Sparsity-constrained optimization is an important and challenging problem that has wide applicability in data mining, machine learning, and statistics. In this paper, we focus on sparsity-constrained optimization in cases where the cost function is a general nonlinear function and, in particular, the sparsity constraint is defined by a graph-structured sparsity model. Existing methods explore this problem in the context of sparse estimation in linear models. To the best of our knowledge, this is the first work to present an efficient approximation algorithm, namely, Graph-structured Matching Pursuit (Graph-Mp), to optimize a general nonlinear function subject to graph-structured constraints. We prove that our algorithm enjoys the strong guarantees analogous to those designed for linear models in terms of convergence rate and approximation accuracy. As a case study, we specialize Graph-Mp to optimize a number of well-known graph scan statistic models for the connected subgraph detection task, and empirical evidence demonstrates that our general algorithm performs superior over state-of-the-art methods that are designed specifically for the task of connected subgraph detection.
Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education（教育部智能感知与图像理解重点实验室）；School of Artificial Intelligence, Xidian University（西安电子科技大学人工智能学院）；Department of Computer Science, The University of Texas at Austin（德克萨斯大学奥斯汀分校计算机科学系）
Large-scale non-convex sparsity-constrained problems have recently gained extensive attention. Most existing deterministic optimization methods (e.g., GraSP) are not suitable for large-scale and high-dimensional problems, and thus stochastic optimization methods with hard thresholding (e.g., SVRGHT) become more attractive. Inspired by GraSP, this paper proposes a new general relaxed gradient support pursuit (RGraSP) framework, in which the sub-algorithm only requires to satisfy a slack descent condition. We also design two specific semi-stochastic gradient hard thresholding algorithms. In part…
École Polytechnique（巴黎综合理工学院）；Athens University of Economics and Business（雅典经济与商业大学）
In text classification, the problem of overfitting arises due to the high dimensionality, making regularization essential. Although classic regularizers provide sparsity, they fail to return highly accurate models. On the contrary, state-of-the-art group-lasso regularizers provide better results at the expense of low sparsity. In this paper, we apply a greedy variable selection algorithm, called Orthogonal Matching Pursuit, for the text classification task. We also extend standard group OMP by introducing overlapping Group OMP to handle overlapping groups of features. Empirical analysis verifi…
The performance of Orthogonal Matching Pursuit (OMP) for variable selection is analyzed for random designs. When contrasted with the deterministic case, since the performance is here measured after averaging over the distribution of the design matrix, one can have far less stringent sparsity constraints on the coefficient vector. We demonstrate that for exact sparse vectors, the performance of the OMP is similar to known results on the Lasso algorithm [\textit{IEEE Trans. Inform. Theory} \textbf{55} (2009) 2183--2202]. Moreover, variable selection under a more relaxed sparsity assumption on th…
ETH Zürich（苏黎世联邦理工学院）；UT Austin（德克萨斯大学奥斯汀分校）；EPFL（洛桑联邦理工学院）
Two of the most fundamental prototypes of greedy optimization are the matching pursuit and Frank-Wolfe algorithms. In this paper, we take a unified view on both classes of methods, leading to the first explicit convergence rates of matching pursuit methods in an optimization sense, for general sets of atoms. We derive sublinear ($1/t$) convergence for both classes on general smooth objectives, and linear convergence on strongly convex objectives, as well as a clear correspondence of algorithm variants. Our presented algorithms and rates are affine invariant, and do not need any incoherence or …
Georgia Institute of Technology（佐治亚理工学院）；Zuse Institute Berlin（柏林祖斯研究所）；Technische Universität Berlin（柏林工业大学）
Matching pursuit algorithms are an important class of algorithms in signal processing and machine learning. We present a blended matching pursuit algorithm, combining coordinate descent-like steps with stronger gradient descent steps, for minimizing a smooth convex function over a linear space spanned by a set of atoms. We derive sublinear to linear convergence rates according to the smoothness and sharpness orders of the function and demonstrate computational superiority of our approach. In particular, we derive linear rates for a wide class of non-strongly convex functions, and we demonstrat…
