text:
Q: One to One Relationship With Shared not Working at Execution time Im using EclipseLink(JPA 2.0) with MySql and I'm having a problem with something in the model.
I have 2 entities: Delincuente, PerfilFisico. They share the Delincuente's Pk.
The DB is working correctly but when i insert a Delincuente and then the Perfil Fisico,  i tried to show the PerfilFisico of that Delincuente (Using Delincuente.PerfilFisico.), it was just empty. I checked the tables at the DB and everything was ok.
If i run the system again (I'm developing in Netbeans 7.2) and i consult the Delincuente that i just registered on the DB, it shows all the fields and everything is perfect!!
There is the code:
Delincuente.java
@OneToOne(cascade = CascadeType.ALL, mappedBy = "delincuente")
private PerfilFisico perfilFisico;

PerfilFisico.java
@JoinColumn(name = "PK_IdDelincuente", referencedColumnName = "PK_Identificacion", insertable = false, updatable = false)
@OneToOne(optional = false)
private Delincuente delincuente;

What is wrong, i've been reading for days and i can't figure out which the problem is. Any tips?
As i said it works, but when you are running it and you register a new Delincuente, you have to re-run the System to succesfully complete the consult and see the PerfilFisico

A: JPA EntityManagers have a first level cache for managed entities, and providers like EclipseLink have a second level cache.  So if you don't maintain both sides of your bidirectional mappings so they are out of sync with the db, it won't show unless you force a refresh, clear the cache or restart.  The app must maintain bidirectional relationships, JPA won't do it for you, as they are meant to be pojos.


Join The Great Gnomeregan Run–October 13!
Are you a gnome? Do you know a gnome? Would you like to BE a gnome? (And who wouldn't?) Then rise UP to the challenge of The Great Gnomeregan Run on Saturday, October 13 from Gnomeregan to Booty Bay.
Whether you're a level 1 gnome (highly encouraged), a level 71 human, or a level 120 dark iron dwarf, you'll be able to join in this fun run beginning in the gnome starting area, winding through Dun Morogh, Ironforge, the Deeprun Tram, Stormwind, Elwynn Forest, Duskwood, and Stranglethorn Vale where the run culminates in Booty Bay.
It's a big big BIG world out there, but don't worry about getting lost, there will be course markers and spectators along the way to help keep you on track. You'll even see a few familiar faces among the onlookers!
Stranglethorn Vale is a dangerous place for low-level characters so you may want to find some muscle to join you along the way for a little extra protection. If you're a higher-level character, think about playing chaperone for a gnome.
Whether short or tall, it will be a good time for all.


text:
#include "metaTypes.h"

#ifndef ITKMetaIO_METASURFACE_H
#define ITKMetaIO_METASURFACE_H

#include "metaUtils.h"
#include "metaObject.h"

#ifdef _MSC_VER
#pragma warning ( disable: 4251 )
#endif

#include <list>

/*!    MetaSurface (.h and .cxx)
 *
 * Description:
 *    Reads and Writes MetaSurfaceFiles.
 *
 * \author Julien Jomier
 *
 * \date July 02, 2002
 *
 */

#if (METAIO_USE_NAMESPACE)
namespace METAIO_NAMESPACE {
#endif

class SurfacePnt
{
public:

  SurfacePnt(int dim);
  ~SurfacePnt();

  unsigned int m_Dim;
  float* m_X;
  float* m_V;
  float  m_Color[4];
};

class METAIO_EXPORT MetaSurface : public MetaObject
  {

  /////
  //
  // PUBLIC
  //
  ////
  public:

   typedef METAIO_STL::list<SurfacePnt*> PointListType;
    ////
    //
    // Constructors & Destructor
    //
    ////
    MetaSurface(void);

    MetaSurface(const char *_headerName);

    MetaSurface(const MetaSurface *_surface);

    MetaSurface(unsigned int dim);

    ~MetaSurface(void) MET_OVERRIDE;

    void PrintInfo(void) const MET_OVERRIDE;

    void CopyInfo(const MetaObject * _object) MET_OVERRIDE;

    //    NPoints(...)
    //       Required Field
    //       Number of points which compose the tube
    void  NPoints(int npnt);
    int   NPoints(void) const;

    //    PointDim(...)
    //       Required Field
    //       Definition of points
    void        PointDim(const char* pointDim);
    const char* PointDim(void) const;

    void  Clear(void) MET_OVERRIDE;

    PointListType & GetPoints(void) {return m_PointList;}
    const PointListType & GetPoints(void) const {return m_PointList;}

    MET_ValueEnumType ElementType(void) const;
    void  ElementType(MET_ValueEnumType _elementType);

  ////
  //
  // PROTECTED
  //
  ////
  protected:

    bool  m_ElementByteOrderMSB;

    void  M_Destroy(void) MET_OVERRIDE;

    void  M_SetupReadFields(void) MET_OVERRIDE;

    void  M_SetupWriteFields(void) MET_OVERRIDE;

    bool  M_Read(void) MET_OVERRIDE;

    bool  M_Write(void) MET_OVERRIDE;

    int m_NPoints;      // "NPoints = "         0

    char m_PointDim[255]; // "PointDim = "       "x y z r"

    PointListType m_PointList;

    MET_ValueEnumType m_ElementType;

  };

#if (METAIO_USE_NAMESPACE)
};
#endif

#endif

redpajama_set_name: RedPajamaGithub


Premier Gladys Berejiklian today announced $100,000 in funding to help the Sydney Institute of Marine Science (SIMS) to expand its Chowder Bay, Mosman, location.
Ms Berejiklian said the funds would help the Institute develop a business plan for its proposed Centre for Urban Marine Innovation, which will be a global hub for integrated urban marine research, teaching and innovation.
"Sydney has the world's best harbour and the NSW Government recognises that the critical research SIMS is undertaking in marine biodiversity and management will benefit future generations of Sydneysiders," Ms Berejiklian said.
Ms Berejiklian said the NSW Government and SIMS would work with the Sydney Harbour Federation Trust on finding an appropriate location for the new Centre within the Trust's lands.
Investing $9.1 million in fisheries science and research to guide the sustainable management of the marine environment, including fisheries.
A $100,000 grant to SIMS World Harbour Project, an extension of the Sydney Harbour Research program.


