2)$ uniformly at random in the square $[0,1]^2$.
Then the input space
$[-\pi,\pi]^2$ was sampled uniformly at random $N$ times to generate
$x_1,\dots,x_N$ and the empirical
estimate ${\hat \rho}(x_i,x_j)$ was constructed using \eqref{rhat} for each of the
$N(N+1)/2$ pairs
$(x_i,x_j)$ for $1\leq i\leq j\leq N$.
A separate cross-validation set was also generated using the same values
of $M$ and $N$ used for the training set.
The network was trained using conjugate gradient
descent---with the gradient computed from \eqref{Ed}---until
the error on the cross-validation set failed to decrease for more
than five iterations in a row.
After the network had been trained, an initial quantization set
${q_1,\dots,q_m}$ of size $m\ll N$ was chosen uniformly at random from
$\{x_1,\dots,x_N\}$ and then the empirical Lloyd algorithm
\cite{lloyd} was used to optimize the positions of the quantization
points. The trained neural network---suitably symmetrised via
\eqref{sym}---was used as the distortion measure.

Several different experiments were performed with different values of $M,N$ and
$m$---the Voronoi regions for $(M,N,m) = (100,100,20)$ are plotted in
Figure \ref{voronoi}, along with the
optimal quantization set and corresponding Voronoi
regions for the true CDM \eqref{rhoex}.
Note that the regions generated by the neural net
approximation are very similar to those generated by the true CDM.
The reconstruction error of the input space was estimated from the
training set $\{x_1,\dots,x_N\}$ and also calculated (nearly) exactly
by quantizing the input space into a $250\times 250$ grid. These
quantities are shown in the first two columns of table \ref{rec} for
several different values of $M,N$ and $m$. Note the good agreement
between the estimate ${\hat E}_{\rho^*}$ and the ``true'' value
$E_{\rho^*}$ in each case, and also that the reconstruction error is
very small in all cases, improving as the training set size ($M,N$)
for the CDM increases, and also as the number of quantization points
$m$ increases.

\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=2.0in\epsfysize=2.0in\epsfbox{v100x100x20i.eps}\hspace{5mm}
\epsfxsize=2.0in\epsfysize=2.0in\epsfbox{v100x100x20p.eps}
\caption{\label{voronoi}
The picture on the left is the Voronoi diagram generated from $20$
quantization points by a neural network approximation to the CDM
($\rho^*$), trained on a sample size of $M=100$ functions and $N=100$
input points. The right-hand-side is the Voronoi diagram generated
using the true CDM.  All pixels with the same colour belong to the
same Voronoi region. The white dots are the quantization points found
by Lloyd's algorithm.}
\end{center}
\end{figure}

\sloppy Recall that by lemma \ref{twoerrs},
$E_{\rho^*}(q_1,\dots,q_m)$ equals the reconstruction error of the
functions in the environment ($E^{\rho^*}_{\cal F}(q_1,\dots,q_m)$).  That is, if
\begin{enumerate}
\item a function $f$ is picked
at random ({\em i.e.\ } the link lengths $r_1$ and $r_2$ are picked
at random),
\item  the values $f(q_i)$ for each $q_i$ in the optimal
quantization set are generated and stored, and
\item the value of $f$ at any novel input
$x\in X$, chosen according to $P$, is estimated by $f(q_{\rho^*}(x))$,
\end{enumerate}
then $E_{\rho^*}(q_1,\dots,q_m)$ is the expected generalisation error:
$E_X\[f(x) - f(q_{\rho^*}(x))\]^2$.  Hence a small value of
$E_{\rho^*}(q_1,\dots,q_m)$ indicates that any function in the
environment is likely to be learnable to high accuracy by this
procedure\footnote{Note that this procedure is just $1$ Nearest
Neighbour, with $\rho^*$ as the distance metric and the quantization
points intelligently placed.}. To demonstrate this,
$E^{\rho^*}_{\cal F}(q_1,\dots,q_m)$ was estimated by generating 100 new
functions at random from the environment and learning them according to the
above procedure. Each function's generalisation error was estimated
using a fine grid on the input space, and then averaged across all
$100$ functions.  The average generalisation error is plotted in the
third column of table \ref{rec} for the various CDMs generated from
different training set sizes and the different numbers of quantization
points. Note the good agreement between the reconstruction error of
the environment (${\hat E}^{\rho^*}_{\cal F}$) and the reconstruction error of
the input space $E_{\rho^*}$.

