text:
Candler Bed and Breakfasts
North Carolina /
Mountains /
Asheville Area /
Candler (2)
Engadine Inn & Cabins
Candler, NC
With a stay at Engadine Inn & Cabins in Candler, you'll be in the mountains, within a 15-minute d...
Pisgah View Ranch
With a stay at Pisgah View Ranch in Candler, you'll be 15 mi (24.1 km) from Bent Creek Experiment...
Arbor House of Black Mountain
Black Mountain, NC (21 miles from Candler)
With a stay at Arbor House of Black Mountain in Black Mountain, you'll be next to a golf course, ...
(21 miles from Candler)
My Place Inn
With a stay at My Place Inn in Black Mountain, you'll be within a 5-minute drive of Lake Tomahawk...
Black Mountain Inn
With a stay at Black Mountain Inn in Black Mountain, you'll be in the mountains, within a 10-minu...
Greenfield Manor Bed & Breakfast
Leicester, NC (6 miles from Candler)
Located in Leicester, Greenfield Manor Bed & Breakfast is within a 15-minute drive of U.S. Cellu...
Leicester, NC
(6 miles from Candler)
Wildberry Lodge Bed & Breakfast
With a stay at Wildberry Lodge Bed & Breakfast in Leicester, you'll be 8.2 mi (13.2 km) from Addi...
At Cumberland Falls Bed & Breakfast Inn
Asheville, NC (8 miles from Candler)
With a stay at At Cumberland Falls Bed & Breakfast Inn in Asheville (Historic Montford), you'll b...
Chestnut Street Inn Bed and Breakfast
With a stay at Chestnut Street Inn Bed and Breakfast in Asheville, you'll be within a 5-minute dr...
Pinecrest Bed & Breakfast
With a stay at Pinecrest Bed & Breakfast in Asheville (Historic Montford), you'll be within a 15-...
Carolina Bed & Breakfast
With a stay at Carolina Bed & Breakfast in Asheville (Historic Montford), you'll be within a 5-mi...
North Lodge on Oakland
Located in Asheville, North Lodge on Oakland is within a 5-minute drive of Grey Eagle and The Ora...
Princess Anne Hotel & Breakfast
A stay at Princess Anne Hotel & Breakfast places you in the heart of Asheville, within a 5-minute...
Oakland Cottage Bed & Breakfast
With a stay at Oakland Cottage Bed & Breakfast in Asheville, you'll be within a 5-minute drive of...
A Bed of Roses Bed & Breakfast
With a stay at A Bed of Roses Bed & Breakfast in Asheville (Historic Montford), you'll be within ...
1889 WhiteGate Inn & Cottage
When you stay at 1889 WhiteGate Inn & Cottage in Asheville, you'll be in the historical district,...
Sweet Biscuit Inn
Located in Asheville (Kenilworth), Sweet Biscuit Inn is within a 5-minute drive of Biltmore Villa...
Biltmore Village Inn Bed and Breakfast
With a stay at Biltmore Village Inn Bed and Breakfast, you'll be centrally located in Asheville, ...
Cedar Crest Inn
With a stay at Cedar Crest Inn in Asheville (Kenilworth), you'll be within a 5-minute drive of Bi...
The Reynolds Mansion
When you stay at The Reynolds Mansion in Asheville, you'll be in the mountains, within a 15-minut...
Bent Creek Lodge Bed & Breakfast
Located in Asheville, Bent Creek Lodge Bed & Breakfast is in the mountains, a 4-minute drive from...
The Applewood Manor
With a stay at The Applewood Manor in Asheville (Historic Montford), you'll be within a 5-minute ...
1899 Wright Inn & Carriage House
With a stay at 1899 Wright Inn & Carriage House in Asheville (Historic Montford), you'll be withi...
1900 Inn on Montford
With a stay at 1900 Inn on Montford, you'll be centrally located in Asheville, within a 5-minute ...
The Lion and the Rose Bed and Breakfast
Located in Asheville (Historic Montford), The Lion and the Rose Bed and Breakfast is within a 5-m...
Hill House Bed and Breakfast
With a stay at Hill House Bed and Breakfast in Asheville (Five Points), you'll be a 5-minute driv...
The Black Walnut B & B Inn
With a stay at The Black Walnut B & B Inn in Asheville (Historic Montford), you'll be within a 5-...
Abbington Green Bed & Breakfast Inn
A stay at Abbington Green Bed & Breakfast Inn places you in the heart of Asheville, within a 5-mi...
Aberdeen Inn
With a stay at Aberdeen Inn in Asheville (Murdock - Linden), you'll be within a 5-minute drive of...
Beaufort House Inn
With a stay at Beaufort House Inn in Asheville (Claxton Community), you'll be within a 5-minute d...
Albemarle Inn
Asheville, NC (10 miles from Candler)
With a stay at Albemarle Inn in Asheville (Grove Park- Sunset), you'll be within a 5-minute drive...
Dry Ridge Inn
Weaverville, NC (13 miles from Candler)
When you stay at Dry Ridge Inn in Weaverville, you'll be in the mountains, within a 15-minute dri...
Weaverville, NC
The Charleston Inn
Hendersonville, NC (19 miles from Candler)
With a stay at The Charleston Inn in Hendersonville, you'll be in a shopping district, just a 4-m...
Melange Bed and Breakfast
A stay at Melange Bed and Breakfast places you in the heart of Hendersonville, a 3-minute drive f...
The Henderson
A stay at The Henderson places you in the heart of Hendersonville, within a 5-minute walk of Mine...
Echo Mountain Inn
Located in Hendersonville, Echo Mountain Inn is in the mountains, within a 10-minute drive of Osc...
1898 Waverly Inn
With a stay at 1898 Waverly Inn, you'll be centrally located in Hendersonville, just a 4-minute w...
Aunt Adeline's Bed and Breakfast
Located in Hendersonville (Hyman Heights Historic District), Aunt Adeline's Bed and Breakfast is ...
The Inn at Brevard
Brevard, NC (21 miles from Candler)
A stay at The Inn at Brevard places you in the heart of Brevard, within a 10-minute walk of Cryst...
Willowbrook Inn
Lake Lure, NC (28 miles from Candler)
When you stay at Willowbrook Inn in Lake Lure, you'll be in the mountains, within a 5-minute driv...
Lake Lure, NC
The Ashe Farm Bed & Breakfast
Marshall, NC (17 miles from Candler)
With a stay at The Ashe Farm Bed & Breakfast in Marshall, you'll be 10.4 mi (16.7 km) from Bailey...
Marshall, NC
Marshall House Inn
With a stay at Marshall House Inn in Marshall, you'll be 7.7 mi (12.5 km) from Sandy Bottom Trail...
Inn on Mill Creek
Old Fort, NC (25 miles from Candler)
With a stay at Inn on Mill Creek in Old Fort, you'll be in a national park, just steps from Pisga...
The Orchard Inn
Saluda, NC (29 miles from Candler)
Located in Saluda, The Orchard Inn is within a 5-minute drive of Heartwood Contemporary Crafts Ga...
Saluda, NC
Almost Heaven Bed & Breakfast
Waynesville, NC (16 miles from Candler)
When you stay at Almost Heaven Bed & Breakfast in Waynesville, you'll be within a 10-minute drive...
Waynesville, NC
The Yellow House on Plott Creek Road
Located in Waynesville, The Yellow House on Plott Creek Road is in a rural location, within a 10-...
Oak Hill on Love Lane
Located in Waynesville, Oak Hill on Love Lane is in the historical district, a 4-minute drive fro...
Andon-Reid Inn Bed and Breakfast
Located in Waynesville, Andon-Reid Inn Bed and Breakfast is in the mountains, within a 15-minute ...
Grandview Lodge
Located in Waynesville, Grandview Lodge is in the mountains, within a 15-minute drive of Smoky Mo...
Brookside Mountain Mist Inn & Cottages
Located in Waynesville, Brookside Mountain Mist Inn & Cottages is in the mountains, within a 10-m...
Broad River Inn
Chimney Rock, NC (26 miles from Candler)
Located in Chimney Rock, Broad River Inn is on a river, just a 4-minute walk from Hickory Nut Gor...
The River House at Chimney Rock
With a stay at The River House at Chimney Rock in Chimney Rock, you'll be within a 5-minute walk ...
Leicester,NC 2 8 mi.
Asheville,NC 24 9 mi.
Waynesville,NC 6 17 mi.
Hendersonville,NC 6 19 mi.
Black Mountain,NC 3 21 mi.
Carmel,CA 11
Cape May,NJ 14
Santa Barbara,CA 7
Asheville,NC 24


Seriously. Read more comics. It's not kid stuff and they're not all about superheroes.
Try it out and go from there. You won't regret it. I promise.
my boyfriend is a HUGE comic collector/reader. but i havent noticed this one in his collection yet...ill suggest it to him!


Crunchy oriental flavored salad. Don't mix it together until just before serving otherwise your cabbage will become soggy.
Crush Ramen noodles before toasting. Toast noodles, almonds and butter in frying pan until golden brown. Set aside.
Chop green onion and cabbage.
Combine oil, sugar, vinegar, soy sauce and Ramen noodle seasoning packets.
Just before serving, mix all ingredients together.
"A sure-fire hit to bring anywhere. Usually you don't have to worry about this since it's all gone anyway, but it doesn't keep once it's been dressed."


text:
Livingston County Business Briefs
Local company cited for growth
While Michigan's economy struggled over the last several years, and manufacturing across the state declined, one Livingston County company has experienced substantial growth.
Total Security Solutions, a manufacturer of bulletproof glass in based in Fowlerville, saw 55 percent growth in revenue since 2010. Just as the company reached its 10th anniversary, Inc. Magazine named the company as one of the 5,000 fastest-growing private companies in the nation. The award is notable, considering the company is headquartered in a state that lost about 323,000 manufacturing jobs between 2000 and 2012 (according to U.S. Census Bureau data).
"There's no doubt that our success comes from the hard work and dedication of our staff, along with their commitment to quality and customer service," said Jim Richards, CEO of Total Security Solutions. "We are honored to make the list this year, and are focused more than ever on our processes and infrastructure to maintain stability for the long term."
Total Security Solutions has increased its workforce by 44 percent to keep up with growth demands. The company is not just a manufacturer of bulletproof glass, but also specializes in the design of bulletproof systems for a variety of industries, from hospitals to convenience stores. Many of its employees commute from the Lansing area and metro Detroit area. The company also added an office in California, which supports its increasingly national client base.
Lush Lawn appoints first female branch manager in Genoa
Lush Lawn at 7041 W. Grand River Ave., Suite 200, in Genoa Township has announced the appointment of Jennifer Terrance to position of branch manager. Terrance is the first woman in the company to hold the position. She distinguished herself throughout her seven-year career at Lush Lawn through her individual performance and management ability.
Terrance began her career at Lush Lawn as a lawn consultant at its Grand Blanc office. She advanced through the company becoming sales manager and senior sales manager before her appointment to branch manager.
Terrence's contribution to the company's sales department through corporate training materials and procedure development are indicative of her dedication to her work and position.
Howell's Foguth to sign books Saturday
Michael Foguth, president of Foguth Financial Group in Howell, is sharing his knowledge about retirement planning as a contributor to financial expert and publishing executive Steve Forbes' new book "Successonomics."
On Saturday, Foguth will sign copies of the book from noon to 2 p.m. at Blue Frog Books, 3615 E. Grand River Ave. in the Grand River Plaza in Genoa Township.