Spike and Slab priors have been of much recent interest in signal processing as a means of inducing sparsity in Bayesian inference. Applications domains that benefit from the use of these priors include sparse recovery, regression and classification. It is well-known that solving for the sparse coefficient vector to maximize these priors results in a hard non-convex and mixed integer programming problem. Most existing solutions to this optimization problem either involve simplifying assumptions/relaxations or are computationally expensive. We propose a new greedy and adaptive matching pursuit …
Massachusetts Institute of Technology（麻省理工学院）；University of Michigan（密歇根大学）；Harvard University（哈佛大学）
We propose a novel diverse feature selection method based on determinantal point processes (DPPs). Our model enables one to flexibly define diversity based on the covariance of features (similar to orthogonal matching pursuit) or alternatively based on side information. We introduce our approach in the context of Bayesian sparse regression, employing a DPP as a variational approximation to the true spike and slab posterior distribution. We subsequently show how this variational DPP approximation generalizes and extends mean-field approximation, and can be learned efficiently by exploiting the …
Basis Pursuit (BP), Basis Pursuit DeNoising (BPDN), and LASSO are popular methods for identifying important predictors in the high-dimensional linear regression model, i.e. when the number of rows of the design matrix X is smaller than the number of columns. By definition, BP uniquely recovers the vector of regression coefficients b if there is no noise and the vector b has the smallest L1 norm among all vectors s such that Xb=Xs (identifiability condition). Furthermore, LASSO can recover the sign of b only under a much stronger irrepresentability condition. Meanwhile, it is known that the mod…
Sparse coding techniques for image processing traditionally rely on a processing of small overlapping patches separately followed by averaging. This has the disadvantage that the reconstructed image no longer obeys the sparsity prior used in the processing. For this purpose convolutional sparse coding has been introduced, where a shift-invariant dictionary is used and the sparsity of the recovered image is maintained. Most such strategies target the $\ell_0$ "norm" or the $\ell_1$ norm of the whole image, which may create an imbalanced sparsity across various regions in the image. In order to …
Level-set optimization formulations with data-driven constraints minimize a regularization functional subject to matching observations to a given error level. These formulations are widely used, particularly for matrix completion and sparsity promotion in data interpolation and denoising. The misfit level is typically measured in the l2 norm, or other smooth metrics. In this paper, we present a new flexible algorithmic framework that targets nonsmooth level-set constraints, including L1, Linf, and even L0 norms. These constraints give greater flexibility for modeling deviations in observation …
In Compressed Sensing and high dimensional estimation, signal recovery often relies on sparsity assumptions and estimation is performed via $\ell_1$-penalized least-squares optimization, a.k.a. LASSO. The $\ell_1$ penalisation is usually controlled by a weight, also called "relaxation parameter", denoted by $λ$. It is commonly thought that the practical efficiency of the LASSO for prediction crucially relies on accurate selection of $λ$. In this short note, we propose to consider the hyper-parameter selection problem from a new perspective which combines the Hedge online learning method by Fre…
We provide a necessary and sufficient condition for the uniqueness of penalized least-squares estimators whose penalty term is given by a norm with a polytope unit ball, covering a wide range of methods including SLOPE, PACS, fused, clustered and classical LASSO as well as the related method of basis pursuit. We consider a strong type of uniqueness that is relevant for statistical problems. The uniqueness condition is geometric and involves how the row span of the design matrix intersects the faces of the dual norm unit ball, which for SLOPE is given by the signed permutahedron. Further consid…
