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Computing the degrees of freedom of rank-regularized estimators and cousins

Rahul Mazumder, Haolei Weng.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 1348--1385.

Abstract:
Estimating a low rank matrix from its linear measurements is a problem of central importance in contemporary statistical analysis. The choice of tuning parameters for estimators remains an important challenge from a theoretical and practical perspective. To this end, Stein’s Unbiased Risk Estimate (SURE) framework provides a well-grounded statistical framework for degrees of freedom estimation. In this paper, we use the SURE framework to obtain degrees of freedom estimates for a general class of spectral regularized matrix estimators—our results generalize beyond the class of estimators that have been studied thus far. To this end, we use a result due to Shapiro (2002) pertaining to the differentiability of symmetric matrix valued functions, developed in the context of semidefinite optimization algorithms. We rigorously verify the applicability of Stein’s Lemma towards the derivation of degrees of freedom estimates; and also present new techniques based on Gaussian convolution to estimate the degrees of freedom of a class of spectral estimators, for which Stein’s Lemma does not directly apply.




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Sparse and low-rank multivariate Hawkes processes

We consider the problem of unveiling the implicit network structure of node interactions (such as user interactions in a social network), based only on high-frequency timestamps. Our inference is based on the minimization of the least-squares loss associated with a multivariate Hawkes model, penalized by $ell_1$ and trace norm of the interaction tensor. We provide a first theoretical analysis for this problem, that includes sparsity and low-rank inducing penalizations. This result involves a new data-driven concentration inequality for matrix martingales in continuous time with observable variance, which is a result of independent interest and a broad range of possible applications since it extends to matrix martingales former results restricted to the scalar case. A consequence of our analysis is the construction of sharply tuned $ell_1$ and trace-norm penalizations, that leads to a data-driven scaling of the variability of information available for each users. Numerical experiments illustrate the significant improvements achieved by the use of such data-driven penalizations.




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Exact Guarantees on the Absence of Spurious Local Minima for Non-negative Rank-1 Robust Principal Component Analysis

This work is concerned with the non-negative rank-1 robust principal component analysis (RPCA), where the goal is to recover the dominant non-negative principal components of a data matrix precisely, where a number of measurements could be grossly corrupted with sparse and arbitrary large noise. Most of the known techniques for solving the RPCA rely on convex relaxation methods by lifting the problem to a higher dimension, which significantly increase the number of variables. As an alternative, the well-known Burer-Monteiro approach can be used to cast the RPCA as a non-convex and non-smooth $ell_1$ optimization problem with a significantly smaller number of variables. In this work, we show that the low-dimensional formulation of the symmetric and asymmetric positive rank-1 RPCA based on the Burer-Monteiro approach has benign landscape, i.e., 1) it does not have any spurious local solution, 2) has a unique global solution, and 3) its unique global solution coincides with the true components. An implication of this result is that simple local search algorithms are guaranteed to achieve a zero global optimality gap when directly applied to the low-dimensional formulation. Furthermore, we provide strong deterministic and probabilistic guarantees for the exact recovery of the true principal components. In particular, it is shown that a constant fraction of the measurements could be grossly corrupted and yet they would not create any spurious local solution.




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Union of Low-Rank Tensor Spaces: Clustering and Completion

We consider the problem of clustering and completing a set of tensors with missing data that are drawn from a union of low-rank tensor spaces. In the clustering problem, given a partially sampled tensor data that is composed of a number of subtensors, each chosen from one of a certain number of unknown tensor spaces, we need to group the subtensors that belong to the same tensor space. We provide a geometrical analysis on the sampling pattern and subsequently derive the sampling rate that guarantees the correct clustering under some assumptions with high probability. Moreover, we investigate the fundamental conditions for finite/unique completability for the union of tensor spaces completion problem. Both deterministic and probabilistic conditions on the sampling pattern to ensure finite/unique completability are obtained. For both the clustering and completion problems, our tensor analysis provides significantly better bound than the bound given by the matrix analysis applied to any unfolding of the tensor data.