redpajama_set_name: RedPajamaCommonCrawl


text:
Wild International is a tribal/experimental band from New York City, USA. Originally formed in 2009 in Long Island, the band is composed of Ryan Camenzuli, Bryan Daly, and Greg Coffey. The band bases its sound on drums, bass and guitar, but also creates tribal sounds by layering in percussion, vocal harmonies and other experimental noises. The band's first self-titled EP was released in 2009, while touring around Long Island. Since then, the band has based itself in New York City, playing shows all over the five boroughs. In 2010, the band took part in The Break Contest, and wound up playing at The Bamboozle festival of that year. In 2012, Wild International released its second album, titled Lake Tones. The self-released album has received positive critical acclaim from such music websites and magazines as The Owl Mag, Bestnewbands.com, and The Deli Magazine.

References

"Review: Wild International - Lake Tones [EP"]. AltSounds, 26 November 2012.
"Wild International – Lake Tones EP". MixDwn,12 December 2012.
Hamel, Alysha. "MMC Bands". The Monitor, 3 May 2011.
Essert, Matt. "Wild International Brings a Controlled Array of Tribal Excitement with their First EP". Downtown. 6 December 2012.
Forsythe, Dan. "Wild International: Lake Tones". Performer Magazine/Top Picks January 2013.

Musical groups from New York City

redpajama_set_name: RedPajamaWikipedia


People want to know what I do but more important they want to know can I help them. This is a time where I learn about you, what you are going through that you would like changed and what you have tried to create the change . I will explain the process of how I work. Then we can determine if this is a good fit and go from there. You are welcome to come to my office in West Knoxville or we can meet virtually, whichever is convenient for you.


Usually this practice is done over two days, with the Mahayana precepts* being taken at the beginning of each day. On the first day it is customary to abstain from all meals apart from lunch. On the second day we engage in a complete fast, abstaining from all food and drink for 24 hours. On both days, we do three sessions of meditation, combining prostrations with chanted meditation. Those who are unable to engage in a complete fast can still participate in the sessions on the second day, by abstaining from all meals apart from lunch.
There is no charge for the retreat. Please email the Education Program Coordinator if you intending to attend the retreat.


text:
Coach Bryan Garoutte, Aidan Cretarolo, Coach Bryan Peterson, Jonah Frago, Owen Scheer and Jacob Hall gather for a team photo at the state meet. (Markos Scheer photo)
CLUB GYMNASTICS
Fantastic season for club gymnasts
Tuesday, May 15, 2018 8:30am
KENNEWICK – The boys of the Bainbridge Island Metro Park & Rec Gymnastics Club had a banner year.
The Bainbridge team went to the Washington State Men's Gymnastics Championships (held in Kennewick the weekend of the March 17-18) and came home with a team gold and enough hardware to fill a trunk.
The Bainbridge Island Gymnastics Club's men's team sent only eight competitors of all levels and ages, to compete against teams from around the state – some teams number in the dozens of members.
Yet, the BIGC team, under the steady and outstanding leadership of coaches Bryan Garoutte and Bryan "Wilbur" Peterson, showcased the quality of the Bainbridge club in across each of the six competition events; floor, pommel, rings, vault, parallel bars and high bar.
As a testament to their individual strength, five of the eight Bainbridge club gymnasts qualified to compete at 2018 Region 2 Men's Championships.
At the Washington state meet, the BIGC Junior Development team led the Bainbridge charge, competing March 17.
Aidan Cretarolo rocked the house, earning all-around gold for the 12-14 year age group, with a combined score of 64.600, setting the bar for the competition.
Cretarolo dominated in the individual events, scoring gold on floor (11.400), rings (11.500), vault (11.100), high bar (10.400) and tying with teammate Jacob Hall for gold on the parallel bars (11.000).
Jacob Hall, also in the 12-14-year group, tumbled, flipped and swung his way into a silver in the all-around.
Hall, in addition to tying with Cretarolo on parallel bars, scored silver on pommel (9.300) and high bar (9.800), and bronze on vault (10.300), floor (11.100), rings (10.700) and tying with teammate Owen Scheer on vault (10.400).
Owen Scheer, after returning from an injury at the 2017 Washington State Men's Gymnastics Championship, returned to form, competed in four of the six events and scored a gold on pommel (10.100), silver on floor (11.300) and bronze on parallel bars (9.700).
Jonah Frago, in only his first year of competition and the sole Bainbridge club member competing in the 15-19-year age group, rounded out the BIGC medal winners with a gold on rings (11.400) and a bronze on high bar (9.600).
The following day brought the BIGC Level 4s and 5s to the championship. Connor Ziemba was the sole Division 1, Level 4 representative, and competed in the 7-year-old group. He landed sixth in vault, only .150 short of the podium.
Kian Guiliano was BIGC's only competitor in the Division 1, Level 4, 10-year-old group. He won silver medals on the rings (9.900) and parallel bars (9.300).
Leo Koch was BIGC's single Level 5, Division 1 age 7-8 competitor, winning bronze medals on the pommel horse (9.700) and parallel bars (9.500).
In Level 5, Division 2, Charlie Phillips took bronze in pommel horse (8.8), vault (8.7) and high bar (8.3) and finished up with fourth in all-around, scoring a grand total of 49.0.
The group of Cretarolo, Hall, Scheer and Frago cruised to a Junior Development team gold with a dominant team score of 190.5, topping both Metropolitan and Mount Baker Gymnastics by more than 20 points. Metropolitan (a much larger team from Kent), took silver with a team score of 169.300, and Mount Baker took home the team bronze with a team score of 156.500.
Cretarolo, Hall, Scheer, Frago and Koch each qualified for Men's Region 2 Championships 2018, which was held the weekend of April 6-8 in Eugene, Oregon. There, the Bainbridge men competed against the best gymnasts from Alaska, Idaho, Montana, Oregon and Washington.
The rain in Eugene did nothing to dampen the spirits and intensity of the Bainbridge team.
At Regionals, against the best that the five states offered, Cretarolo continued his outstanding season, competing in the Junior Development Division 1, age 12-14, group and winning gold in the All-Around, with a score of 65.650.
Cretarolo took home gold on vault (11.350), high bar (11.200), tied for silver on rings (11.150) and bronze on parallel bars (11.200).
Scheer completed his successful season with a silver on floor (11.500), bronze on pommel (10.200), fourth on vault, sixth on parallel bars and a seventh place (62.550) in the all-around.
Hall, always steady and consistent, finished fourth on pommel horse, fifth on high bar and ninth overall (61.000).
Frago was the only BIGC competitor in the Men's Junior Development, Division 1 age 15-19 group.
For his first year of competition and first regionals meet, Frago finished his season strongly and consistently, averaging more than 10 points in every event, and placing eighth on pommel horse and the all-around with a score of 61.800.
Leo Koch, also in his first Level 5 regional competition, Division 1, 7-8 year old group and finished seventh on high bar and in the all-around with a score of 56.500, besting 24 other competitors from around the region.
The Regionals competition concluded the BIGC 2017-18 competition season on a very high note. Under the leadership of the fine coaching and strength of the returning team, the Bainbridge club expects another strong team next season.
Owen Scheer impresses during his rings routine at the Region 2 meet. (Markos Scheer photo)
Jacob Hall competes on pommel during the Region 2 meet. (Markos Scheer photo)
Tough regatta for BI sailors
Spartans split in Seattle
A speedy start: Year's first All-Comers Track Meet draws eager crowd | Photo gallery
A reportedly record-breaking crowd turned out for the first Kiwanis All-Comers Track… Continue reading
Bainbridge baseball team wins District 2 championship
Bainbridge Island's comeback kids won the Washington Little League District 2 Championship… Continue reading
Planning for new pickleball courts at Battle Point Park proceeds
Those looking to play the Bainbridge-born sport of pickleball on its home… Continue reading
Slainte survives scare in third win
Just three teams remain undefeated as men's rec softball entered the third… Continue reading
Signups start for annual Fun Run
Online registration for the 40th annual Bainbridge Youth Services (BYS) Fun Run… Continue reading
Bainbridge divers compete at Zone F qualifier
The Bainbridge Island Diving Club sent two divers to Midland, Texas to… Continue reading
Bainbridge seniors win big over Bremerton
The BI Park and Rec Senior Softball team defeated the Bremerton Park… Continue reading
Hot hitting for start of softball
The 2019 mens rec softball season started with an abundance of offense,… Continue reading
New coaches join Spartan JV teams
Kara Lindberg-Baker will take over as the coach for Bainbridge High junior… Continue reading
Gear Grinders claim eight wins at state championship race
The Bainbridge Island Gear Grinders, a high school and middle school mountain… Continue reading
BI Rowing alumna nabs fourth national championship
Bainbridge High School Class of 2015 graduate and Bainbridge Island Rowing vet… Continue reading
Bainbridge Junior Football joins Greater Eastside association
The ongoing resurgence of youth football on Bainbridge Island continued recently, as… Continue reading


Shannon Watts is the founder of Moms Demand Action for Gun Sense in America --- the leading force for gun violence prevention, with chapters in all 50 states and a powerful grassroots network of moms that has successfully effected change at the local, state and national level. In 2013, Moms Demand Action partnered with Mayors Against Illegal Guns to form Everytown for Gun Safety, a nationwide movement of millions of Americans working together to end the epidemic of gun violence. She is an active board member of Emerge America, one of the nation's leading organizations for recruiting and training women to run for office. Her writing has appeared in Vogue, Refinery 29, Time, The Washington Post, The Hill, and others. She is a mother of five.
FIGHT LIKE A MOTHER is the incredible account how one mother's cry for change became the driving force behind gun safety progress. Along with stories of perseverance, courage, and compassion, Watts shines a light on the unique power of women --- starting with what they have, leading with their maternal strengths, and doubling down instead of backing down. While not everyone can be on the front lines lobbying congress, every mom is already a multi-tasking organizer, and Shannon explains how to go from amateur activist to having a real impact in your community and beyond.


