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Télé Lumière is the first Christian television station in Lebanon and the Arab world and was founded in 1991. Since 2003, it also broadcasts satellite programming worldwide under the name Noursat.

History

Télé Lumière meaning the "Television of Light" was founded in 1991 by a group of committed lay people: Late Charles Helou, a former President of Lebanon, Brother Nour, Jack Kallassi, Late George Frem (ex. MP and minister), George Moawad, Rola and Sana Nassar, Dr. Antoine Saad and Christiane Debbaneh.

Télé Lumière is a non-profit organization, its mission is ecumenical, it is not associated to any political party or movement, do not broadcast any political views and do not accept or promote any commercial endorsements. As a non profit broadcasting station, it has no lucrative purposes and survives only from various donations given by its supporters and friends.
The Church cooperated with this television station since the beginning when she found it one of the best means to promote the principles of justice, love, freedom and human rights; values called for by Christianity everywhere.
Télé Lumière is supervised by the Assembly of Catholic Patriarchs and Bishops in Lebanon and directed by a committee involving religious leaders from various denominations and a group of laity eager to support this distinguished project. The relationship between the Church and Télé Lumière is organized through a "cooperation protocol".
As for the Government, it considers Télé Lumière, through the Assembly of Catholic Patriarchs and Bishops in Lebanon, a Christian station, financially independent and having freedom of management and programming controlled only by the authority of the Church.

Noursat

In June 2003, on Pentecost, Télé Lumière celebrated its 12th anniversary as well as the launching of its satellite station Noursat covering European Union, the Southwest Asia and North Africa. Actually (2020) the station is located at the A05 transponder of Eutelsat 7West-A at 7,3°W. One year later, at September 8th, 2004, Noursat began its official broadcast in North and South America, United States, Canada and Australia (platforms Mysat, MysatGo).

Board of directors

Board of Directors
Télé Lumière/Noursat board of directors:
Archbishop Roland Abou Jaoudé (President)
Jack Kallassi (General Manager)
Brother Nour (General Supervisor)
Rola Nassar (Financial Manager)
Dr. Antoine Saad (General Secretary)
Bishop George Riachi (member)
Father Khalil Alwan (member)
Neemat Frem (member)
George Moawad (member)
Sana Nassar (member)
Raymond Nader (Noursat Executive Director)

See also
Catholic television
Catholic television channels
Catholic television networks
Television in Lebanon

External links

1991 establishments in Lebanon
Christianity in Beirut
Christian media
Television channels and stations established in 1991
Mass media in Beirut
Television networks in Lebanon
Television stations in Lebanon
Arabic-language television stations


As an accomplished woman in a historically male dominated profession, what advice would you give your younger self? How could you better prepare for the unique challenges you'll face? As a man, how can you be the best coworker, ally, or mentor?
On International Women's Day, Local First Arizona's For(u)m is highlighting a few of the valley's most accomplished women in development to share stories of challenges and triumphs, plus some of their lessons learned and best practices.
Join us at the Historic Campbell House, renovated and operated by Ashley Harder of Harder Development to celebrate, learn from, and grow with women leading charge and building our cities.
Tickets are $10 each and can be purchased by registering below.


Description: Fully enclosed fruit stand offering seasonal produce, jarred goods, deli lunch menu, espresso, hard ice cream, baked goods, pies. Northwest wines, hard cider and micro brews. Growler fill station, garden center and free Wi-Fi.


While out collecting his morning food, a farmer basically tells the Buddha that if he wants to eat he should grow his own food. The Buddha responds with a deep Dhamma teaching.
1. "You profess to be a plowman, yet your plow we do not see; asked about your plow and the rest, tell us of them that we may know."
2. "Faith is my seed, austerity the rain, wisdom my yoke and plow, modesty is the pole, mind the strap, mindfulness is my plowshare and goad.
3. "Controlled in speech and conduct, guarded in deed and speech, abstemious in food, I make truth my weed cutter; arahantship, my deliverance complete.
4. "Exertion, my team in yoke, draws me to Nibbana's security, and on it goes without stopping, wither gone one does not suffer.
5. "Thuswise is this plowing plowed which bears the fruit of Deathlessness; having plowed this plowing one is freed from every ill."


Ever wanted to develop and print your own films and photographs?? Now you can! Well you still need chems, darkroom and an enlarger! but this is most of the gear.
Contains all the essential equipment for processing both film and photographic paper.


The impact and loss associated with natural disasters can be devastating and the cost to recover from them continues to rise significantly each year. It is impossible to predict exactly when these disasters will occur, or the extent to which they will affect the City of Beaverton. However, with careful planning and collaboration among public agencies, private sector organizations, and citizens within the community, it is possible to minimize the losses that can result from natural disasters. Natural hazard mitigation is the responsibility of individuals, private businesses and industries, state and local governments, and the federal government.
Natural hazard mitigation is defined as a method of permanently reducing or alleviating the losses of life, property, and injuries resulting from natural hazards through long and short-term strategies. Example strategies include planning, policy changes, programs, projects, and other activities.
The City of Beaverton developed a Natural Hazards Mitigation Plan in an effort to reduce future loss of life and property resulting from natural disasters. The Natural Hazard Mitigation Plan (NHMP) is intended to assist the City of Beaverton in reducing its risk from natural hazards by identifying resources, information and strategies for risk reduction. It will also help guide and coordinate mitigation activities throughout the City.
People who live in or visit Oregon enjoy some of the most beautiful scenic places in the United States; this beauty however has an underlying cost; earthquakes. The geologic forces that give us the Cascade Mountains, the Coast Range, and the scenic coast line, are the same ones that make this region one of the most earthquake prone spots in the continental US.
The potential impacts posed by earthquakes include property damage and bodily injuries. There are no guarantees of safety during earthquakes, but properly constructed and strengthened homes are far less likely to collapse or be damaged during earthquakes.
There are structural and non-structural mitigation activities you can undertake to minimize the potential of damage and injury. The non-structural activities can typically be done by the homeowner, whereas the structural projects may require a licensed and experienced contractor to complete.


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CITIC to complete $464m investment in Ivanhoe Mines
CITIC Metal has received its internal approvals required for its subsidiary CITIC Metal Africa Investments (CITIC Metal Africa) to complete its second investment in Ivanhoe Mines for an additional C$612m ($464m).
Ivanhoe's Mount Elliott smelter in Queensland. Credit: Heritage branch staff
CITIC Metal has received the internal approvals required for its subsidiary CITIC Metal Africa Investments (CITIC Metal Africa) to complete its second investment in Ivanhoe Mines for an additional C$612m ($464m).
Ivanhoe Mines will issue 15.3 million common shares to CITIC Metal Africa, with each share priced at C$3.98 ($3.01). This is a premium of 29% over Ivanhoe's 24 April 2019 closing price, under the terms of the agreement signed in April.
This investment is CITIC Metal's second major financing in less than a year. With this move, CITIC's total investment in Ivanhoe reaches approximately $1bn. It is expected to close no later than 7 September.
In May, Zijin Mining Group, Ivanhoe's joint venture partner in the project, exercised its anti-dilution rights which is expected to generate C$67m ($51m) for the latter. The amount is expected to be secured at the same time as the CITIC Metal funds.
After completion of the financing, Citic will own 29.4% of Ivanhoe's issued and outstanding common shares while Zijin Mining will own around 9.8%.
The total funds of more than C$679m ($514m) from the CITIC Metal and Zijin Mining investments will enable Ivanhoe to completely finance its share of the capital costs required to bring the Kakula Mine into commercial production.
The first copper concentrate production from the Kakula Mine is expected in the third quarter of 2021.
Ivanhoe Mines is a Canadian mining firm focusing on three assets in Southern Africa. In The Democratic Republic of the Congo (DRC), it includes the Kamoa-Kakula copper discoveries and the historic Kipushi zinc-copper-germanium-silver mine.
Ivanhoe also partnered with La Générale des Carrières et des Mines to upgrade the Kipushi zinc-copper-germanium-silver mine in the DRC.
Additionally, it is developing the Platreef palladium-platinum-nickel-copper-gold discovery in South Africa.
allmineral
Separation and Beneficiation Systems for Sand, Gravel, Ore, Coal, Lignite and Slag
Mine Environment Management
Geoscience and Environmental Engineering Consultancy Solutions
Powering and Digitizing the Mining industry