For comparison purposes the same 100 functions were learnt
using a 10-hidden node neural network, without the assistance of the
CDM. The average generalisation error across all $100$ functions is
displayed in Figure  \ref{avgen} along with the same quantity for the
piecewise-constant approximations. The piecewise-constant approach
based on the estimated CDM is clearly superior to the normal approach
approach in this case.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=3.0in\epsfbox{plot.eps}
\caption{\label{avgen}Average generalisation error as a function of
training samples for the robot arm environment: standard approach vs.
training using the estimated CDM.
``Standard'' means learning each function from scratch with a one-hidden
layer, 10 hidden node neural net. ``$100\times 100$'' and
``$200\times 200$'' refer to the number of functions and input examples
used to train the approximation to the CDM.}
\end{center}
\end{figure}

\begin{table}[h]
\caption{\label{rec}Empirical (${\hat E}_{\rho^*}$) and true
($E_{\rho^*}$) reconstruction error of the input space using Voronoi
regions and quantization points generated by the neural network
approximation ($\rho^*$) to the CDM. The final column is the estimated
reconstruction error of the functions in the environment.  $M$ is the
number of functions sampled from the environment and $N$ is the number
of points sampled from the input space to train $\rho^*$.  $m$ is the
number of quantization points.}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}ccccc}
\hline
$(M,N)$ & $m$ & ${\hat E}_{\rho^*}$ & $E_{\rho^*}$ & ${\hat E}^{\rho^*}_{\cal F}$ \cr
\hline $(100,100)$ & $5$ & $8.3\times10^{-3}$&$1.0\times 10^{-2}$&$1.2\times 10^{-2}$ \cr
&$20$ & $6.5\times 10^{-4}$&$2.4\times 10^{-3}$&$6.9\times 10^{-3}$ \cr
\hline $(200,200)$ &$5$&$5.2\times 10^{-3}$&$5.0 \times 10^{-3}$&$7.9\times 10^{-3}$ \cr
&$20$&$6.0\times 10^{-4}$&$1.0\times 10^{-3}$&$1.8\times 10^{-3}$\cr
\hline
\end{tabular*}
\end{table}

\section{Conclusion}

It has been shown that the existence of an {\em environment} of
functions for a quantization process generates a {\em canonical
distortion measure} (CDM) $\rho$ on the input space.  It has been
proven that generating an optimal quantization set for the input space
using $\rho$ as the distortion measure automatically produces Voronoi
regions that are optimal for forming piecewise-constant approximations
to the functions in the environment.  The optimality theorem shows
that the CDM contains {\em all} the information necessary for learning
piecewise constant approximations to the functions in the
environment. Hence learning the CDM is a process of {\em learning to
learn}.

The CDM was calculated in closed form for several simple environments.
A surprising result is that the squared Euclidean distortion measure
is the CDM for a linear environment, and hence is optimal only if we
are interested in approximating linear functions.

Techniques for estimating the CDM and training a neural network to
implement it have been presented, and the results of several promising
experiments on a toy environment have been reported.  It remains to be
seen whether these techniques work for more complex domains such as
speech and character recognition.

One may be tempted to ask ``Why bother going to all the trouble of
learning the CDM first. Why not just learn the functions in the
environment directly?'' The answer to this question goes to the core
of what distinguishes learning to learn from ordinary learning. Of
course, if one is only interested in learning at most a handful of
functions from the same environment then learning the CDM is
overkill. Standard statistical learning techniques will be much more
efficient. However, if one is interested in in learning a large number
of functions from the environment, then learning the CDM will be a
big advantage, because by theorem \ref{optimal}, the CDM contains
{\em all} the information necessary for {\em optimal} learning of
piecewise constant approximations to the functions in the environment.

