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Boaz Nadler

Publications and source records attributed to Boaz Nadler.

At least 37 records · Page 2Linked to original sources
arXiv2021

The Trimmed Lasso: Sparse Recovery Guarantees and Practical Optimization by the Generalized Soft-Min Penalty

We present a new approach to solve the sparse approximation or best subset selection problem, namely find a $k$-sparse vector ${\bf x}\in\mathbb{R}^d$ that minimizes the $\ell_2$ residual $\lVert A{\bf x}-{\bf y} \rVert_2$. We consider a regularized approach, whereby this residual is penalized by the non-convex $\textit{trimmed lasso}$, defined as the $\ell_1$-norm of ${\bf x}$ excluding its $k$ largest-magnitude entries. We prove that the trimmed lasso has several appealing theoretical properties, and in particular derive sparse recovery guarantees assuming successful optimization of the penalized objective. Next, we show empirically that directly optimizing this objective can be quite challenging. Instead, we propose a surrogate for the trimmed lasso, called the $\textit{generalized soft-min}$. This penalty smoothly interpolates between the classical lasso and the trimmed lasso, while taking into account all possible $k$-sparse patterns. The generalized soft-min penalty involves summation over $\binom{d}{k}$ terms, yet we derive a polynomial-time algorithm to compute it. This, in turn, yields a practical method for the original sparse approximation problem. Via simulations, we demonstrate its competitive performance compared to current state of the art.

cs.LG↗
arXiv2020

Rank $2r$ iterative least squares: efficient recovery of ill-conditioned low rank matrices from few entries

We present a new, simple and computationally efficient iterative method for low rank matrix completion. Our method is inspired by the class of factorization-type iterative algorithms, but substantially differs from them in the way the problem is cast. Precisely, given a target rank $r$, instead of optimizing on the manifold of rank $r$ matrices, we allow our interim estimated matrix to have a specific over-parametrized rank $2r$ structure. Our algorithm, denoted R2RILS for rank $2r$ iterative least squares, has low memory requirements, and at each iteration it solves a computationally cheap sparse least-squares problem. We motivate our algorithm by its theoretical analysis for the simplified case of a rank-1 matrix. Empirically, R2RILS is able to recover ill conditioned low rank matrices from very few observations -- near the information limit, and it is stable to additive noise.

math.OC↗
arXiv2020

Spectral neighbor joining for reconstruction of latent tree models

A common assumption in multiple scientific applications is that the distribution of observed data can be modeled by a latent tree graphical model. An important example is phylogenetics, where the tree models the evolutionary lineages of a set of observed organisms. Given a set of independent realizations of the random variables at the leaves of the tree, a key challenge is to infer the underlying tree topology. In this work we develop Spectral Neighbor Joining (SNJ), a novel method to recover the structure of latent tree graphical models. Given a matrix that contains a measure of similarity between all pairs of observed variables, SNJ computes a spectral measure of cohesion between groups of observed variables. We prove that SNJ is consistent, and derive a sufficient condition for correct tree recovery from an estimated similarity matrix. Combining this condition with a concentration of measure result on the similarity matrix, we bound the number of samples required to recover the tree with high probability. We illustrate via extensive simulations that in comparison to several other reconstruction methods, SNJ requires fewer samples to accurately recover trees with a large number of leaves or long edges.

stat.ML↗
arXiv2020

Tight Recovery Guarantees for Orthogonal Matching Pursuit Under Gaussian Noise

Orthogonal Matching pursuit (OMP) is a popular algorithm to estimate an unknown sparse vector from multiple linear measurements of it. Assuming exact sparsity and that the measurements are corrupted by additive Gaussian noise, the success of OMP is often formulated as exactly recovering the support of the sparse vector. Several authors derived a sufficient condition for exact support recovery by OMP with high probability depending on the signal-to-noise ratio, defined as the magnitude of the smallest non-zero coefficient of the vector divided by the noise level. We make two contributions. First, we derive a slightly sharper sufficient condition for two variants of OMP, in which either the sparsity level or the noise level is known. Next, we show that this sharper sufficient condition is tight, in the following sense: for a wide range of problem parameters, there exist a dictionary of linear measurements and a sparse vector with a signal-to-noise ratio slightly below that of the sufficient condition, for which with high probability OMP fails to recover its support. Finally, we present simulations which illustrate that our condition is tight for a much broader range of dictionaries.