redpajama_set_name: RedPajamaCommonCrawl


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Home»Interviews»2.7 Million Trump Ballots Deleted; Alleged Ballot Fraud in Dominion Voting Systems
2.7 Million Trump Ballots Deleted; Alleged Ballot Fraud in Dominion Voting Systems
Herland Report: Ballot Fraud in Dominion Voting Systems? With the American election still undecided, lawyers are working to determine whether the reported media victory for Joe Biden was determined by voter fraud.
This voter fraud was allegedly made possible through backdating mail in ballots, allowing unregistered voters to vote, counting days past the election, and it all systematized by the Dominion Smartmatic Voting System, which deleted millions of Trump votes.
2.7 Million Trump Ballots Deleted; Alleged Ballot Fraud in Dominion Voting Systems.
Rudy Giuliani, Donald Trump's lawyer, asserts that hundreds of workers that participated in facilitating the fraud – among them many Democrats who were uneased by what was going on – are now coming forward to testify.
Whether the courts will establish that voter fraud happened or not, remains to be seen.
There are three main voting software companies used in U.S. elections—Election Systems & Software (ES&S), Hart InterCivic, and Dominion Voting Systems.
The Dominion Voting System donations was administered by Act Blue – a fundraiser that has given hundreds of millions to Democrats running for President. Act Blue was the organization that received the Black Lives Matter donations into its accounts.
While the Black Lives Matter funding was said to go to those in need, donations went directly to Act Blue. Act Blue seems to have funded Bernie Sanders, Biden, Elisabeth Warren, which is clear from its financial reports that are available at OpenSecrets.org.
Rudy Giuliani, President Trump's lawyer, asserts that the Dominion Smartmatic Voting System was created by Hugo Chavez and was banned in US decades ago, but came back as a sub-contractor in the 2020 elections.
"This Dominion company is a radical left company. One of the people there is a big supporter of Antifa and has written horrible things about the President for the last three or four years. And the software that they used is done by a company called Smartmatic" said Giuliani.
2.7 Million Trump Ballots Deleted; Alleged Ballot Fraud in Dominion Voting Systems: GNews writes: On Nov. 12th, 2020, President Donald Trump posted a tweet proving that Dominion voting machines across the nation deleted 2.7 million Trump ballots. "Data analysis finds 221,000 Pennsylvania votes switched from President Trump to Biden," he added.
Records from the Federal Election Commission showed that the company received massive donations from Democratic donors. According to The National Pulse, "Of the 96 donations, 92 in total – or 95.8 percent – went to Democratic candidates, predominantly via ActBlue."
This probably explains why those machines "mysteriously" counted Trump ballots toward Biden. Federal Election Commission Records.
The Dominion Voting System was designed for Hugo Chavez, the former President of Venezuela. Smartmatic was founded by Chavez in 2004, teaming up with a partly Chavez's government-owned software company, Bitza, to provide election voting services.
In 2004, Chavez used Smartmatic with a "redesigned" code to run for the election. According to Giuliani, the Smartmatic Voting System was banned in US decades ago, however, it came back as a sub-contractor this time to the country.
More ironically, just one day before its statement, Rudy Giuliani claimed that the Trump Campaign had found insider witnesses from Dominion alleging its fraud system.
On November 11th EST, Rudy Giuliani stated that the attorneys of the Trump Campaign Thor Hearn and David Kallman have obtained abundant evidence alleging ballot fraud in Michigan on Steve Bannon's War Room.
Specifically, he emphasized that witnesses in the Dominion Voting Systems disclosed solid evidence to the Trump Campaign regarding the voter fraud.
"There are Democrats who are testifying against the democratic effort because it was so disgusting," Giuliani alleged in the interview and confirmed that Thor Hearne obtained more than 100 sworn affidavits witnessing the Democratic Party's voter fraud behaviors.
According to Giuliani, another attorney, David Kallman has obtained evidence from the "community people and the Dominion People."
He continued, "[among] the observers who were excluded and were lied to, two of them stayed behind after all republicans left out. They are the ones who got the evidence of hundreds of thousands of votes coming in… and they have some photographs also."
Also check out these topics at CNN or FOX News. Other sources may be New York Times, USA Today, the Washington Post or from the British angle, BBC, The Guardian, The Telegraph or Financial Times.
Led by Scandinavian bestselling author, Hanne Nabintu Herland, The Herland Report news and opinion website provides independent analysis from leading Western intellectuals and ground breaking YouTube interviews, cutting through the mainstream media rhetoric. It is a great place to watch interviews and read the articles of leading intellectuals, thought leaders, authors and activists from across the political spectrum. The Herland Report believes in freedom of speech and its editorial policy resides above the traditional Left vs Right paradigm which we believe has lost its relevance and ability to describe the current driving forces in Western politics.
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48% Of US Small Businesses May Be Forced To Shut Down Permanently
The US Banana Republic Mess: Voter fraud, Dominion Voting System
Few Discuss the Massive Corona Redistribution of Wealth #ElonMusk #BillGates
Rudy Giuliani Legal Team Presents Overview of Election Fraud Evidence
Herland Report: Yesterday in a press conference, Rudy Giuliani legal team, Sidney Powell and …
US elections still not over: The Vote Fraud is Overwhelming, so massive it is suffocating
Herland Report: US elections still not over: The Vote Fraud is Overwhelming. Remember: The …


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\section{Introduction}

Inference tasks encountered in natural language processing, graph inference and manifold learning employ the singular value decomposition (SVD) as a first step to reduce dimensionality while retaining \emph{useful} structure in the input. Such spectral embeddings go under various guises: Principle Component Analysis (PCA), Latent Semantic Indexing (natural language processing), Kernel Principal Component Analysis, commute time and diffusion embeddings of graphs, to name a few.  In this paper, we present a {\it compressive} approach for accomplishing SVD-based dimensionality reduction, or embedding, without actually performing the computationally expensive SVD step.

The setting is as follows. The input is represented in matrix form. This matrix could represent the adjacency matrix or the Laplacian of a graph, the probability transition matrix of a random walker on the graph, a bag-of-words representation of documents, the action of a kernel on a set of $l$ points $\{\mathbf{x}(p) \in\Rline{d}:p = 1,\dots,m\}$ (kernel PCA)\cite{scholkopf_kernel_1997}\cite{mika_kernel_1999} such as
\begin{align}
A(p,q) = e^{\left.-\norm{\mathbf{x}(p) - \mathbf{x}(q)}^2\middle/\left.2\alpha^2\right.\right.} ~~ \text{(or)}~~ A(p,q) = \indicator{\norm{\mathbf{x}(p) - \mathbf{x}(q)} < \alpha},~~1\leq p,q\leq l,\label{eq:kernel-matrix}
\end{align}
where $\indicator{\cdot}$ denotes the indicator function or matrices derived from $K$-nearest-neighbor graphs constructed from $\{\mathbf{x}(p)\}$. We wish to compute a {transformation} of the rows of this $m\times n$ matrix $A$ which succinctly captures the global structure of $A$ via \emph{euclidean} distances (or similarity metrics derived from the $\ell_2$-norm, such as normalized correlations). A common approach is to compute a partial SVD of $A$, $\sum_{l = 1}^{l = k} \sigma_l \mathbf{u}_l \mathbf{v}_l^T,~k\ll n$, and to use it to embed the rows of $A$ into a $k$-dimensional space using the rows of $ {E} = \left[f(\sigma_1)\mathbf{u}_1~f(\sigma_2)\mathbf{u}_2~\cdots~f(\sigma_k)\mathbf{u}_k\right]$, for some function $f(\cdot)$. The embedding of the variable corresponding to the $l$-th row of the matrix $A$ is the $l$-th row of ${E}$. For example, $f(x) = x $ corresponds to Principal Component Analysis (PCA): the $k$-dimensional rows of ${E}$ are projections of the $n$-dimensional rows of $A$ along the first $k$ principal components, $\{\mathbf{v}_l,~l=1,\dots,k\}$. Other important choices include $f(x) = \text{constant}$ used to cut graphs \cite{white_spectral_2005} and $f(x) = \left.{1}\middle/{\sqrt{1-x}}\right.$ for commute time embedding of graphs \cite{gobel_random_1974}. Inference tasks such as (unsupervised) clustering and (supervised) classification are performed using $\ell_2$-based pairwise similarity metrics on the embedded coordinates (rows of ${E}$) instead of the ambient data (rows of $A$).

Beyond the obvious benefit of dimensionality reduction from $n$ to $k$, embeddings derived from the \emph{leading} partial-SVD can often be interpreted as denoising, since the ``noise'' in matrices arising from real-world data manifests itself via the smaller singular vectors of $A$ (e.g., see \cite{newman_graph_dist_2012}, which analyzes graph adjacency matrices). This is often cited as a motivation for choosing PCA over ``isotropic'' dimensionality reduction techniques such as random embeddings, which, under the setting of the Johnson-Lindenstrauss (JL) lemma, can also preserve structure.