In ordinary learning we are interested primarily in aquiring the
information necessary to solve a particular learning problem, whereas
in learning to learn we want to acquire the information necessary to
solve a whole class of learning problems. As many real world problems
are members of large classes of similar problems (speech, face and
character recognition to name a few), tackling these problems from a
learning to learn perspective should be a fruitful approach.

redpajama_set_name: RedPajamaArXiv


How do architects show the historical repurposing of architectural structures? Called spolia, what's its relevance to modern architecture?
Undergraduate Oonagh Davis '20 participates in the Cornell Spolia project, preparing her work for exhibition at the Detroit Center for Design and Technology, summer 2018.
Davis' project is to analyze the Arles Amphitheater, a Roman amphitheater in the southern French town, originally built in 90 AD as an entertainment space and later repurposed as a housing community.
The experience of studying at a boarding school broadened Davis' educational aspirations. She began considering institutions beyond those in the immediate vicinity of Cincinnati. Consequently, Cornell emerged as her primary choice for an undergraduate degree in architecture.
Creativity, Davis contends, should be the primary focus of an architectural project. During her time at Cornell, Davis has constantly sought projects and academic ventures that require her to be creative and to find innovative solutions to architectural problems. In the spring semester of 2018, Davis was recruited for a project led by Aleksandr Mergold, Architecture.
Davis had been a keen admirer of Mergold's earlier works, which she characterizes as having sparked conversations on a variety of relevant issues—ranging from cultural history to green architecture. One of the program's initial student members introduced her to the project. She was immediately drawn in by the focus on what she describes as an increasingly important element of modern architecture.
The project centers around spolia, a Latin term Davis defines as the repurposing aspect of architecture throughout history. As part of the project, Davis and three additional students have been assigned a structure, for which each student repurposes elements unique to each structure for an art exhibition in Detroit, Michigan—the Detroit Center for Design and Technology—during the summer of 2018.
Davis' task is to analyze the Arles Amphitheater, a Roman amphitheater located in the southern French town of Arles. The structure was initially constructed in 90 AD purely as an entertainment space and played host to a variety of events, including chariot races and gladiatorial battles.
This phenomenon represents an archetypal example of the organic repurposing of a historical structure; the amphitheater gradually manifested an impressively dynamic relationship with the local community. Davis says, however, that it has proven especially difficult to document this process of spolia in the amphitheater, due to the lack of relevant historical documentation about the structure. Even after sourcing the assistance of experts in medieval French architecture, Davis was unable to determine the exact structures built inside the amphitheater. Today, the arena has reverted to its initial appearance because the inner structures were destroyed over the centuries.
The primary method Mergold chose to represent the repurposing of the structures is through columns constructed from concrete, wood, aluminum, and PVC. Each column is characterized by unique features, representing the spolia-associated characteristics of its corresponding historical structure. Davis, for example, is deliberating over the spacing of holes which she will drill into her column.
Aware that the decidedly abstract nature of the columns may render it slightly difficult to draw an interpretive link between the historical structures and their corresponding columns, Davis and her team have decided to exhibit a series of drawings, highlighting the repurposing aspect of each structure over time—as informational supplements to the columns themselves.
In addition to her research and curricular projects, Davis has been working as a teaching assistant at the Architecture, Art, and Planning Material Practice Facilities' woodworking and metal shop since her freshman year. She also took a teaching assistant role in the laser-cutting lab in the spring semester of 2018. The chance to collaborate with and mentor students drew Davis to the teaching roles. Davis says that she is glad to assist younger students with several of the issues she faced herself during her initial years at Cornell. She views teaching as a way of giving back to the community.
Following the spolia exhibition in Detroit, Davis will be interning at Skidmore, Owings, and Merrill LLP, an architecture firm in San Francisco. Upon the completion of her undergraduate degree, Davis plans to begin a professional career as a licensed architect. She says, however, that she may choose to pursue a graduate architecture degree in the future.


text:
change your view?
images images and text text only
12 24 36 48 60 72 all
Dealers A-Z
Print Finder
Marino Marini (1901-1980)
at F. L. Braswell Fine Art
view 49 of 49
Click on one of the following views:
Collector View
Professional View
Jongleur et Chevaux, Fond Vert, Bordure Jaune
* Approximate price based on current exchange rates
Born in Pistoia in 1901 Marion Marini attended the Academia di Belle Arte in Florence. Marini moved to Milan and taught sculpture and he won the Grand Prize for sculpture at the Rome Quadriennale in 1935. After a time in Switzerland Marino Marini returned to Milan in 1946. In 1950 Marini had a one man exhibition at the Curt Valentin Gallery in New York. Two years later Marino Marini won the Grand Prize for scuplture at the Venice Biennale. Marini had two main themes, the horse and rider which allowed him to explore interdependence between man and nature. At the end of the second world war Marini's work changed. He carried on working with the equestrian theme but his work now conveyed a sense of tragedy. Marino Marini died in 1980.
© 2004 - 2014 OriginalPrints Ltd - webmaster@originalprints.com - E&OE - Terms of Use - Privacy Statement - Labelled with ICRA


