The recovery of sparse data is at the core of many applications in machine learning and signal processing. While such problems can be tackled using $\ell_1$-regularization as in the LASSO estimator and in the Basis Pursuit approach, specialized algorithms are typically required to solve the corresponding high-dimensional non-smooth optimization for large instances. Iteratively Reweighted Least Squares (IRLS) is a widely used algorithm for this purpose due its excellent numerical performance. However, while existing theory is able to guarantee convergence of this algorithm to the minimizer, it does not provide a global convergence rate. In this paper, we prove that a variant of IRLS converges with a global linear rate to a sparse solution, i.e., with a linear error decrease occurring immediately from any initialization, if the measurements fulfill the usual null space property assumption. We support our theory by numerical experiments showing that our linear rate captures the correct dimension dependence. We anticipate that our theoretical findings will lead to new insights for many other use cases of the IRLS algorithm, such as in low-rank matrix recovery.
Recovering Simultaneously Structured Data via Non-Convex Iteratively Reweighted Least Squares
University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；Ludwig-Maximilians-Universität München（慕尼黑大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）
Efficient Low-rank Identification via Accelerated Iteratively Reweighted Nuclear Norm Minimization
School of Information Science and Technology, ShanghaiTech University（上海科技大学信息科学与技术学院）；Center for Applied Mathematics of Henan Province, Henan University（河南大学河南省应用数学中心）
Alternating Iteratively Reweighted $\ell_1$ and Subspace Newton Algorithms for Nonconvex Sparse Optimization
School of Information Science and Technology, ShanghaiTech University（上海科技大学信息科学与技术学院）；School of Mathematics and Statistics, Henan University（河南大学数学与统计学院）；Center for Applied Mathematics of Henan Province, Henan University（河南大学河南省应用数学中心）
Avoiding strict saddle points of nonconvex regularized problems
School of Information Science and Technology, ShanghaiTech University（上海科技大学信息科学与技术学院）；School of Mathematical Sciences, Shenzhen University（深圳大学数学科学学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
In this paper, we focus on the local convergence rate analysis of the proximal iteratively reweighted $\ell_1$ algorithms for solving $\ell_p$ regularization problems, which are widely applied for inducing sparse solutions. We show that if the Kurdyka-Lojasiewicz (KL) property is satisfied, the algorithm converges to a unique first-order stationary point; furthermore, the algorithm has local linear convergence or local sublinear convergence. The theoretical results we derived are much stronger than the existing results for iteratively reweighted $\ell_1$ algorithms.
The iteratively reweighted l1 algorithm is a widely used method for solving various regularization problems, which generally minimize a differentiable loss function combined with a nonconvex regularizer to induce sparsity in the solution. However, the convergence and the complexity of iteratively reweighted l1 algorithms is generally difficult to analyze, especially for non-Lipschitz differentiable regularizers such as nonconvex lp norm regularization. In this paper, we propose, analyze and test a reweighted l1 algorithm combined with the extrapolation technique under the assumption of Kurdyka…
For solving a wide class of nonconvex and nonsmooth problems, we propose a proximal linearized iteratively reweighted least squares (PL-IRLS) algorithm. We first approximate the original problem by smoothing methods, and second write the approximated problem into an auxiliary problem by introducing new variables. PL-IRLS is then built on solving the auxiliary problem by utilizing the proximal linearization technique and the iteratively reweighted least squares (IRLS) method, and has remarkable computation advantages. We show that PL-IRLS can be extended to solve more general nonconvex and nons…
National University of Defense Technology（国防科技大学）；Hunan University（湖南大学）
In this paper, we consider a class of nonconvex problems with linear constraints appearing frequently in the area of image processing. We solve this problem by the penalty method and propose the iteratively reweighted alternating minimization algorithm. To speed up the algorithm, we also apply the continuation strategy to the penalty parameter. A convergence result is proved for the algorithm. Compared with the nonconvex ADMM, the proposed algorithm enjoys both theoretical and computational advantages like weaker convergence requirements and faster speed. Numerical results demonstrate the effi…
