E18-304

Abstract: Properties of random projections of high-dimensional probability measures are of interest in a variety of fields, including asymptotic convex geometry, and potential applications to high-dimensional statistics and data analysis. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been well studied over the past decade, we describe more recent work on the tail behavior of such projections, and various implications. This talk…

Abstract: We introduce the jackknife+, a novel method for constructing predictive confidence intervals that is robust to the distribution of the data. The jackknife+ modifies the well-known jackknife (leaveoneout cross-validation) to account for the variability in the fitted regression function when we subsample the training data. Assuming exchangeable training samples, we prove that the jackknife+ permits rigorous coverage guarantees regardless of the distribution of the data points, for any algorithm that treats the training points symmetrically (in contrast, such guarantees…

Abstract: We introduce the diffusion K-means clustering method on Riemannian submanifolds, which maximizes the within-cluster connectedness based on the diffusion distance. The diffusion K-means constructs a random walk on the similarity graph with vertices as data points randomly sampled on the manifolds and edges as similarities given by a kernel that captures the local geometry of manifolds. Thus the diffusion K-means is a multi-scale clustering tool that is suitable for data with non-linear and non-Euclidean geometric features in mixed dimensions. Given…

Abstract: Privacy-preserving data analysis has been put on a firm mathematical foundation since the introduction of differential privacy (DP) in 2006. This privacy definition, however, has some well-known weaknesses: notably, it does not tightly handle composition. This weakness has inspired several recent relaxations of differential privacy based on the Renyi divergences. We propose an alternative relaxation we term “f-DP”, which has a number of nice properties and avoids some of the difficulties associated with divergence based relaxations. First, f-DP preserves…

Abstract: Bulk sequencing of tumor DNA is a popular strategy for uncovering information about the spectrum of mutations arising in the tumor, and is often supplemented by multi-region sequencing, which provides a view of tumor heterogeneity. The statistical issues arise from the fact that bulk sequencing makes the determination of sub-clonal frequencies, and other quantities of interest, difficult. In this talk I will discuss this problem, beginning with its setting in population genetics. The data provide an estimate of the…

Abstract: Many algorithms take prohibitively long to run on modern, large data sets. But even in complex data sets, many data points may be at least partially redundant for some task of interest. So one might instead construct and use a weighted subset of the data (called a “coreset”) that is much smaller than the original dataset. Typically running algorithms on a much smaller data set will take much less computing time, but it remains to understand whether the output…

Abstract: The affinity between statistical physics and machine learning has long history, this is reflected even in the machine learning terminology that is in part adopted from physics. Current theoretical challenges and open questions about deep learning and statistical learning call for unified account of the following three ingredients: (a) the dynamics of the learning algorithm, (b) the architecture of the neural networks, and (c) the structure of the data. Most existing theories are not taking in account all of those…

Abstract: We consider the problems of learning discrete structures from network data under statistical settings. Popular examples include various block models, Z2 synchronization and mixture models. Semidefinite programming (SDP) relaxation has emerged as a versatile and robust approach to these problems. We show that despite being a relaxation, SDP achieves the optimal Bayes error rate in terms of distance to the target solution. Moreover, SDP relaxation is provably robust under the so-called semirandom model, which frustrates many existing algorithms. Our…

Abstract: Accurate estimation and inference in finite sample is important for decision making in many experimental and social fields, especially when the available data are complex, like when they include mixed types of measurements, they are dependent in several ways, there are missing data, outliers, etc. Indeed, the more complex the data (hence the models), the less accurate are asymptotic theory results in finite samples.  This is in particular the case, for example, with logistic regression, with possibly also random effects…

Abstract: Real-world data typically do not fit statistical models or satisfy assumptions underlying the theory exactly, hence reducing the number and strictness of these assumptions helps to lessen the gap between the “mathematical” world and the “real” world. The concept of robustness, in particular, robustness to outliers, plays the central role in understanding this gap. The goal of the talk is to introduce the principles and robust algorithms based on these principles that can be applied in the general framework of statistical…

Abstract: What happens when an optimization problem has a good solution built into it, but which is partly obscured by randomness? Here we revisit a classic polynomial-time problem, the minimum perfect matching problem on bipartite graphs. If the edges have random weights in , Mézard and Parisi — and then Aldous, rigorously — showed that the minimum matching has expected weight zeta(2) = pi^2/6. We consider a “planted” version where a particular matching has weights drawn from an exponential distribution…

Abstract: What are the most expressive classes of neural networks that can be learned, provably, in polynomial-time in a distribution-free setting? In this talk we give the first efficient algorithm for learning neural networks with two nonlinear layers using tools for solving isotonic regression, a nonconvex (but tractable) optimization problem. If we further assume the distribution is symmetric, we obtain the first efficient algorithm for recovering the parameters of a one-layer convolutional network. These results implicitly make use of a…