math.ST↗
arXiv2020

Beyond Trees: Classification with Sparse Pairwise Dependencies

Several classification methods assume that the underlying distributions follow tree-structured graphical models. Indeed, trees capture statistical dependencies between pairs of variables, which may be crucial to attain low classification errors. The resulting classifier is linear in the log-transformed univariate and bivariate densities that correspond to the tree edges. In practice, however, observed data may not be well approximated by trees. Yet, motivated by the importance of pairwise dependencies for accurate classification, here we propose to approximate the optimal decision boundary by a sparse linear combination of the univariate and bivariate log-transformed densities. Our proposed approach is semi-parametric in nature: we non-parametrically estimate the univariate and bivariate densities, remove pairs of variables that are nearly independent using the Hilbert-Schmidt independence criteria, and finally construct a linear SVM on the retained log-transformed densities. We demonstrate using both synthetic and real data that our resulting classifier, denoted SLB (Sparse Log-Bivariate density), is competitive with popular classification methods.

stat.ML↗
arXiv2018

Robust Sparse Covariance Estimation by Thresholding Tyler's M-Estimator

Estimating a high-dimensional sparse covariance matrix from a limited number of samples is a fundamental problem in contemporary data analysis. Most proposals to date, however, are not robust to outliers or heavy tails. Towards bridging this gap, in this work we consider estimating a sparse shape matrix from $n$ samples following a possibly heavy tailed elliptical distribution. We propose estimators based on thresholding either Tyler's M-estimator or its regularized variant. We derive bounds on the difference in spectral norm between our estimators and the shape matrix in the joint limit as the dimension $p$ and sample size $n$ tend to infinity with $p/n\toγ>0$. These bounds are minimax rate-optimal. Results on simulated data support our theoretical analysis.

math.ST↗
arXiv2018

SpectralNet: Spectral Clustering using Deep Neural Networks

Spectral clustering is a leading and popular technique in unsupervised data analysis. Two of its major limitations are scalability and generalization of the spectral embedding (i.e., out-of-sample-extension). In this paper we introduce a deep learning approach to spectral clustering that overcomes the above shortcomings. Our network, which we call SpectralNet, learns a map that embeds input data points into the eigenspace of their associated graph Laplacian matrix and subsequently clusters them. We train SpectralNet using a procedure that involves constrained stochastic optimization. Stochastic optimization allows it to scale to large datasets, while the constraints, which are implemented using a special-purpose output layer, allow us to keep the network output orthogonal. Moreover, the map learned by SpectralNet naturally generalizes the spectral embedding to unseen data points. To further improve the quality of the clustering, we replace the standard pairwise Gaussian affinities with affinities leaned from unlabeled data using a Siamese network. Additional improvement can be achieved by applying the network to code representations produced, e.g., by standard autoencoders. Our end-to-end learning procedure is fully unsupervised. In addition, we apply VC dimension theory to derive a lower bound on the size of SpectralNet. State-of-the-art clustering results are reported on the Reuters dataset. Our implementation is publicly available at https://github.com/kstant0725/SpectralNet .

stat.ML↗
arXiv2018

Newton correction methods for computing real eigenpairs of symmetric tensors

Real eigenpairs of symmetric tensors play an important role in multiple applications. In this paper we propose and analyze a fast iterative Newton-based method to compute real eigenpairs of symmetric tensors. We derive sufficient conditions for a real eigenpair to be a stable fixed point for our method, and prove that given a sufficiently close initial guess, the convergence rate is quadratic. Empirically, our method converges to a significantly larger number of eigenpairs compared to previously proposed iterative methods, and with enough random initializations typically finds all real eigenpairs. In particular, for a generic symmetric tensor, the sufficient conditions for local convergence of our Newton-based method hold simultaneously for all its real eigenpairs.

