In this paper, we analyze the generalization performance of the Iterative Hard Thresholding (IHT) algorithm widely used for sparse recovery problems. The parameter estimation and sparsity recovery consistency of IHT has long been known in compressed sensing. From the perspective of statistical learning, another fundamental question is how well the IHT estimation would predict on unseen data. This paper makes progress towards answering this open question by introducing a novel sparse generalization theory for IHT under the notion of algorithmic stability. Our theory reveals that: 1) under natural conditions on the empirical risk function over $n$ samples of dimension $p$, IHT with sparsity level $k$ enjoys an $\mathcal{\tilde O}(n^{-1/2}\sqrt{k\log(n)\log(p)})$ rate of convergence in sparse excess risk; 2) a tighter $\mathcal{\tilde O}(n^{-1/2}\sqrt{\log(n)})$ bound can be established by imposing an additional iteration stability condition on a hypothetical IHT procedure invoked to the population risk; and 3) a fast rate of order $\mathcal{\tilde O}\left(n^{-1}k(\log^3(n)+\log(p))\right)$ can be derived for strongly convex risk function under proper strong-signal conditions. The results have been substantialized to sparse linear regression and sparse logistic regression models to demonstrate the applicability of our theory. Preliminary numerical evidence is provided to confirm our theoretical predictions.
Iterative thresholding algorithms seek to optimize a differentiable objective function over a sparsity or rank constraint by alternating between gradient steps that reduce the objective, and thresholding steps that enforce the constraint. This work examines the choice of the thresholding operator, and asks whether it is possible to achieve stronger guarantees than what is possible with hard thresholding. We develop the notion of relative concavity of a thresholding operator, a quantity that characterizes the worst-case convergence performance of any thresholding operator on the target optimiza…
Microsoft Research（微软研究院）；University of Michigan, Ann Arbor（密歇根大学安娜堡分校）
The use of M-estimators in generalized linear regression models in high dimensional settings requires risk minimization with hard $L_0$ constraints. Of the known methods, the class of projected gradient descent (also known as iterative hard thresholding (IHT)) methods is known to offer the fastest and most scalable solutions. However, the current state-of-the-art is only able to analyze these methods in extremely restrictive settings which do not hold in high dimensional statistical models. In this work we bridge this gap by providing the first analysis for IHT-style methods in the high dimens…
Microsoft Corporation（微软公司）；Conduent Labs India（Conduent印度实验室）；University of Illinois Chicago（伊利诺伊大学芝加哥分校）；University of California Irvine（加利福尼亚大学尔湾分校）；Xerox Research Centre India（施乐印度研究中心）
The robust PCA problem, wherein, given an input data matrix that is the superposition of a low-rank matrix and a sparse matrix, we aim to separate out the low-rank and sparse components, is a well-studied problem in machine learning. One natural question that arises is that, as in the inductive setting, if features are provided as input as well, can we hope to do better? Answering this in the affirmative, the main goal of this paper is to study the robust PCA problem while incorporating feature information. In contrast to previous works in which recovery guarantees are based on the convex rela…
Rutgers University（罗格斯大学）；Nanjing University of Information Science & Technology（南京信息工程大学）
Iterative Hard Thresholding (IHT) is a class of projected gradient descent methods for optimizing sparsity-constrained minimization models, with the best known efficiency and scalability in practice. As far as we know, the existing IHT-style methods are designed for sparse minimization in primal form. It remains open to explore duality theory and algorithms in such a non-convex and NP-hard problem setting. In this paper, we bridge this gap by establishing a duality theory for sparsity-constrained minimization with $\ell_2$-regularized loss function and proposing an IHT-style algorithm for dual…
Adaptive thresholding methods have proved to yield high SNRs and fast convergence in finding the solution to the Compressed Sensing (CS) problems. Recently, it was observed that the robustness of a class of iterative sparse recovery algorithms such as Iterative Method with Adaptive Thresholding (IMAT) has outperformed the well-known LASSO algorithm in terms of reconstruction quality, convergence speed, and the sensitivity to the noise. In this paper, we introduce a new method towards solving the CS problem. The logic of this method is based on iterative projections of the thresholded signal on…