One of the ways that one can succeed in inbound marketing is when one focuses on a target audiences needs. A brand should convince their target audience that the products and services that they selling are the best and this can be done by using inbound marketing. One of the ways to share this information about products and services with one's potential customers is through quality content. Inbound marketing enables one to work on the perspective of a customer in order to cause them to change their thinking about one's products and services and to see the benefits of purchasing a brand's products and services.
One of the tools that is used for inbound marketing includes search engine optimisation. Products and services that are provided by a brand can easily gain popularity when one uses social media to advertise their products and services. Brands who want to attract potential customers through social media may benefit if the current customers share their products and services with some of their friends and family members. Social media can be used to put up content such as videos, graphics, among others that can be able to capture the attention of potential customers. Social media is also a place where one can decide to pay for advertising to make a brand known to potential customers.
Brands also need to have websites where they can show more of their products and services and also put useful information for audiences. When a client chooses to use a website for inbound marketing, they should make sure that they get a good website design. Through email marketing, one can send additional information to customers after one gets their email addresses as a strategy to make people one's customers. One can hire a marketing company which does inbound marketing and they will come up with a good strategy for one's business.
One should not spend a lot of time trying to do inbound marketing when one does not have the skills for this since one can be able to hire experts at inbound marketing. Experience is one of the things that one should look at when one is looking for experts who offer inbound marketing services. Every business has different needs and one may need a tailor made solutions for their business when they need inbound marketing services so one should look for experts who can be able to provide tailor-made solutions. In case one is interested to get inbound marketing services for their business, it can be beneficial to look for a marketing company with a track record of success from their previous clients. Before hiring a team of experts to do inbound marketing, one may need to consider the cost of their services.


text:
Q: Click on a navbar button and go to the actual div? Imagine a nav bar with different options (Home, Contact, Products, etc.), how can i do to click on Contact, and it goes to Contact on the website ? I mean, it goes down to find the right div.
Thanks a lot for your answers, if you need more infos, tell me !

A: On your contact div add following link:

Contact

and on div element you scrolling to give it same id:

its called Anchor

redpajama_set_name: RedPajamaStackExchange


text:
Academic story
Quantum Computers Will Change Everything. Including Our Security Needs.
When large scale quantum computers become a reality, they will outperform today's most powerful supercomputers and rapidly advance fields like artificial intelligence, drug development, finance. They will also render our current data and communications security techniques obsolete.
"Certain quantum algorithms have been proposed that will be able to break all of our security today," says Professor Máire O'Neill, one of Europe's leading cyber security experts and Regius Professor in Electronics and Computer Engineering at Queen's University Belfast. "We know from past experience that it takes about 15 to 20 years to properly integrate new security techniques so we can't wait until there's a breakthrough in quantum technology. We need to be working on the alternative security system now."
Professor Máire O'Neill (image courtesy of Queen's University Belfast)
Máire is one of the researchers working to develop new security approaches that can protect our electronic information from the threat of quantum computers, a field called post-quantum cryptography. She recently led a European consortium of eight academic and industry partners that looked at how lattice-based cryptography could be adapted to withstand quantum attacks. The group wanted to find out if it would be possible to plug and replace today's security techniques with more complex ones. One of the consortium partners succeeded in developing several algorithms, three of which are now in the running to be standardised by the National Institute of Standards and Technology. Máire is confident that the consortium's algorithms will be among those standardised to protect everything from microchips to satellite communications from quantum hackers. Currently, she's working alongside the UK Quantum Communications Hub and the National Cyber Security Center to investigate how lattice-based cryptographic algorithms can be scaled to protect more types of electronic information.
Máire has been a proponent of commercialising research for societal benefit since her undergraduate days when she did a summer placement at a local cybersecurity startup in Belfast. Now as a professor, she continues to pursue research that can be successfully translated into useful mainstream technology. She is the director of the Institute of Electronics, Communications & Information Technology (ECIT) at Queen's, which was established to conduct high-quality research as well as promote the transfer of research to benefit society. The world-class institute has really put Belfast on the map in terms of digital innovation. It provides researchers with state-of-the-art equipment including numerous test beds for hardware security research, the latest equipment for side channel analysis, and a newly-developed cyber range lab.
But what really makes ECIT stand out in Máire's mind compared to other UK universities is the industry-experienced engineers who work at the institute and help facilitate the commercialisation of research. "The engineers bridge the gap between what we do and economic and societal benefit from research. They develop proof-of-concept demonstrators from our research. Rather than handing an industry partner a paper and saying, 'this is a great idea, go sell it,' it's much more convincing to show them the idea actually working in practice on a demonstration device," says Máire. ECIT also has a dedicated business development team who have deep domain knowledge and can help engage industry partners, as well as an innovation manager who assists students and staff who are considering starting spin outs.
Máire and Professor Ian Greer, the Vice-Chancellor of Queen's University Belfast, who awarded her a rare and prestigious Regius Professorship in January 2020. (image courtesy of Queen's University Belfast)
The next few years will be exciting ones for Queen's and ECIT. As part of a Belfast region city deal, ECIT will be developed and scaled into a £50M Global Innovation Institute. It will bring together security, wireless communications, and data science, as well as health science and global food security, to help solve large-scale societal problems. "One of our focus areas will be the 'One Health' agenda, so looking at how we can track and trace health data right through from health in soil, to animal and human for better nutrition and environment. This has the potential to bring personalised medicine or precision medicine to a whole new level," Máire explains. The new institute will be a key part of creating a new Innovation District in Belfast. "It's a huge investment for the city. It's a really exciting time to be at Queen's."
Header image courtesy of Queen's University Belfast
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QUB is a member of the Russell Group of 24 leading UK research-intensive universities, providing world-class education underpinned by world-class research.
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Prof. Máire O'Neill
Prof. O'Neill is the Regius Professor in Electronics and Computer Engineering at Queen's University Belfast. She was the youngest person to be made a Professor of Engineering at Queen's and youngest person to be inducted into the Irish Academy of Engineering. She has received numerous awards, including a Blavatnik Engineering and Physical Sciences medal, a Royal Academy of Engineering Silver Medal, and British Female Inventor of the Year.
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redpajama_set_name: RedPajamaCommonCrawl


The official website for Digimon Adventure announced on Wednesday that the " Digimon Thanksgiving 2018" event on July 29 will have a "Special Meeting and Latest Game Announcement" presentation in which the staff will unveil a new Digimon game project. The presentation will begin at 6:00 p.m. JST.
The presentation will also have an anime part that will include a talkshow with anime cast members. It will stream live on YouTube and Niconico.
Bandai Namco Entertainment producer Kazumasa Habu revealed in a December interview that a new Digimon Story game was in the works for PlayStation 4. The latest game in the series, Digimon Story Cyber Sleuth - Hacker's Memory , shipped in Japan in December 2017 and in the West in January. The game is available for PS4 and PlayStation Vita, but was developed for PS Vita using the previous Digimon Story Cyber Sleuth game as a base.


Hello, yoga mat! How I have missed you! It feels amazing to get back onto the mat! I took a short break while doing Whole30. Too much stuff to focus on, and I don't want to stress myself out with so many challenges.
Backbends are the ultimate worst thing for me. I am not a fan, but that is my goal in 2015. I want to get better at leaning back - way back. So, I was practicing my forearm balance yesterday, and decided to try a little Scorpion Pose. This felt AMAZING on my back. I love using the wall for support - it helps me with balancing, and it's always there when I fall out of the pose!
Can't wait to keep practicing this, and getting bendy with this!