Abstract: The problem of transfer and domain adaptation is ubiquitous in machine learning and concerns situations where predictive technologies, trained on a given source dataset, have to be transferred to a new target domain that is somewhat related. For example, transferring voice recognition trained on American English accents to apply to Scottish accents, with minimal retraining. A first challenge is to understand how to properly model the ‘distance’ between source and target domains, viewed as probability distributions over a feature…

Abstract: We first study the rate of convergence for learning distributions with the adversarial framework and Generative Adversarial Networks (GANs), which subsumes Wasserstein, Sobolev, and MMD GANs as special cases. We study a wide range of parametric and nonparametric target distributions, under a collection of objective evaluation metrics. On the nonparametric end, we investigate the minimax optimal rates and fundamental difficulty of the implicit density estimation under the adversarial framework. On the parametric end, we establish a theory for general…

Abstract: We consider the problem of efficient sampling from the hard-core and Potts models from statistical physics. On certain families of graphs, phase transitions in the underlying physics model are linked to changes in the performance of some sampling algorithms, including Markov chains. We develop new sampling and counting algorithms that exploit the phase transition phenomenon and work efficiently on lattices (and bipartite expander graphs) at sufficiently low temperatures in the phase coexistence regime. Our algorithms are based on Pirogov-Sinai…

Abstract: Chao Gao will discuss the problem of statistical estimation with contaminated data. In the first part of the talk, I will discuss depth-based approaches that achieve minimax rates in various problems. In general, the minimax rate of a given problem with contamination consists of two terms: the statistical complexity without contamination, and the contamination effect in the form of modulus of continuity. In the second part of the talk, I will discuss computational challenges of these depth-based estimators. An…

Logistic regression is a fundamental task in machine learning and statistics. For the simple case of linear models, Hazan et al. (2014) showed that any logistic regression algorithm that estimates model weights from samples must exhibit exponential dependence on the weight magnitude. As an alternative, we explore a counterintuitive technique called improper learning, whereby one estimates a linear model by fitting a non-linear model. Past success stories for improper learning have focused on cases where it can improve computational complexity.…

Abstract: This talk will present a class of exponential bounds for the probability that a martingale sequence crosses a time-dependent linear threshold. Our key insight is that it is both natural and fruitful to formulate exponential concentration inequalities in this way. We will illustrate this point by presenting a single assumption and a single theorem that together strengthen many tail bounds for martingales, including classical inequalities (1960-80) by Bernstein, Bennett, Hoeffding, and Freedman; contemporary inequalities (1980-2000) by Shorack and Wellner,…

Abstract: The talk will focus on optimization on the high-dimensional sphere when the objective function is a linear combination of homogeneous polynomials with standard Gaussian coefficients. Such random processes are called spherical spin glasses in physics, and have been extensively studied since the 80s. I will describe certain geometric properties of spherical spin glasses unique to the full-RSB case, and explain how they can be used to design a polynomial time algorithm that finds points within small multiplicative error from…

Abstract: In this work we consider bootstrap-based inference methods for functions of the parameter vector in the presence of many moment inequalities where the number of moment inequalities, denoted by p, is possibly much larger than the sample size n. In particular this covers the case of subvector inference, such as the inference on a single component associated with a treatment/policy variable of interest. We consider a min-max of (centered and non-centered) Studentized statistics and study the properties of the…

Abstract: Total variation denoising (TVD) is a popular technique for nonparametric function estimation. I will first present a theoretical optimality result for univariate TVD for estimating piecewise constant functions. I will then present related results for various extensions of univariate TVD including adaptive risk bounds for higher-order TVD (also known as trend filtering) as well as a multivariate extension via the Hardy-Krause Variation which avoids the curse of dimensionality to some extent. I will also mention connections to shape restricted…

Abstract: Over 500K scientific articles have been published since 1999 with the word “network” in the title. And the vast majority of these report network summary statistics of one type or another. However, these numbers are rarely accompanied by any quantification of uncertainty. Yet any error inherent in the measurements underlying the construction of the network, or in the network construction procedure itself, necessarily must propagate to any summary statistics reported. Perhaps surprisingly, there is little in the way of…

Abstract: The perceptron is a toy model of a simple neural network that stores a collection of given patterns. Its analysis reduces to a simple problem in high-dimensional geometry, namely, understanding the intersection of the cube (or sphere) with a collection of random half-spaces. Despite the simplicity of this model, its high-dimensional asymptotics are not well understood. I will describe what is known and present recent results. This is joint work with Jian Ding. Biography: Nike Sun is a faculty…

Abstract: We consider the Sherrington-Kirkpatrick model of spin glasses with ferromagnetically biased couplings. For a specific choice of the couplings mean, the resulting Gibbs measure is equivalent to the Bayesian posterior for a high-dimensional estimation problem known as "Z2 synchronization". Statistical physics suggests to compute the expectation with respect to this Gibbs measure (the posterior mean in the synchronization problem), by minimizing the so-called Thouless-Anderson-Palmer (TAP) free energy, instead of the mean field (MF) free energy. We prove that this identification…