The proposed smooth blockwise iterative thresholding estimator (SBITE) is a model selection technique defined as a fixed point reached by iterating a likelihood gradient-based thresholding function. The smooth James-Stein thresholding function has two regularization parameters $λ$ and $ν$, and a smoothness parameter $s$. It enjoys smoothness like ridge regression and selects variables like lasso. Focusing on Gaussian regression, we show that SBITE is uniquely defined, and that its Stein unbiased risk estimate is a smooth function of $λ$ and $ν$, for better selection of the two regularization p…
We consider the problem of low rank matrix recovery in a stochastically noisy high dimensional setting. We propose a new estimator for the low rank matrix, based on the iterative hard thresholding method, and that is computationally efficient and simple. We prove that our estimator is efficient both in terms of the Frobenius risk, and in terms of the entry-wise risk uniformly over any change of orthonormal basis. This result allows us, in the case where the design is Gaussian, to provide the limiting distribution of the estimator, which is of great interest for constructing tests and confidenc…
In this article, we developed and analyzed a thresholding method in which soft thresholding estimators are independently expanded by empirical scaling values. The scaling values have a common hyper-parameter that is an order of expansion of an ideal scaling value that achieves hard thresholding. We simply call this estimator a scaled soft thresholding estimator. The scaled soft thresholding is a general method that includes the soft thresholding and non-negative garrote as special cases and gives an another derivation of adaptive LASSO. We then derived the degree of freedom of the scaled soft …
We consider the matrix completion problem where the aim is to esti-mate a large data matrix for which only a relatively small random subset of its entries is observed. Quite popular approaches to matrix completion problem are iterative thresholding methods. In spite of their empirical success, the theoretical guarantees of such iterative thresholding methods are poorly understood. The goal of this paper is to provide strong theo-retical guarantees, similar to those obtained for nuclear-norm penalization methods and one step thresholding methods, for an iterative thresholding algorithm which is…
Compressed sensing (CS) or sparse signal reconstruction (SSR) is a signal processing technique that exploits the fact that acquired data can have a sparse representation in some basis. One popular technique to reconstruct or approximate the unknown sparse signal is the iterative hard thresholding (IHT) which however performs very poorly under non-Gaussian noise conditions or in the face of outliers (gross errors). In this paper, we propose a robust IHT method based on ideas from $M$-estimation that estimates the sparse signal and the scale of the error distribution simultaneously. The method h…
The problem of recovering a low $n$-rank tensor is an extension of sparse recovery problem from the low dimensional space (matrix space) to the high dimensional space (tensor space) and has many applications in computer vision and graphics such as image inpainting and video inpainting. In this paper, we consider a new tensor recovery model, named as minimum $n$-rank approximation (MnRA), and propose an appropriate iterative hard thresholding algorithm with giving the upper bound of the $n$-rank in advance. The convergence analysis of the proposed algorithm is also presented. Particularly, we s…
The L1-regularized maximum likelihood estimation problem has recently become a topic of great interest within the machine learning, statistics, and optimization communities as a method for producing sparse inverse covariance estimators. In this paper, a proximal gradient method (G-ISTA) for performing L1-regularized covariance matrix estimation is presented. Although numerous algorithms have been proposed for solving this problem, this simple proximal gradient method is found to have attractive theoretical and numerical properties. G-ISTA has a linear rate of convergence, resulting in an O(log…
Recently, sparsity-based algorithms are proposed for super-resolution spectrum estimation. However, to achieve adequately high resolution in real-world signal analysis, the dictionary atoms have to be close to each other in frequency, thereby resulting in a coherent design. The popular convex compressed sensing methods break down in presence of high coherence and large noise. We propose a new regularization approach to handle model collinearity and obtain parsimonious frequency selection simultaneously. It takes advantage of the pairing structure of sine and cosine atoms in the frequency dicti…