What is an emotionally intelligent business? It's a business that serves the needs of all its people. Where employees care about their jobs and are internally motivated to create the best outcomes. Where management communicates effectively and encourages creativity. It's a business that succeeds and grows through fostering emotional intelligence in the workplace.
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Bringing out the best of everyone boosts employee productivity and the bottom line.
How can my business become emotionally intelligent? By embracing the idea that all the elements of a business must seamlessly work together toward one goal, and that organizational structure change must be accompanied by personal change.
Achieving excellence means having the right people in the right jobs. Our top-of-the-line career assessments tools and hiring processes will help you hire people who are committed to helping your business grow.
Your organizational structure impacts your business performance. Our team of industrial psychologists and business experts will help you determine what type of organizational structure is most effective for your business.
Real business improvement needs the right type of personal change. Our proprietary brain training and development system - Brain Upgrade - combines innovative emotional intelligence training practices and unique brain development techniques that bring results in record time.


text:
Luxembourg jewellery thieves caught in Cologne
A group of jewellery thieves, suspects in robberies in Luxembourg, Germany and France, has been apprehended in Cologne, Germany.
(CS) A group of jewellery thieves, suspects in robberies in Luxembourg, Germany and France, has been apprehended in Cologne, Germany.
According to Cologne newspaper "Kölner Stadtanzeiger" the criminals from Lithuania stole jewellery and watches worth around 300,000 euros from luxury store Wempe in the city centre – only one of three burglaries committed and eight other failed attempts, the daily states on its website.
After two initial arrests were made in Germany in March, police caught up with the group when two of its members, aged 18 and 19, were recently caught stealing a bottle of Fanta at a department store.
Last week on Friday, four further men were arrested after a special unit stormed their hotel room.
German police commented to the "Kölner Stadtanzeiger" that they cannot rule out that further members of the gang are at large. However, they said that they had in any case delivered a considerable blow to the criminal cell, with eight people now in custody.
The German newspaper offered no further information about the incident in Luxembourg. Grand Ducal police refused to comment, saying the investigation is still ongoing.
Luxembourg media have, however, raised the question whether the group is connected to an armed robbery at Cartier in March this year.


text:
\section{Introduction}\label{sec:intro}
One of the most basic tasks in data analysis is to identify correlations between data sources, modeled as random variables. Discovered correlations are used to remove unnecessary features, to build predictive models,
and to identify unexpected behaviors and dependencies. In this paper, we consider the most common measure of correlation: the Pearson product-moment correlation coefficient, which describes the linear relationship between a pair of random variables. This measure is simple to state and interpret: it is computed as the (sample) covariance of the two variables, divided by the product of the corresponding standard deviations. It ranges from $-1$ (strong negative correlation) through $0$ (no correlation) to $+1$ (strong positive correlation). Hence, we are typically interested only in attribute pairs with correlation close to $1$ in magnitude.

For large numbers of variables, it can quickly become infeasible to
compute the correlations of all of the quadratically many
pairs. However, our observation is that most correlations are
uninteresting: for many kinds of data, we expect that most pairs of
variables would \emph{not} display any (strong) correlation.
For example, if we consider the activity profiles of users of a large
web service, then we do not expect many pairs to be strongly
correlated (there may be weak correlations due to similar time-of-day
and day-of-week behavior) - any strong correlation between a pair
would be unusual, indicating potentially
nefarious activity worthy of further investigation.
We model this by assuming that the number of correlated pairs is
asymptotically smaller than the quadratically many possible pairs.

With this in mind, we can ask the following questions: given a stream of observation data, can we identify all correlation outliers (unusually large correlation coefficients, defined by being greater in magnitude than some parameter $\phi$) with query time cost sub-quadratic in the number of variables and sub-linear in the number of observations?

This can be done using a combination of a Fast Johnson-Lindenstrauss Transform (FJLT, to compress the rows of the input matrix) and Locality Sensitive Hashing (LSH, to efficiently find the outlier pairs). However, for small $\phi$, the query time of this strategy looks like $n^{2-\Theta(\phi)}$, even as we shrink all the non-outlier correlations down to $0$. Can we improve the query time in this case?
This paper answers this question in the positive, by describing an algorithm which takes sketches of the rows and uses fast matrix multiplication to quickly transform them into an approximation of a sketch of the correlation matrix. We then remove the $1$'s along the diagonal, and use a heavy hitters recovery technique to pull out the outliers. For fixed Frobenius
norm\footnote{For matrix $\mathbf{A}$, the Frobenius norm is
  $\textstyle \lVert \mathbf{A} \rVert_F = (\sum\limits_{i,j}\mathbf{A}_{i,j}^2)^\frac{1}{2}$.} of the non-outlier non-diagonal correlations, this query process can be performed in time $\tilde{O}(\phi^{-2}n^{5/3})$, assymptotically better than LSH for $\phi < 1/3$. However, this comes at the cost of requiring much larger sketches of the input matrix rows.

\section{Preliminaries}\label{sec:prelims}
\subsection{Models}\label{sec:models}
We treat the observation data as defining an $n \times p$ matrix of
reals, $\mathbf{M}$.
Here, $n$ denotes the number of attributes, while $p$ indexes the
different observations.
Hence, each of the $p$ columns represents an independent observation
of some $n$-dimensional random variable.
We label the columns (observations) as $\mathbf{x^{(i)}}$ for $i \in [p]$.
For this data, we can apply standard definitions of covariance and
correlation.

\begin{definition}\label{def:estimators}
Recall that:
\begin{itemize}
\item
  The \emph{sample mean} is given by $\mathbf{\overline{x}} = \frac{1}{p} \sum_{i=1}^p \mathbf{x^{(i)}}$.
\item
  The \emph{sample covariance} is given by
  $\mathbf{V} = \frac{1}{p-1}(\mathbf{M} - \mathbf{\overline{x}} \mathbf{e}^T) (\mathbf{M} - \mathbf{\overline{x}} \mathbf{e}^T)^T \text{,}$
  where $\mathbf{e}$ is the $p$-dimensional vector with entries all ones.
\item
  The \emph{sample correlation} is given by $\mathbf{C} = \mathbf{\Sigma}^{-\frac{1}{2}} \mathbf{V} \mathbf{\Sigma}^{-\frac{1}{2}}$, where $\mathbf{\Sigma}$ is the diagonal matrix consisting of the diagonal entries of $\mathbf{V}$.
\end{itemize}
\end{definition}

Essentially, the covariance is found by shifting the rows of
$\mathbf{M}$ to have mean $0$ and then taking inner products between
them normalized by a factor of $\frac{1}{p-1}$.
The definition of the correlations is similar, but further normalized by diving out the standard deviations.

It will be useful for our analysis to have notations for the rows of $\mathbf{M}$ with the shift normalization applied.

\begin{definition}\label{def:standardized}
  For each row vector $\mathbf{y^{(i)}}$:
  \begin{itemize}
  \item
    Let the standardized row vector $\mathbf{\hat{y}^{(i)}}$ be given by $\mathbf{\hat{y}^{(i)}} = \frac{\mathbf{\hat{y}^{(i)}} - \mathbf{\bar{x}_ie}^T}{\| \mathbf{\hat  {y}^{(i)}} - \mathbf{\bar{x}_ie}^T\|_2}$.
  \end{itemize}
\end{definition}

The observation matrix $\mathbf{M}$ is input as a stream of $m$
updates $\langle u_1, u_2, \cdots u_m \rangle$ arriving one at a
time.
Starting from the zero matrix $\mathbf{M^{(0)}} = \mathbf{0}$, each update $u_s$ describes a change to be made to $\mathbf{M^{(s-1)}}$ in order to determine $\mathbf{M^{(s)}}$. By the end of the stream, we have $\mathbf{M^{(m)}} = \mathbf{M}$. The format of the updates depends on the exact choice of stream model---we will consider three variants: row-wise permutation, column-wise permutation, and turnstile.

\para{Row-Wise Permutation Stream (\textsf{RPS}\xspace).}
In this model, the updates are simply a list of the entries of
$\mathbf{M}$, one row at a time. With each step from $\mathbf{M^{(s-1)}}$ to
$\mathbf{M^{(s)}}$, one entry is changed from $0$ to its final
value.
Entries in the same row arrive contiguously, so each row is
filled out one after the other.
Without loss of generality, we can assume that rows arrive in index
order, so that
$\mathbf{M}_{i,j} \gets u_{ip+j} $
Since each entry is set exactly once, the stream has length $m =
np$.
The arrival of each new row corresponds to adding a new
attribute to the data set.

\para{Column-Wise Permutation Stream (\textsf{CPS}\xspace).}
This model works the same as the row-wise version, but with entries
arriving as contiguous columns.
Again, $m = np$ but now
$\mathbf{M}_{i,j} \gets u_{jn + i} $.
The arrival of a new column corresponds to adding a new observation
(e.g. from a new time step).

\para{Turnstile Stream (\textsf{TS}\xspace).}
The turnstile model is the most general that we consider. Here updates
are of the form $u_t = (\alpha, i, j)$ indicating that the $(i,
j)$\textsuperscript{th} entry should be incremented by $\alpha \in
\mathbb{R}$.
That is, $\mathbf{M}^{(s)}_{i,j} \gets \mathbf{M}^{(s-1)}_{i,j} + \alpha$,
while all other entries remain the same.
Changes happen in any order, and entries can change any number of
times as long as the correct state is reached by the end of the
stream.
Hence, the stream length $m$ is arbitrary.

Both \textsf{RPS}\xspace and \textsf{CPS}\xspace are then special cases of this model. \textsf{TS}\xspace
represents the situation where each of the observed values needs to be
aggregated from a variety of sources.
For example: suppose the entries in our observation matrix represent the number of requests for a specific resource (indexed by rows) at a specific site (indexed by columns) in a distributed system. Then, the number of requests at each node will need to be accumulated to produce the actual observation data.