Abstract: We present an algorithm for creating high resolution anatomically plausible images that are consistent with acquired clinical brain MRI scans with large inter-slice spacing. Although large databases of clinical images contain a wealth of information, medical acquisition constraints result in sparse scans that miss much of the anatomy. These characteristics often render computational analysis impractical as standard processing algorithms tend to fail when applied to such images. Our goal is to enable application of existing algorithms that were originally…

Andrea Montanari Professor, Department of Electrical Engineering, Department of Statistics Stanford University This lecture is in conjunction with the LIDS Student Conference. Abstract: Let A be n × n symmetric random matrix with independent and identically distributed Gaussian entries above the diagonal. We consider the problem of maximizing xT Ax over binary vectors with ±1 entries. In the language of statistical physics, this amounts to finding the ground state of the Sherrington-Kirkpatrick model of spin glasses. The asymptotic value of this…

Abstract: In 1973 Erdos asked whether there are n-vertex partial Steiner triple systems with arbitrary high girth and quadratically many triples. (Here girth is defined as the smallest integer g \ge 4 for which some g-element vertex-set contains at least g-2 triples.) We answer this question, by showing existence of approximate Steiner triple systems with arbitrary high girth. More concretely, for any fixed \ell \ge 4 we show that a natural constrained random process typically produces a partial Steiner triple…

Abstract: The prototypical high-dimensional statistics problem entails finding a structured signal in noise. Many of these problems exhibit an intriguing phenomenon: the amount of data needed by all known computationally efficient algorithms far exceeds what is needed for inefficient algorithms that search over all possible structures. A line of work initiated by Berthet and Rigollet in 2013 has aimed to explain these gaps by reducing from conjecturally hard problems in computer science. However, the delicate nature of average-case reductions has…

Abstract: We discuss a recent approach to bias reduction in a problem of estimation of smooth functionals of high-dimensional parameters of statistical models. In particular, this approach has been developed in the case of estimation of functionals of covariance operator Σ : Rd → Rd of the form f(Σ), B based on n i.i.d. observations X1, . . . , Xn sampled from the normal distribution with mean zero and covariance Σ, f : R → R being a sufficiently…

Abstract: Many contemporary large-scale applications, from genomics to advertising, involve linking a response of interest to a large set of potential explanatory variables in a nonlinear fashion, such as when the response is binary. Although this modeling problem has been extensively studied, it remains unclear how to effectively select important variables while controlling the fraction of false discoveries, even in high-dimensional logistic regression, not to mention general high-dimensional nonlinear models. To address such a practical problem, we propose a new…

Abstract: Inference in the framework of Ising models has received significant attention in Statistics and Machine Learning in recent years. In this talk we study joint estimation of the inverse temperature parameter β, and the magnetization parameter B, given one realization from the Ising model, under the assumption that the underlying graph of the Ising model is completely specified. We show that if the graph is either irregular or sparse, then both the parameters can be estimated at rate n−1/2…

Abstract Concentration of measure refers to a collection of tools and results from analysis and probability theory that have been used in many areas of pure and applied mathematics. Arguably, the first data science application of measure concentration (under the name ‘‘blowing-up lemma’’) is the proof of strong converses in multiuser information theory by Ahlswede, G'acs and K"orner in 1976. Since then, measure concentration has found applications in many other information theoretic problems, most notably the converse (impossibility) results in…

Abstract: We give a solution to the following question from manifold learning. Suppose data belonging to a high dimensional Euclidean space is drawn independently, identically distributed from a measure supported on a low dimensional twice differentiable embedded compact manifold M, and is corrupted by a small amount of i.i.d gaussian noise. How can we produce a manifold $M_o$ whose Hausdorff distance to M is small and whose reach (normal injectivity radius) is not much smaller than the reach of M?…

Abstract: Spectral methods have been widely used for a large class of challenging problems, ranging from top-K ranking via pairwise comparisons, community detection, factor analysis, among others. Analyses of these spectral methods require super-norm perturbation analysis of top eigenvectors. This allows us to UNIFORMLY approximate elements in eigenvectors by linear functions of the observed random matrix that can be analyzed further. We first establish such an infinity-norm pertubation bound for top eigenvectors and apply the idea to several challenging problems…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…

The Stochastics and Statistics Seminar is a weekly meeting organized by the Statistics and Data Science Center (SDSC). It consists of a series of one-hour presentations by worldwide leaders making cutting edge contributions to methodological and theoretical advances in data science. These fields include probability, statistics, optimization, and applied mathematics. The seminar also regularly features experts in applications domains such as biology or engineering. This intellectual diversity has contributed to the organic assembly of a dynamic and diverse audience articulated…