Principal component analysis (PCA) is a classical dimension reduction method which projects data onto the principal subspace spanned by the leading eigenvectors of the covariance matrix. However, it behaves poorly when the number of features p is comparable to, or even much larger than, the sample size n. In this paper, we propose a new iterative thresholding approach for estimating principal subspaces in the setting where the leading eigenvectors are sparse. Under a spiked covariance model, we find that the new approach recovers the principal subspace and leading eigenvectors consistently, an…
Over the past decade, various matrix completion algorithms have been developed. Thresholded singular value decomposition (SVD) is a popular technique in implementing many of them. A sizable number of studies have shown its theoretical and empirical excellence, but choosing the right threshold level still remains as a key empirical difficulty. This paper proposes a novel matrix completion algorithm which iterates thresholded SVD with theoretically-justified and data-dependent values of thresholding parameters. The estimate of the proposed algorithm enjoys the minimax error rate and shows outsta…
Sparse coding is a core building block in many data analysis and machine learning pipelines. Typically it is solved by relying on generic optimization techniques, such as the Iterative Soft Thresholding Algorithm and its accelerated version (ISTA, FISTA). These methods are optimal in the class of first-order methods for non-smooth, convex functions. However, they do not exploit the particular structure of the problem at hand nor the input data distribution. An acceleration using neural networks, coined LISTA, was proposed in Gregor and Le Cun (2010), which showed empirically that one could ach…
Sparse coding strategies have been lauded for their parsimonious representations of data that leverage low dimensional structure. However, inference of these codes typically relies on an optimization procedure with poor computational scaling in high-dimensional problems. For example, sparse inference in the representations learned in the high-dimensional intermediary layers of deep neural networks (DNNs) requires an iterative minimization to be performed at each training step. As such, recent, quick methods in variational inference have been proposed to infer sparse codes by learning a distrib…
Quadratic regression involves modeling the response as a (generalized) linear function of not only the features $x^{j_1}$ but also of quadratic terms $x^{j_1}x^{j_2}$. The inclusion of such higher-order "interaction terms" in regression often provides an easy way to increase accuracy in already-high-dimensional problems. However, this explodes the problem dimension from linear $O(p)$ to quadratic $O(p^2)$, and it is common to look for sparse interactions (typically via heuristics). In this paper, we provide a new algorithm - Interaction Hard Thresholding (IntHT) which is the first one to prova…
Ecole Polytechnique Federale de Lausanne (EPFL)（洛桑联邦理工学院）；The University of Edinburgh（爱丁堡大学）
Classifiers based on sparse representations have recently been shown to provide excellent results in many visual recognition and classification tasks. However, the high cost of computing sparse representations at test time is a major obstacle that limits the applicability of these methods in large-scale problems, or in scenarios where computational power is restricted. We consider in this paper a simple yet efficient alternative to sparse coding for feature extraction. We study a classification scheme that applies the soft-thresholding nonlinear mapping in a dictionary, followed by a linear cl…
In this paper, we consider the problem of compressed sensing where the goal is to recover almost all the sparse vectors using a small number of fixed linear measurements. For this problem, we propose a novel partial hard-thresholding operator that leads to a general family of iterative algorithms. While one extreme of the family yields well known hard thresholding algorithms like ITI (Iterative Thresholding with Inversion) and HTP (Hard Thresholding Pursuit), the other end of the spectrum leads to a novel algorithm that we call Orthogonal Matching Pursuit with Replacement (OMPR). OMPR, like th…
Mitsubishi Electric Research Laboratories（三菱电机研究实验室）
Iterative shrinkage/thresholding algorithm (ISTA) is a well-studied method for finding sparse solutions to ill-posed inverse problems. In this letter, we present a data-driven scheme for learning optimal thresholding functions for ISTA. The proposed scheme is obtained by relating iterations of ISTA to layers of a simple deep neural network (DNN) and developing a corresponding error backpropagation algorithm that allows to fine-tune the thresholding functions. Simulations on sparse statistical signals illustrate potential gains in estimation quality due to the proposed data adaptive ISTA.