Orthogonal matching pursuit (OMP) is a widely used compressive sensing (CS) algorithm for recovering sparse signals in noisy linear regression models. The performance of OMP depends on its stopping criteria (SC). SC for OMP discussed in literature typically assumes knowledge of either the sparsity of the signal to be estimated $k_0$ or noise variance $σ^2$, both of which are unavailable in many practical applications. In this article we develop a modified version of OMP called tuning free OMP or TF-OMP which does not require a SC. TF-OMP is proved to accomplish successful sparse recovery under…
Sparsity-based subspace clustering algorithms have attracted significant attention thanks to their excellent performance in practical applications. A prominent example is the sparse subspace clustering (SSC) algorithm by Elhamifar and Vidal, which performs spectral clustering based on an adjacency matrix obtained by sparsely representing each data point in terms of all the other data points via the Lasso. When the number of data points is large or the dimension of the ambient space is high, the computational complexity of SSC quickly becomes prohibitive. Dyer et al. observed that SSC-OMP obtai…
This paper proposes an Adaptive Stochastic Model Predictive Control (MPC) strategy for stable linear time-invariant systems in the presence of bounded disturbances. We consider multi-input, multi-output systems that can be expressed by a Finite Impulse Response (FIR) model. The parameters of the FIR model corresponding to each output are unknown but assumed sparse. We estimate these parameters using the Recursive Least Squares algorithm. The estimates are then improved using set-based bounds obtained by solving the Basis Pursuit Denoising [1] problem. Our approach is able to handle hard input …
Stanford University（斯坦福大学）；Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）；The University of Texas at Dallas（德克萨斯大学达拉斯分校）
In this paper, we study the problem of compressed sensing using binary measurement matrices and $\ell_1$-norm minimization (basis pursuit) as the recovery algorithm. We derive new upper and lower bounds on the number of measurements to achieve robust sparse recovery with binary matrices. We establish sufficient conditions for a column-regular binary matrix to satisfy the robust null space property (RNSP) and show that the associated sufficient conditions % sparsity bounds for robust sparse recovery obtained using the RNSP are better by a factor of $(3 \sqrt{3})/2 \approx 2.6$ compared to the s…
This article develops a framework for testing general hypothesis in high-dimensional models where the number of variables may far exceed the number of observations. Existing literature has considered less than a handful of hypotheses, such as testing individual coordinates of the model parameter. However, the problem of testing general and complex hypotheses remains widely open. We propose a new inference method developed around the hypothesis adaptive projection pursuit framework, which solves the testing problems in the most general case. The proposed inference is centered around a new class…
The goal of Sparse Convex Optimization is to optimize a convex function $f$ under a sparsity constraint $s\leq s^*γ$, where $s^*$ is the target number of non-zero entries in a feasible solution (sparsity) and $γ\geq 1$ is an approximation factor. There has been a lot of work to analyze the sparsity guarantees of various algorithms (LASSO, Orthogonal Matching Pursuit (OMP), Iterative Hard Thresholding (IHT)) in terms of the Restricted Condition Number $κ$. The best known algorithms guarantee to find an approximate solution of value $f(x^*)+ε$ with the sparsity bound of $γ= O\left(κ\min\left\{\l…
Orthogonal matching pursuit (OMP) is a widely used algorithm for recovering sparse high dimensional vectors in linear regression models. The optimal performance of OMP requires \textit{a priori} knowledge of either the sparsity of regression vector or noise statistics. Both these statistics are rarely known \textit{a priori} and are very difficult to estimate. In this paper, we present a novel technique called residual ratio thresholding (RRT) to operate OMP without any \textit{a priori} knowledge of sparsity and noise statistics and establish finite sample and large sample support recovery gu…
Sparse Bayesian Learning (SBL) is a powerful framework for attaining sparsity in probabilistic models. Herein, we propose a coordinate ascent algorithm for SBL termed Relevance Matching Pursuit (RMP) and show that, as its noise variance parameter goes to zero, RMP exhibits a surprising connection to Stepwise Regression. Further, we derive novel guarantees for Stepwise Regression algorithms, which also shed light on RMP. Our guarantees for Forward Regression improve on deterministic and probabilistic results for Orthogonal Matching Pursuit with noise. Our analysis of Backward Regression on dete…