\subsection{Problem Statement}\label{sec:problem}

In what we term the \emph{correlation outliers} problem, we are given
a stream describing $\mathbf{M}$ (according to one of the three
models), and three parameters: $k$, $\phi$, and $R$.
We make use of the following concepts:

\begin{definition}\label{def:large}
For the sample correlation matrix $\mathbf{C}$ (of $\mathbf{M}$):
\begin{itemize}
\item
  Let $\textsc{Large}_\phi \subset [n]^2$ refer to the set of index pairs of off-diagonal entries of $\mathbf{C}$ which have magnitude at least $\phi$.
\item
  Let $\mathbf{C}_{-k}$ refer to the matrix obtained by taking $\mathbf{C}$, removing all the diagonal entries, and removing the $k$ largest magnitude off-diagonal entries (replacing them with $0$'s).
\end{itemize}
\end{definition}

The problem is then to maintain a summay of the stream so that
all index pairs contained in $\textsc{Large}_\phi$ can be retrieved
with high probability ($o(\frac1n)$ chance of failure), provided
 that $|\textsc{Large}_\phi| \leq k$ and $\lVert \mathbf{C}_{-k}
 \rVert_F \leq R$.
Since the full input can be trivially maintained in $O(np)$ space, we
seek solutions with space cost that is $o(np)$.
Further, the summary should be quick to update (taking polylogarithmic time) and, at the end of the stream, the query routine should run in time $o(n^2)$.
We argue that the assumption that $k$, the number of highly correlated
pairs, is $o(n^2)$ is a reasonable one: otherwise, simply reporting
all the correlated pairs would take quadratic time, and naive exhausitve
solutions would suffice.
A similar assumption is made in prior work considering Boolean vectors~\cite{BooleanCorrelation,FasterBooleanCorrelation}.

\subsection{Related Work}\label{sec:related}

\para{Locality Sensitive Hashing.}
Asking for high correlation is equivalent to looking for small Euclidean distance between the standardized (normalized and centered) row vectors. A correlation of $\phi$ corresponds to a distance on the sphere of $\sqrt{2 - 2\phi}$. Hence, this problem can be solved using Euclidean Locality Sensitive Hashing (LSH). Negative correlation outlier pairs can be found by simply considering every row and its negation.

Assuming outlier correlations are greater than $\phi_0$, and non-outlier correlations are smaller than $\phi_1$. We can use a Fast Johnson-Lindenstrauss Transformation to compress input rows to length $O(\epsilon^{-2}\log{n})$, distorting the pairwise distances by at most $(1 \pm \epsilon)$. Then we can do LSH with $c^2 = \left(\frac{1-\epsilon}{1+\epsilon}\right)^2 \left(\frac{1-\phi_0}{1-\phi_1}\right)$.

The best known Euclidean LSH algorithms have $\rho = \frac{1}{c^2} + o(1)$ (data independent, \cite{LSH}) and $\rho = \frac{1}{2c^2 - 1} + o(1)$ (data dependent, \cite{LSH2}). This gives us a time and space cost of $O(n^{2-\Theta(\phi_0)})$ for fixed $\phi_0$, even if $\epsilon$ goes to $1$ and $\phi_1$ goes to 0.

\noindent{\bf Compressed Matrix Multiplication.}
Pagh~\cite{OuterProducts} considered the problem of efficiently
computing sparse or approximate matrix products. The key idea is that
by choosing a particular structure for the sketching functions, it is
possible to quickly compute a sketch of the outer product
$\mathbf{xy}^T$, from sketches of $\mathbf{x}$ and $\mathbf{y}$,
through the use of FFTs (in time $O(b\log{b})$ for length $b$
sketches).
Since a matrix product $\mathbf{AB}^T$ can be decomposed into a sum of
such outer products between corresponding columns, this allows for
efficient computation of matrix products from sketches of columns.

As the algorithm only requires access to matched columns of
$\mathbf{A}$ and $\mathbf{B}$ one at a time, in the special case of
$\mathbf{A} = \mathbf{B}$ this approach can be used in the \textsf{CPS}\xspace model
to build a sketch of $\mathbf{AA}^T$.
In particular, we can build a sketch of the covariance matrix $\mathbf{V}$ in this streaming model, from input observation matrix $\mathbf{M}$, with update time cost $O(b\log{b})$ ($O(1)$ amortized, since $n$ dominates $b\log{b}$) and space usage $O(b)$.
To recover dominant entries from these sketches, Pagh describes an
approach (building on Gilbert et al.~\cite{SparseRecovery}) that uses
$O(\log^2{n})$ sketches of sub-matrices of $\mathbf{AB}$, along with
error correcting codes, to discover the identity of a small number of
entries which dominate the Frobenius norm of the product, with high
probability.
This process runs in $O(b\log^2{n})$ time and space.
Putting these pieces together provides a solution to a
{\em covariance outliers} version of our problem in the \textsf{CPS}\xspace model.

Unfortunately, this approach cannot be adapted directly to the {\em
  correlation outliers} problem.
Large correlations between low variance signals would be drowned out
by the contribution from high variance signals that are much more
weakly correlated. To apply this technique, we would need to record
the whole of $\mathbf{M}$ (perhaps feasible for small $np$), or
perform two passes over the stream---using the first to determine the
variances, and then using the covariance solution on the rescaled
inputs with the second pass.
Instead, we will adapt the recovery process to work on different kinds of sketches.

\para{Boolean Vectors.}
Karppa et al.~\cite{FasterBooleanCorrelation} (improving on work by Valiant~\cite{BooleanCorrelation}) considered the problem of identifying a small number of highly correlated pairs of \emph {Boolean} vectors (entries are $\pm1$) from a collection of vectors which mostly have low pairwise correlation. Several tricks are used to beat LSH for small $\phi$:

\begin{itemize}
\item
  Calculate tensor powers of each signal vector to amplify the relative difference between the small and large correlations.
\item
  Sub-sample from the tensor power vectors to reduce dependence on the blown up dimension of these vectors. This is possible because Boolean vectors have a uniform distribution of weight among their entries.
\item
  Sum together groups of the sub-sampled vectors (pre-multiplied by random signs) to reduce dependence on the number of vectors. This is possible because only a few of the pairwise correlations are large.
\item
  Use fast matrix multiplication to quickly compute the inner products between pairs of groups. These are then inspected for unusually large entries, which indicates the presence of a large pair in the group.
\end{itemize}

The first two points depend strongly on the vectors being Boolean and
hence do \emph{not} apply in our more general real-valued setting.
Note however, that our approach shares the use of fast matrix
multiplication and the sketch-like idea of signed group
aggregation for increasing speed.

\subsection{Our Contributions}\label{sec:contributions}
We describe an algorithm which answers the correlation outliers
problem in the turnstile streaming model.
We analyze its space and time costs, and show that they meet the
desiderata above.
Our algorithm stores a separate Fast AMS sketch of each row of
$\mathbf{M}$ (described in Section~\ref{sec:sketch}).
Comparing these directly would still take time $\Theta(n^2)$ to
perform an all-pairs comparison.
Instead, we achieve an improved query time with the following three ideas:
\begin{itemize}
\item
  By randomly assigning variables into $\Pi$ groups, and linearly
  combining the (sketched) information of all variables in the same
  group we can go from having to consider $n^2$ pairs of variables to $\Pi^2$ pairs of groups. This can be seen as a second level of sketching.
\item
  Error correcting codes are composed with the grouping step (including/excluding variables from groups based on code bits) in a way that allows us to recover the identities of large entries in a particular group pair using the decoder.
\item
  Fast matrix multiplication algorithms allow us to quickly generate batches of sketch estimates of inner products. This speeds up the evaluation of the inner products between pairs of groups. Then checking whether the results of these computations exceeds a given threshold produces the strings of bits for the decoder.
\end{itemize}

Our main result is stated in full in Theorem~\ref{thm:main}.
As an example, setting an internal parameter $\theta = 2/3$ and for fixed $k, R$,
we obtain subquadratic space and time
costs summarized in the below table.

\begin{center}
  \begin{tabular}{| c | c | c | c | c |}
    \hline
    \textbf{Technique} & \textbf{Models} & \textbf{Sketch Size} & \textbf{Query Space} & \textbf{Query Time}\\
    \hline
    Fast AMS Sketches & All & $\tilde{O}(\phi^{-2}n)$ & $\tilde{O}(\phi^{-2}n)$ & $\tilde{O}(\phi^{-2}n^2)$\\
    FJLT + LSH & All & $\tilde{O}(n)$ & $\tilde{O}(n^{2-\Theta(\phi)})$ & $\tilde{O}(n^{2-\Theta(\phi)})$\\
    Our Approach & All & $\tilde{O}(\phi^{-2}n^{5/3})$ & $\tilde{O}(\phi^{-2}n^{5/3})$ & $\tilde{O}(\phi^{-2}n^{5/3})$\\
    \hline
  \end{tabular}
\end{center}

This space usage is $o(np)$ for $p \in \Omega(n^{2/3 + \epsilon})$.

\subsection{Sketches of Vectors}
\label{sec:sketch}
Our results make use of \emph{sketches} of vectors. These can be thought of as random projections from the original high-dimensional space down to a lower dimensional space, such that geometric properties of the vectors are (approximately) preserved. In particular, given vectors $\mathbf{x}$ and $\mathbf{y}$, sketches exist that can estimate:
\begin{align}
&\text{
  Squared Euclidean length $\lVert \mathbf{x} \rVert_2^2$ up to error
  $\epsilon \lVert \mathbf{x} \rVert_2^2$.}\label{eq:euclid} \\
&\text{
  Inner product $\langle \mathbf{x} , \mathbf{y} \rangle$ up to error $\epsilon \lVert \mathbf{x} \rVert_2 \lVert \mathbf{y} \rVert_2$.}
\label{eq:inner}
\end{align}

Many results for such sketches are known, from the earliest (non-constructive) results based on the Johnson-Lindenstrauss lemma \cite{Johnson:Lindenstrauss:84}, the tug-of-war sketches due to Alon, Matthias, Szegedy and Gibbons~\cite{Alon:Matias:Szegedy:96,Alon:Gibbons:Matias:Szegedy:99}, and several more~\cite{Achlioptas:01,Li:Hastie:Church:07,Kane:Nelson:14}. For concreteness, we will adopt the so-called (fast) AMS sketches (explained in~\cite{Cormode:11}). These create a sketch of size $O(\epsilon^{-2} \log{1/\delta} )$ so that any query obtains the above claimed $\epsilon$ guarantee with probability at least $1-\delta$, where the probability is over the random choices used to determine the random projection.