In my 'career' of a tea lover, there is one question I have been asked more often than any other: what is my favourite tea? Well, a mother cannot favour one of her children, so of course, NO I cannot pick (only) one, and I do really love them all :). HOWEVER, in the top 20 of all time, there would definitely be a spot for matcha & matcha latte.
HA? Matcha tea? What on earth is that? – you'd ask. Sadly, this incredible type of green tea is still fairly unkown in the West, despite being probably one of, if not the healthiest & fairest of them all. So, as this is a NEW start for my tea blog at the beginning of a NEW year, I have decided to hopefully make you all familiar with this super-delicious yet – till now – NEW to you form of green tea & chose it to be the subject of my very first 2016 post.
Matcha (meaning 'powdered tea'), in a quickest nutshell, is… powdered green tea :). And matcha latte is the only officially accepted combination of green tea & milk (though of course if you enjoy adding anything incl.milk to your cup of green tea, you are free and welcome to do so any time 🙂 ).
Oh but there is so much more about it! You could write hollywood blockbuster scripts about matcha & its preparation. Traditional famous Japanese tea ceremony centers around matcha giving it indeed a very special place in the culture of tea. Not surprising, considering that due to the way it is grown & processed, it is probably indeed the healthiest tea you could ever drink.
When drinking tea from other leaves, you get to consume, according to certain sources, about 5-10% of all the tea goodness. The rest, in your teabag or strainer, gets thrown away & lost for ever. Matcha is tea leaves, without stems, ground on stones to a fine powder so drinking it, you get to taste & keep the whole of what green tea has to offer. And to achieve the very special subtle taste that only matcha has, together with better flavour and texture, tea plants are grown under a shade for about three weeks before the harvest. This gives the leaves a chance to produce even more chlorophyll and amino acids, and the end result – a cup of matcha – a lovely brighter green color (so positive – how not to love it? 🙂 ). One cup of matcha has as many antioxidants as 10!!! cups of green tea!, and in comparison to spinach for example, it can have up to 150 times more of the good stuff!
Thanks to all its properties, matcha is proven to boost your immune system, improve concentration, & help you detoxify! Many sources say it has also been said to help regulate blood sugar and metabolism. You can see a whole beautiful list of matcha's benefits here.
AND – as if it all wasn't enough – matcha is not only limited to tea, but thanks to its powder form, is often used in food & drink recipes. I strongly recommend trying matcha ice cream… be careful though, as it is addictive. Most recently, matcha gets more and used in many delicious non & yes-alcoholic cocktails.
For some gorgeous green cocktails & mocktails head on to Matcha cocktails – so yummy!
Matcha is so popular in Asia, I remember having it in every corner Starbucks in Malaysia, when I worked there a few years ago. I was told in Japan, every coffee / tea shop has an on site matcha grinding machine which on the spot prepares fresh matcha powder, just like all espresso machines grinding coffee beans in every coffee shop in London. I cannot wait to visit that part of Asia & see it & TASTE it one day! In London, matcha latte can be found in lovely Yumchaa tea shops all around central London, but also in the independent and lovely Tea Box cafe in Richmond (which also organises music events combining tea & music!).


text:
Микілківська сільська рада — колишній орган місцевого самоврядування у Котелевському районі Полтавської області з центром у селі Микілка.

Населені пункти

Сільраді були підпорядковані населені пункти:

 c. Микілка
 с. Гетьманка
 с. Зайці
 с. Терни
 с. Шевченкове

Посилання

 Микілківська сільська рада на сайті Верховної Ради України

redpajama_set_name: RedPajamaWikipedia


In several ways, a CM Punk-Daniel Bryan WWE Championship Match was always fated to be. The two Superstars are among the purest athletes and finest technicians WWE has ever seen. Their championship reigns were some of the most dominant in late 2011 and early 2012, and they come from a hardscrabble, old-school wrestling background that has endeared them to the WWE Universe in a way that is not typically seen amongst modern Superstars. And now, at Over the Limit, they will finally meet with the supreme prize on the line.
The clash is sure to be a memorable one: each time these two have met in the ring, the results have been electric. The two wrestled a fierce sequence of Champion vs. Champion Matches earlier this year, when Bryan still held the World Heavyweight Title, and their paths have crossed numerous times over 10 years in smaller, independent promotions all over the world.
Bryan seemed to be out of the world title picture after a pair of losses to Sheamus – the infamous 18-second defeat that cost him the World Heavyweight Championship at WrestleMania, and a grueling 2-out-of-3-Falls rematch a few weeks later at Extreme Rules. But the upstart Superstar was actually emboldened by his effort in the rematch, which saw him trap The Great White in the "Yes!" Lock and render him unconscious in the process. Bryan's bravado carried him through a Beat the Clock Challenge on the following night's Raw, when he beat times set by Randy Orton and The Miz to earn the right to challenge for Punk's mantle.
Immediately after learning of his new No. 1 contender, The Second City Saint made his presence known. Punk sized up his future opponent from the stage and offered a round of applause in approval for his opponent before holding his title aloft in defiance of his challenger. Bryan responded with a mocking gesture of his own, which is particularly fitting given the rivalry between these two: neither will budge, neither will back down. But only time will tell who is left standing with the ultimate prize in hand.
At WWE Over the Limit, those paths intersected.
With arguably WWE's two most technically proficient grapplers clashing head-on, the match was projected from the outset to be a wrestling purist's dream, and it more than lived up to those lofty expectations. Despite their obvious scientific wrestling prowess, neither the 6-foot-2 Punk nor the 5-foot-11 Bryan has ever been confused for anything resembling a WWE poster boy.
Instead of boasting larger-than-life chiseled physiques, Punk and Bryan have established themselves as ring generals, and at WWE Over the Limit, vindication for their hard work – in the form of the ultimate symbol of excellence in sports-entertainment, the WWE Championship – was up for grabs.
Although the title bout seemed forged by fate, it actually originated out of circumstance, when Bryan won a Beat the Clock Challenge on Raw SuperShow to claim his spot as the No. 1 contender. Since then, Bryan made every attempt to score an advantage on The Second City Saint. Bryan's latest offense, an act of trickery that caused an irate Kane to brutalize Punk with a steel chair and two chokeslams following a match on SmackDown, appeared to have done the trick.
Punk's words, it turned out, would prove prescient.
At the sound of the opening bell, Punk and Bryan began the contest with a feeling-out process in which they peppered each other with sharp kicks. The Second City Saint quickly went on the offensive, using submission holds – including an Indian Death Lock – against the self-proclaimed "submission specialist." Throughout the bout, Punk would return again and again to Bryan's left leg.
After avoiding a Punk baseball slide, however, Bryan rammed the WWE Champion back-first into the ringside barrier, further weakening Punk's ribs, which appeared to have not yet healed from Kane's attack Friday.
From there, Bryan attacked with pit bull-like tenacity. He kneed Punk's midsection and used a torso-wrenching abdominal stretch to great effect.
Never one to be counted down, The Straight Edge Superstar would again turn the tables on his storied foe, reversing a Bryan kick into a dragon-screw leg whip before locking him into a seemingly inescapable figure-four – causing more damage to Bryan's already smarting knee.
"How many near falls have there been in this match, not to mention the submission predicaments?" Jerry "The King" Lawler wondered aloud, as Punk and Bryan warred on to the sounds of the WWE Universe's dueling chants of "Daniel Bryan" and "CM Punk" (and, occasionally, "Yes!" and "No!").
Punk and Bryan's intimate knowledge of each other's move sets was obvious, as they reversed signature holds and tried valiantly to check kicks and block elbow strikes. As Punk built momentum, he nailed his trademark high running knee in the corner.
Yet, when Punk tried for the second half of his patented combo, the running bulldog, his effort was blocked by Bryan and reversed into a "Yes!" Lock.
Although the lock was clamped on for several moments, the WWE Champion nonetheless had the wherewithal to use Bryan's momentum against him. As Bryan applied more pressure and leaned back on the hold, Punk turned the dangerous lock into a cradle for a stunning pinfall at the 24-minute mark.
The win, however, was not without controversy, as almost simultaneous with the pinfall, Punk visibly tapped out.
With Punk barely able to hold his championship overhead in an exhausted celebration, Bryan – who clearly believed he was wronged – jawed angrily with the referee.