Trick or Treat! Our Strange Halloween episode begins with a crash-course on Doctor Strange, from the original 1963 comic to the brand new Marvel film out November 4th.
It's almost Halloween night when we all spend the better part of the evening knocking on neighbors doors asking for candy. We want to make sure everyone is safe this holiday season so C-3PO is back to give some safety tips on trick or treating. In fact, some Skywalkers may just be trick-or-treating early on this episode! Tweet at them, if you dare.
Things We Want To Share – Our Star Wars / Disney Masquerade Ball Wedding, October 25, 2009.


text:
Discussions of neo-liberalism at Crooked Timber.
Henry Farrell
Doug Henwood has a go at Matthew Yglesias.
Duncan Black, in a game of telephone, linking to someone linking to CT.
Leaving aside policy disagreements, and for sake of discussion assuming good intentions, I think the group being discussed are basically not fans of democracy.
Atrios seems not to understand they're talking about a friend of his.
Posts continue with Henry again, responding to Yglesias, and then Bertram
Well not content with his inaccurate digs at Henry, Brad DeLong is having another go at me.
For discussions of Crooked Timber as neoliberal, go here, and here and (why not) here
Chris Bertram from the second of the three above
The right frame, in my view, is to think of the state as "we, the people" and to ask what conditions need to be in place for the people, and for each citizen, to play their role in effective self-government. Once you look at things like that then various speech restrictions naturally suggest themselves.
Various references to Yglesias on this site over the years
My inbox (email): "And for the record (don't post this), Yglesias as an individual has a great, self-aware sense of humor and is much more starkly honest (if also unapologetic) about his own elitism than most liberals. Take him out for a beer and I think you'd find that."
Yglesias: "At the appropriate level of abstraction, the neocons couldn't be more right about this stuff, but when it comes to actually getting it done their policies have been a miserable failure."
Yglesias: "After the last depressing news from the Middle East I think we have to start asking just how inhumane it would be for Israel to just expel the Palestinians from the occupied terroritories. [sic] The result would probably be out-and-out war with the neighboring Arab states, but Israel could win that.
All forced population transfers are humanitarian disasters, of course, but so is the current situation. It's not like there's not any room in the whole Arab world for all these Palestinian Arabs to go live in, it's just that the other Arab leaders don't want to cooperate."
Doug Henwood recently (on Facebook)
There's a Marc Jacobs boutique in Ho Chi Minh City??
20 years since I first quipped that Nike won the Vietnam War. Henwood's ignorance surprised me. It shouldn't have.
The choices above are between the authoritarianism of the market, the authoritarianism of intellectuals, and utopian fantasy.
Tyler Cowen and DeLong are no longer favorites at Crooked Timber, though they were once. Allegiances have drifted, as they have for Duncan Black, but though there's change, none of them, Atrios, Henry Farrell, Bertram and the rest, are capable of describing that change. They'd put it up it to a reasoning process, but the data isn't new, even to them. Change in their awareness came slowly because they had to acculturate the data, and that takes time. Liberalism, neoliberalism, and modernism itself have no concept of time, more specifically of its effects. Authors may model the world as dynamic but they model themselves as static. As I've said, individualism is textual originalism read into the self, as author and actor, and read forward. They all claim to write as Scalia claims to read.
Someone googling "academic neoliberalism" found this: the jackass Bourdieu. Click through to T.J. Clark.
All summer repeats. And repeats of repeats.
Labels: Atrios, Brad DeLong, Culture, Determinism, Drift, Make it Idiot-Proof, Middle East, More Reasons to Read Oswald Spengler, Philosophy, Politics, The Press