The number of singular vectors $k$ needed to capture the structure of an $m \times n$ matrix grows with its size, and two bottlenecks emerge as we scale: (a)~The computational effort required to extract a large number of singular vectors using conventional iterative methods such as Lanczos or simultaneous iteration or approximate algorithms like Nystr{o}m  \cite{fowlkes_spectral_2004}, \cite{Mahoney_2005_nystrom} and Randomized SVD \cite{Halko_Tropp_2011} for computation of partial SVD becomes prohibitive (scaling as $\Omega(k\mathcal{T})$, where $\mathcal{T}$ is the number of non-zeros in $A$) (b)~the resulting $k$-dimensional embedding becomes unwieldy for use in subsequent inference steps.

\noindent
{\bf Approach and Contributions:}
In this paper, we tackle these scalability bottlenecks by focusing on what embeddings are actually used for: computing $\ell_2$-based \emph{pairwise} similarity metrics typically used for supervised or unsupervised learning. For example, K-means clustering uses pairwise Euclidean distances, and SVM-based classification uses pairwise inner products. We therefore ask the following question: ``Is it possible to compute an embedding which captures the pairwise euclidean distances between the rows of the spectral embedding $E = \left[f(\sigma_1)\mathbf{u}_1~\cdots~f(\sigma_k)\mathbf{u}_k\right]$, while sidestepping the computationally expensive partial SVD?'' We answer this question in the affirmative by presenting a \emph{compressive} algorithm which directly computes a low-dimensional embedding.

There are two key insights that drive our algorithm:\\
\noindent ~$\bullet$~By approximating $f(\sigma)$ by a low-order ($L\ll\min\{m,n\}$) polynomial, we can compute the embedding iteratively using matrix-vector products of the form $A\mathbf{q}$ or $A^T\mathbf{q}$.\\
\noindent ~$\bullet$~The iterations can be computed \emph{compressively:} by virtue of the celebrated JL lemma, the embedding geometry is approximately captured by a small number $d = O(\log (m + n))$ of randomly picked starting vectors.

The number of passes over $A$, $A^T$ and time complexity of the algorithm are $L$, $L$ and $O( L (\mathcal{T} + m +n ) \log (m + n) )$ respectively. These are all \emph{independent} of the {number of singular vectors} $k$ whose effect we wish to capture via the embedding. This is in stark contrast to embedding directly based on the partial SVD.  Our algorithm lends itself to parallel implementation as a sequence of $2L$ matrix-vector products interlaced with vector additions, run in parallel across $d = O( \log (m + n) )$ randomly chosen starting vectors. This approach significantly reduces both computational complexity and embedding dimensionality relative to partial SVD. A freely downloadable Python implementation of the proposed algorithm that exploits this inherent parallelism can be found in \cite{code}.

\section{Related work}

As discussed in Section~\ref{sec:random}, the concept of compressive measurements forms a key ingredient in our algorithm, and is based on the JL lemma \cite{Achlioptas_2001}.  The latter, which provides probabilistic guarantees on approximate preservation of the Euclidean geometry for a finite collection of points under random projections, forms the basis for many other applications, such as compressive sensing \cite{compressive_sampling_tut}.

We now mention a few techniques for exact and approximate SVD computation, before discussing algorithms that sidestep the SVD as we do. The time complexity of the full SVD of an $m \times n$ matrix is $O(mn^2)$ (for $m > n$). Partial SVDs are computed using iterative methods for eigen decompositions of symmetric matrices derived from $A$ such as $A\bA^T$ and $\left[ {0} ~ {A}^T; {A} ~ {0} \right]$ \cite{trefethen_numerical_1997}. The complexity of standard iterative eigensolvers such as simultaneous iteration\cite{mccormick_simultaneous_1977} and the Lanczos method scales as $\Omega(k\mathcal{T})$  \cite{trefethen_numerical_1997}, where $\mathcal{T}$ denotes the number of non-zeros of $A$.

The leading $k$ singular value, vector triplets $\{(\sigma_l, \mathbf{u}_l,\mathbf{v}_l),~l = 1,\dots,k\}$ minimize the matrix \emph{reconstruction} error under a rank $k$ constraint: they are a solution to the optimization problem ${\arg\min} \Vert{A - \sum_{l=1}^{l=k}\sigma_l \mathbf{u}_l \mathbf{v}_l^T }\Vert_F^2$, where $\Vert\cdot\Vert_F$ denotes the Frobenius norm. Approximate SVD algorithms strive to reduce this error while also placing constraints on the computational budget and/or the number of passes over $A$. A commonly employed approximate eigendecomposition algorithm is the Nystr{o}m  method \cite{fowlkes_spectral_2004}, \cite{Mahoney_2005_nystrom} based on random sampling of $s$ columns of $A$, which has time complexity $O(k s n + s^3)$.  A number of variants of the Nystr{o}m  method for kernel matrices like \eqref{eq:kernel-matrix} have been proposed in the literature. These aim to improve accuracy using preprocessing steps such as $K$-means clustering \cite{Zhang_k_means_2008_anchors} or random projection trees \cite{Jordan_09_Fast_Approx_Spectral}. Methods to reduce the complexity of the Nystr{o}m  algorithm to $O(k s n + k^3)$\cite{liang_icml_10_nystrom_and_randomSVD},  \cite{kumar_ensemble_nystrom_nips2009} enable Nystr{o}m  sketches that see more columns of $A$. The complexity of all of these grow as $\Omega( k s n)$.  Other randomized algorithms, involving iterative computations, include the Randomized SVD \cite{Halko_Tropp_2011}. Since all of these algorithms set out to recover $k$-leading eigenvectors (exact or otherwise), their complexity scales as $\Omega(k \mathcal{T})$.

We now turn to algorithms that sidestep SVD computation. In \cite{lin_cohen_10_power_iteration}, \cite{lin_thesis_2012_scalable}, vertices of a graph are embedded based on diffusion of probability mass in random walks on the graph, using the power iteration run independently on random starting vectors, and stopping ``prior to convergence.''  While this approach is specialized to probability transition matrices (unlike our general framework) and does not provide explicit control on the nature of the embedding as we do, a feature in common with the present paper is that the time complexity of the algorithm and the dimensionality of the resulting embedding are independent of the number of eigenvectors $k$ captured by it. A parallel implementation of this algorithm was considered in  \cite{Yan_Power_Iteration_implementation_2013}; similar parallelization directly applies to our algorithm. Another specific application that falls within our general framework is the commute time embedding on a graph, based on the normalized adjacency matrix and weighing function $f(x) = 1/\sqrt{1-x}$  \cite{gobel_random_1974}, \cite{random_walk_survey_lovasz1993}.  Approximate commute time embeddings have been computed using Spielman-Teng solvers \cite{spielman_graph_sparsification_2004}, \cite{spielman_linear_systems_2014} and the JL lemma in \cite{spielman_comm_time_approx_2011}. The complexity of the latter algorithm and the dimensionality of the resulting embedding are comparable to ours, but the method is specially designed for the normalized adjacency matrix and the weighing function $f(x) = 1/\sqrt{1-x}$. Our more general framework would, for example, provide the flexibility of suppressing small eigenvectors from contributing to the embedding (e.g, by setting $f(x) = \indicator{x > \epsilon}/\sqrt{1-x}$).

Thus, while randomized projections are extensively used in the embedding literature, to the best of our knowledge, the present paper is the first to develop a general compressive framework for spectral embeddings derived from the SVD. It is interesting to note that methods similar to ours have been used in a different context, to estimate the \emph{empirical distribution} of eigenvalues of a large hermitian matrix \cite{silver_kernel_1996}, \cite{di_napoli_efficient_2013}. These methods use a polynomial approximation of indicator functions $f(\lambda) = \indicator{a\leq \lambda \leq b}$ and random projections to compute an approximate histogram of the number of eigenvectors across different bands of the spectrum: $[a,b]\subseteq [\lambda_{\min}, \lambda_{\max}]$.

\section{Algorithm}\label{sec:algo}

\looseness=0

We first present the algorithm for a symmetric $n \times n$ matrix $S$. Later, in Section~\ref{sec:map_to_SVD}, we show how to handle a general $m\times n$ matrix by considering a related $(m+n)\times(m+n)$ symmetric matrix. Let $\lambda_l$ denote the eigenvalues of $S$ sorted in descending order and $\mathbf{v}_l$ their corresponding unit-norm eigenvectors (chosen to be orthogonal in case of repeated eigenvalues). For any function $g(x):\Rline{}\mapsto\Rline{}$, we denote by $g(S)$ the $n\times n$ symmetric matrix $g(S) = \sum_{l = 1}^{l = n} g(\lambda_l) \mathbf{v}_l \mathbf{v}_l^T$. We now develop an $O(n \log n)$ algorithm to compute a $d=O(\log n)$ dimensional embedding which approximately captures pairwise euclidean distances between the rows of the embedding $E = \left[f\left(\lambda_1\right)\mathbf{v}_1~f\left(\lambda_2\right)\mathbf{v}_2~\cdots~f\left(\lambda_n\right)\mathbf{v}_n\right]$.