math.NA↗
arXiv2018

Learning Binary Latent Variable Models: A Tensor Eigenpair Approach

Latent variable models with hidden binary units appear in various applications. Learning such models, in particular in the presence of noise, is a challenging computational problem. In this paper we propose a novel spectral approach to this problem, based on the eigenvectors of both the second order moment matrix and third order moment tensor of the observed data. We prove that under mild non-degeneracy conditions, our method consistently estimates the model parameters at the optimal parametric rate. Our tensor-based method generalizes previous orthogonal tensor decomposition approaches, where the hidden units were assumed to be either statistically independent or mutually exclusive. We illustrate the consistency of our method on simulated data and demonstrate its usefulness in learning a common model for population mixtures in genetics.

stat.ML↗
arXiv2017

On Detection of Faint Edges in Noisy Images

A fundamental question for edge detection in noisy images is how faint can an edge be and still be detected. In this paper we offer a formalism to study this question and subsequently introduce computationally efficient multiscale edge detection algorithms designed to detect faint edges in noisy images. In our formalism we view edge detection as a search in a discrete, though potentially large, set of feasible curves. First, we derive approximate expressions for the detection threshold as a function of curve length and the complexity of the search space. We then present two edge detection algorithms, one for straight edges, and the second for curved ones. Both algorithms efficiently search for edges in a large set of candidates by hierarchically constructing difference filters that match the curves traced by the sought edges. We demonstrate the utility of our algorithms in both simulations and applications involving challenging real images. Finally, based on these principles, we develop an algorithm for fiber detection and enhancement. We exemplify its utility to reveal and enhance nerve axons in light microscopy images.

cs.CV↗
arXiv2017

Unsupervised Ensemble Regression

Consider a regression problem where there is no labeled data and the only observations are the predictions $f_i(x_j)$ of $m$ experts $f_{i}$ over many samples $x_j$. With no knowledge on the accuracy of the experts, is it still possible to accurately estimate the unknown responses $y_{j}$? Can one still detect the least or most accurate experts? In this work we propose a framework to study these questions, based on the assumption that the $m$ experts have uncorrelated deviations from the optimal predictor. Assuming the first two moments of the response are known, we develop methods to detect the best and worst regressors, and derive U-PCR, a novel principal components approach for unsupervised ensemble regression. We provide theoretical support for U-PCR and illustrate its improved accuracy over the ensemble mean and median on a variety of regression problems.

stat.ML↗
arXiv2017

Minimax-optimal semi-supervised regression on unknown manifolds

We consider semi-supervised regression when the predictor variables are drawn from an unknown manifold. A simple two step approach to this problem is to: (i) estimate the manifold geodesic distance between any pair of points using both the labeled and unlabeled instances; and (ii) apply a k nearest neighbor regressor based on these distance estimates. We prove that given sufficiently many unlabeled points, this simple method of geodesic kNN regression achieves the optimal finite-sample minimax bound on the mean squared error, as if the manifold were known. Furthermore, we show how this approach can be efficiently implemented, requiring only O(k N log N) operations to estimate the regression function at all N labeled and unlabeled points. We illustrate this approach on two datasets with a manifold structure: indoor localization using WiFi fingerprints and facial pose estimation. In both cases, geodesic kNN is more accurate and much faster than the popular Laplacian eigenvector regressor.

stat.ML↗
arXiv2016

The discrete sign problem: uniqueness, recovery algorithms and phase retrieval applications

In this paper we consider the following real-valued and finite dimensional specific instance of the 1-D classical phase retrieval problem. Let ${\bf F}\in\mathbb{R}^N$ be an $N$-dimensional vector, whose discrete Fourier transform has a compact support. The sign problem is to recover ${\bf F}$ from its magnitude $|{\bf F}|$. First, in contrast to the classical 1-D phase problem which in general has multiple solutions, we prove that with sufficient over-sampling, the sign problem admits a unique solution. Next, we show that the sign problem can be viewed as a special case of a more general piecewise constant phase problem. Relying on this result, we derive a computationally efficient and robust to noise sign recovery algorithm. In the noise-free case and with a sufficiently high sampling rate, our algorithm is guaranteed to recover the true sign pattern. Finally, we present two phase retrieval applications of the sign problem: (i) vectorial phase retrieval with three measurement vectors; and (ii) recovery of two well separated 1-D objects.