LISTING IS FOR ONE STICK.
Selenite is not only a protective stone that will keep the negative energies away, but it can also heal your body with the healing energies that it possesses, especially if you use it with some of the Orange Stones.
It's also a great stone to have for the attraction of wealth, prosperity, and abundance.
Selenite crystals help to remove all energy blockages in the body. Place a selenite wand on any area of the body that feels sore or tight, for about 20 minutes, and you will feel the difference.
To cleanse your aura/energy - take a selenite wand and use it like it's an eraser. Comb the crystal up and down your body, visualizing it sucking all the negativity, anxiety and worries from your day away. Make sure you start at the top of your head, moving to the bottom of your feet, about 5 inches away from the body, and let the Selenite healing crystal energetically cleanse you.
This simple technique is extremely powerful to do, in less than 5 minutes, if you are around large groups of people or find yourself around negative people or environments. This energy clearing technique helps to cleanse and purify your aura, making sure that you don't pick up on other peoples' energy and unwanted "stuff".


text:
Violin Sonatas Vol. 1
Rued Langgaard (1893-1952) is unique in the history of Danish music - a loner whose music, which has only become generally known in recent years, ranges stylistically from Romantic pastiche to inter-war Modernism. The uniqueness consists of the expressiveness of the music, the visionary ideas and the experimental and sometimes extreme musical forms, which in some cases have turned out to point far beyond the composer's own time. Most stylistic aspects are represented in Langgaard's works for violin and piano, two important representatives of which are presented here: the youthful, classically oriented Sonata No. 1, and the experimental one-movement Sonata No. 2 - one of the composer's major works - with a religious-philosophical programme.
Rued Langgaard
Violin Sonata No. 1 "Viole", BVN 94 (1915, rev. 1945)
I Maestoso fiero espressivo 13:33
II Molto allegro scherzando 6:41
III Lento cogitoso 8:23
IV Molto allegro marcato 10:06
Violin Sonata No. 2 "Den store Mester kommer", BVN 16
I Poco animato con devozione - Allegro - 5:54
II Frenetico - Poco più tranquillo - 4:24
III Lento rigoroso - 2:28
IV Grave - Poco tranquillo - 5:49
V Femminile e virile e con lustro 7:41
The String Quartets
Heralded the arrival of a major new ensemble and won the maverick Danish composer thousands of new fans
Complete Works for Violin and Piano, Vol. 2
Violin Sonata No. 1 accompanied by Langgaard's last violin work, the Andante Religioso, and the Sunday Sonate, one of his most bizarre works
Rued Langgaard, Jacob Gade
Symphonies 2 & 6
With the tradition-conscious Vienna Philharmonic, Langgaard's music 'returns home' to a central European musical culture
Poul Ruders, Per Nørgård, Pelle Gudmundsen-Holmgreen, Rued Langgaard, Vagn Holmboe
Vildtvoksende fantasier (Imagination run riot)
Thirteen stories for all children with a love of adventure.
Complete Works for Violin and Piano Vol. 1
The early 'Unfinished sonata' and three late, incitingly Romantic works performed by Gunvor Sihm and Berit Johansen Tange


text:
Q: What is time complexity of recursive level order traversal of a binary tree In THIS geeksforgeeks link they are describing the time complexity of recursive level order traversal as O(n^2).
Time Complexity: O(n^2) in worst case. For a skewed tree, printGivenLevel() takes O(n) time where n is the number of nodes in the skewed tree. So time complexity of printLevelOrder() is O(n) + O(n-1) + O(n-2) + .. + O(1) which is O(n^2).This is not clear to me.
Can someone please help me understand.

A: For a skewed tree like this:
1
 \
  2
   \
   ...
     \
      N

The depth of this tree is N, so the below function will run from 1 to N,
printLevelorder(tree)
for d = 1 to height(tree)
   printGivenLevel(tree, d);

That is,
printGivenLevel(tree, 1);
printGivenLevel(tree, 2);
...
printGivenLevel(tree, N);

And printGivenLevel(tree, depth) takes O(depth) time since it begins from the root every time.
The time complexity is:
O(1) + O(2) + ... + O(N) = O(N^2)

A: Sure.
Notice that this is an arithmetic progression. We know that it will sum as follows:
n + (n - 1) + (n - 2) + ... + 1 = n * (n - 1) / 2

But:
n * (n - 1) / 2 = (n^2 - n) / 2

However, we know that the quadratic (squared) term dominates the expression compared to the linear term and that the 1 / 2 is a constant factor, both of which simplify the expression as follows:
First, drop the constant factor:
O((n^2 - n) / 2) = O(n^2 - n)

Next, keep the dominant term:
O(n^2 - n) = O(n^2)

This is how you arrive at this complexity.


text:
\section{Introduction}
Central to any form of learning is an inductive bias that induces some
sort of capacity control (i.e.~restricts or encourages predictors to
be ``simple'' in some way), which in turn allows for generalization.
The success of learning then depends on how well the inductive bias
captures reality (i.e.~how expressive is the hypothesis class of
``simple'' predictors) relative to the capacity induced, as well as on
the computational complexity of fitting a ``simple'' predictor to the
training data.

Let us consider learning with feed-forward networks from this
perspective.  If we search for the weights minimizing the training
error, we are essentially considering the hypothesis class of
predictors representable with different weight vectors, typically for
some fixed architecture.  Capacity is then controlled by the size
(number of weights) of the network\footnote{The exact correspondence
  depends on the activation function---for hard thresholding
  activation the pseudo-dimension, and hence sample complexity, scales
  as $O(S \log S)$, where $S$ is the number of weights in the network.
  With sigmoidal activation it is between $\Omega(S^2)$ and
  $O(S^4)$ \citep{anthony1999}.}.  Our justification for using such networks is
then that many interesting and realistic functions can be represented
by not-too-large (and hence bounded capacity) feed-forward networks.
Indeed, in many cases we can show how specific architectures can
capture desired behaviors.  More broadly, any $O(T)$ time computable
function can be captured by an $O(T^2)$ sized network, and so the
expressive power of such networks is indeed great~\cite[Theorem~9.25]{sipser2012}.