Sparse modeling is a powerful framework for data analysis and processing. Traditionally, encoding in this framework is performed by solving an L1-regularized linear regression problem, commonly referred to as Lasso or Basis Pursuit. In this work we combine the sparsity-inducing property of the Lasso model at the individual feature level, with the block-sparsity property of the Group Lasso model, where sparse groups of features are jointly encoded, obtaining a sparsity pattern hierarchically structured. This results in the Hierarchical Lasso (HiLasso), which shows important practical modeling a…
We consider high dimensional sparse regression, and develop strategies able to deal with arbitrary -- possibly, severe or coordinated -- errors in the covariance matrix $X$. These may come from corrupted data, persistent experimental errors, or malicious respondents in surveys/recommender systems, etc. Such non-stochastic error-in-variables problems are notoriously difficult to treat, and as we demonstrate, the problem is particularly pronounced in high-dimensional settings where the primary goal is {\em support recovery} of the sparse regressor. We develop algorithms for support recovery in s…
The theory of Compressed Sensing (CS) asserts that an unknown signal $x\in\mathbb{R}^p$ can be accurately recovered from an underdetermined set of $n$ linear measurements with $n\ll p$, provided that $x$ is sufficiently sparse. However, in applications, the degree of sparsity $\|x\|_0$ is typically unknown, and the problem of directly estimating $\|x\|_0$ has been a longstanding gap between theory and practice. A closely related issue is that $\|x\|_0$ is a highly idealized measure of sparsity, and for real signals with entries not equal to 0, the value $\|x\|_0=p$ is not a useful description …
Given a limited number of entries from the superposition of a low-rank matrix plus the product of a known fat compression matrix times a sparse matrix, recovery of the low-rank and sparse components is a fundamental task subsuming compressed sensing, matrix completion, and principal components pursuit. This paper develops algorithms for distributed sparsity-regularized rank minimization over networks, when the nuclear- and $\ell_1$-norm are used as surrogates to the rank and nonzero entry counts of the sought matrices, respectively. While nuclear-norm minimization has well-documented merits wh…
Motivated by distributed machine learning settings such as Federated Learning, we consider the problem of fitting a statistical model across a distributed collection of heterogeneous data sets whose similarity structure is encoded by a graph topology. Precisely, we analyse the case where each node is associated with fitting a sparse linear model, and edges join two nodes if the difference of their solutions is also sparse. We propose a method based on Basis Pursuit Denoising with a total variation penalty, and provide finite sample guarantees for sub-Gaussian design matrices. Taking the root o…
Efficiently representing real world data in a succinct and parsimonious manner is of central importance in many fields. We present a generalized greedy pursuit framework, allowing us to efficiently solve structured matrix factorization problems, where the factors are allowed to be from arbitrary sets of structured vectors. Such structure may include sparsity, non-negativeness, order, or a combination thereof. The algorithm approximates a given matrix by a linear combination of few rank-1 matrices, each factorized into an outer product of two vector atoms of the desired structure. For the non-c…
University of California Los Angeles（加利福尼亚大学洛杉矶分校）；University of Kentucky（肯塔基大学）
Traditional machine learning techniques require centralizing all training data on one server or data hub. Due to the development of communication technologies and a huge amount of decentralized data on many clients, collaborative machine learning has become the main interest while providing privacy-preserving frameworks. In particular, federated learning (FL) provides such a solution to learn a shared model while keeping training data at local clients. On the other hand, in a wide range of machine learning and signal processing applications, the desired solution naturally has a certain structu…
In this paper we present a linear programming solution for sign pattern recovery of a sparse signal from noisy random projections of the signal. We consider two types of noise models, input noise, where noise enters before the random projection; and output noise, where noise enters after the random projection. Sign pattern recovery involves the estimation of sign pattern of a sparse signal. Our idea is to pretend that no noise exists and solve the noiseless $\ell_1$ problem, namely, $\min \|β\|_1 ~ s.t. ~ y=G β$ and quantizing the resulting solution. We show that the quantized solution perfect…