\noindent
{\it Rotations are inconsequential:} We first observe that rotation of basis does not alter $\ell_2$-based similarity metrics. Since ${V} = [\mathbf{v}_1\cdots \mathbf{v}_n]$ satisfies ${V}\bV^T = {V}^T{V} = \eye{n}$, pairwise distances between the rows of $E$ are equal to corresponding pairwise distances between the rows of $E{V}^T = \sum_{l = 1}^{l = n} f(\lambda_l) \mathbf{v}_l\mathbf{v}_l^T = f(S)$. We use this observation to compute embeddings of the rows of $f(S)$ rather than those of $E$.

\subsection{Compressive embedding}\label{sec:random}

Suppose now that we know $f(S)$.  This constitutes an $n$-dimensional embedding, and
similarity queries between two ``vertices'' (we refer to the variables corresponding to rows of $S$ as vertices, as we would for matrices derived from graphs) requires $O(n)$ operations. However, we can reduce this time to $O(\log n)$ by using the JL lemma, which informs us that pairwise distances can be approximately captured by compressive projection onto $d = O(\log n)$ dimensions.

Specifically, for $d > \left.(4+2\beta)\log n\middle/\left(\epsilon^2/2 - \epsilon^3/3\right)\right.$,
let  $\Omega$ denote an $n\times d$ matrix with i.i.d. entries drawn uniformly at random from $\{\pm 1/\sqrt{d}\}$. According to the JL lemma, pairwise distances between the rows of $f(S)\Omega$ approximate pairwise distances between the rows of $f(S)$ with high probability.
In particular, the following statement holds with probability at least $1- n^{-\beta}$: $ (1-\epsilon)\norm{\mathbf{u} - \mathbf{v}}^2 \leq \norm{\left(\mathbf{u} - \mathbf{v}\right)\Omega}^2 \leq (1+\epsilon)\norm{\mathbf{u}-\mathbf{v}}^2$, for any two rows $\mathbf{u},\mathbf{v}$ of $f(S)$.

The key take-aways are that (a) we can reduce the embedding dimension to $d = O( \log n)$, since we are only interested in pairwise similarity measures, and (b) We do not need to compute $f(S )$.  We only need to compute $f(S)\Omega$.  We now discuss how to accomplish the latter efficiently.

\subsection{Polynomial approximation of embedding}

Direct computation of $E^{\prime} = f(S)\Omega$ from the eigenvectors and eigenvalues of $S$, as $f(S) = \sum f(\lambda_l) \mathbf{v}_l\mathbf{v}_l^T$ would suggest, is expensive ($O(n^3)$). However, we now observe that computation of $\psi(S)\Omega$ is easy when $\psi ( \cdot )$ is a polynomial. In this case, $\psi(S) = \sum_{p = 0}^{p = L} b_p S^p$ for some $b_p\in\Rline{}$, so that $\psi(S)\Omega$ can be computed as a sequence of $L$ matrix-vector products interlaced with vector additions run in \emph{parallel} for each of the $d$ columns of $\Omega$. Therefore, they only require $Ld\mathcal{T} + O(Ldn)$ flops. Our strategy is to approximate $E^{\prime} = f(S)\Omega$ by $\widetilde{E} = \widetilde{f}_L(S)\Omega$, where $\widetilde{f}_L(x)$ is an $L$-th order polynomial approximation of $f(x)$. We defer the details of computing a ``good'' polynomial approximation to Section~\ref{sec:poly_fun}. For now, we assume that one such approximation $\widetilde{f}_L(\cdot)$ is available and give bounds on the loss in fidelity as a result of this approximation.

\subsection{Performance guarantees}\label{sec:perf:guarantees}

The spectral norm of the ``error matrix'' $Z = f(S) - \widetilde{f}(S) = \sum_{r=1}^{r=n} (f(\lambda_r) - \widetilde{f}_L(\lambda_r))\mathbf{v}_r\mathbf{v}_r^T$ satisfies $\norm{Z} = \delta =  {\max}_l |{f(\lambda_l) - \widetilde{f}_L(\lambda_l)}| \leq {\max} \{|{f(x) - \widetilde{f}_L(x)}|\}$, where the spectral norm of a matrix $B$, denoted by $\VertB\Vert$ refers to the induced $\ell_2$-norm. For symmetric matrices, $\norm{B} \leq \alpha \iff \abs{\lambda_l} \leq \alpha ~\forall l$, where $\lambda_l$ are the eigenvalues of $B$. Letting $\mathbf{i}_p$ denote the unit vector along the $p$-th coordinate of $\Rline{n}$, the distance between the $p,q$-th rows of $\widetilde{f}(S)$ can be written as
\begin{equation}\label{eq:bound_poly_app}
\Vert {\widetilde{f}_L(S) \left(\mathbf{i}_p - \mathbf{i}_q\right)} \Vert  = \norm{f(S) \left(\mathbf{i}_p - \mathbf{i}_q\right) - Z\left(\mathbf{i}_p - \mathbf{i}_q\right)} \leq  \Vert {E^T \left(\mathbf{i}_p - \mathbf{i}_q\right)}\Vert + \delta\sqrt{2}.
\end{equation}
Similarly, we have that $\Vert{\widetilde{f}_L(S) \left(\mathbf{i}_p - \mathbf{i}_q\right)}\Vert\geq \Vert {E^T \left(\mathbf{i}_p - \mathbf{i}_q\right)} \Vert - \delta\sqrt{2}$. Thus pairwise distances between the rows of $\widetilde{f}_L(S)$ approximate those between the rows of $E$. However, the distortion term $\delta\sqrt{2}$ is \emph{additive} and must be controlled by carefully choosing $\widetilde{f}_L(\cdot)$, as discussed in Section~\ref{sec:implement}.

Applying the JL lemma \cite{Achlioptas_2001} to the rows of $\widetilde{f}_L(S)$, we have that when $d > O\left(\epsilon^{-2} \log n \right)$ with i.i.d. entries drawn uniformly at random from $\{\pm 1/\sqrt{d}\}$, the embedding $\widetilde{E} = \widetilde{f}_L(S)\Omega$ captures pairwise distances between the rows of $\widetilde{f}_L(S)$ up to a multiplicative distortion of $1\pm\epsilon$ with high probability:
\begin{align*}
\norm{\widetilde{E}^T \left(\mathbf{i}_p - \mathbf{i}_q\right)}  = \norm{\Omega^T\widetilde{f}_L(S) \left(\mathbf{i}_p - \mathbf{i}_q\right)}
 \leq \sqrt{1+\epsilon}\norm{\widetilde{f}_L(S) \left(\mathbf{i}_p - \mathbf{i}_q\right)}
\end{align*}
Using \eqref{eq:bound_poly_app}, we can show that $\Vert{\widetilde{E}^T \left(\mathbf{i}_p - \mathbf{i}_q\right)}\Vert  \leq \sqrt{1+\epsilon} \left(\Vert{E^T \left(\mathbf{i}_p - \mathbf{i}_q\right)}\Vert + \delta\sqrt{2}\right)$. Similarly, $ \Vert{\widetilde{E}^T \left(\mathbf{i}_p - \mathbf{i}_q\right)}\Vert  \geq \sqrt{1-\epsilon} \left(\Vert{E^T \left(\mathbf{i}_p - \mathbf{i}_q\right)}\Vert - \delta\sqrt{2}\right)$. We state this result in Theorem~\ref{thm:guarantees}.

\begin{theorem}
\label{thm:guarantees}

Let $\widetilde{f}_L(x)$ denote an $L$-th order polynomial such that: $\delta = {\max}_l |{f(\lambda_l) - \widetilde{f}_L(\lambda_l)}| \leq {\max} |{f(x) - \widetilde{f}_L(x)}|$ and $\Omega$ an $n\times d$ matrix with entries drawn independently and uniformly at random from $\{\pm 1/\sqrt{d}\}$, where $d$ is an integer satisfying $d > \left.{(4+2\beta)}\log n\middle/{(\epsilon^2/2 - \epsilon^3/3)}\right.$. Let $g:~\Rline{p}\rightarrow\Rline{d}$ denote the mapping from the $i$-th row of $E = \left[f\left(\lambda_1\right)\mathbf{v}_1~\cdots~f\left(\lambda_n\right)\mathbf{v}_n\right]$ to the $i$-th row of $\widetilde{E}=\widetilde{f}_L(S) \Omega$. The following statement is true with probability at least $1-n^{-\beta}$:
\begin{equation*}
\sqrt{1-\epsilon}(\norm{u - v}-\delta\sqrt{2}) \leq \norm{g(u) - g(v)} \leq \sqrt{1+\epsilon}(\norm{u - v}+\delta\sqrt{2})
\end{equation*}
for any two rows $u,v$ of $E$. Furthermore, there exists an algorithm to compute each of the $d=O(\log n)$ columns of $\widetilde{E}$ in $O(L (\mathcal{T} + n))$ flops independent of its other columns which makes $L$ passes over $S$ ($\mathcal{T}$ is the number of non-zeros in $S$).
\end{theorem}