At the same time, we also know that learning even moderately sized
networks is computationally intractable---not only is it NP-hard to
minimize the empirical error, even with only three hidden units, but
it is hard to learn small feed-forward networks using {\em any}
learning method (subject to cryptographic assumptions).  That is, even
for binary classification using a network with a single hidden layer and a logarithmic (in the
input size) number of hidden units, and even if we know the true
targets are {\em exactly} captured by such a small network, there is
likely no efficient algorithm that can ensure error better than 1/2
\citep{klivans2006cryptographic,Daniely14}---not if the algorithm
tries to fit such a network, not even if it tries to fit a much larger
network, and in fact no matter how the algorithm represents
predictors (see the Appendix).  And so, merely knowing that some not-too-large
architecture is excellent in expressing reality does {\em not} explain
why we are able to learn using it, nor using an even larger network.
Why is it then that we succeed in learning using multilayer
feed-forward networks?  Can we identify a property that makes them
possible to learn?  An alternative inductive bias?

Here, we make our first steps at shedding light on this question by
going back to our understanding of network size as the capacity
control at play.

Our main observation, based on empirical experimentation with
single-hidden-layer networks of increasing size (increasing number of
hidden units), is that size does {\em not} behave as a capacity
control parameter, and in fact there must be some other, implicit,
capacity control at play.  We suggest that this hidden capacity
control might be the real inductive bias when learning with deep
networks.

In order to try to gain an understanding at the possible inductive
bias, we draw an analogy to matrix factorization and understand
dimensionality versus norm control there.  Based on this analogy we
suggest that implicit norm regularization might be central also for
deep learning, and also there we should think of infinite-sized
bounded-norm models.  We then also demonstrate how (implicit) $\ell_2$
weight decay in an infinite two-layer network gives rise to a ``convex
neural net'', with an infinite hidden layer and $\ell_1$ (not $\ell_2$)
regularization in the top layer.

\section{Network Size and Generalization}\label{sec:netsize}

Consider training a feed-forward network by finding the weights
minimizing the training error.  Specifically, we will consider a
network with $d$ real-valued inputs $\boldsymbol{x}=(x[1],\ldots,x[d])$, a single
hidden layer with $H$ rectified linear units, and $k$ outputs
$y[1],\ldots,y[k]$,
\begin{equation}
  \label{eq:yk}
  y[j] = \sum_{h=1}^H v_{hj} [\inner{\boldsymbol{u}_h}{\boldsymbol{x}}]_+
\end{equation}
where $[z]_{+}:=\max(z,0)$ is the rectified linear activation function
and $\boldsymbol{u}_h\in\mathbb{R}^d, v_{hj}\in\mathbb{R}$ are the weights learned by minimizing
a (truncated) soft-max cross entropy loss\footnote{When using soft-max
  cross-entropy, the loss is never exactly zero for correct
  predictions with finite margins/confidences.  Instead, if the data
  is seperable, in order to minimize the loss the weights need to be
  scaled up toward infinity and the cross entropy loss goes to zero,
  and a global minimum is never attained.  In order to be able to say
  that we are actually reaching a zero loss solution, and hence a
  global minimum, we use a slightly modified soft-max which does not
  noticeably change the results in practice.  This truncated loss
  returns the same exact value for wrong predictions or correct
  prediction with confidences less than a threshold but returns zero
  for correct predictions with large enough margins: Let
  $\{s_i\}_{i=1}^k$ be the scores for $k$ possible labels and $c$ be
  the correct labels. Then the soft-max cross-entropy loss can be
  written as $\ell(s,c) = \ln \sum_{i} \exp(s_i - s_c)$ but we instead
  use the differentiable loss function $\hat{\ell}(s,c) = \ln \sum_{i}
  f(s_i-s_c)$ where $f(x)=\exp(x)$ for $x\geq -11$ and $f(x)
  =\exp(-11) [x+13]_+^2/4$ otherwise. Therefore, we only deviate from
  the soft-max cross-entropy when the margin is more than $11$, at
  which point the effect of this deviation is negligible (we always
  have $\abs{\ell(s,c)-\hat{\ell}(s,c)}\leq 0.000003k$)---if there are
  any actual errors the behavior on them would completely dominate
  correct examples with margin over $11$, and if there are no errors
  we are just capping the amount by which we need to scale up the
  weights.} on $n$ labeled training examples.  The total number of
weights is then $H(d+k)$.

What happens to the training and test errors when we increase the
network size $H$? The training error will necessarily decrease.  The
test error might initially decrease as the approximation error is
reduced and the network is better able to capture the targets.
However, as the size increases further, we loose our capacity control
and generalization ability, and should start overfitting.  This is the
classic approximation-estimation tradeoff behavior.

Consider, however, the results shown in Figure \ref{fig:intro}, where
we trained networks of increasing size on the MNIST and CIFAR-10
datasets.  Training was done using stochastic gradient descent with
momentum and diminishing step sizes, on the training error and without
any explicit regularization.  As expected, both training and test
error initially decrease.  More surprising is that if we increase the
size of the network past the size required to achieve zero training
error, the test error continues decreasing!  This behavior is not at
all predicted by, and even contrary to, viewing learning as fitting a
hypothesis class controlled by network size.  For example for MNIST, 32 units
are enough to attain zero training error.  When we allow more units,
the network is not fitting the training data any better, but the
estimation error, and hence the generalization error, should increase
with the increase in capacity.  However, the test error goes down.  In
fact, as we add more and more parameters, even beyond the number
of training examples, the generalization error does not go up.

\begin{figure*}[t!]
\hbox{ \centering
\setlength{\epsfxsize}{0.455\textwidth}
\epsfbox{line-mnist-1.eps}
\hspace{0.1in}
\setlength{\epsfxsize}{0.45\textwidth}
\epsfbox{line-cifar-1.eps}
}
\begin{picture}(0,0)(0,0)
\put(85, 155){MNIST}
\put(280, 155){CIFAR-10}
\end{picture}
\vspace{0.1in}
\caption{\small The training error and the test error based on different stopping
  criteria when 2-layer NNs with different number of hidden units
  are trained on MNIST and CIFAR-10. Images in both datasets are downsampled
  to 100 pixels. The size of the training set is 50000 for MNIST and 40000 for CIFAR-10.
  The early stopping is based on the error on a validation set
  (separate from the training and test sets) of size 10000. The training was
  done using stochastic gradient descent with momentum and mini-batches
  of size 100. The network was initialized with weights generated randomly from
  the Gaussian distribution. The initial step size and momentum were set to 0.1 and 0.5
  respectively. After each epoch, we used the update rule $\mu^{(t+1)}=0.99\mu^{(t)}$
  for the step size and $m^{(t+1)}=\min\{0.9,m^{(t)}+0.02\}$ for the momentum.
  \label{fig:intro}
}
\end{figure*}