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Estimation of a Low-rank Topic-Based Model for Information Cascades

We consider the problem of estimating the latent structure of a social network based on the observed information diffusion events, or cascades, where the observations for a given cascade consist of only the timestamps of infection for infected nodes but not the source of the infection. Most of the existing work on this problem has focused on estimating a diffusion matrix without any structural assumptions on it. In this paper, we propose a novel model based on the intuition that an information is more likely to propagate among two nodes if they are interested in similar topics which are also prominent in the information content. In particular, our model endows each node with an influence vector (which measures how authoritative the node is on each topic) and a receptivity vector (which measures how susceptible the node is for each topic). We show how this node-topic structure can be estimated from the observed cascades, and prove the consistency of the estimator. Experiments on synthetic and real data demonstrate the improved performance and better interpretability of our model compared to existing state-of-the-art methods.




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A rank-based Cramér–von-Mises-type test for two samples

Jamye Curry, Xin Dang, Hailin Sang.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 3, 425--454.

Abstract:
We study a rank based univariate two-sample distribution-free test. The test statistic is the difference between the average of between-group rank distances and the average of within-group rank distances. This test statistic is closely related to the two-sample Cramér–von Mises criterion. They are different empirical versions of a same quantity for testing the equality of two population distributions. Although they may be different for finite samples, they share the same expected value, variance and asymptotic properties. The advantage of the new rank based test over the classical one is its ease to generalize to the multivariate case. Rather than using the empirical process approach, we provide a different easier proof, bringing in a different perspective and insight. In particular, we apply the Hájek projection and orthogonal decomposition technique in deriving the asymptotics of the proposed rank based statistic. A numerical study compares power performance of the rank formulation test with other commonly-used nonparametric tests and recommendations on those tests are provided. Lastly, we propose a multivariate extension of the test based on the spatial rank.




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Adaptive estimation of the rank of the coefficient matrix in high-dimensional multivariate response regression models

Xin Bing, Marten H. Wegkamp.

Source: The Annals of Statistics, Volume 47, Number 6, 3157--3184.

Abstract:
We consider the multivariate response regression problem with a regression coefficient matrix of low, unknown rank. In this setting, we analyze a new criterion for selecting the optimal reduced rank. This criterion differs notably from the one proposed in Bunea, She and Wegkamp ( Ann. Statist. 39 (2011) 1282–1309) in that it does not require estimation of the unknown variance of the noise, nor does it depend on a delicate choice of a tuning parameter. We develop an iterative, fully data-driven procedure, that adapts to the optimal signal-to-noise ratio. This procedure finds the true rank in a few steps with overwhelming probability. At each step, our estimate increases, while at the same time it does not exceed the true rank. Our finite sample results hold for any sample size and any dimension, even when the number of responses and of covariates grow much faster than the number of observations. We perform an extensive simulation study that confirms our theoretical findings. The new method performs better and is more stable than the procedure of Bunea, She and Wegkamp ( Ann. Statist. 39 (2011) 1282–1309) in both low- and high-dimensional settings.




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Active ranking from pairwise comparisons and when parametric assumptions do not help

Reinhard Heckel, Nihar B. Shah, Kannan Ramchandran, Martin J. Wainwright.

Source: The Annals of Statistics, Volume 47, Number 6, 3099--3126.

Abstract:
We consider sequential or active ranking of a set of $n$ items based on noisy pairwise comparisons. Items are ranked according to the probability that a given item beats a randomly chosen item, and ranking refers to partitioning the items into sets of prespecified sizes according to their scores. This notion of ranking includes as special cases the identification of the top-$k$ items and the total ordering of the items. We first analyze a sequential ranking algorithm that counts the number of comparisons won, and uses these counts to decide whether to stop, or to compare another pair of items, chosen based on confidence intervals specified by the data collected up to that point. We prove that this algorithm succeeds in recovering the ranking using a number of comparisons that is optimal up to logarithmic factors. This guarantee does depend on whether or not the underlying pairwise probability matrix, satisfies a particular structural property, unlike a significant body of past work on pairwise ranking based on parametric models such as the Thurstone or Bradley–Terry–Luce models. It has been a long-standing open question as to whether or not imposing these parametric assumptions allows for improved ranking algorithms. For stochastic comparison models, in which the pairwise probabilities are bounded away from zero, our second contribution is to resolve this issue by proving a lower bound for parametric models. This shows, perhaps surprisingly, that these popular parametric modeling choices offer at most logarithmic gains for stochastic comparisons.