\subsection{Choosing the polynomial approximation}\label{sec:poly_fun}

We restrict attention to matrices which satisfy $\norm{S}\leq 1$, which implies that $ |\lambda_l| \leq 1$. We observe that we can trivially center and scale the spectrum of any matrix to satisfy this assumption when we have the following bounds: $\lambda_l\leq \sigma_{\max}$ and $\lambda_l\geq \sigma_{\min}$ via the rescaling and centering operation given by: $S^{\prime} = \left. 2 S\middle/\left(\sigma_{\max}-\sigma_{\min}\right)\right. - \left.\left(\sigma_{\max}+\sigma_{\min}\right)\eye{n}\middle/ \left(\sigma_{\max}-\sigma_{\min}\right)\right.$ and by modifying $f(x)$ to $f^{\prime}(x) = f\left( \left. x \left(\sigma_{\max}-\sigma_{\min}\right)\middle/2\right. + \left. \left(\sigma_{\max}+\sigma_{\min}\right)\middle/2\right.\right)$.

In order to compute a polynomial approximation of $f(x)$, we need to define the notion of ``good'' approximation. We showed in Section~\ref{sec:perf:guarantees} that the errors introduced by the polynomial approximation can be summarized by furnishing a bound on the spectral norm of the error matrix $Z = f(S) - \widetilde{f}_L(S)$: Since $\norm{Z} = \delta = \max_{l} |f(\lambda_l) - \widetilde{f}_L(\lambda_l)|$, what matters is how well we approximate the function $f(\cdot)$ \emph{at} the eigenvalues $\{\lambda_l\}$ of $S$. Indeed, if we know the eigenvalues, we can minimize $\norm{Z}$ by minimizing $\max_{l} |f(\lambda_l) - \widetilde{f}_L(\lambda_l)|$. This is not a particularly useful approach, since computing the eigenvalues is expensive. However, we can use our prior knowledge of the domain from which the matrix $S$ comes from to penalize deviations from $f(\lambda)$ differently for different values of $\lambda$. For example, if we know the distribution $p(x)$ of the eigenvalues of $S$, we can minimize the average error $\Delta_L = \int_{-1}^{1} p(\lambda) |f(\lambda) - \widetilde{f}_L(\lambda)|^2\der{x}$.  In our examples, for the sake of concreteness, we assume that the eigenvalues are uniformly distributed over $[-1,1]$ and give a procedure to compute an $L$-th order polynomial approximation of $f(x)$ that minimizes $\Delta_L = (1/2)\int_{-1}^{1}|f(x) - \widetilde{f}_L(x)|^2 \der{x}$.

A numerically stable procedure to generate finite order polynomial approximations of a function over $[-1,1]$ with the objective of minimizing $\int_{-1}^{1}|f(x) - \widetilde{f}_L(x)|^2 \der{x}$ is via Legendre polynomials $p(r, x),~r=0,1,\dots,L$. They satisfy the recursion $ p(r, x) = ( 2 - 1/r ) x p(r-1, x) - ( 1 - 1/r ) p(r-2, x) $ and are orthogonal: $\int_{-1}^{1} p(k,x)p(l,x)\der{x} = \left.2\indicator{k=l}\middle/(2r+1)\right.$. Therefore we set $\widetilde{f}_L(x) = \sum_{r=0}^{r = L}a(r)p(r,x)$ where $a(r) = (r+1/2) \int_{-1}^{1} p(r,x)f(x)\der{x}$. We give a method in Algorithm~\ref{alg:proposed-embed-matrices} that uses the Legendre recursion to compute $p(r,S)\Omega,~r=0,1,\dots,L$ using $Ld$ matrix-vector products and vector additions. The coefficients $a(r)$ are used to compute $\widetilde{f}_L(S)\Omega$ by adding weighted versions of $p(r,S)\Omega$.

\begin{algorithm}
  \caption{Proposed algorithm to compute approximate $d$-dimensional eigenvector \emph{embedding} of a $n\times n$ symmetric matrix $S$ (such that $\norm{S} \leq 1$) using the $n\times d$ random projection matrix $\Omega$.
    \label{alg:proposed-embed-matrices}}
  \begin{algorithmic}[1]

    \Function{FastEmbedEIG}{$S$, $f(x)$, $L$, $\Omega$}
   	\State\Comment{Compute polynomial approximation $\widetilde{f}_L(x)$ which minimizes $\int_{-1}^{1}|f(x) - \widetilde{f}_L(x)|^2 \der{x}$}
	\For{$r = 0,\dots,L$}
		\Let{$a(r)$}{$(r+1/2)\int_{x = -1}^{x=1}f(x)p(r,x)\der{x}$}\qquad \Comment{$p(r, x)$: Order $r$ Legendre polynomial}
	\EndFor
	\vspace{0.075in}

	\State ${Q}(0)\gets\Omega$, ${Q}(-1)\gets$ {$0$}, {$\widetilde{E}\gets a(0){Q}(0)$}
	\For{$r = 1,2,\dots,L$}
		\Let{${Q}(r)$}{$\left.(2-1/r) S {Q}({r-1}) - (1-1/r) {Q}({r-2})\right.$}\quad\qquad\Comment{${Q}(r) = p(r,S)\Omega$}
		\Let{$\widetilde{E}$}{$\widetilde{E} + a(r){Q}(r)$}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Comment{$\widetilde{E}$ now holds $\widetilde{f}_r(S)\Omega$}
	\EndFor
	\vspace{0.075in}
	\State \Return{$\widetilde{E}$} \qquad\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Comment{$\widetilde{E} = \widetilde{f}_L(S)\Omega$}
	\EndFunction
  \end{algorithmic}
\end{algorithm}

As described in Section~\ref{sec:implement}, if we have prior knowledge of the distribution of eigenvalues (as we do for many commonly encountered large matrices), then we can ``boost'' the performance of the generic Algorithm~\ref{alg:proposed-embed-matrices} based on the assumption of eigenvalues uniformly distributed over $[-1,1]$.

\subsection{Embedding general matrices}\label{sec:map_to_SVD}

We complete the algorithm description by generalizing to any $m\times n$ matrix $A$ (not necessarily symmetric) such that $\norm{A} \leq 1$. The approach is to utilize Algorithm~\ref{alg:proposed-embed-matrices} to compute an approximate $d$-dimensional embedding of the symmetric matrix $S=[0~A^T;A~0]$. Let $\left\{\left(\sigma_l,\mathbf{u}_l,\mathbf{v}_l\right):~l=1,\dots,\min\{m,n\}\right\}$ be an SVD of $A = \sum_l\sigma_l\mathbf{u}_l\mathbf{v}_l^T$ ($\norm{A} \leq 1 \iff \sigma_l \leq 1$). Consider the following spectral mapping of the rows of $A$ to the rows of $E_{\textrm{row}} = \left[f(\sigma_1)\mathbf{u}_1~\cdots~f(\sigma_m)\mathbf{u}_m\right]$ and the columns of $A$ to the rows of $E_{\textrm{col}} = \left[f(\sigma_1)\mathbf{v}_1~\cdots~f(\sigma_n)\mathbf{v}_n\right]$. It can be shown that the unit-norm orthogonal eigenvectors of $S$ take the form $\left.\left[\mathbf{v}_l;\mathbf{u}_l\right]\middle/\sqrt{2}\right.$ and $\left.\left[\mathbf{v}_l;-\mathbf{u}_l\right]\middle/\sqrt{2}\right.$, $l=1,\dots,\min\{m,n\}$, and their corresponding eigenvalues are $\sigma_l$ and $-\sigma_l$ respectively. The remaining $|m-n|$ eigenvalues of $S$ are equal to $0$. Therefore, we call $\widetilde{E}_{\textrm{all}}\gets~$\textproc{FastEmbedEIG}$(S,f^{\prime}(x), L, \Omega)$ with $f^{\prime}(x) = f(x) \indicator{x \geq 0} - f(-x) \indicator{x<0}$ and $\Omega$ is an $(m+n)\times d, d = O( \log(m+n))$ matrix (entries drawn independently and uniformly at random from $\{\pm 1/\sqrt{d}\}$). Let $\widetilde{E}_{\textrm{col}}$ and $\widetilde{E}_{\textrm{row}}$ denote the first $n$ and last $m$ rows of $\widetilde{E}_{\textrm{all}}$. From Theorem~\ref{thm:guarantees}, we know that, with overwhelming probability, pairwise distances between any two rows of $\widetilde{E}_{\textrm{row}}$ approximates those between corresponding rows of $E_{\textrm{row}}$. Similarly, pairwise distances between any two rows of $\widetilde{E}_{\textrm{col}}$ approximates those between corresponding rows of $E_{\textrm{col}}$.