The AMS sketching procedure maps (linearly and randomly) the space of $p$-dimensional vectors to the space of $d \times b$ matrices. Each row of the output sketch is obtained by pre-multiplying the input vector by a diagonal matrix whose entries are Rademacher (uniformly random $\pm 1$), and then pre-multiplying by a $b \times p$ sparse matrix where each column has a single $1$, with $0$ everywhere else\footnote{The construction does not require the entries to be chosen fully independently at random, so it is common to describe the sketch transformations in terms of hash functions drawn from limited independence families. This allows the transform to be stored in polylogarithmic space.}. This process generates one row of the sketch, and is repeated independently $d$ times to generate all rows. The stated $(\epsilon$, $\delta)$ guarantee can be achieved for $d \in \Theta(\log{1/\delta})$ and $b \in \Theta(\epsilon^{-2})$. As they are a sparse linear transformation of their input, any addition to an entry in the sketched vector can be applied to the sketch in time $O(d) = O(\log 1/\delta)$.

\begin{definition}\label{def:ams}
Let:
\begin{itemize}
\item
  $\textsc{AMS}_{\epsilon, \delta}$ refer to a distribution of random
  linear maps corresponding to fast AMS sketches with the stated
  $(\epsilon, \delta)$ norm and inner product approximation guarantees
  (\eqref{eq:euclid} and \eqref{eq:inner}).
\item
  \emph{$(\epsilon, \delta)$-sketch transformation} $\mathbf{S}$ be a linear map drawn from $\text{AMS}_{\epsilon, \delta}$.
\item
  The symbol $\odot$ represent the binary operation of performing the inner product query between two sketches. So
  $\mathbf{S}(\mathbf{x}) \odot \mathbf{S}(\mathbf{y}) \approx \langle \mathbf{x}, \mathbf{y} \rangle \text{, and }\, \mathbf{S}(\mathbf{x}) \odot \mathbf{S}(\mathbf{x}) \approx \|\mathbf{x}\|_2^2 \text{.}$
\end{itemize}
\end{definition}

One application of sketches is to estimate the value of a particular index in a vector. This can be achieved as a special case of an inner product query: we use the sketch to estimate $\langle \mathbf{x} , \mathbf{e}_i \rangle$, where $\mathbf{e}_i$ is the vector that is $1$ at location $i$ and $0$ elsewhere. The guarantee ensures that we obtain an estimate with error at most $\epsilon \lVert \mathbf{x} \rVert_2$. This use of sketches is referred to as a Count sketch~\cite{Charikar:Chen:Farach-Colton:02}.

\section{Algorithm and Analysis}\label{sec:theory}
\subsection{Algorithm Overview}\label{sec:overview}
Our algorithm works in the most general stream model we considered, the turnstile model. At a high level, our algorithm consists of:

\begin{itemize}
\item
  An \emph{initialization} procedure to set up the sketch data structure.
\item
  An \emph{update} procedure to process updates from the stream.
\item
  A \emph{query} procedure to recover the suspected elements of $\textsc{Large}_{\phi, k}$.
\end{itemize}

Our sketch structure is built on top of a collection of AMS sketches with standard initialization and update procedures, plus some additional variables to keep running totals. We will briefly review these procedures in Section \ref{sec:rowsketch}, as well as discussing some basic properties and routines required for the query algorithm.

The query process itself is based on two main ideas. First, we can
take linear combinations of AMS sketches and then perform inner
product queries between them in order to estimate certain kinds of
linear combinations of entries of $\mathbf{C}$. Further, we can
perform batches of such queries quickly using fast matrix
multiplication. And secondly, we can utilize error correcting codes to
identify the large magnitude entries in these linear combinations of
entries
even with the error introduce by the AMS sketches.

Rather than fast matrix multiplication between combinations of rows, we could have hoped to employ Pagh's compressed matrix multiplication for our second layer of sketching. However, to produce a $w$-bucket sketch would take time $\frac{w\log{w}}{\epsilon^2}$ for each row of the AMS sketches. Then to control the variance of the output buckets, we would need $w = n^2\epsilon^2$, giving a time cost of $\Omega(n^2)$ to build the secondary sketch.

The outline and the discussion of the query algorithm is therefore
broken up into four parts. In Section \ref{sec:cartesian} we describe
a ``Cartesian sketch'' which compresses a matrix by applying a pair of
independent transformations (each akin to the Count sketch), one row-wise and one column-wise. Then, in Section \ref{sec:recovery} we show that we can use the error correcting code technique to recover large entries from Cartesian sketches. Further, we show that the recovery technique is robust to additional sources of noise per entry of the sketch. Next, in Section \ref{sec:approximation} we show how the AMS sketches in our structure can be used to build good enough approximations of the Cartesian sketches to satisfy the noise limits. Finally, in Section \ref{sec:analysis} we analyze the overall space and time costs and discuss how to amplify the probability of success.
For brevity, full proofs are deferred to the Appendix, and we present
informal proofs in the main body to convey the high level ideas.

\subsection{Row Sketching}\label{sec:rowsketch}
For our data structure we will keep an AMS sketch of each row of the observation matrix along with a running total. The choice of sketch parameters $(\epsilon, \delta)$ will be made in the final analysis in Section~\ref{sec:analysis}.

To initialize the structures, we randomly pick an $(\epsilon,
\delta)$-sketch transformation $S$, and initialize
$n$ sketches $\mathbf{r^{(i)}}$ to all zeros.
We also create $n$ counters $t^{(i)}$, initialized to zero.
Algorithm \ref{alg:update} shows how to apply a received update in the
\textsf{TS}\xspace model.
We use $\mathbf{e_j}$ to indicate the length $p$-vector consisting of
a $1$ in entry $j$ and $0$ everywhere else.
The update simply updates the $i$th sketch with index $j$, and updates
the corresponding sum of weights, $t^{(i)}$.
Let $\mathbf{y^{(i)}}$ to refer to the $i$\textsuperscript{th} row of $\mathbf{M}$ for $i \in [n]$. By following these procedures we will have $\mathbf{r^{(i)}} = \mathbf{S}(\mathbf{y^{(i)}})$ and $t^{(i)} = \sum_{j \in [p]} \mathbf{y^{(i)}}_j$ at the conclusion of the stream.

\begin{figure}[t]
  \begin{minipage}{2.75in}
   \begin{algorithm}[H]
  \DontPrintSemicolon
  \KwIn{TS model update $u_s = (\alpha, i, j)$}
  \BlankLine
    $\mathbf{r^{(i)}} \gets \mathbf{r^{(i)}} + \alpha \cdot \mathbf{S}(\mathbf{e_j})$\;
    $t^{(i)} \gets t^{(i)} + \alpha$\;
  \caption{\textsc{Update}\label{alg:update}}
  \end{algorithm}
  \end{minipage}
  \hfill
  \begin{minipage}{2.5in}
\begin{algorithm}[H]
  \DontPrintSemicolon
    \For{$i \in [n]$}{
      $\mathbf{r^{(i)}} \gets \mathbf{r^{(i)}} - (t^{(i)}/p) \cdot \mathbf{S}(\mathbf{e})$\;
      $\mathbf{r^{(i)}} \gets (\mathbf{r^{(i)}} \odot \mathbf{r^{(i)}})^{-1/2} \cdot \mathbf{r^{(i)}}$\;
    }
  \caption{\textsc{Standardize}\label{alg:standardize}}
\end{algorithm}
  \end{minipage}
\end{figure}

An important operation we will need to be able to perform on these row sketches is to \emph{standardize} them. Recalling $\mathbf{\overline{x}}$ and $\mathbf{e}$ from Definition~\ref{def:estimators}, we define:
\begin{definition}
  For a given row vector $\mathbf{y^{(i)}}$, the \emph{standardized vector} $\mathbf{\hat{y}^{(i)}}$ is given by:
  \[ \mathbf{\hat{y}^{(i)}} = (\mathbf{y^{(i)}} -
  \mathbf{\overline{x}}_i \mathbf{e}^T) \big/ \|\mathbf{y^{(i)}} - \mathbf{\overline{x}}_i \mathbf{e}^T\|_2 \text{.} \]
  If we have a sketch $\mathbf{S}(\mathbf{y^{(i)}})$ we will refer to $\mathbf{S}(\mathbf{\hat{y}^{(i)}})$ as the \emph{standardized sketch}.
\end{definition}

In the \textsf{RPS}\xspace and \textsf{CPS}\xspace models we could keep track of the running sums of $\alpha^2$ for each row, allowing us to compute the exact rescaling factor required to standardize the sketches. However, in the more general \textsf{TS}\xspace setting, the best we can do is an approximation.
Algorithm~\ref{alg:standardize} describes the procedure for computing the approximately standardized sketches.

Initially, it may appear that to perform this standardization at query
time, we need to spend $\Omega(pd)$ time building the sketch
$\mathbf{S}(\mathbf{e})$.
However, we can amortize this cost during the update phase.
As long as at least $p$ entries of the final $\mathbf{M}$ are
non-zero, then we can build up $\mathbf{S}(\mathbf{e})$ one entry per
update by using a single counter to track which entries have been
added.
In the atypical case that $\mathbf{M}$ is extremely sparse, we will need to add $O(pd)$ to the query time to complete the construction of this sketch.

\begin{proposition}\label{prop:standardized}
After performing the \textsc{standardize} routine, the inner product query between sketches $\mathbf{r^{(i)}}$ and $\mathbf{r^{(j)}}$ produces an estimate of $\mathbf{C}_{i,j}$ having $4\epsilon$ additive error with probability at least $1 - 3\delta$, for $\epsilon < 1/2$.
\end{proposition}
\begin{informal}
Each sketch approximates the sketch of a standardized row
($\mathbf{r^{(i)}} \approx \mathbf{S}(\mathbf{\hat{y}^{(i)}}$) and the
inner product query between sketches of standardized rows approximates
the correlation ($\mathbf{S}(\mathbf{\hat{y}^{(i)}}) \odot
\mathbf{S}(\mathbf{\hat{y}^{(j)}}) \approx \mathbf{C}_{i,j}$). To get
a small additive error on our estimates, we then just need both
sketches to be approximated well and the inner product query between
them to give a good results. Each of the three events occurs with probability $(1-\delta)$.