We also further tested this phenomena under some artificial
mutilations to the data set.  First, we wanted to artificially ensure
that the approximation error was indeed zero and does not decrease as
we add more units.  To this end, we first trained a network with a
small number $H_0$ of hidden units ($H_0=4$ on MNIST and $H_0=16$ on
CIFAR) on the entire dataset (train+test+validation).  This network
did have some disagreements with the correct labels, but we then
switched all labels to agree with the network creating a ``censored''
data set.  We can think of this censored data as representing an
artificial source distribution which can be exactly captured by a
network with $H_0$ hidden units.  That is, the approximation error is zero
for networks with at least $H_0$ hidden units, and so does not
decrease further. Still, as can be seen in the middle row of
Figure~\ref{fig:netsize}, the test error continues decreasing even
after reaching zero training error.

Next, we tried to force overfitting by adding random label noise to
the data.  We wanted to see whether now the network will use its
higher capacity to try to fit the noise, thus hurting generalization.
However, as can be seen in the bottom row of Figure~\ref{fig:netsize},
even with five percent random labels, there is no significant
overfitting and test error continues decreasing as network size
increases past the size required for achieving zero training error.

\begin{figure*}[t!]
\hbox{ \centering
\setlength{\epsfxsize}{0.455\textwidth}
\epsfbox{line-mnist-21.eps}
\hspace{0.1in}
\setlength{\epsfxsize}{0.45\textwidth}
\epsfbox{line-cifar-21.eps}
}

\hbox{ \centering
\setlength{\epsfxsize}{0.455\textwidth}
\epsfbox{line-mnist-22.eps}
\hspace{0.1in}
\setlength{\epsfxsize}{0.45\textwidth}
\epsfbox{line-cifar-22.eps}
}

\hbox{ \centering
\setlength{\epsfxsize}{0.455\textwidth}
\epsfbox{line-mnist-23.eps}
\hspace{0.1in}
\setlength{\epsfxsize}{0.45\textwidth}
\epsfbox{line-cifar-23.eps}
}
\begin{picture}(0,0)(0,0)
\put(70, 445){Small MNIST}
\put(260, 445){Small CIFAR-10}
\end{picture}
\caption{\small The training error and the test error based on different stopping
  criteria when 2-layer NNs with different number of hidden units
  are trained on small subsets of MNIST and CIFAR-10. Images in both
  datasets are downsampled to 100 pixels. The sizes of the training
  and validation sets are 2000 for both MNIST and CIFAR-10 and the early stopping
  is based on the error on the validation set. The top plots are
  the errors for the original datasets with and without explicit regularization.The best
  weight decay parameter is chosen based on the validation error.
  The middle plots are on the censored data set that is constructed
  by switching all the labels to agree with the predictions of a trained network
  with a small number $H_0$ of hidden units  $H_0=4$ on MNIST and $H_0=16$
  on CIFAR-10) on the  entire dataset (train+test+validation).
  The plots on the bottom are also for the censored data except we also add 5 percent
  noise to the labels by randomly changing 5 percent of the labels.
  The optimization method is the same as the in Figure 1.  The results in this figure are the average error over 5 random repetitions.
\label{fig:netsize}
}
\end{figure*}

What is happening here?  A possible explanation is that the
optimization is introducing some implicit regularization.  That is, we are
implicitly trying to find a solution with small ``complexity'', for
some notion of complexity, perhaps norm.  This can explain why we do
not overfit even when the number of parameters is huge.  Furthermore,
increasing the number of units might allow for solutions that actually
have lower ``complexity'', and thus generalization better.  Perhaps an
ideal then would be an infinite network controlled only through this
hidden complexity.

We want to emphasize that we are not including any explicit
regularization, neither as an explicit penalty term nor by modifying
optimization through, e.g., drop-outs, weight decay, or with
one-pass stochastic methods.  We are using a stochastic method, but we
are running it to convergence---we achieve zero surrogate loss and zero training error. In fact, we also tried training using batch conjugate
gradient descent and observed almost identical behavior.  But it seems
that even still, we are not getting to some random global
minimum---indeed for large networks the vast majority of the many
global minima of the training error would horribly overfit.  Instead,
the optimization is directing us toward a ``low complexity'' global
minimum.

Although we do not know what this hidden notion of complexity is, as a
final experiment we tried to see the effect of adding explicit
regularization in the form of weight decay.  The results are shown in the top
row of figure~\ref{fig:netsize}. There is a slight improvement in generalization
but we still see that increasing the network size helps generalization.

\section{A Matrix Factorization Analogy}

To gain some understanding at what might be going on, let us consider
a slightly simpler model which we do understand much better.  Instead of
rectified linear activations, consider a feed-forward network with a
single hidden layer, and {\em linear} activations, i.e.:
\begin{equation}
  \label{eq:ykl}
  y[j] = \sum_{h=1}^H v_{hj} \inner{\boldsymbol{u}_h}{\boldsymbol{x}}.
\end{equation}
This is of course simply a matrix-factorization model, where $\boldsymbol{y}=\boldsymbol{W}\boldsymbol{x}$
and $\boldsymbol{W}=\boldsymbol{V}\boldsymbol{U}^\top$.  Controlling capacity by limiting the number of hidden
units exactly corresponds to constraining the rank of $\boldsymbol{W}$,
i.e.~biasing toward low dimensional factorizations.  Such a low-rank
inductive bias is indeed sensible, though computationally intractable
to handle with most loss functions.

However, in the last decade we have seen much success for learning
with low {\em norm} factorizations.  In such models, we do not
constrain the inner dimensionality $H$ of $\boldsymbol{U},\boldsymbol{V}$, and instead only
constrain, or regularize, their norm.  For example, constraining the
Frobenius norm of $\boldsymbol{U}$ and $\boldsymbol{V}$ corresponds to using the {\em
  trace-norm} as an inductive bias \citep{Srebro04}:
\begin{equation}\label{eq:tracenorm}
  \norm{\boldsymbol{W}}_{tr} = \min_{\boldsymbol{W}=\boldsymbol{V}\boldsymbol{U}^\top} \frac{1}{2}(\norm{\boldsymbol{U}}_F^2+\norm{\boldsymbol{V}}_F^2).
\end{equation}
Other norms of the factorization lead to different regularizers.

Unlike the rank, the trace-norm (as well as other factorization norms)
is convex, and leads to tractable learning problems
\citep{Fazel01,Srebro04}.  In fact, even if learning is done by a
local search over the factor matrices $\boldsymbol{U}$ and $\boldsymbol{V}$ (i.e.~by a local
search over the weights of the network), if the dimensionality is high
enough and the norm is regularized, we can ensure convergence to a
global minima \citep{Burer06}. This is in stark contrast to the
dimensionality-constrained low-rank situation, where the limiting
factor is the number of hidden units, and local minima are abundant
\citep{Srebro03}.