Iteratively reweighted $\ell_1$ algorithm is a popular algorithm for solving a large class of optimization problems whose objective is the sum of a Lipschitz differentiable loss function and a possibly nonconvex sparsity inducing regularizer. In this paper, motivated by the success of extrapolation techniques in accelerating first-order methods, we study how widely used extrapolation techniques such as those in [4,5,22,28] can be incorporated to possibly accelerate the iteratively reweighted $\ell_1$ algorithm. We consider three versions of such algorithms. For each version, we exhibit an expl…
We propose a new Iteratively Reweighted Least Squares (IRLS) algorithm for the problem of completing or denoising low-rank matrices that are structured, e.g., that possess a Hankel, Toeplitz or block-Hankel/Toeplitz structure. The algorithm optimizes an objective based on a non-convex surrogate of the rank by solving a sequence of quadratic problems. Our strategy combines computational efficiency, as it operates on a lower dimensional generator space of the structured matrices, with high statistical accuracy which can be observed in experiments on hard estimation and completion tasks. Our expe…
Iteratively Linearized Reweighted Alternating Direction Method of Multipliers for a Class of Nonconvex Problems
Department of Mathematics, National University of Defense Technology（国防科技大学数学系）；College of Computer, National University of Defense Technology（国防科技大学计算机学院）；The State Key Laboratory for High Performance Computation, National University of Defense Technology（国防科技大学高性能计算国家重点实验室）；Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University（湖南计算与模拟科学工程重点实验室，湘潭大学数学与计算科学学院）
IIT Kanpur（印度理工学院坎普尔分校）；Microsoft Research India（微软印度研究院）
We provide the first global model recovery results for the IRLS (iteratively reweighted least squares) heuristic for robust regression problems. IRLS is known to offer excellent performance, despite bad initializations and data corruption, for several parameter estimation problems. Existing analyses of IRLS frequently require careful initialization, thus offering only local convergence guarantees. We remedy this by proposing augmentations to the basic IRLS routine that not only offer guaranteed global recovery, but in practice also outperform state-of-the-art algorithms for robust regression. …
The affine rank minimization (ARM) problem is well known for both its applications and the fact that it is NP-hard. One of the most successful approaches, yet arguably underrepresented, is iteratively reweighted least squares (IRLS), more specifically $\mathrm{IRLS}$-$0$. Despite comprehensive empirical evidence that it overall outperforms nuclear norm minimization and related methods, it is still not understood to a satisfying degree. In particular, the significance of a slow decrease of the therein appearing regularization parameter denoted $γ$ poses interesting questions. While commonly equ…
We propose a new algorithmic framework for constrained compressed sensing models that admit nonconvex sparsity-inducing regularizers including the log-penalty function as objectives, and nonconvex loss functions such as the Cauchy loss function and the Tukey biweight loss function in the constraint. Our framework employs iteratively reweighted $\ell_1$ and $\ell_2$ schemes to construct subproblems that can be efficiently solved by well-developed solvers for basis pursuit denoising such as SPGL1 [6]. We propose a new termination criterion for the subproblem solvers that allows them to return an…
We propose a method, called ACQUIRE, for the solution of constrained optimization problems modeling the restoration of images corrupted by Poisson noise. The objective function is the sum of a generalized Kullback-Leibler divergence term and a TV regularizer, subject to nonnegativity and possibly other constraints, such as flux conservation. ACQUIRE is a line-search method that considers a smoothed version of TV, based on a Huber-like function, and computes the search directions by minimizing quadratic approximations of the problem, built by exploiting some second-order information. A classica…
Johns Hopkins University（约翰斯·霍普金斯大学）；Technical University of Munich（慕尼黑工业大学）
We propose an iterative algorithm for low-rank matrix completion that can be interpreted as both an iteratively reweighted least squares (IRLS) algorithm and a saddle-escaping smoothing Newton method applied to a non-convex rank surrogate objective. It combines the favorable data efficiency of previous IRLS approaches with an improved scalability by several orders of magnitude. Our method attains a local quadratic convergence rate already for a number of samples that is close to the information theoretical limit. We show in numerical experiments that unlike many state-of-the-art approaches, ou…