\section{Implementation considerations} \label{sec:implement}

We now briefly go over implementation considerations before presenting numerical results in Section~\ref{sec:results}.

\noindent\textbf{Spectral norm estimates}
In order to ensure that the eigenvalues of $S$ are within $[-1,1]$ as we have assumed, we scale the matrix by its spectral norm ($\Vert S\Vert = \max |\lambda_l|$). To this end, we obtain a tight lower bound (and a good approximation) on the spectral norm using power iteration ($20$ iterates on $6\log n$ randomly chosen starting vectors), and then scale this up by a small factor ($1.01$) for our estimate (typically an upper bound) for $\Vert S\Vert$.

{\noindent\textbf{Polynomial approximation order}~$L$:
The error in approximating $f(\lambda)$ by $\widetilde{f}_L(\lambda)$, as measured by $\Delta_L = \int_{-1}^{1}|{f(x) - \widetilde{f}_L(x)}|^2\der{x}$ is a non-increasing function of the polynomial order $L$. Reduction in $\Delta_L$ often corresponds to a reduction in $\delta$ that appears as a bound on distortion in Theorem~\ref{thm:guarantees}. ``Smooth'' functions generally admit a lower order approximation for the same target error $\Delta_L$, and hence yield considerable savings in algorithm complexity, which scales linearly with $L$.}

\noindent\textbf{Polynomial approximation method}: The rate at which $\delta$ decreases as we increase $L$ depends on the function $p(\lambda)$ used to compute $\widetilde{f}_L(\lambda)$ (by minimizing $\Delta_L= \int p(\lambda) |f(\lambda) - \widetilde{f}_L(\lambda)|^2\der{x}$). The choice $p(\lambda) \propto 1$ yields the Legendre recursion used in Algorithm~\ref{alg:proposed-embed-matrices}, whereas $p(\lambda) \propto 1/\sqrt{1-\lambda^2}$ corresponds to the Chebyshev recursion, which is known to result in fast convergence. We defer to future work a detailed study of the impact of alternative choices for $p(\lambda)$ on $\delta$.

\noindent\textbf{Denoising by cascading}
In large-scale problems, it may be necessary to drive the contribution from certain singular vectors to zero. In many settings, singular vectors with smaller singular values correspond to noise. The number of such singular values can scale as fast as $O(\min\{m, n\})$. Therefore, when we place nulls (zeros) in $f(\lambda)$, it is desirable to ensure that these nulls are \emph{pronounced} after we approximate $f(\lambda)$ by $\widetilde{f}_L(\lambda)$. We do this by computing $\left(\widetilde{g}_{L/b}(S)\right)^b\Omega$, where $\widetilde{g}_{L/b}(\lambda)$ is an $L/b$-th order approximation of $g(\lambda) = f^{1/b}(\lambda)$. The small values in the polynomial approximation of $f^{1/b}(\lambda)$ which correspond to $f(\lambda) = 0$ (nulls which we have set) get amplified when we pass them through the $x^b$ non-linearity.

\section{Numerical results}\label{sec:results}

While the proposed approach is particularly useful for large problems in which exact eigendecomposition is computationally infeasible, for the purpose of comparison, our results are restricted to smaller settings where the exact solution can be computed. We compute the exact partial eigendecomposition using the ARPACK library (called from MATLAB). For a given choice of weighing function $f(\lambda)$, the associated embedding ${E}=[f(\lambda_1)\mathbf{v}_1~\cdots~f(\lambda_n)\mathbf{v}_n]$ is compared with the compressive embedding $\widetilde{E}$ returned by Algorithm~\ref{alg:proposed-embed-matrices}. The latter was implemented in Python using the Scipy's sparse matrix-multiplication routines and is available for download from \cite{code}.

We consider two real world undirected graphs in \cite{SNAP_ground_truth_2012} for our evaluation, and compute embeddings for the normalized adjacency matrix $\bAtilde$ ($ = D^{-1/2}A D^{-1/2}$, where $D$ is a diagonal matrix with row sums of the adjacency matrix $A$; the eigenvalues of $\bAtilde$ lie in $[-1, 1]$) for graphs. We study the accuracy of embeddings by comparing pairwise normalized correlations between $i,j$-th rows of ${E}$ given by $\left.<{E}(i,:),{E}(j,:)>\middle/\Vert{E}(i,:)\Vert\Vert{E}(j,:)\Vert\right.$ with those predicted by the approximate embedding $\left.<\widetilde{E}(i,:),\widetilde{E}(j,:)>/\Vert\widetilde{E}(i,:)\Vert\Vert\widetilde{E}(j,:)\Vert\right.$ (${E}(i,:)$ is short-hand for the $i$-th row of ${E}$).

\noindent\textbf{DBLP collaboration network} {\cite{SNAP_ground_truth_2012}~} is an undirected graph on $n=317080$ vertices with $1049866$ edges. We compute the leading $500$ eigenvectors of the normalized adjacency matrix $\bAtilde$. The smallest of the five hundred eigenvalues is $0.98$, so we set $f(\lambda) = \indicator{\lambda \geq 0.98}$ and $S = \bAtilde$ in Algorithm~\ref{alg:proposed-embed-matrices} and compare the resulting embedding $\widetilde{E}$ with ${E} = [\mathbf{v}_1~\cdots~\mathbf{v}_{500}]$. We demonstrate the dependence of the quality of the embedding $\widetilde{E}$ returned by the proposed algorithm on two parameters: (i) number of random starting vectors $d$, which gives the dimensionality of the embedding and (ii) the boosting/cascading parameter $b$ using this dataset.

{\noindent\textit{Dependence on the number of random projections $d$:}~In Figure~\eqref{fig:effect_dims_distortions}, $d$ ranges from $1$ to $120 \approx 9 \log n$ and plot the 1-st, 5-th, 25-th, 50-th, 75-th, 95-th and 99-th percentile values of the deviation between the compressive normalized correlation (from the rows of $\widetilde{E}$) and the corresponding exact normalized correlation (rows of ${E}$). The deviation decreases with increasing $d$, corresponding to $\ell_2$-norm concentration (JL lemma), but this payoff saturates for large values of $d$ as polynomial approximation errors start to dominate. From the $5$-th and $95$-th percentile curves, we see that a significant fraction (90\%) of pairwise normalized correlations in $\widetilde{E}$ lie within $\pm 0.2$ of their corresponding values in ${E}$ when $d = 80 \approx 6 \log n $. For Figure~\eqref{fig:effect_dims_distortions}, we use $L = 180$ matrix-vector products for each randomly picked starting vector and set cascading parameter $b = 2$ for the algorithm in Section~\ref{sec:implement}. }

\begin{figure}
        \centering
        \begin{subfigure}[b]{0.425\columnwidth}
        \centering
\includegraphics[width=.95\columnwidth]{effect_of_dimensionality_boost_2}
\caption{Effect of dimensionality $d$ of embedding}
\label{fig:effect_dims_distortions}
        \end{subfigure}%
        \quad
        \begin{subfigure}[b]{0.54\columnwidth}
\centering
\includegraphics[width=.475\columnwidth]{effect_of_boost_1_L180}~~~~\includegraphics[width=.475\columnwidth]{effect_of_boost_2_L90}
\caption{Effect of cascading: $b=1$(left) and $b=2$ (right)}
\label{fig:effect_boosting_distortions}
        \end{subfigure}
        \caption{DBLP collaboration network normalized correlations}
\end{figure}

{\noindent\textit{Dependence on cascading parameter $b$:}~In Section~\ref{sec:implement} we described how cascading can help suppress the contribution to the embedding $\widetilde{E}$ of the eigenvectors whose eigenvalues lie in regions where we have set $f(\lambda)=0$. We illustrate the importance of this boosting procedure by comparing the quality of the embedding $\widetilde{E}$ for $b=1$ and $b=2$ (keeping the other parameters of the algorithm in Section~\ref{sec:implement} fixed: $L=180$ matrix-vector products for each of $d = 80$ randomly picked starting vectors). We report the results in Figure~\eqref{fig:effect_boosting_distortions} where we plot percentile values of compressive normalized correlation (from the rows of $\widetilde{E}$) for different values of the exact normalized correlation (rows of ${E}$). For $b = 1$, the polynomial approximation of $f(\lambda)$ does not suppress small eigenvectors. As a result, we notice a deviation (bias) of the 50-percentile curve (green) from the ideal $y=x$ dotted line drawn (Figure~\ref{fig:effect_boosting_distortions} left). This disappears for $b=2$ (Figure~\ref{fig:effect_boosting_distortions} right).}

The running time for our algorithm on a standard workstation was about two orders of magnitude smaller than partial SVD using off-the-shelf sparse eigensolvers (e.g., the $80$ dimensional embedding of the leading $500$ eigenvectors of the DBLP graph took 1 minute whereas their exact computation took $105$ minutes). A more detailed comparison of running times is beyond the scope of this paper, but it is clear that the promised gains in computational complexity are realized in practice.