Furthermore, the trace-norm and other factorization norms are
well-justified as sensible inductive biases.  We can ensure
generalization based on having low trace-norm, and a low-trace norm
model corresponds to a realistic factor model with many factors of
limited overall influence.  In fact, empirical evidence suggests that
in many cases low-norm factorization are a more appropriate inductive
bias compared to low-rank models.

We see, then, that in the case of linear activations (i.e.~matrix
factorization), the norm of the factorization is in a sense a better
inductive bias than the number of weights: it ensures generalization,
it is grounded in reality, and it explain why the models can be learned
tractably.

Let us interpret the experimental results of Section \ref{sec:netsize} in this
light.  Perhaps learning is succeeding not because there is a good
representation of the targets with a small number of units, but rather
because there is a good representation with small overall norm, and
the optimization is implicitly biasing us toward low-norm models.
Such an inductive bias might potentially explain both the
generalization ability {\em and} the computational tractability of
learning, even using local search.

Under this interpretation, we really should be using infinite-sized
networks, with an infinite number of hidden units.  Fitting a finite
network (with implicit regularization) can be viewed as an
approximation to fitting the ``true'' infinite network.  This
situation is also common in matrix factorization: e.g., a very
successful approach for training low trace-norm models, and other
infinite-dimensional bounded-norm factorization models, is to
approximate them using a finite dimensional representation
\cite{rennie2005fast,srebro2010collaborative}.  The finite
dimensionality is then not used at all for capacity (statistical
complexity) control, but purely for computational reasons.  Indeed, increasing
the allowed dimensionality generally improves generalization performance, as it
allows us to better approximate the true infinite model.

\section{Infinite Size, Bounded Norm Networks}

In this final section, we consider a possible model for infinite sized
norm-regularized networks.  Our starting point is that of global weight decay,
i.e.~adding a regularization term that penalizes the sum of squares of
all weights in the network, as might be approximately introduced by
some implicit regularization.  Our result in this Section is that this
global $\ell_2$ regularization is equivalent to a  {\em Convex Neural
Network} (Convex NN; \cite{Bengio05})---an infinite network with $\ell_1$
regularization on the top layer.  Note that such models are rather
different from infinite networks with $\ell_2$ regularization on the
top layer, which reduce to linear methods with some specific
kernel~\citep{Cho09,Bach14}. Note also that our aim here is to explain what
neural networks are doing instead of trying to match the performance
of deep models with a known shallow model as done by, e.g., \citet{Lu14}.

For simplicity, we will focus on single output networks ($k=1$), i.e.~networks
which compute a function $f:\mathbb{R}^d\rightarrow\mathbb{R}$.  We first consider
finite two-layer networks (with a single hidden layer) and show that
$\ell_2$ regularization on both layers is equivalent to and $\ell_2$
constraint on each unit in the hidden layer, and $\ell_1$
regularization on the top unit:
\begin{theorem}\label{thm:L2toL1}
Let $L$ be a loss function and $D=(\boldsymbol{x}_t,y_t)_{t=1}^{n}$ be training examples.
\begin{align}\label{eq:weight-decay-net}
 \argmin{\boldsymbol{v}\in\mathbb{R}^{H},(\boldsymbol{u}_h)_{h=1}^{H}}\left(
\sum_{t=1}^{n}L\left(y_t, \sum\nolimits_{h=1}^{H}v_h[\inner{\boldsymbol{u}_h}{\boldsymbol{x}_t}]_+\right)+\frac{\lambda}{2}\sum_{h=1}^{H}\left(\norm{\boldsymbol{u}_h}^2+|v_h|^2\right)
\right),
\end{align}
is the same as
\begin{align}
\label{eq:learned-L1}
 \argmin{\boldsymbol{v}\in\mathbb{R}^{H},(\boldsymbol{u}_h)_{h=1}^{H}}\Biggl(
&\sum_{t=1}^{n}L\left(y_t,
 \sum\nolimits_{h=1}^{H}v_h[\inner{\boldsymbol{u}_h}{\boldsymbol{x}_t}]_+\right)+\lambda\sum_{h=1}^{H}|v_h|\Biggr),\\
& \text{subject to}\quad \|\boldsymbol{u}_h\|\leq 1\quad (h=1,\ldots,H).\notag
\end{align}
\end{theorem}
\begin{proof}
By the inequality between the arithmetic and
geometric means, we have
\begin{align*}
\frac{1}{2}\sum_{h=1}^{H}\left(\|\boldsymbol{u}_h\|^2+|v_h|^2\right)\geq \sum_{h=1}^{H}\|\boldsymbol{u}_h\|\cdot|v_h|.
\end{align*}
The right-hand side can always be attained without changing the
 input-output mapping by the rescaling
$\tilde{\boldsymbol{u}}_h=\sqrt{|v_h|/\|\boldsymbol{u}_h\|}\cdot\boldsymbol{u}_h$
 and $\tilde{v}_h=\sqrt{\|\boldsymbol{u}_h\|/|v_h|}\cdot v_h$.
The reason we can rescale the weights without changing
the input-output mapping is that the rectified linear unit is
 piece-wise linear and the piece a hidden unit is on is invariant to
rescaling of the weights.
Finally, since the right-hand side of the
 above inequality is invariant to rescaling, we can always choose
the norm $\|\boldsymbol{u}_h\|$ to be bounded by one.
\end{proof}

Now we establish a connection between the $\ell_1$-regularized network
\eqref{eq:learned-L1}, in which we learn the input to hidden
connections, to convex NN \citep{Bengio05}.