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Semiparametrically point-optimal hybrid rank tests for unit roots

Bo Zhou, Ramon van den Akker, Bas J. M. Werker.

Source: The Annals of Statistics, Volume 47, Number 5, 2601--2638.

Abstract:
We propose a new class of unit root tests that exploits invariance properties in the Locally Asymptotically Brownian Functional limit experiment associated to the unit root model. The invariance structures naturally suggest tests that are based on the ranks of the increments of the observations, their average and an assumed reference density for the innovations. The tests are semiparametric in the sense that they are valid, that is, have the correct (asymptotic) size, irrespective of the true innovation density. For a correctly specified reference density, our test is point-optimal and nearly efficient. For arbitrary reference densities, we establish a Chernoff–Savage-type result, that is, our test performs as well as commonly used tests under Gaussian innovations but has improved power under other, for example, fat-tailed or skewed, innovation distributions. To avoid nonparametric estimation, we propose a simplified version of our test that exhibits the same asymptotic properties, except for the Chernoff–Savage result that we are only able to demonstrate by means of simulations.




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Spectral method and regularized MLE are both optimal for top-$K$ ranking

Yuxin Chen, Jianqing Fan, Cong Ma, Kaizheng Wang.

Source: The Annals of Statistics, Volume 47, Number 4, 2204--2235.

Abstract:
This paper is concerned with the problem of top-$K$ ranking from pairwise comparisons. Given a collection of $n$ items and a few pairwise comparisons across them, one wishes to identify the set of $K$ items that receive the highest ranks. To tackle this problem, we adopt the logistic parametric model—the Bradley–Terry–Luce model, where each item is assigned a latent preference score, and where the outcome of each pairwise comparison depends solely on the relative scores of the two items involved. Recent works have made significant progress toward characterizing the performance (e.g., the mean square error for estimating the scores) of several classical methods, including the spectral method and the maximum likelihood estimator (MLE). However, where they stand regarding top-$K$ ranking remains unsettled. We demonstrate that under a natural random sampling model, the spectral method alone, or the regularized MLE alone, is minimax optimal in terms of the sample complexity—the number of paired comparisons needed to ensure exact top-$K$ identification, for the fixed dynamic range regime. This is accomplished via optimal control of the entrywise error of the score estimates. We complement our theoretical studies by numerical experiments, confirming that both methods yield low entrywise errors for estimating the underlying scores. Our theory is established via a novel leave-one-out trick, which proves effective for analyzing both iterative and noniterative procedures. Along the way, we derive an elementary eigenvector perturbation bound for probability transition matrices, which parallels the Davis–Kahan $mathop{mathrm{sin}} olimits Theta $ theorem for symmetric matrices. This also allows us to close the gap between the $ell_{2}$ error upper bound for the spectral method and the minimax lower limit.




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Measuring human activity spaces from GPS data with density ranking and summary curves

Yen-Chi Chen, Adrian Dobra.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 409--432.

Abstract:
Activity spaces are fundamental to the assessment of individuals’ dynamic exposure to social and environmental risk factors associated with multiple spatial contexts that are visited during activities of daily living. In this paper we survey existing approaches for measuring the geometry, size and structure of activity spaces, based on GPS data, and explain their limitations. We propose addressing these shortcomings through a nonparametric approach called density ranking and also through three summary curves: the mass-volume curve, the Betti number curve and the persistence curve. We introduce a novel mixture model for human activity spaces and study its asymptotic properties. We prove that the kernel density estimator, which at the present time, is one of the most widespread methods for measuring activity spaces, is not a stable estimator of their structure. We illustrate the practical value of our methods with a simulation study and with a recently collected GPS dataset that comprises the locations visited by 10 individuals over a six months period.