The correlation between two rows can be expressed as the inner product
of the corresponding standardized vectors. The sketches output by \textsc{standardize} approximate the true standardized sketches. To get a small additive error on our estimate, we then just need both sketches to be approximated well and the inner product query between them to give a good result. Each of the three occurs with probability $(1 - \delta)$.
\end{informal}

\subsection{Cartesian Sketches} \label{sec:cartesian}
\begin{definition}\label{def:cartesian}
  For an $n \times n$ matrix $\mathbf{A}$, we call $\textsc{Cart}(\mathbf{A})$ a $\Pi \times \Pi$ Cartesian sketch of $\mathbf{A}$ if for each $(h, g) \in [\Pi]^2$ we have
  \[\textsc{Cart}(\mathbf{A})_{h,g} = \sum\limits_{P_1(x)=h}\sum\limits_{P_2(y)=g} \left( s_1(x) s_2(y) \mathbf{A}_{x, y} \right) \text{,}\]
  where $s_1$ and $s_2$ are independently selected from a pairwise independent family of random sign functions $[n] \to \lbrace -1, +1 \rbrace$, and where $P_1$ and $P_2$ are functions $[n] \to [\Pi]$ selected independently and uniformly at random from the set of functions:
  \[\lbrace f : [n] \to [\Pi] \text{ s.t. } |f^{-1}(i)| = n/\Pi \text{ for each } i \in [\Pi] \rbrace \text{.}\]
\end{definition}

From this definition, we can see that a Cartesian sketch transformation is very similar to a pair of independent Count sketch transformations (one performed row-wise, one column-wise). The  difference is the use of fully random partitioning functions which produce exactly equal buckets. If we were performing exactly a pair of Count sketches, we would also have $P_1$ and $P_2$ expressed as limited independence hash functions. However, the $O(n\log{n})$ space needed to store fully random permutations will not impact our asymptotic space usage and makes the subsequent analysis simpler.
The entries of the sketches will be referred to as buckets, and the $(i,j)$\textsuperscript{th} entry of the original matrix $\mathbf{A}$ is said to be mapped to the $(h,g)$\textsuperscript{th} bucket (for a given choice of sketch functions) if $P_1(i) = h$ and $P_2(j) = g$.

\begin{definition}\label{def:bucket}
 Let $\mathcal{B}_{h,g} = \lbrace (i, j) \in [n]^2 \text{ s.t. } P_1(i) = h$  and  $P_2(j) = g \rbrace \text{,}$
  i.e. the set of index pairs mapped to bucket $(h, g)$.
\end{definition}

\subsection{Recovery Process}\label{sec:recovery}
Now we will describe how to apply the recovery process to a series of
Cartesian sketches of a given matrix. We will describe an algorithm
which gives a constant probability of finding any given element of
$\textsc{Large}_{\phi, k}$ (Definition~\ref{def:large}) and argue that it works.

For this procedure, we apply an error correcting code to encode row
and column indices (which take values in $[n]$) into a longer binary
codeword.
We will assume access to some family of functions (over choices of
$n$) with the desired properties to perform the encoding and
decoding.
For a fixed $n$, let:
\[\mathcal{E}: [n] \to \lbrace 0, 1 \rbrace^{L\log{n}}\, \text{ and }\, \mathcal{D} : \lbrace 0, 1 \rbrace^{L \log{n}} \to [n]\]

\noindent
where $L > 1$ indicates how much bigger the codeword is compared to the
input size.
Here, we write $\mathcal{E}$ and $\mathcal{D}$ for the encoder and
decoder functions (respectively) of a scheme which can recover from up
to $\lambda L \log{n}$ bit flip errors --- i.e. an error rate of
$\lambda$.
That is, for any length $L\log{n}$ binary word $\mathbf{w}$ with at most $\lambda L \log{n}$ bits set to $1$, we have\footnote{Here $\oplus$ represents the ``exclusive-or'' bitwise operation between binary words.} $\mathcal{D}(\mathcal{E}(i) \oplus \mathbf{w}) = i$ for every $i \in [n]$.

Error correcting codes are known to exist for $L \in O(1)$ and $\lambda \in \Omega(1)$, which can be implemented to perform encoding and decoding in $O(\log{n})$ time and $O(\text{polylog}\,n)$ space (for example \cite{FastECC}).

\begin{definition}\label{def:maskingmatrix}
For each $l \in [L\log{n}]$ (each bit in the code words), we define a
\emph{masking matrix} $\mathbf{E^{(l)}}$. This is a diagonal binary
matrix where entry $\mathbf{E^{(l)}}_{i,i}$ is the
$l$\textsuperscript{th} bit of the code word $\mathcal{E}(i)$.
That is, $\mathbf{E^{(l)}}_{i,i} = \mathcal{E}(i)_l$.
\end{definition}

These masking matrices can be pre- or post-multiplied with $\mathbf{C}$ to mask rows or columns (respectively) based on bits of their index encodings.

The recovery process is described in Algorithm \ref{alg:simplerecovery}. It takes as input sketches $\mathbf{L^{(l)}} = \textsc{Cart}(\mathbf{CE^{(l)}})$ and $\mathbf{R^{(l)}} = \textsc{Cart}(\mathbf{E^{(l)}C})$ for each $l \in [L\log{n}]$, for a randomly selected Cartesian sketch transformation $\textsc{Cart}$.

\begin{algorithm}[t]
  \caption{\textsc{RecoveryStep}\label{alg:simplerecovery}}
  \KwIn{Cartesian sketches $\mathbf{L^{(l)}}$ and $\mathbf{R^{(l)}}$ for $l \in [L\log{n}]$, and Cartesian sketch transformation $\textsc{Cart}$}
  \KwOut{Index multiset $\Omega$ of suspected large entries}
  \DontPrintSemicolon
  \BlankLine
    Create new empty multiset $\Omega$\;
    \For{$(h,g) \in [\Pi]^2$}{
      Create new empty strings $\mathcal{I}$ and $\mathcal{J}$\;
      \For{$l \in [L\log{n}]$}{
        \lIf{$|\mathbf{L^{(l)}} - \textsc{Cart}(\mathbf{E^{(l)}})| \geq \phi/2$}{
          Append $1$ to $\mathcal{I}$
          \textbf{else}
          Append $0$ to $\mathcal{I}$
          }
        \lIf{$|\mathbf{R^{(l)}} - \textsc{Cart}(\mathbf{E^{(l)}})| \geq \phi/2$}{
          Append $1$ to $\mathcal{J}$
          \textbf{else}          Append $0$ to $\mathcal{J}$
        }
      }
      Append $(\mathcal{D}(\mathcal{I}), \mathcal{D}(\mathcal{J}))$ to $\Omega$\;
    }
    \Return{$\Omega$}
\end{algorithm}

To understand why this process should work, consider the special case
where $\mathbf{C}$ is $0$ on all the non-large, off-diagonal
entries. That is, the only non-zero entries are the diagonals (which
must be $1$) and the entries corresponding to elements of
$\textsc{Large}_{\phi, k}$. In this situation, the only entries
contributing to the Cartesian sketches are entries of
$\textsc{Large}_{\phi, k}$ corresponding to unmasked rows and
columns. Now, consider what happens in a bucket with a single large
entry mapped to it. Whenever the row or column of the large entry is
masked, the corresponding bucket value will be $0$; and when the row
and column are not masked, the bucket value will have magnitude at
least $\phi$ --- in particular, greater than $\phi/2$. This means
that, in the algorithm, on the outer loop corresponding to this
bucket, $\mathcal{I}$ and $\mathcal{J}$ will be exactly the code words
corresponding to the row and column indices of the large entry. Hence,
the index pair of the large entry is added to $\Omega$.
So, isolated large entries will be correctly recovered in this special case, and as long as $k$ is sufficiently smaller than $\Pi$ we have a good chance of any given large entry being isolated.
To formalize this argument, and extend it to the more general case, we define a few different events.

\begin{definition} \label{def:events} For fixed $\mathbf{C}$ and a fixed coding scheme define the following random events, over the random choice of sketch functions:
  \begin{itemize}
    \item Let $\textsc{CorrectDecode}_{h,g}$ be the event that the recovery process returns the ``correct'' index for bucket $(h,g)$. If there is exactly one of $\textsc{Large}_{\phi, k}$ in the bucket, then the correct result is the index pair of that entry. Otherwise, any returned value is considered correct.

    \item Let $\textsc{SmallError}_{h,g,l}$ be the event that the non-large entries of $\mathbf{CE^{(l)}} - \mathbf{E^{(l)}}$ and $\mathbf{E^{(l)}C} - \mathbf{E^{(l)}}$ each contribute less than $\phi/4$ to bucket $(h,g)$ of their corresponding sketches. That is, $\textsc{Cart}(\mathbf{E^{(l)}C_{-k}})_{h,g}$ has magnitude smaller than $\phi/4$.
  \end{itemize}
\end{definition}

We begin with a proposition explaining the circumstances we are looking for to successfully find a large entry.

\begin{proposition} \label{prop:fulldecode} If we have that:
  $
  \Prob{\textsc{CorrectDecode}_{h,g}} \geq 1-x \text{
      for all } (h,g) \in [\Pi]^2 \text{,}

  $
  then for any given $(i, j) \in \textsc{Large}_{\phi, k}$, we have that $(i, j)$ is in the list of index pairs produced by \textsc{RecoveryStep} with probability at least $1 - x - 2k/\Pi$.
\end{proposition}

From this proposition, we can see that if we can get a lower bound on
the probability of $\textsc{CorrectDecode}_{h,g}$ for every $(h,g) \in
[\Pi]^2$, then we can get an overall guarantee for the recovery process.

\begin{proposition} \label{prop:correctdecode} If we have that:
  $
    \Prob{\textsc{SmallError}_{h,g,l}} \geq 1 - y \text{
      for all } l \in [L\log{n}] \text{,}$
  then $\Prob{\textsc{CorrectDecode}_{h,g}} \geq 1 - y/\lambda$.
\end{proposition}

The last piece we need is a lower bound on the probability of $\textsc{SmallError}_{h,g,l}$.

\begin{proposition} \label{prop:smallerror}
  Recalling Definition~\ref{def:large}, we have:
  $\Prob{\textsc{SmallError}_{h,g,l}} \geq (1 - 32 \|\mathbf{C_{-k}}\|_F^2 / (\Pi^2 \phi^2)) \text{.}$
\end{proposition}

Now we have all the pieces we need to show that the recovery process works.

\begin{proposition} \label{prop:recoverycorrect}
  If we have that
  $\Pi \geq \max\lbrace 18k, 18 \|\mathbf{C_{-k}}\|_F / (\phi \lambda^{1/2}) \rbrace \text{,}$
  then the output of $\textsc{RecoveryStep}$ will include any fixed index pair in $\textsc{Large}_{\phi, k}$ with probability at least $\frac23$.
\end{proposition}

Observe that we chose the definition of the event $\textsc{SmallError}_{h,g,l}$ to leave room for an additional source of noise of similar size $\phi/4$. We will need this robustness later.

\begin{corollary} \label{prop:recoveryrobust}
  Lemma \ref{prop:recoverycorrect} holds even when there is additional noise applied to each bucket entry, provided it has magnitude smaller than $\phi/4$ with probability at least $(1 - \lambda/18)$ on any fixed bucket.
\end{corollary}

In the next subsection, we will show that an approximate Cartesian sketch can be constructed from row sketches within these tolerances.