First we recall the definition of a convex NN. Let $\mathcal{U}$ be a fixed
``library'' of possible weight vectors $\{\boldsymbol{u}\in\mathbb{R}^{d}: \|\boldsymbol{u}\|\leq
1\}$ and $\mu_+$ and $\mu_-$ be positive (unnormalized) measures over
$\mathcal{U}$ representing the positive and negative part of the weights of
each unit. Then a ``convex neural net'' is given by predictions of the form
\begin{equation}
  \label{eq:predconvNN}
  y=\int_{\mathcal{U}}
\left({\rm d}\mu_{+}(\boldsymbol{u})-
{\rm d}\mu_{-}(\boldsymbol{u})\right)[\inner{\boldsymbol{u}}{\boldsymbol{x}}]_+
\end{equation}
with regularizer (i.e.~complexity measure) $\mu_{+}(\mathcal{U})+\mu_{-}(\mathcal{U})$.  This is simply an infinite generalization of network with hidden units $\mathcal{U}$ and $\ell_1$ regularization on the second layer: if $\mathcal{U}$ is finite, \eqref{eq:predconvNN} is equivalent to
\begin{equation}
  \label{eq:predconvNNfinite}
  y = \sum_{\boldsymbol{u}\in\mathcal{U}} v(\boldsymbol{u})[\inner{\boldsymbol{u}}{\boldsymbol{x}}]_+
\end{equation}
with $v(\boldsymbol{u}):=\mu_{+}(\boldsymbol{u})-\mu_{-}(\boldsymbol{u})$ and regularizer $\norm{v}_1=\sum_{\boldsymbol{u}\in\mathcal{U}}|v(\boldsymbol{u})|$.  Training a convex NN is then given by:
\begin{align}
\label{eq:fixed-L1}
 \argmin{\mu_+,\mu_-}\left(
\sum_{t=1}^{n}L\Bigl(y_t,\int_{\mathcal{U}}
\left({\rm d}\mu_{+}(\boldsymbol{u})-
{\rm d}\mu_{-}(\boldsymbol{u})\right)[\inner{\boldsymbol{u}}{\boldsymbol{x}_t}]_+\Bigr)+\lambda
\left(\mu_{+}(\mathcal{U})+\mu_{-}(\mathcal{U})\right)
\right).
\end{align}
Moreover, even if $\mathcal{U}$ is infinite and even continuous, there will always be an optimum of \eqref{eq:fixed-L1} which is a discrete measure with support at most $n+1$ \citep{Rosset07}.  That is, \eqref{eq:fixed-L1} can be equivalently written as:
\begin{align}\label{eq:fixed-L1-finite}
\argmin{\boldsymbol{u}_1,\ldots,\boldsymbol{u}_{n+1}\in\mathcal{U},\boldsymbol{v}\in\mathbb{R}^{n+1}}\left(
\sum_{t=1}^{n}L\Bigl(y_t,\sum_{h=1}^{n+1}
v_h[\inner{\boldsymbol{u}_h}{\boldsymbol{x}_t}]_+\Bigr)+\lambda \norm{\boldsymbol{v}}_1
\right),
\end{align}
which is the same as \eqref{eq:learned-L1}, with $\mathcal{U}=\left\{ \boldsymbol{u}\in\mathbb{R}^d \,\middle|\, \norm{\boldsymbol{u}}\leq 1 \right\}$.

The difference between the network \eqref{eq:learned-L1}, and the
infinite network \eqref{eq:fixed-L1} is in {\em learning} versus {\em
  selecting} the hidden units, and in that in \eqref{eq:fixed-L1} we
have no limit on the number of units used.  That is, in
\eqref{eq:fixed-L1} we have all possible units in $\mathcal{U}$ available to
us, and we merely need to select which we want to use, without any
constraint on the {\em number} of units used, only the over $\ell_1$
norm.  But the equivalence of \eqref{eq:fixed-L1} and
\eqref{eq:fixed-L1-finite} establishes that as long as the number of
allowed units $H$ is large enough, the two are equivalent:

\begin{corollary}
  As long as $H>n$, the weight-decay regularized network
  \eqref{eq:weight-decay-net} is equivalent to the convex neural net \eqref{eq:fixed-L1} with $\mathcal{U}=\left\{ \boldsymbol{u}\in\mathbb{R}^d \,\middle|\, \norm{\boldsymbol{u}}\leq 1 \right\}$.
\end{corollary}

In summary, learning and selecting is equivalent if we have sufficiently
many hidden units and Theorem \ref{thm:L2toL1} gives an alternative
justification for employing $\ell_1$ regularization when the input to
hidden weights are fixed and normalized to have unit norm, namely, it is
equivalent to $\ell_2$ regularization, which can be achieved by weight
decay or implicit regularization via stochastic gradient descent.

The above equivalence holds also for networks with multiple output
units, i.e. $k>1$, where the $\ell_1$ regularization on $v$ is
replaced with the group lasso regularizer
$\sum_{h=1}^{H}\norm{\boldsymbol{v}_h}$. Indeed, for matrix factorizations
(i.e.~with linear activations), such a group-lasso regularized
formulation is known to be equivalent to the trace norm
\eqref{eq:tracenorm} \citep[see][]{ArgEvgPon07}.

redpajama_set_name: RedPajamaArXiv


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The Northwest Airlines is a part of history in the aviation sector and is a pioneer in the field of aviation in the United States making a humble start by providing an airmail service to become a major brand and a international player in the aviation and airfreight industry.
Northwest Airlines was incepted in the year 1926 with a corporation formed in Detroit. Initially the airline operated airmail contracts between the cities of Minneapolis, Detroit, & Chicago and was not in the passenger service. The passenger service was commenced in the year 1927 and all its fleet was converted to cabin aircraft in the year 1928. The rising demand in the passenger service leads to the expansion in services to Winnipeg, Canada which was its first ever international destination. Subsequently the operations were expanded in Midwest with destinations like Fargo, Madison, North Dakota, and Wisconsin. Sensing the tremendous growth in the airlines, the investors from Minneapolis purchased the airline in 1929 and shifted the headquarters to Twin Cities the same year.
The airlines had tasted blood with expansion and sensed a tremendous potential in Japan. As a expansion strategy to explore the skies contacted Charles Lindbergh to initiate routes to Japan with logistical support from stops in Canada & Alaska. During those times, all the flights from the United States to Japan used to stop at Hawaii & the Philippines. The changed route was named "GREAT CIRCLE Route". When World War II began in full force, the airlines had to postpone the implementation of this new route. But the research that was done by the airline in the year 1930 was fruitful in the war effort and placed the company in a strong expansion mode in to the routes to the east once the hostilities stopped.
The aviation routes during that era was controlled and handled by The Federal Aviation Administration. In the year 1933 , the FAA granted Northwest Airlines the "Northern Transcontinental Route'' extending from the city of Seattle , Washington in the west , to the city of New York in the east . This route opened new vistas for the airlines to expand their course adding extra services to communities in Montana, North Dakota, Idaho & Washington. Besides this the airline also designed feeder routes which terminated at cities besides transcontinental route further making forays in to new destinations. The company was rehabilitated with the name Northwest Airlines during this period.
Till 1947, the airlines had cemented its position in the aviation industry and there were regular flights to Japan taking the Great Circle route and its flights made fuelling halts in Canada, Alaska & the Aleutians while in a course to Tokyo. The airlines were in the expansion mode with new flights being added frequently to destinations like China, Korea & Philippines. The airlines expanded its fleet by inducting new state of art aircrafts and in the process reducing the travel times and multiplying the passenger capacity. At this stage the airlines started promoting itself as Northwest Airlines, even though the corporate name remained unchanged.
Though Northwest Airlines was making profits and generating revenues, the unfortunate downturn in the aviation sector turned its fortunes. For the first time in the Northwest Airlines history, the company was forced to file for bankruptcy protection in the year 2005. The airline was able to emerge from its bankruptcy protection in the year 2007 and the following year the company as a business strategy and to revive its fortunes merged with Delta Air and thus forming the world's grandest commercial air lines.