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Cliques in rank-1 random graphs: The role of inhomogeneity

Kay Bogerd, Rui M. Castro, Remco van der Hofstad.

Source: Bernoulli, Volume 26, Number 1, 253--285.

Abstract:
We study the asymptotic behavior of the clique number in rank-1 inhomogeneous random graphs, where edge probabilities between vertices are roughly proportional to the product of their vertex weights. We show that the clique number is concentrated on at most two consecutive integers, for which we provide an expression. Interestingly, the order of the clique number is primarily determined by the overall edge density, with the inhomogeneity only affecting multiplicative constants or adding at most a $log log (n)$ multiplicative factor. For sparse enough graphs the clique number is always bounded and the effect of inhomogeneity completely vanishes.




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From Wends we came : the story of Johann and Maria Huppatz & their descendants / compiled by Frank Huppatz and Rone McDonnell.

Huppatz (Family).




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Frank Rich: The Rage Won't End on Election Day




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Enjoy Free Video Tours of Frank Lloyd Wright Buildings Across America

The 20th-century architect defined a uniquely American style that used nature-inspired motifs and horizontal lines




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Why the Anne Frank House Is Reimagining the Young Diarist as a Vlogger

The controversial series stems from the museum's desire to reach a younger generation by telling history in new ways




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How New York Made Frank Lloyd Wright a Starchitect

The Wisconsin-born architect's buildings helped turn the city he once called an 'inglorious mantrap' into the center of the world




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The Last Dance: Ranking every Michael Jordan playoff opponent – NBA IN

The Last Dance: Ranking every Michael Jordan playoff opponent  NBA IN



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Richard Cockerill looks for homegrown heroes as Edinburgh replenish ranks

Edinburgh Rugby have announced the signing of three Scottish qualified youngsters in Nathan Chamberlain, Ben Muncaster and Dan Gamble on academy deals which will turn into full-time contracts next summer, and head coach Richard Cockerill has promised that three current academy prospects in Rory Darge, Connor Boyle and Sam Grahamslaw will soon be elevated to the senior squad ahead of next season.




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Herald Diary: Bagpipes and bad boy Ian Rankin

Batty idea




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Philanthropist Frank Giustra Donates </br>$1 Million for Crisis Group Fellows

The International Crisis Group is honoured to announce the creation of the Giustra Fellowship for Conflict Prevention, made possible by a generous gift of $1 million from Canadian businessman and philanthropic leader Frank Giustra through The Radcliffe Foundation. Mr. Giustra has been a long-time advocate for Crisis Group, providing transformational financial support since joining its Board of Trustees in 2005.




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Fin24.com | Govt ranks firms for investment

Nedbank, FirstRand and Anglo Platinum have received the highest scores in the Public Investment Corporation's new corporate governance rating system.




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Salzburg v Eintracht Frankfurt facts

Defeated 4-1 at 2018/19 semi-finalists Eintracht Frankfurt, UEFA Europa League stalwarts Salzburg have it all to do on home soil.




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Eintracht Frankfurt v Basel facts

Two former semi-finalists have been paired together in the round of 16 as Eintracht Frankfurt go head to head for the first time with Basel.




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Basel v Eintracht Frankfurt facts

Basel have one foot in the quarter-finals after overwhelming 2018/19 semi-finalists Eintracht Frankfurt 3-0 away in the first leg of their round of 16 tie.




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Fin24.com | Top 200's return on equity rankings

Many happy returns - on equity and assets.




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State Treasurer to Celebrate National Library Week at Frankford Public Library

State Treasurer Colleen C. Davis will join the Frankford Public Library for their Community Connect Night on Monday, April 8 to celebrate National Library Week. National Library Week is an annual celebration that began in 1958 to highlight the valuable role libraries, librarians and library workers play in transforming lives and communities. “Libraries in Delaware […]




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'Mehendi Prank' with Rohit at Bachelor party | Kahaan Hum Kahaan Tum

Sonakshi, a renowned television actress of a popular show, and Rohit, a highly successful cardiologist, cross paths and develop a unique bond even as they struggle with their demanding careers.   




