We focus on the problem of estimating the graph structure associated with a discrete Markov random field. We describe a method based on 1- regularized logistic regression, in which the neighborhood of any given node is estimated by performing logistic regression subject to an1-constraint. Our framework applies to the high-dimensional setting, in which both the number of nodes p and maximum neighborhood sizes d are allowed to grow as a function of the number of observations n. Our main result is to estab- lish sufficient conditions on the triple (n, p, d) for the method to succeed in consistently estimating the neighborhood of every node in the graph simul- taneously. Under certain mutual incoherence conditions analogous to those imposed in previous work on linear regression, we prove that consistent neighborhood selection can be obtained as long as the number of observa- tions n grows more quickly than 6d6 log d 2d5 log p, thereby establishing that logarithmic growth in the number of samples n relative to graph size p is sufficient to achieve neighborhood consistency.

We consider the following instance of transfer learning: given a pair of regression problems, suppose that the regression coefficients share a partially common support, parameterized by the overlap fraction \overlap between the two supports. This set-up suggests the use of 1, \infty -regularized linear regression for recovering the support sets of both regression vectors. Our main contribution is to provide a sharp characterization of the sample complexity of this 1,\infty relaxation, exactly pinning down the minimal sample size n required for joint support recovery as a function of the model dimension \pdim, support size \spindex and overlap \overlap \in [0,1] . For measurement matrices drawn from standard Gaussian ensembles, we prove that the joint 1,\infty -regularized method undergoes a phase transition characterized by order parameter \orpar( umobs, \pdim, \spindex, \overlap) umobs{(4 - 3 \overlap) s \log(p-(2-\overlap)s)} . More precisely, the probability of successfully recovering both supports converges to 1 for scalings such that \orpar 1, and converges to 0 to scalings for which \orpar 1 .

This paper uses information-theoretic techniques to determine minimax rates for estimating nonparametric sparse additive regression models under high-dimensional scaling. The first term reflects the difficulty of performing \emph{subset selection} and is independent of the Hilbert space \Hilb; the second term \LowerRateSq is an \emph{\s-dimensional estimation} term, depending only on the low dimension \s but not the ambient dimension \pdim, that captures the difficulty of estimating a sum of \s univariate functions in the Hilbert space \Hilb . The minimax rates are compared with rates achieved by an \ell_1 -penalty based approach, it can be shown that a certain \ell_1 -based approach achieves the minimax optimal rate.

We study the problem of learning a sparse linear regression vector under additional conditions on the structure of its sparsity pattern. We present a family of convex penalty functions, which encode this prior knowledge by means of a set of constraints on the absolute values of the regression coefficients. This family subsumes the \ell_1 norm and is flexible enough to include different models of sparsity patterns, which are of practical and theoretical importance. We establish some important properties of these functions and discuss some examples where they can be computed explicitly. Moreover, we present a convergent optimization algorithm for solving regularized least squares with these penalty functions.

Principal Components Analysis (PCA) is often used as a feature extraction procedure. Given a matrix X \in \mathbb{R} {n \times d}, whose rows represent n data points with respect to d features, the top k right singular vectors of X (the so-called \textit{eigenfeatures}), are arbitrary linear combinations of all available features. The eigenfeatures are very useful in data analysis, including the regularization of linear regression. Enforcing sparsity on the eigenfeatures, i.e., forcing them to be linear combinations of only a \textit{small} number of actual features (as opposed to all available features), can promote better generalization error and improve the interpretability of the eigenfeatures. We present deterministic and randomized algorithms that construct such sparse eigenfeatures while \emph{provably} achieving in-sample performance comparable to regularized linear regression.

This paper examines the possibility of a reject option' in the context of least squares regression. It is shown that using rejection it is theoretically possible to learn selective' regressors that can \epsilon -pointwise track the best regressor in hindsight from the same hypothesis class, while rejecting only a bounded portion of the domain. Moreover, the rejected volume vanishes with the training set size, under certain conditions. We then develop efficient and exact implementation of these selective regressors for the case of linear regression. Empirical evaluation over a suite of real-world datasets corroborates the theoretical analysis and indicates that our selective regressors can provide substantial advantage by reducing estimation error.

The Covid-19 pandemic drastically changed urban mobility, both during the height of the pandemic with government lockdowns, but also in the longer term with the adoption of working-from-home policies. To understand its effects on rail public transport ridership, we propose a dedicated Regression Mixture Model able to perform both the clustering of public transport stations and the segmentation of time periods, while ignoring variations due to additional variables such as the official lockdowns or non-working days. Each cluster is thus defined by a series of segments in which the effect of the exogenous variables is constant. As each segment within a cluster has its own regression coefficients to model the impact of the covariates, we analyze how these coefficients evolve to understand the changes in the cluster. We present the regression mixture model and the parameter estimation using the EM algorithm, before demonstrating the benefits of the model on both simulated and real data. Thanks to a five-year dataset of the ridership in the Paris public transport system, we analyze the impact of the pandemic, not only in terms of the number of travelers but also on the weekly commute. We further analyze the specific changes that the pandemic caused inside each cluster.

We consider the functional linear regression model with a scalar response and a Hilbert space-valued predictor, a well-known ill-posed inverse problem. We propose a new formulation of the functional partial least-squares (PLS) estimator related to the conjugate gradient method. We shall show that the estimator achieves the (nearly) optimal convergence rate on a class of ellipsoids and we introduce an early stopping rule which adapts to the unknown degree of ill-posedness. Some theoretical and simulation comparison between the estimator and the principal component regression estimator is provided.

Building upon score-based learning, new interest in stochastic localization techniques has recently emerged. In these models, one seeks to noise a sample from the data distribution through a stochastic process, called observation process, and progressively learns a denoiser associated to this dynamics. Apart from specific applications, the use of stochastic localization for the problem of sampling from an unnormalized target density has not been explored extensively. This work contributes to fill this gap. We consider a general stochastic localization framework and introduce an explicit class of observation processes, associated with flexible denoising schedules. We provide a complete methodology, $\textit{Stochastic Localization via Iterative Posterior Sampling}$ (SLIPS), to obtain approximate samples of this dynamics, and as a by-product, samples from the target distribution. Our scheme is based on a Markov chain Monte Carlo estimation of the denoiser and comes with detailed practical guidelines. We illustrate the benefits and applicability of SLIPS on several benchmarks, including Gaussian mixtures in increasing dimensions, Bayesian logistic regression and a high-dimensional field system from statistical-mechanics.

We study the use of linear regression for multiclass classification in the over-parametrized regime where some of the training data is mislabeled. In such scenarios it is necessary to add an explicit regularization term, $\lambda f(w)$, for some convex function $f(\cdot)$, to avoid overfitting the mislabeled data. In our analysis, we assume that the data is sampled from a Gaussian Mixture Model with equal class sizes, and that a proportion $c$ of the training labels is corrupted for each class. Under these assumptions, we prove that the best classification performance is achieved when $f(\cdot) = \|\cdot\|^2_2$ and $\lambda \to \infty$. We then proceed to analyze the classification errors for $f(\cdot) = \|\cdot\|_1$ and $f(\cdot) = \|\cdot\|_\infty$ in the large $\lambda$ regime and notice that it is often possible to find sparse and one-bit solutions, respectively, that perform almost as well as the one corresponding to $f(\cdot) = \|\cdot\|_2^2$.