text:
By : D'Anne Witkowski
Tags:apology, cancer, column, creep, dern homosexers, First Amendment, gay, give, God, good, Indiana, Iowa, Jesus, lgbt, man, Senator, soul-searching, teaching, ted cruz
Before I begin this column I'd like to first offer an apology.
You see, I have a bad habit of referring to anti-gay politicians and talking heads using pejorative terms like "ass," "asshole," "dick" and "dickhead." I've done some soul-searching and realized that I can no longer do this in good conscience and I sincerely apologize.
After all, butts and penises serve a legitimate purpose in the lives of so many people. In other words: there are plenty of people, straight and gay alike, who like backsides and wieners. Not only for the sexing, mind you, but also for the very essential bodily functions for which they allow. I hope that each and every booty and schlong and the people who love them can forgive me.
Which leads me to this week's Creep: United States Senator Ted Cruz (R-Texas), the first Republican to throw his hate in the ring (no, I did not spell "hat" wrong) to run for president.
On April 9, Cruz spoke in Iowa to a gathering of The Network of Iowa Christian Home Educators and—surprise!—said something anti-gay.
More specifically, he expressed his support for the so-called "religious liberty" laws being passed across the country, the most notably in Indiana. He also expressed his frustration that anyone would dare challenge such laws just because they can be used to discriminate against LGBT people in states where there are no anti-discrimination protections for LGBTs (like in Indiana).
"We look at the jihad that is being waged right now, in Indiana, and Arkansas," he said, "going after people of faith who respect the biblical teaching that marriage is the union of one man and one woman."
It is, of course, a mischaracterization, at best, to describe the opposition to so-called religious freedom laws as "jihad." But there's no reason to give Cruz the benefit of the doubt here. He's clearly using "jihad" intentionally in order to arouse his audience's inherent dislike and distrust in Muslims in order to make them even more mad at those "dern homosexers."
And while the definition of "jihad" is open to some debate, no doubt Cruz and his ilk believe that it means "gay married terrorists killing Jesus lovers for sport." And since he's speaking to a group of homeschoolers, said gay married terrorists were clearly trained in the public schools.
Because Cruz is apparently unaware of irony, after he laments the fact that LGBT people are kicking up a fuss over being singled out for discrimination, he then issues a call for unity—against gay people.
"We need to bring people together to the religious liberty values that built this country. It wasn't long ago when this was an area of bipartisan agreement," he said. "It used to be Democrats and Republicans, when it came to the First Amendment, we all stood together and said, 'Of course every one of us has a God-given right to seek out and worship God and to live according to our faith and our conscience and our belief.'"
Mind you, when he's talking about the First Amendment, he means the right to the free exercise of religion, which specifically means the right to discriminate against icky LGBT people because Jesus said so.
He's claiming discrimination as a God given right.
And this is where the old me would have called Cruz a dick. But the new me will not stoop so low. The new me declares that Cruz is a cancer. He is a tumor in the asshole of America, full of shit and recklessly multiplying and growing without regard to his host's life or well-being. The good news is that he's been detected fairly early. The bad news is that he's convinced a lot of Americans, for whom religion trumps science, that he's actually good for them. And so he grows.
D'Anne Witkowski
My View: From a Broad – Left Out
4.23.15 Editor's Desk
Homo Erectus: Not Show Ready
The Wonderful World of Wanzie: WAVEs of My Own
Guest Column: Transgender people need non-trans advocates


We need a framework for interactive content in Wikipedia.
Add support for an interactive web framework to MediaWiki. This framework would preferrably be open, making OpenLaszlo (, based on Flash) and Java () the most promising candidates.
Which framework best fits the interactivity requirements of an encyclopedia?
How can interactivity be used to improve encyclopedic content?
For an example proving that interactivity can greatly enhance encyclopedic content, see  (Java applets used to provide rotatable models of the polyhedron).
See Proposal:Foundations_for_Interactive_Articles for a more theoretical proposal with the same general idea of adding interactivity.
Do you have a thought about this proposal? A suggestion? Discuss this proposal by going to Proposal talk:Add OpenLaszlo/Java support to make interactive content possible.


text:
Parafia Zaśnięcia Matki Bożej – prawosławna parafia katedralna w Wilnie, w eparchii wileńskiej i litewskiej.

Powołana po ponownym przejęciu przez prawosławnych soboru Zaśnięcia Matki Bożej w Wilnie w 1865. Działała do wejścia Armii Czerwonej na Litwę w 1940. Ponownie zarejestrowana po II wojnie światowej w 1948, wówczas należały do niej 424 osoby.

W 2012 r. z parafii Zaśnięcia Matki Bożej wydzielono parafię św. Paraskiewy.

Przypisy

Bibliografia
 Opis  i dane teleadresowe na stronie eparchii wileńskiej i litewskiej

Wilno
Zaśnięcia

redpajama_set_name: RedPajamaWikipedia