\noindent\textbf{Application to graph clustering for the Amazon co-purchasing network} {\cite{SNAP_ground_truth_2012}~}: This is an undirected graph on $n=334863$ vertices with $925872$ edges.  We illustrate the potential downstream benefits of our algorithm by applying $K$-means clustering on embeddings (exact and compressive) of this network. For the purpose of our comparisons, we compute the first 500 eigenvectors for $\bAtilde$ explicitly using an exact eigensolver, and use an $80$-dimensional compressive embedding $\widetilde{E}$ which captures the effect of these, with $f(\lambda) = I(\lambda \geq \lambda_{500})$, where $\lambda_{500}$ is the 500th eigenvalue. We compare this against the usual spectral embedding using the first $80$ eigenvectors of $\bAtilde$: ${E}=[\mathbf{v}_1~\cdots~\mathbf{v}_{80}]$. We keep the dimension fixed at $80$ in the comparison because $K$-means complexity scales linearly with it, and quickly becomes the bottleneck. Indeed, our ability to embed a large number of eigenvectors directly into a low dimensional space ($d \approx 6 \log n$) has the added benefit of dimensionality reduction within the subspace of interest (in this case the largest $500$ eigenvectors).

We consider $25$ instances of $K$-means clustering with $K=200$ throughout, reporting the median of a commonly used graph clustering score, modularity \cite{fortunato_community_2010} (larger values translate to better clustering solutions).  The median modularity for clustering based on our embedding $\widetilde{E}$ is $0.87$. This is significantly better than that for ${E}$, which yields median modularity of $0.835$.  In addition, the computational cost for $\widetilde{E}$ is one-fifth that for ${E}$ (1.5 minutes versus 10 minutes). When we replace the exact eigenvector embedding ${E}$ with approximate eigendecomposition using Randomized SVD \cite{Halko_Tropp_2011} (parameters: power iterates $q = 5$ and excess dimensionality $l=10$), the time taken reduces from 10 minutes to $17$ seconds, but this comes at the expense of inference quality: median modularity drops to $0.748$. On the other hand, the median modularity increases to $0.845$ when we consider exact partial SVD embedding with $120$ eigenvectors. This indicates that our compressive embedding yields better clustering quality because it is able to concisely capture more eigenvectors($500$ in this example, compared to $80$ and $120$ with conventional partial SVD). {It is worth pointing out that, even for known eigenvectors, the number of dominant eigenvectors $k$ that yields the best {\it inference performance} is often unknown {\it a priori}, and is treated as a hyper-parameter. For compressive spectral embedding $\widetilde{E}$, an elegant approach for implicitly optimizing over $k$ is to use the embedding function  $f(\lambda) = I(\lambda \geq c)$, with $c$ as a hyper-parameter.}

\section{Conclusion}

We have shown that random projections and polynomial expansions provide a powerful approach for spectral embedding of large matrices: for an $m \times n$ matrix $A$, our $O((\mathcal{T} + m + n) \log (m+n))$ algorithm computes an $O(\log (m+n))$-dimensional compressive embedding that provably approximates pairwise distances between points in the desired spectral embedding.  Numerical results for several real-world data sets show that our method provides good approximations for embeddings based on partial SVD, while incurring much lower complexity.  Moreover, our method can also approximate spectral embeddings which depend on the entire SVD, since its complexity does not depend on the number of dominant vectors whose effect we wish to model.  A glimpse of this potential is provided by the example of $K$-means based clustering for estimating sparse-cuts of the Amazon graph, where our method yields much better performance (using graph metrics) than a partial SVD with significantly higher complexity. This motivates further investigation into applications of this approach for improving downstream inference tasks in a variety of large-scale problems.

\section*{Acknowledgments}
\vspace{-0.15in}

{This work is supported in part by DARPA GRAPHS (BAA-12-01) and by Systems on Nanoscale Information fabriCs (SONIC), one of the six SRC STARnet Centers, sponsored by MARCO and DARPA. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding agencies.}

\vspace{-0.15in}
\bibliographystyle{IEEEtran}
\small{

redpajama_set_name: RedPajamaArXiv


text:
Online Sports Betting Secrets: How to Spot an Upset
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Most likely. Minnesota is a lot more grounded at home, going 13-7 SU and 12-8 ATS while just playing .500 ball out and about during a similar period. At the end of the day, the Vikings look like season finisher competitors at whatever point they play at home, however transform into a normal group out and about.
In the interim, SF has been awful this year, however they've actually had the option to pull out two games straight up at home, beating Oakland and the Rams.
Yet, the genuine fascinating detail identifying with this game is that Minnesota has generally played inadequately against the NFC West groups, going just 7-13 ATS. I search for Minnesota to be in a fight down the last seconds of this game, with SF possibly ending up as the winner, 30-27.
Furthermore, what do you think the last score in this game was? What about this: San Francisco won by and large, similarly as I anticipated, 9-3.
What amount of cash do you think sports wagering stars made on these games?
Posted on November 17, 2020 / 0	Categories	Uncategorized
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SCHOLARSHIP PROGRAM PROVIDED BY UNIVERSITÉ PARIS-SACLAY INTENDS TO PROVIDE OUTSTANDING CANDIDATES TO ENROLL IN THE UNIVERSITY, AND THOSE WHO WISH TO CONTINUE THEIR ACADEMIC JOURNEY. STUDENTS WILL BE FUNDED BASED ON ACADEMIC MERIT.
How to apply: you need to receive the school admission first before applying for scholarship. Completing the online application, you will require providing the names of two references who would be willing to submit a reference for you.


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The 2019 Lady Dutch Volleyball season
Daisy Eguia
The Pella Dutch volleyball team has had a challenging season. According to senior leaders Courtney Thompson and Emma Pleima the Lady Dutch had highs and lows throughout the season.
"We had a lot of good games, we beat the #9 ranked Knoxville at our home tournament which was really good for us, and we also beat some other big teams, but we did lose key games that would've been helpful for us," said Thompson.
Every team faces difficulties and Pleima believed one of those for them this season was losing their senior players who graduated last spring.
"I'd say we're a pretty young team. This year we have a lot of sophomores, freshmen, juniors and even some seniors that didn't see any playing time last year because we had such an amazing senior class last year; so we're all pretty new and fresh to the varsity scene," said Pleima. "I think that is a difficulty we've had to overcome all season; just not being as experienced as other teams, but it made us grow together and become more of a team."
Along with difficulties come strengths and team chemistry is one that these girls have. Team chemistry is proven to make a team more productive because they understand what each member brings.
"I think that Team chemistry was pretty big. We were also very close as a team and we got along a lot; team bonding was important to us, we had a camp out at the beginning of summer and throughout the season we had different things that are bonding for our team,'' Thompson.
Playoffs started Wednesday, October 23, 2019, against Washington, and they took the win in 4 sets. Then they played Marion on Tuesday, October 29 and lost 0-3.
"Our best goal is to go to state," said Pleima "but even if that doesn't happen, we've built friendships and made so many memories throughout the season, that it probably won't matter to us either way if we win or lose."
Girls' Take to the Mat
Basketball: Back in Action
Lady Dutch Leadership
After a loss against Washington, Pella does not advance to playoffs
Pella Dutch Cheer: Behind the Scenes
Softball Starting Up
On track to a state title, Pella boys track and field
All Business, Boys Win 3A Basketball State Title
State Wrestling 2021
Freshman Spotlight: Jackson Van Kley


A combined team of Ministry of Health officers and Zambia Police Service has apprehended a 30-year-old businessman of Liumena area of Sikongo District for selling assorted drugs suspected to have been stolen from a government hospital.
Acting Sikongo District Medical Officer Kawana Mulonda confirmed the development to ZANIS in Kalabo on Monday.
Mr Mulonda explained that the businessman was allegedly found selling assorted drugs with GRZ labels.
He disclosed that a number of boxes containing drugs with GRZ tags were found in the suspect's shop and home.
Mr Mulonda was however, not in the position to determine the exact names, amount and quantity of the drugs.
When questioned, the suspect maintained he bought the drugs from Lusaka's Kamwala market.
On Sunday the Zambia Medicines Regulatory Authority arrested two foreign nations for operating an illegal drug store.