math.NA↗
arXiv2016

On the exact Berk-Jones statistics and their p-value calculation

Continuous goodness-of-fit testing is a classical problem in statistics. Despite having low power for detecting deviations at the tail of a distribution, the most popular test is based on the Kolmogorov-Smirnov statistic. While similar variance-weighted statistics, such as Anderson-Darling and the Higher Criticism statistic give more weight to tail deviations, as shown in various works, they still mishandle the extreme tails. As a viable alternative, in this paper we study some of the statistical properties of the exact $M_n$ statistics of Berk and Jones. We derive the asymptotic null distributions of $M_n, M_n^+, M_n^-$, and further prove their consistency as well as asymptotic optimality for a wide range of rare-weak mixture models. Additionally, we present a new computationally efficient method to calculate $p$-values for any supremum-based one-sided statistic, including the one-sided $M_n^+,M_n^-$ and $R_n^+,R_n^-$ statistics of Berk and Jones and the Higher Criticism statistic. We illustrate our theoretical analysis with several finite-sample simulations.

stat.ME↗
arXiv2016

Unsupervised Ensemble Learning with Dependent Classifiers

In unsupervised ensemble learning, one obtains predictions from multiple sources or classifiers, yet without knowing the reliability and expertise of each source, and with no labeled data to assess it. The task is to combine these possibly conflicting predictions into an accurate meta-learner. Most works to date assumed perfect diversity between the different sources, a property known as conditional independence. In realistic scenarios, however, this assumption is often violated, and ensemble learners based on it can be severely sub-optimal. The key challenges we address in this paper are:\ (i) how to detect, in an unsupervised manner, strong violations of conditional independence; and (ii) construct a suitable meta-learner. To this end we introduce a statistical model that allows for dependencies between classifiers. Our main contributions are the development of novel unsupervised methods to detect strongly dependent classifiers, better estimate their accuracies, and construct an improved meta-learner. Using both artificial and real datasets, we showcase the importance of taking classifier dependencies into account and the competitive performance of our approach.

cs.LG↗
arXiv2016

A Deep Learning Approach to Unsupervised Ensemble Learning

We show how deep learning methods can be applied in the context of crowdsourcing and unsupervised ensemble learning. First, we prove that the popular model of Dawid and Skene, which assumes that all classifiers are conditionally independent, is {\em equivalent} to a Restricted Boltzmann Machine (RBM) with a single hidden node. Hence, under this model, the posterior probabilities of the true labels can be instead estimated via a trained RBM. Next, to address the more general case, where classifiers may strongly violate the conditional independence assumption, we propose to apply RBM-based Deep Neural Net (DNN). Experimental results on various simulated and real-world datasets demonstrate that our proposed DNN approach outperforms other state-of-the-art methods, in particular when the data violates the conditional independence assumption.

stat.ML↗
arXiv2015

Detecting the large entries of a sparse covariance matrix in sub-quadratic time

The covariance matrix of a $p$-dimensional random variable is a fundamental quantity in data analysis. Given $n$ i.i.d. observations, it is typically estimated by the sample covariance matrix, at a computational cost of $O(np^{2})$ operations. When $n,p$ are large, this computation may be prohibitively slow. Moreover, in several contemporary applications, the population matrix is approximately sparse, and only its few large entries are of interest. This raises the following question, at the focus of our work: Assuming approximate sparsity of the covariance matrix, can its large entries be detected much faster, say in sub-quadratic time, without explicitly computing all its $p^{2}$ entries? In this paper, we present and theoretically analyze two randomized algorithms that detect the large entries of an approximately sparse sample covariance matrix using only $O(np\text{ poly log } p)$ operations. Furthermore, assuming sparsity of the population matrix, we derive sufficient conditions on the underlying random variable and on the number of samples $n$, for the sample covariance matrix to satisfy our approximate sparsity requirements. Finally, we illustrate the performance of our algorithms via several simulations.

stat.CO↗