University of Rouen（鲁昂大学）；LITIS Lab（LITIS实验室）；Aix-Marseille University（艾克斯-马赛大学）；LIF（LIF实验室）
Several sparsity-constrained algorithms such as Orthogonal Matching Pursuit or the Frank-Wolfe algorithm with sparsity constraints work by iteratively selecting a novel atom to add to the current non-zero set of variables. This selection step is usually performed by computing the gradient and then by looking for the gradient component with maximal absolute entry. This step can be computationally expensive especially for large-scale and high-dimensional data. In this work, we aim at accelerating these sparsity-constrained optimization algorithms by exploiting the key observation that, for these…
Samsung Research Institute（三星研究院）；Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
Simultaneous orthogonal matching pursuit (SOMP) and block OMP (BOMP) are two widely used techniques for sparse support recovery in multiple measurement vector (MMV) and block sparse (BS) models respectively. For optimal performance, both SOMP and BOMP require \textit{a priori} knowledge of signal sparsity or noise variance. However, sparsity and noise variance are unavailable in most practical applications. This letter presents a novel technique called generalized residual ratio thresholding (GRRT) for operating SOMP and BOMP without the \textit{a priori} knowledge of signal sparsity and noise…
Shenzhen University（深圳大学）；University of York（约克大学）
Space-time adaptive processing (STAP) algorithms with coprime arrays can provide good clutter suppression potential with low cost in airborne radar systems as compared with their uniform linear arrays counterparts. However, the performance of these algorithms is limited by the training samples support in practical applications. To address this issue, a robust two-stage reduced-dimension (RD) sparsity-aware STAP algorithm is proposed in this work. In the first stage, an RD virtual snapshot is constructed using all spatial channels but only $m$ adjacent Doppler channels around the target Doppler…
We propose a batchwise monotone algorithm for dictionary learning. Unlike the state-of-the-art dictionary learning algorithms which impose sparsity constraints on a sample-by-sample basis, we instead treat the samples as a batch, and impose the sparsity constraint on the whole. The benefit of batchwise optimization is that the non-zeros can be better allocated across the samples, leading to a better approximation of the whole. To accomplish this, we propose procedures to switch non-zeros in both rows and columns in the support of the coefficient matrix to reduce the reconstruction error. We pr…
Parsimony, including sparsity and low rank, has been shown to successfully model data in numerous machine learning and signal processing tasks. Traditionally, such modeling approaches rely on an iterative algorithm that minimizes an objective function with parsimony-promoting terms. The inherently sequential structure and data-dependent complexity and latency of iterative optimization constitute a major limitation in many applications requiring real-time performance or involving large-scale data. Another limitation encountered by these modeling techniques is the difficulty of their inclusion i…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Eindhoven University of Technology（埃因霍温理工大学）
Sparse Neural Networks (SNNs) have received voluminous attention predominantly due to growing computational and memory footprints of consistently exploding parameter count in large-scale models. Similar to their dense counterparts, recent SNNs generalize just as well and are equipped with numerous favorable benefits (e.g., low complexity, high scalability, and robustness), sometimes even better than the original dense networks. As research effort is focused on developing increasingly sophisticated sparse algorithms, it is startling that a comprehensive benchmark to evaluate the effectiveness o…
From Flat to Hierarchical: Extracting Sparse Representations with Matching Pursuit
EPFL（瑞士联邦理工学院）；Kempner Institute, Harvard University（哈佛大学凯普纳研究所）；CBS-NTT Program in Physics of Intelligence, Harvard University（哈佛大学物理智能CBSTNTT项目）；Physics of Artificial Intelligence Group, NTT Research, Inc., Sunnyvale, CA, USA（NTT研究公司人工智能物理研究组）；University of Alberta（阿尔伯塔大学）；Alberta Machine Intelligence Institute (Amii)（阿尔伯塔人工智能研究所）；SEAS, Harvard University（哈佛大学工程与应用科学学院）