I just finished reading David Fitch's take on sustainable ministry for missional leaders and pastors. You can read it here. He explained that the temptation to get pragmatic in our ministries (i.e., tell the people what they want to hear in order to pay for programs/salaries) is drastically lessoned when a missional community (1) keeps building expenses minimal, (2) maintains a multiple bi-vocational/bi-ministerial pastorate, and (3) develops an economy of thrift (free-cycling, meal sharing, etc.) among members. What do you think? Is he being realistic?
So, to all my church friends and leaders…do not fret! Let's get our hands in the dirt so to speak and quietly walk away from anxious reactions. Let's make a more sustainable ministry for our communities and then see the fruit of joining the lasting missio dei. How else could we expect our lives to become the change we hope to see?
p.s. If I didn't already mention this, you have to read David's post. I did? Well, just read it…pretty please!
I guess I'm a bit of a prude (or dogmatic) when it comes to economy and sin. Why do we put up with evil when it's attached to our paychecks?
Friends, please read this piece I wrote for Jesus Manifesto. I hoped, in it, to express the discontent I have often felt when facing the underbelly of global-consumer-capitalism, and how maddening it can be to find oneself compelled to live within nationalized and politicized states of greed. All that to say, I'm hopeful (and more easily de-scripted from the empire's imagination) every time some small-scale, local community begins to take shape–friends and family sharing with one another–working off the grid, so to speak! So, take a look at the article and tell me what you think.
My boss at work, Eric, sent me an article with an interesting take on the subject as well. You can find it here. It's definitely worth the read. Also, Josh Brown highlighted the absurdity of AIG taking a $443,000 luxury retreat in the midst of its own "economic bailout"/bankruptcy. They haven't got a clue!
Feel free to leave comments about any of the above topics. Agree/disagree with my take on things?


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Sufe Bradshaw (* 16. April 1986 in Chicago, Illinois) ist eine US-amerikanische Schauspielerin. Bekanntheit erlangte sie durch die Rolle der Sue Wilson aus der Dramedy-Serie Veep – Die Vizepräsidentin.

Leben und Karriere
Sufe Bradshaw wurde als eines von neun Kindern eines US-Amerikaners und einer Mutter ghanaischer Abstammung in Chicago geboren. Nach der Schule besuchte sie die Los Angeles City Theatre Academy und erlangte einen Abschluss in Theater. Sie ist seit 2006 als Schauspielerin aktiv. Nachdem sie zunächst Gastrollen in Serien, wie Cold Case – Kein Opfer ist je vergessen, Emergency Room – Die Notaufnahme, Southland, Prison Break oder FlashForward übernahm, wurde sie ab 2012 auch einem größeren Publikum als Sue Wilson, eine Hauptrolle in der Dramedy-Serie Veep – Die Vizepräsidentin, bekannt.

Zu ihren Filmauftritten gehören Star Trek und Overnight.

Filmografie (Auswahl)
 2006: Yo Momma (Fernsehserie, Episode 3x09)
 2006: Speechless
 2008: Mind of Mencia (Fernsehserie, Episode 4x02)
 2008: Cold Case – Kein Opfer ist je vergessen (Cold Case, Fernsehserie, Episode 5x18)
 2009: Trust Me (Fernsehserie, Episode 1x04)
 2009: Emergency Room – Die Notaufnahme (ER, Fernsehserie, Episode 15x22)
 2009: Star Trek
 2009: Southland (Fernsehserie, Episode 1x07)
 2009: Dance Flick – Der allerletzte Tanzfilm (Dance Flick)
 2009: Prison Break (Fernsehserie, 2 Episoden)
 2009: FlashForward (Fernsehserie, Episode 1x02)
 2009: Bones – Die Knochenjägerin (Bones, Fernsehserie, Episode 5x06)
 2012: Overnight
 2012–2016, 2019: Veep – Die Vizepräsidentin (Veep, Fernsehserie, 49 Episoden)
 2014: Crossroads
 2014: Rizzoli & Isles (Fernsehserie, Episode 4x15)
 2014: Imperial Dreams
 2019: Murder Mystery
 2020: Lovecraft Country (Fernsehserie, Episode 1x07)
 2021: Together Together
 2021: Hidden Canyons (Fernsehserie)
 2022: Gasoline Alley
 2022: Metal Lords
 2023: Little Dixie

Weblinks

Einzelnachweise

Filmschauspieler
US-Amerikaner
Geboren 1986
Frau

redpajama_set_name: RedPajamaWikipedia


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VFW Salutes Chairman Akaka
'The senator will always hold a place of honor among the 1.7 million VFW and Auxiliary members'
KANSAS CITY, Mo. - The Veterans of Foreign Wars of the U.S. extends its heartfelt condolences and sympathy to the family of Senator Daniel K. Akaka.
"From his brave service during World War II and significant Congressional tenure, to his leadership as Chairman of the Senate Veterans' Affairs Committee and efforts toward maintaining strong national defense and homeland security, Senator Akaka was a true American hero and veterans' advocate," said VFW National Commander Keith Harman. "He served our country with pride, and I salute him for all he has given us through his service, and for what he has contributed to our nation. The senator will always hold a place of honor among the 1.7 million VFW and Auxiliary members, and will be missed by many."


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Section Menu Undergraduate coursesDegree with Foundation YearDegree enhancement opportunitiesGet a prospectusLife at HopeHow to applyFees and fundingParents and supportersOutreach ActivitiesNetwork of HopeVisit usContact us
Undergraduate coursesDegree with Foundation YearDegree enhancement opportunitiesGet a prospectusLife at HopeHow to applyFees and fundingParents and supportersOutreach ActivitiesNetwork of HopeVisit usContact us
CHILDHOOD & YOUTH AND EDUCATION BA (HONS)
UCAS Code: LX93|Duration: 3 years years|Full Time|Hope Park|UCAS Campus Code: L46
Work placement opportunities|International students can apply
Childhood and Youth is an exciting, multidisciplinary subject in which you will explore a range of issues and challenges faced by children and young people in contemporary society, both here in the UK and on a global level. Drawing on key ideas from sociology, psychology, politics, history and geography, the degree examines important questions about children and young people's development, their life chances and their opportunities for participation in wider society.
Topics studied include how social class, poverty, gender and ethnicity impact upon young people's lives; how the media influence how children and young people are perceived by adults, and how they view themselves; and why young people take part in risky activities. In examining such questions, you will also explore various political debates and policy initiatives, as well as learning about various ways of researching with children and young people that aim to help them overcome the many challenges that they face.
With its strong emphasis on social justice and welfare, the degree will enable you to develop as a critical social scientist who values your role within the academic community at Hope and who, as a graduate, will be able to use your skills and knowledge to the benefit of your local community and society more broadly.
For more details and information about this course visit:
A degree in Education is a dynamic and stimulating opportunity to study the individual, social, and political forces that shape education, in all its forms. If you are interested in how and why we learn, both formally and informally, in the UK or abroad, then studying Education will challenge and excite you in equal measure.
Studying Education at Liverpool Hope will help you to develop academically as you engage with evidence and research, professionally as you link theory to practice, preparing you for a wide variety of work contexts, and personally as you think more deeply about your own values and aspirations. Our degree will bring education to life, tackling important questions such as 'Can education help overcome inequality in society?' and 'How does modern technology transform the learning and teaching experience?'. It is structured around the four key disciplines of education: psychology, sociology, history, and philosophy. Each discipline provides a different lens through which key ideas and issues in education can be understood.
By studying Education at Liverpool Hope University, you will become a Hope Education Professional: inspired to put educational theory in to practice, able to critically evaluate current policy, and willing to make a difference in the world. Single Honours students complete a placement as part of the course and Combined Honours students will be offered optional placement opportunities. The degree is an ideal foundation for those who decide to go on to train to become a teacher, and a variety of other professions, at Liverpool Hope University or elsewhere.
You may also be interested in studying the
To view all the Education undergraduate programmes we offer, click here to download the brochure
Department of Education Studies
Combined Honours:
Please note that Combined Honours degrees at Liverpool Hope University are split 50/50. This means both subjects will be studied equally.
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