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Delaware Forest Service to conduct 74-acre controlled burn near Frankford

The Delaware Forest Service plans to conduct a prescribed fire on 74 acres west of U.S. Route 113 near the Town of Frankford, Sussex County. The burning could begin as early as Monday, October 27, though officials estimate the date might be closer to mid-week. The actual date of ignition will depend on local weather and fuel conditions.




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Franklin Templeton debacle to hit investors very hard

The regulator had also observed that mutual funds should not depend totally on credit rating agencies but should do their own assessments.




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Coronavirus fallout: Rs 25,800 crore investments at risk as Franklin Templeton shutters six schemes

Debt schemes of mutual funds for long have been seen as low risk, but shuttering six schemes and leaving investors in a lurch will impact investor sentiment.




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Franklin Templeton says will return money to investors

Franklin Templeton MF also said that their commitment to India remains steadfast as they have been early as patient investors in India.




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How to plan the best April Fools' day prank

  Considering this is April, the most obvious thing to do this month would be to sharpen your funny bone and hone your 'pranking' technique. A good prank is like making lasagne. You need layers of past...




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Covid-19 impact: Real estate sentiment at historic low, says Knight Frank-Ficci-Naredco survey

Even while the government and Reserve Bank of India (RBI) have provided some stimulus, further support may be required to help real estate and for the economy to stay afloat during the crisis.




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This Volkswagen first model to celebrate world premiere at Frankfurt Motor Show

The ID.3 is the first fully electric model based on the new MEB platform




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Top 5 cars unveiling at 2019 IAA Frankfurt: From electric Hyundai to Lamborghini plug-in hybrid hypercar




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2019 Frankfurt Motor Show in images: From Honda’s electric hatchback to Lamborghini’s hybrid hypercar




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ICC ODI rankings: Virat Kohli continues to dominate batsmen ranking; Jasprit Bumrah loses top spot among bowlers

ICC ODI rankings: In the all-rounders' category, Ravindra Jadeja, moved up three places above to be placed at seventh position .




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Virat Kohli loses top spot in ICC Test Player rankings

In the ICC World Test Championship, India continue to be placed at the top with 360 points, followed by Australia.




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Teen sensation Shafali Verma rises to top in ICC women’s T20 rankings

The 16-year-old Verma takes over from New Zealand's Suzie Bates, who had been the top batter since October 2018 after wresting the spot from West Indies captain Stafanie Taylor.




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ICC Test Rankings: India lose top spot to Australia, drop to third spot

India dropped in the ladder largely because the record of 12 Tests victories and just one Test defeat in 2016-17 was removed in the latest chart, the ICC said in a statement.




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What brought Franklin Templeton in SEBI’s crosshairs? Here’s why the fund house apologised

From wounding up six debt mutual fund schemes with an AUM of over Rs 30,000 crore, to dealing with the ire of investors, the fund house has been busy fire-fighting ever since the nationwide lockdown was announced.




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College basketball transfer rankings for 2020-21 and 2021-22

Here are the top names seeking new homes (and those sitting out).




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Ranking the bottom 10 jerseys in NBA history

We dug deep into the NBA's closet to find the jerseys that truly put the "dud" in "duds."




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Ranking the top 74 jerseys in NBA history

Sartorial splendor abounds as we count down the loveliest looks.




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Ranking the top 74 sneakers in NBA history

Ranking sneakers from the Converse Chuck Taylor All-Star to the many Air Jordans and everything in between.




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Kiper's 2021 NFL draft rankings: Way-too-early Big Board, top prospects at every position

Quarterbacks at the top. Elite wide receiver talent. And a top tier of offensive linemen. The Class of 2021 could be special.




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Ranking the best offensive teams in college football for the next 3 years

Here's why Clemson, Ohio State, Oklahoma and 22 more teams will have the top offenses through the next three seasons.




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Updated rankings for the 2021 ESPN 300 college football prospects

Which high school football prospects could be the next big-time college football players?




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NFL Power Rankings: 1-32 poll, plus post-draft winners for every team

Ben Roethlisberger and Chandler Jones got some support in the NFL draft, while Alvin Kamara's importance was cemented even more.