Compressed sensing aims to undersample certain high-dimensional signals, yet accurately reconstruct them by exploiting signal characteristics. Accurate reconstruction is possible when the object to be recovered is sufficiently sparse in a known basis. Currently, the best known sparsity-undersampling tradeoff is achieved when reconstructing by convex optimization -- which is expensive in important large-scale applications. Fast iterative thresholding algorithms have been intensively studied as alternatives to convex optimization for large-scale problems. Unfortunately known fast algorithms offe…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of California at Berkeley（加州大学伯克利分校）；Rice University（莱斯大学）
Iterative hard thresholding (IHT) is a projected gradient descent algorithm, known to achieve state of the art performance for a wide range of structured estimation problems, such as sparse inference. In this work, we consider IHT as a solution to the problem of learning sparse discrete distributions. We study the hardness of using IHT on the space of measures. As a practical alternative, we propose a greedy approximate projection which simultaneously captures appropriate notions of sparsity in distributions, while satisfying the simplex constraint, and investigate the convergence behavior of …
Sparse reconstruction approaches using the re-weighted l1-penalty have been shown, both empirically and theoretically, to provide a significant improvement in recovering sparse signals in comparison to the l1-relaxation. However, numerical optimization of such penalties involves solving problems with l1-norms in the objective many times. Using the direct link of reweighted l1-penalties to the concave log-regularizer for sparsity, we derive a simple prox-like algorithm for the log-regularized formulation. The proximal splitting step of the algorithm has a closed form solution, and we call the a…
University of Science and Technology of China（中国科学技术大学）
Rank minimization (RM) is a wildly investigated task of finding solutions by exploiting low-rank structure of parameter matrices. Recently, solving RM problem by leveraging non-convex relaxations has received significant attention. It has been demonstrated by some theoretical and experimental work that non-convex relaxation, e.g. Truncated Nuclear Norm Regularization (TNNR) and Reweighted Nuclear Norm Regularization (RNNR), can provide a better approximation of original problems than convex relaxations. However, designing an efficient algorithm with theoretical guarantee remains a challenging …
ETH Zurich（苏黎世联邦理工学院）；Institute of Science and Technology Austria（奥地利科学技术研究所）
Modern scientific instruments produce vast amounts of data, which can overwhelm the processing ability of computer systems. Lossy compression of data is an intriguing solution, but comes with its own drawbacks, such as potential signal loss, and the need for careful optimization of the compression ratio. In this work, we focus on a setting where this problem is especially acute: compressive sensing frameworks for interferometry and medical imaging. We ask the following question: can the precision of the data representation be lowered for all inputs, with recovery guarantees and practical perfo…
University of Washington（华盛顿大学）；Applied Physics Laboratory, University of Washington（华盛顿大学应用物理实验室）
In this paper, we propose a novel recurrent neural network architecture for speech separation. This architecture is constructed by unfolding the iterations of a sequential iterative soft-thresholding algorithm (ISTA) that solves the optimization problem for sparse nonnegative matrix factorization (NMF) of spectrograms. We name this network architecture deep recurrent NMF (DR-NMF). The proposed DR-NMF network has three distinct advantages. First, DR-NMF provides better interpretability than other deep architectures, since the weights correspond to NMF model parameters, even after training. This…
We consider the demixing problem of two (or more) high-dimensional vectors from nonlinear observations when the number of such observations is far less than the ambient dimension of the underlying vectors. Specifically, we demonstrate an algorithm that stably estimate the underlying components under general \emph{structured sparsity} assumptions on these components. Specifically, we show that for certain types of structured superposition models, our method provably recovers the components given merely $n = \mathcal{O}(s)$ samples where $s$ denotes the number of nonzero entries in the underlyin…
NTT DATA Mathematical Systems Inc.（NTT数据数学系统公司）；The University of Tokyo（东京大学）；RIKEN（理化学研究所）
We develop new stochastic gradient methods for efficiently solving sparse linear regression in a partial attribute observation setting, where learners are only allowed to observe a fixed number of actively chosen attributes per example at training and prediction times. It is shown that the methods achieve essentially a sample complexity of $O(1/\varepsilon)$ to attain an error of $\varepsilon$ under a variant of restricted eigenvalue condition, and the rate has better dependency on the problem dimension than existing methods. Particularly, if the smallest magnitude of the non-zero components o…
Stochastic optimization algorithms update models with cheap per-iteration costs sequentially, which makes them amenable for large-scale data analysis. Such algorithms have been widely studied for structured sparse models where the sparsity information is very specific, e.g., convex sparsity-inducing norms or $\ell^0$-norm. However, these norms cannot be directly applied to the problem of complex (non-convex) graph-structured sparsity models, which have important application in disease outbreak and social networks, etc. In this paper, we propose a stochastic gradient-based method for solving gr…
Recovery of low-rank matrices from a small number of linear measurements is now well-known to be possible under various model assumptions on the measurements. Such results demonstrate robustness and are backed with provable theoretical guarantees. However, extensions to tensor recovery have only recently began to be studied and developed, despite an abundance of practical tensor applications. Recently, a tensor variant of the Iterative Hard Thresholding method was proposed and theoretical results were obtained that guarantee exact recovery of tensors with low Tucker rank. In this paper, we uti…