\subsection{Approximation from Row Sketches}\label{sec:approximation}
We need a way of quickly approximating $\textsc{Cart}(\mathbf{E^{(l)}C})$ and $\textsc{Cart}(\mathbf{CE^{(l)}})$ for each $l \in [L\log{n}]$ for a randomly chosen Cartesian sketch transformation $\textsc{Cart}$, from the row sketches described in Section \ref{sec:rowsketch}. This is done by the procedure described in Algorithm \ref{alg:approximation}.

\begin{algorithm*}[t]
  \caption{\textsc{Approximate}\label{alg:approximation}}
  \KwIn{Approximately standardized row sketches $\mathbf{r^{(1)}},\, \cdots,\, \mathbf{r^{(n)}}$ and functions $P_1$, $P_2$, $s_1$, $s_2$ corresponding to a Cartesian sketch transformation $\textsc{Cart}$}
  \KwOut{$\mathbf{L^{(l)}}$ and $\mathbf{R^{(l)}}$, estimates of $\textsc{Cart}(\mathbf{E^{(l)}C})$ and $\textsc{Cart}(\mathbf{CE^{(l)}})$ respectively, for $l \in [L\log{n}]$}
  \DontPrintSemicolon
  \BlankLine
    \For{$l \in [L\log{n}]$}{
      Initialize empty matrices $\mathbf{L^{(l)}}$ and $\mathbf{R^{(l)}}$\;
      \For{$h \in [\Pi]$}{
        Initialize $\textsc{left}[h]$, $\textsc{right}[h]$, $\textsc{leftMasked}[h]$, $\textsc{rightMasked}[h]$ as zero sketches $\mathbf{S}(\mathbf{0}) = \mathbf{0}$\;
      }
      \For{$i \in [n]$}{
        $\textsc{left}[P_1(i)] \gets \textsc{left}[P_1(i)] + s_1(i) \cdot \mathbf{r^{(i)}}$\;
        $\textsc{leftMasked}[P_1(i)] \gets \textsc{leftMasked}[P_1(i)] + \mathbf{E^{(l)}}_{i,i} \cdot s_1(i) \cdot \mathbf{r^{(i)}}$\;
        $\textsc{right}[P_2(i)] \gets \textsc{right}[P_2(i)] + s_2(i) \cdot \mathbf{r^{(i)}}$\;
        $\textsc{rightMasked}[P_2(i)] \gets \textsc{rightMasked}[P_2(i)] + \mathbf{E^{(l)}}_{i,i} \cdot s_2(i) \cdot \mathbf{r^{(i)}}$\;
      }
      \For{$(h, g) \in [\Pi]^2$}{
        $\mathbf{L^{(l)}}_{h,g} \gets \textsc{leftMasked}[h] \odot \textsc{right}[g]$\;
        $\mathbf{R^{(l)}}_{h,g} \gets \textsc{left}[h] \odot \textsc{rightMasked}[g]$\;
      }
    }
    \Return{$\mathbf{L^{(1)}},\, \cdots,\, \mathbf{L^{(L\log{n})}}$ and $\mathbf{R^{(1)}},\, \cdots,\, \mathbf{R^{(L\log{n})}}$}
\end{algorithm*}

For each $l \in [L\log{n}]$, the returned $\mathbf{L^{(l)}}$ is our approximate $\textsc{Cart}(\mathbf{E^{(l)}C})$ and $\mathbf{R^{(l)}}$ is our approximate $\textsc{Cart}(\mathbf{CE^{(l)}})$.

The algorithm works by observing that $\mathbf{C}$ can be approximated from the row sketches by performing all the possible inner product queries between pairs of sketches and placing the results in the corresponding positions of the matrix. We could then apply \textsc{Cart} to the result.
However, we make the further observation that since \textsc{Cart} can be broken up into pieces that look like pre- and post-multiplication by matrices, we can rearrange the order of operation. We can perform the \textsc{Cart} sketch first, directly on the row sketches, and then perform the all-pairs inner product query second.
This simple change results in the main performance bottle-neck (the all-pairs inner product query) happening on a much smaller matrix, greatly speeding up the entire query process.

We will show that this process produces a good enough approximation of the \textsc{Cart} sketch to act as the input to \textsc{RecoveryStep}.

\begin{proposition}\label{prop:approximationbound}
  At the end of \textsc{Approximate}, for any given $(h,g) \in [\Pi]^2$, we have:
  \begin{multline*}
    |\mathbf{L^{(l)}}_{h,g} - \textsc{Cart}(\mathbf{E^{(l)}C})_{h,g}| \leq \epsilon n \Pi^{-1} 207 \lambda^{-1/2} \text{,}\\
    \text{and } |\mathbf{R^{(l)}}_{h,g} - \textsc{Cart}(\mathbf{CE^{(l)}})_{h,g}| \leq \epsilon n \Pi^{-1} 207 \lambda^{-1/2} \text{,}
  \end{multline*}
  with probability at least $1 - \lambda/27 - \delta(2 + 12n/\Pi)$, as long as $\epsilon < 1/2$.
\end{proposition}

To meet the requirements for \textsc{Recovery} to work on these approximations, we need to set limits on the choices of $\epsilon$ and $\delta$.

\begin{proposition}\label{prop:approximation}
  If we have that:
 $\delta \leq \lambda/(54(2 + 12n/\Pi))$, and
$\epsilon \leq \min\lbrace 1/2, (\phi \Pi \lambda^{1/2}) / (828n)\rbrace \text{,}$
  then \textsc{Approximate} produces approximations which are within the noise tolerance of \textsc{RecoveryStep}.
\end{proposition}

\subsection{Analysis of Algorithm}\label{sec:analysis}
Putting together the previous subsections, we can make the full recovery algorithm. The outline is listed in Algorithm \ref{alg:full}.

\begin{algorithm*}[t]
  \caption{\textsc{Recover}\label{alg:full}}
  \KwIn{Row sketches $\mathbf{r^{(1)}},\, \cdots,\, \mathbf{r^{(n)}}$ and totals $t^{(1)},\, \cdots,\, t^{(n)}$}
  \KwOut{Index set $\Omega$ of entries we are confident are large}
  \DontPrintSemicolon
  \BlankLine
    Create empty multiset $\Omega$\;
    \For{$\gamma \in [\Gamma]$}{
      Randomly generate functions $P_1$, $P_2$, $s_1$, $s_2$ for a Cartesian sketch $\textsc{Cart}$\;
      Run \textsc{Approximate}, passing it $P_1$, $P_2$, $s_1$, $s_2$ and the row sketches\;
      Run \textsc{RecoveryStep}, passing it \textsc{Cart} and the result of \textsc{Approximate}\;
      Append the result of \textsc{RecoveryStep} to $\Omega$\;
    }
    Remove entries from $\Omega$ appearing fewer than $\Gamma/2$ times\;
    \Return{$\Omega$ (as a set)}

\end{algorithm*}

\begin{proposition}\label{prop:final}
If we have that:
\[\delta \leq \lambda/(54(2 + 12n/\Pi))\text{, }
\epsilon \leq \min\lbrace 1/2, (\phi \Pi \lambda^{1/2}) / (828n)\rbrace
\text{,}
\text{ and }
       \Pi \geq \max\lbrace 18k, 18 \|\mathbf{C_{-k}}\|_F / (\phi \lambda^{1/2}) \rbrace \text{,}\]
then we can choose a $\Gamma \in O(\log{n})$ such that \textsc{Recover} returns every element of $\textsc{Large}_{\phi, k}$ with probability at least $1 - n^{-3}$.
\end{proposition}

\begin{proposition}\label{prop:spacetime}
\textsc{Recover} can be implemented to run in time $\tilde{O}(\Gamma (\Pi^2 + \log{(1/\delta)} (n\epsilon^{-2} + \mathcal{M}(\Pi, \epsilon^{-2}))))$, and space $\tilde{O}(\Pi^2 + n \epsilon^{-2} \log{(1/\delta)})$, where $\tilde{O}(\cdot)$ is the $O(\cdot)$ cost with $\log{n}$ factors suppressed.
\end{proposition}

\begin{theorem}\label{thm:main}
  For every $\theta \in [0, 1]$, there exists a sketch of size
  $\displaystyle \tilde{O}\left( n^{2\theta}(k^2+\frac{R^2}{\phi^2}) + n^{3-2\theta}(\frac{\phi^2}{k^2\phi^2 + R^2}) \right)$
from which we can extract the (up to $k$) entries with magnitude at
least $\phi$ in time \newline
$\displaystyle \tilde{O}\left (n^{2\theta}(k^2+\frac{R^2}{\phi^2}) + n^{3-2\theta}(\frac{\phi^2}{k^2\phi^2 + R^2}) + \mathcal{M} \left( n^\theta(k+\frac{R}{\phi}), n^{2-2\theta}(\frac{\phi^2}{k^2\phi^2 + R^2} \right) \right)$
with high probability.
\end{theorem}

\begin{corollary}\label{prop:example}
In particular, for $\theta = 2/3$ and constant $k$, $R$, we can build a sketch of size $\tilde{O}\left( \phi^{-2}n^{5/3} \right)$ with query time $\tilde{O}\left( \phi^{-2}n^{5/3} \right)$.
\end{corollary}

\section{Concluding Remarks}\label{sec:conclusions}
We have shown how to guarantee accurate recovery of
correlation outliers using a sketch-based method, beating LSH on query time for small correlation outliers and vanishing correlation non-outliers.
A key part of our approach is to use sketching and coding ideas
repeatedly: as well as using sketching to reduce the initial
dimensionality of the data, we use a second ``layer'' of sketching
when we combine subsets of signals, in order to speed up queries over
many pairs of sketches at the cost of increased error. Where LSH tries to hash the correlated elements together, we try to separate them and then recover them from the noise.
This produces a trade-off between the size of the underlying sketches and the final query time. This general approach could work in other situations where a large number of sub-queries need to be evaluated to search for large values, for example with measures of similarity/distance other than correlation.