ShanghaiTech University（上海科技大学）；University of Washington（华盛顿大学）；Beihang University（北京航空航天大学）
This paper investigates the optimality conditions for characterizing the local minimizers of the constrained optimization problems involving an $\ell_p$ norm ($0<p<1$) of the variables, which may appear in either the objective or the constraint. This kind of problems have strong applicability to a wide range of areas since usually the $\ell_p$ norm can promote sparse solutions. However, the nonsmooth and non-Lipschtiz nature of the $\ell_p$ norm often cause these problems difficult to analyze and solve. We provide the calculation of the subgradients of the $\ell_p$ norm and the normal cones of…
ShanghaiTech University（上海科技大学）；Shenzhen University（深圳大学）；University of Chinese Academy of Sciences（中国科学院大学）；Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences（中国科学院上海微系统与信息技术研究所）
We propose a general formulation of nonconvex and nonsmooth sparse optimization problems with convex set constraint, which can take into account most existing types of nonconvex sparsity-inducing terms, bringing strong applicability to a wide range of applications. We design a general algorithmic framework of iteratively reweighted algorithms for solving the proposed nonconvex and nonsmooth sparse optimization problems, which solves a sequence of weighted convex regularization problems with adaptively updated weights. First-order optimality condition is derived and global convergence results a…
Sparse learning is an important topic in many areas such as machine learning, statistical estimation, signal processing, etc. Recently, there emerges a growing interest on structured sparse learning. In this paper we focus on the $\ell_q$-analysis optimization problem for structured sparse learning ($0< q \leq 1$). Compared to previous work, we establish weaker conditions for exact recovery in noiseless case and a tighter non-asymptotic upper bound of estimate error in noisy case. We further prove that the nonconvex $\ell_q$-analysis optimization can do recovery with a lower sample complexity …
ShanghaiTech University（上海科技大学）；University of Washington（华盛顿大学）
We propose a general framework of iteratively reweighted l1 methods for solving lp regularization problems. We prove that after some iteration k, the iterates generated by the proposed methods have the same support and sign as the limit points, and are bounded away from 0, so that the algorithm behaves like solving a smooth problem in the reduced space. As a result, the global convergence can be easily obtained and an update strategy for the smoothing parameter is proposed which can automatically terminate the updates for zero components. We show that lp regularization problems are locally equ…
The log-sum penalty is often adopted as a replacement for the $\ell_0$ pseudo-norm in compressive sensing and low-rank optimization. The hard-thresholding operator, i.e., the proximity operator of the $\ell_0$ penalty, plays an essential role in applications; similarly, we require an efficient method for evaluating the proximity operator of the log-sum penalty. Due to the nonconvexity of this function, its proximity operator is commonly computed through the iteratively reweighted $\ell_1$ method, which replaces the log-sum term with its first-order approximation. This paper reports that the pr…
We consider unconstrained randomized optimization of convex objective functions. We analyze the Random Pursuit algorithm, which iteratively computes an approximate solution to the optimization problem by repeated optimization over a randomly chosen one-dimensional subspace. This randomized method only uses zeroth-order information about the objective function and does not need any problem-specific parametrization. We prove convergence and give convergence rates for smooth objectives assuming that the one-dimensional optimization can be solved exactly or approximately by an oracle. A convenient…
We recover jump-sparse and sparse signals from blurred incomplete data corrupted by (possibly non-Gaussian) noise using inverse Potts energy functionals. We obtain analytical results (existence of minimizers, complexity) on inverse Potts functionals and provide relations to sparsity problems. We then propose a new optimization method for these functionals which is based on dynamic programming and the alternating direction method of multipliers (ADMM). A series of experiments shows that the proposed method yields very satisfactory jump-sparse and sparse reconstructions, respectively. We highlig…