We propose a simple modification to the iterative hard thresholding (IHT) algorithm, which recovers asymptotically sparser solutions as a function of the condition number. When aiming to minimize a convex function $f(x)$ with condition number $κ$ subject to $x$ being an $s$-sparse vector, the standard IHT guarantee is a solution with relaxed sparsity $O(sκ^2)$, while our proposed algorithm, regularized IHT, returns a solution with sparsity $O(sκ)$. Our algorithm significantly improves over ARHT which also finds a solution of sparsity $O(sκ)$, as it does not require re-optimization in each iter…
We investigate an existing distributed algorithm for learning sparse signals or data over networks. The algorithm is iterative and exchanges intermediate estimates of a sparse signal over a network. This learning strategy using exchange of intermediate estimates over the network requires a limited communication overhead for information transmission. Our objective in this article is to show that the strategy is good for learning in spite of limited communication. In pursuit of this objective, we first provide a restricted isometry property (RIP)-based theoretical analysis on convergence of the …
Motivated by the learned iterative soft thresholding algorithm (LISTA), we introduce a general class of neural networks suitable for sparse reconstruction from few linear measurements. By allowing a wide range of degrees of weight-sharing between the layers, we enable a unified analysis for very different neural network types, ranging from recurrent ones to networks more similar to standard feedforward neural networks. Based on training samples, via empirical risk minimization we aim at learning the optimal network parameters and thereby the optimal network that reconstructs signals from their…
Sparse Canonical Correlation Analysis (CCA) has received considerable attention in high-dimensional data analysis to study the relationship between two sets of random variables. However, there has been remarkably little theoretical statistical foundation on sparse CCA in high-dimensional settings despite active methodological and applied research activities. In this paper, we introduce an elementary sufficient and necessary characterization such that the solution of CCA is indeed sparse, propose a computationally efficient procedure, called CAPIT, to estimate the canonical directions, and show…
A genome-wide association study (GWAS) correlates marker variation with trait variation in a sample of individuals. Each study subject is genotyped at a multitude of SNPs (single nucleotide polymorphisms) spanning the genome. Here we assume that subjects are unrelated and collected at random and that trait values are normally distributed or transformed to normality. Over the past decade, researchers have been remarkably successful in applying GWAS analysis to hundreds of traits. The massive amount of data produced in these studies present unique computational challenges. Penalized regression w…
We analyze the sample complexity of the thresholding bandit problem, with and without the assumption that the mean values of the arms are increasing. In each case, we provide a lower bound valid for any risk $δ$ and any $δ$-correct algorithm; in addition, we propose an algorithm whose sample complexity is of the same order of magnitude for small risks. This work is motivated by phase 1 clinical trials, a practically important setting where the arm means are increasing by nature, and where no satisfactory solution is available so far.
We develop mask iterative hard thresholding algorithms (mask IHT and mask DORE) for sparse image reconstruction of objects with known contour. The measurements follow a noisy underdetermined linear model common in the compressive sampling literature. Assuming that the contour of the object that we wish to reconstruct is known and that the signal outside the contour is zero, we formulate a constrained residual squared error minimization problem that incorporates both the geometric information (i.e. the knowledge of the object's contour) and the signal sparsity constraint. We first introduce a m…
In this work, we derive a generic overcomplete frame thresholding scheme based on risk minimization. Overcomplete frames being favored for analysis tasks such as classification, regression or anomaly detection, we provide a way to leverage those optimal representations in real-world applications through the use of thresholding. We validate the method on a large scale bird activity detection task via the scattering network architecture performed by means of continuous wavelets, known for being an adequate dictionary in audio environments.
University at Albany（奥尔巴尼大学）；Stony Brook University（石溪大学）；University of Birmingham（伯明翰大学）
In this paper, we aim to develop stochastic hard thresholding algorithms for the important problem of AUC maximization in imbalanced classification. The main challenge is the pairwise loss involved in AUC maximization. We overcome this obstacle by reformulating the U-statistics objective function as an empirical risk minimization (ERM), from which a stochastic hard thresholding algorithm (\texttt{SHT-AUC}) is developed. To our best knowledge, this is the first attempt to provide stochastic hard thresholding algorithms for AUC maximization with a per-iteration cost $Ø(b d)$ where $d$ and $b$ ar…
The problem of estimating a high-dimensional sparse vector $\boldsymbolθ \in \mathbb{R}^n$ from an observation in i.i.d. Gaussian noise is considered. The performance is measured using squared-error loss. An empirical Bayes shrinkage estimator, derived using a Bernoulli-Gaussian prior, is analyzed and compared with the well-known soft-thresholding estimator. We obtain concentration inequalities for the Stein's unbiased risk estimate and the loss function of both estimators. The results show that for large $n$, both the risk estimate and the loss function concentrate on deterministic values clo…