Further, as the technique produces a linear intermediate sketch, this approach is easily adapted to recover pairs whose correlation deivates from some expected correlation matrix, or has changed comapred with some previous point in time (simply perform the heavy hitters recover on the difference between two intermediate sketches built using the same permuations, signs, and codes).

Future directions would include finding ways to use alternative primitives to fast matrix multiplication (such as fast convolution via FFT, as adopted by Pagh) and trying to combine the advantages of LSH-based methods and heavy-hitters-based methods.

\subsection*{Acknowledgements}
We thank Milan Vojnovic for several discussions about this work.

redpajama_set_name: RedPajamaArXiv


>> I just tried the new Qt 5.2 on my mac, but alas, still no "native"
>> having the mac extras between my legs.
>> Is *still* the qt mac extra a requirement for native toolbars?
> There are two things here that people tend to mix. If you want a native Mac toolbar, i.e. all the features of NSToolbar including the configuration panel and so on, then yes, Qt Mac Extras is the way to go.
Is the native NSToolbar in the 5.2 MacExtras supposed to work?
and last time I tried to compile an earlier version of MacExtras (from some separate Git repository a couple of weeks ago) it failed to compile.
In other words, I should be able to come up with a native looking (and behaving) toolbar solution already /now/, using the (undocumented?) class in the QMacExtras module of Qt 5.2.


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Abstract: This paper discusses different rammed earth construction technique in Asia. Rammed earth construction techniques from China, Indian, Nepal and Bhutan are examined. It is shown that these techniques are demonstrably different from each other, and argued that the techniques may have developed independently. Case study structures are discussed and it is shown that with care it is possible to chart the development of both techniques both chronologically and geographically.
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Jaquin, P. Influence of Arabic and Chinese Rammed Earth Techniques in the Himalayan Region. Sustainability 2012, 4, 2650-2660.
Jaquin P. Influence of Arabic and Chinese Rammed Earth Techniques in the Himalayan Region. Sustainability. 2012; 4(10):2650-2660.
Jaquin, Paul. 2012. "Influence of Arabic and Chinese Rammed Earth Techniques in the Himalayan Region." Sustainability 4, no. 10: 2650-2660.
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Rafa Benitez has been criticised by a local journalist about his team selections.
The criticism came after the FA Cup quarter-final matches this weekend.
As well as Manchester City, Brighton, Watford and Wolves all went through into the semi-finals.
Newcastle fans have instantly came to the manager's defence.
When Newcastle played Watford in the fourth round, seven players didn't play who were then named in the following Premier League match.
However, as fans have pointed out below, Watford's manager changed his entire team. The simple fact being that Watford have a far stronger squad than Newcastle.
By far the biggest factor though is that deep in a relegation fight, Rafa Benitez simply couldn't afford to risk key players such as Salomon Rondon, especially with facing Manchester City only three days later AND any injury or lack of full fitness meaning Joselu would lead the line instead.
At that time, Newcastle were only two points above the relegation zone and had won only one of their last six PL games.
By far the killer fact though is that, Rafa's team selections played a key part in the stat that Newcastle United remain the only team to have beaten Manchester City in 2019.
Indeed their last 20 matches (all competitions) have seen Man City win all of the other 19 games.
"Watford, Wolves and Brighton in the FA Cup semis.
"Watford changed their full 11.
"We didn't have the luxury of safety and a better all round squad so made some changes and still got beat.
"Yes it hurts we're not one of the last 4 in world's most prestigious club competition but so what?
"Javi Gracia changed his whole team for the 4th round game against Newcastle.
"Or… It'd have dented our confidence even further and sent us on a much worse run.
"Surely the first priority is to stay in the premier league.


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The game is set in the 20th century, and you can choose from over 190 countries to play as. You can also choose from 11 different civilizations, each with its own unique units and abilities. The goal of the game is to achieve world domination.
Age of Civilization 2 has been released on Android devices and is available for free download. The game offers stunning graphics and addictive gameplay that will keep you entertained for hours on end. Age of Civilization 2 is a strategy game that lets you control your armies and build your empire from scratch.
You can choose from a variety of civilizations to play as, each with its own unique strengths and weaknesses. The game offers six different campaigns to play through, each with its own objectives and challenges. You can also compete against other players online in real-time matches or asynchronous duels.
Age of Civilization 2 APK is a turn-based strategy game that takes you through the ages. You can choose to play as one of many civilizations, and each has a unique set of bonuses and abilities. The gameplay is smooth and fluid, making it easy to control your armies and navigate the map. The graphics are stunning, and the sound effects are realistic, making the game immersive and enjoyable.
Age of Civilizations II is a strategy board game for Android and iOS devices, based on a game of the same name. The game allows players to compete against each other in order to conquer the world. Players can choose from a variety of civilizations, including America, China, India, and Persia. Age of Civilizations II is available as a free download from the link given above. The game requires Android 4.1 or later and iOS 9.0 or later.
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Age of Civilizations 2 is finally available on Android devices! The popular PC game has been ported over to mobile and offers players an in-depth experience like no other. If you're new to the game, or just need a refresher on the basics, this guide is for you.
To get started, download the Age of Civilizations 2 APK file onto your device. Once it's installed, open the app and click on the new game. You'll be prompted to choose a civilization. There are 18 different civilizations to choose from, each with its own unique bonuses and units. Once you've chosen your civilization, you'll need to set the game up for your specific device. Click on SETTINGS and then RESOLUTION.
The graphics are too good, but the gameplay is amazing. You can play with up to four people, and you can chat with them while you're playing. I don't know about you, but I always loved chatting with my friends while I was playing games on my computer. So, if you're looking for a game that will keep you occupied for hours on end, then Ages of Empires is the game for you.
A recent update to the popular Age of Civilization 2 APK has given users a detailed map of the world that includes many historical borders. The map is based on data from the Geographic Information System (GIS), and it covers all of human history from Pre-History to the present day. The map contains all details, and it includes information on things like political borders, physical features, and major cities.
It also includes information on historical events, such as the fall of Rome and the American Revolution. The Age of Civilization 2 is a free Android app that allows users to explore world history in detail. Millions of people downloaded this app, and it has received rave reviews from critics and users alike.
The game is usually won by either militarily conquering all other civilizations, or achieving the highest score possible. Players can choose to play against AI opponents or other people online. One of the newest editions of Civilization is the ability to form diplomatic ties with other civilizations. This goes beyond just declaring war or peace, and allows for more complex interactions between different civilizations. For example, a player could offer to trade resources with another civilization in order to get them on their side.
In the early days of civilization, peace treaties were created to document the terms of an armistice agreement between two warring factions. These treaties were often drawn up in a very formal and legalistic manner, with each party laying out exactly what they expected from the other. The most famous example of a peace treaty is the Treaty of Westphalia, which ended the Thirty Years' War in 1648.
Age of Civilization 2 APK is a strategy game that comes with an inbuilt editor which lets you create your own scenarios, maps and history. This is a great way to keep the game interesting as you can create new challenges and goals for yourself. The editor is easy to use and lets you add a variety of objects to the map, including buildings, characters and resources.
You can also set up victory conditions and control the weather and time. The editor makes it possible to create detailed maps with lots of different features, making the game more challenging and fun.
There are many different types of people in the world. This is not a new revelation, but it is something that has been studied and talked about more in recent years. With the advent of social media and the ability to share information quickly and easily, people are more aware of the differences between themselves and others.
This is a good thing, as it helps to break down barriers and allows for better understanding. One area where this is particularly evident is in the study of populations. Researchers have found that there is great diversity among populations, even within the same country.
Ages ago, there was an Age of Civilization. It was a time when great leaders led their people to glory. The battles were fought, and the empires were born. However, that was then, and this is now. All you need to do is download the Age of Civilization 2 APK and create your own world.
With the Age of Civilization 2, you can set up your own game and play however you want. Whether you want to be a ruthless conqueror or a benevolent ruler, the choice is yours. You can also choose your own allies and enemies, making for a more personal experience. Best of all, the Age of Civilization 2 APK is free to download and play. So why not give it a try?
Age of Civilization 2 APK is an excellent game that provides hours of entertainment. It is an excellent game that everyone can enjoy. The graphics are stunning, and the gameplay is smooth and engaging. I highly recommend this game to anyone who enjoys strategy games.
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Pick up your fetching racing gear (overalls and helmet).
Introduction and quick safety briefing.
Practice session to get used to the kart and track.
The Grand Prix starts off with 5 heats of 6 laps.
The start-of-the-art computerised timing system will display lap times, race positions and most importantly... the leaderboard!
Competitive hens can watch all the races from the cafe in between races.
The top drivers will progress on to the 10 lap final.
A presentation ceremony for the elite, top 3 drivers.
Our Edinburgh hen party go karting experience is the perfect way to add a little va va voom to your hen weekend. For hens looking for more than the usual genteel hen party activities, this is a huge rush that speed queens will absolutely love!
Competitive hens are going to really get a buzz too as you tackle one of Scotland's best go kart tracks. You'll be hitting speeds of up to 40mph (which feels even faster when your bum is just inches from the tarmac) over the whopping 600 metre track in 200cc karts. Our Edinburgh hen do karting Grand Prix event is truly something special, not only do you get to race your mates in the top karts in the land, but the circuit itself is the first in the country to have an amazing dual fly-over!
Chicanes, hairpin turns, electronic race timing, straights, fly-over, tunnel, all help add to the incredible adrenalin, white knuckle, wheel to wheel action. After a few practise laps to get used to the karts you'll then go head to head in a series of heats. Perfect to bring out your competitive sides as all the action is tracked and recorded by the integrated F1 Analysis System which times drivers to within 100th of a second, so no need for an action reply. You can then see your lap times to see how well you're doing and if you're improving.
You can then watch the rest of the heats from the comfort of the cafe before (hopefully) pulling on your helmet for the grand final. The top 3 races will get to enjoy a podium presentation. Although all the hens can all pile on for a great hen party team photo.