We consider the problem of recovering off-the-grid spikes from Fourier measurements. Successful methods such as sliding Frank-Wolfe and continuous orthogonal matching pursuit (OMP) iteratively add spikes to the solution then perform a costly (when the number of spikes is large) descent on all parameters at each iteration. In 2D, it was shown that performing a projected gradient descent (PGD) from a gridded over-parametrized initialization was faster than continuous orthogonal matching pursuit. In this paper, we propose an off-the-grid over-parametrized initialization of the PGD based on OMP th…
Compressed sensing is a central topic in signal processing with myriad applications, where the goal is to recover a signal from as few observations as possible. Iterative re-weighting is one of the fundamental tools to achieve this goal. This paper re-examines the iteratively reweighted least squares (IRLS) algorithm for sparse recovery proposed by Daubechies, Devore, Fornasier, and Güntürk in \emph{Iteratively reweighted least squares minimization for sparse recovery}, {\sf Communications on Pure and Applied Mathematics}, {\bf 63}(2010) 1--38. Under the null space property of order $K$, the a…
We introduce a convex approach for mixed linear regression over $d$ features. This approach is a second-order cone program, based on L1 minimization, which assigns an estimate regression coefficient in $\mathbb{R}^{d}$ for each data point. These estimates can then be clustered using, for example, $k$-means. For problems with two or more mixture classes, we prove that the convex program exactly recovers all of the mixture components in the noiseless setting under technical conditions that include a well-separation assumption on the data. Under these assumptions, recovery is possible if each cla…
We propose an iterative algorithm for low-rank matrix completion that can be interpreted as an iteratively reweighted least squares (IRLS) algorithm, a saddle-escaping smoothing Newton method or a variable metric proximal gradient method applied to a non-convex rank surrogate. It combines the favorable data-efficiency of previous IRLS approaches with an improved scalability by several orders of magnitude. We establish the first local convergence guarantee from a minimal number of samples for that class of algorithms, showing that the method attains a local quadratic convergence rate. Furthermo…
This note uses the Total Least-Squares (TLS) line-fitting problem as a canvas to explore some modern optimization tools. The contribution is meant to be tutorial in nature. The TLS problem has a lot of mathematical similarities to important problems in robotics and computer vision but is easier to visualize and understand. We demonstrate how to turn this problem into a Quadratically Constrained Quadratic Program (QCQP) so that it can be cast either as an eigenproblem or a Semi-Definite Program (SDP). We then turn to the more challenging situation where a Geman-McClure cost function and M-estim…
We advance both the theory and practice of robust $\ell_p$-quasinorm regression for $p \in (0,1]$ by using novel variants of iteratively reweighted least-squares (IRLS) to solve the underlying non-smooth problem. In the convex case, $p=1$, we prove that this IRLS variant converges globally at a linear rate under a mild, deterministic condition on the feature matrix called the \textit{stable range space property}. In the non-convex case, $p\in(0,1)$, we prove that under a similar condition, IRLS converges locally to the global minimizer at a superlinear rate of order $2-p$; the rate becomes qua…
The problem of the distributed recovery of jointly sparse signals has attracted much attention recently. Let us assume that the nodes of a network observe different sparse signals with common support; starting from linear, compressed measurements, and exploiting network communication, each node aims at reconstructing the support and the non-zero values of its observed signal. In the literature, distributed greedy algorithms have been proposed to tackle this problem, among which the most reliable ones require a large amount of transmitted data, which barely adapts to realistic network communica…
Sparsity is one of the key concepts that allows the recovery of signals that are subsampled at a rate significantly lower than required by the Nyquist-Shannon sampling theorem. Our proposed framework uses arbitrary multiscale transforms, such as those build upon wavelets or shearlets, as a sparsity promoting prior which allow to decompose the image into different scales such that image features can be optimally extracted. In order to further exploit the sparsity of the recovered signal we combine the method of reweighted $\ell^1$, introduced by Candès et al., with iteratively updated weights a…
