This paper proposes the Proximal Iteratively REweighted (PIRE) algorithm for solving a general problem, which involves a large body of nonconvex sparse and structured sparse related problems. Comparing with previous iterative solvers for nonconvex sparse problem, PIRE is much more general and efficient. The computational cost of PIRE in each iteration is usually as low as the state-of-the-art convex solvers. We further propose the PIRE algorithm with Parallel Splitting (PIRE-PS) and PIRE algorithm with Alternative Updating (PIRE-AU) to handle the multi-variable problems. In theory, we prove that our proposed methods converge and any limit solution is a stationary point. Extensive experiments on both synthesis and real data sets demonstrate that our methods achieve comparative learning performance, but are much more efficient, by comparing with previous nonconvex solvers.
Another related work is @cite_16 which aims to solve In each iteration, @math is efficiently updated by solving a series of problem But their solver is only for problem ) which is a special case of ). The convergence proofs also depend on the special property of the @math -norm, thus is not general.
@cite_16
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Many conjugate gradient-like methods for solving linear systems @math use recursion formulas for updating residual vectors instead of computing the residuals directly. For such methods it is shown that the difference between the actual residuals and the updated approximate residual vectors generated in finite precision arithmetic depends on the machine precision @math and on the maximum norm of an iterate divided by the norm of the true solution. It is often observed numerically, and can sometimes be proved, that the norms of the updated approximate residual vectors converge to zero or, at least, become orders of magnitude smaller than the machine precision. In such cases, the actual residual norm reaches the level @math times the maximum ratio of the norm of an iterate to that of the true solution. Using exact arithmetic theory to bound the size of the iterates, we give a priori estimates of the size of the final residual for a number of algorithms. We propose an approximate Newton method for solving the coupled nonlinear system @math and @math where @math , @math , @math and @math . The method involves applying the basic iteration S of a general solver for the equation @math , withtfixed. It is therefore well suited for problems for which such a solver already exists or can be implemented more efficiently than a solver for the coupled system. We derive conditions for S under which the method is locally convergent. Basically, if S is sufficiently contractive for G, then convergence for the coupled system is guaranteed. Otherwise, we show how to construct an @math from S for which convergence is assured. These results are applied to continuation methods where N represents a pseudo-arclength condition. We show that under certain conditions the algorithm converges if S is convergent for G. Numerical results are given for a two-level nonlinear multi-grid solver applied ... We propose simple and extremely efficient methods for solving the basis pursuit problem @math which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of instances of the unconstrained problem @math for given matrix @math and vector @math . We show analytically that this iterative approach yields exact solutions in a finite number of steps and present numerical results that demonstrate that as few as two to six iterations are sufficient in most cases. Our approach is especially useful for many compressed sensing applications where matrix-vector operations involving @math and @math can be computed by fast transforms. Utilizing a fast fixed-point continuation solver that is based solely on such operations for solving the above unconstrained subproblem, we were able to quickly solve huge instances of compressed sensing problems on a standard PC. For numerical computation of the minimal Euclidean norm (least-squares) solution of overdetermined linear systems, usually direct solvers are used (like QR decomposition, see [4]). The iterative methods for such kind of problems need special assumptions about the system (consistency, full rank of the system matrix, some parameters they use or they give not the minimal length solution, [2,3,5,8,10,13]). In the present paper we purpose two iterative algorithms which generate sequences convergent to the minimal Euclidean length solution in the general case (inconsistent system and rank deficient matrix). The algorithms use only some combinations and properties of the well-known Kaczmarz iterative method ([13]) and need no special assumptions about the system. This paper studies the Tensor Robust Principal Component (TRPCA) problem which extends the known Robust PCA ( 2011) to the tensor case. Our model is based on a new tensor Singular Value Decomposition (t-SVD) (Kilmer and Martin 2011) and its induced tensor tubal rank and tensor nuclear norm. Consider that we have a 3-way tensor @math such that @math , where @math has low tubal rank and @math is sparse. Is that possible to recover both components? In this work, we prove that under certain suitable assumptions, we can recover both the low-rank and the sparse components exactly by simply solving a convex program whose objective is a weighted combination of the tensor nuclear norm and the @math -norm, i.e., @math , where @math . Interestingly, TRPCA involves RPCA as a special case when @math and thus it is a simple and elegant tensor extension of RPCA. Also numerical experiments verify our theory and the application for the image denoising demonstrates the effectiveness of our method. In this paper, we consider conic programming problems whose constraints consist of linear equalities, linear inequalities, a nonpolyhedral cone, and a polyhedral cone. A convenient way for solving this class of problems is to apply the directly extended alternating direction method of multipliers (ADMM) to its dual problem, which has been observed to perform well in numerical computations but may diverge in theory. Ideally, one should find a convergent variant which is at least as efficient as the directly extended ADMM in practice. We achieve this goal by designing a convergent semiproximal ADMM (called sPADMM3c for convenience) for convex programming problems having three separable blocks in the objective function with the third part being linear. At each iteration, the proposed sPADMM3c takes one special block coordinate descent (BCD) cycle with the order @math , instead of the usual @math Gauss--Seidel BCD cycle used in the nonconvergent directly extended 3-block ADMM, for updating the variable blocks. Our numerical experiments demonstrate that the convergent method is at least 20 faster than the directly extended ADMM with unit step-length for the vast majority of about 550 large-scale doubly nonnegative semidefinite programming problems with linear equality and or inequality constraints. This confirms that at least for conic convex programming, one can design a convergent and efficient ADMM with a special BCD cycle of updating the variable blocks. The nuclear norm is widely used as a convex surrogate of the rank function in compressive sensing for low rank matrix recovery with its applications in image recovery and signal processing. However, solving the nuclear norm-based relaxed convex problem usually leads to a suboptimal solution of the original rank minimization problem. In this paper, we propose to use a family of nonconvex surrogates of @math -norm on the singular values of a matrix to approximate the rank function. This leads to a nonconvex nonsmooth minimization problem. Then, we propose to solve the problem by an iteratively reweighted nuclear norm (IRNN) algorithm. IRNN iteratively solves a weighted singular value thresholding problem, which has a closed form solution due to the special properties of the nonconvex surrogate functions. We also extend IRNN to solve the nonconvex problem with two or more blocks of variables. In theory, we prove that the IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthesized data and real images demonstrate that IRNN enhances the low rank matrix recovery compared with the state-of-the-art convex algorithms. The minimax problem is an unconstrained optimization problem whose objective function is not differentiable everywhere, and hence cannot be solved efficiently by standard techniques for unconstrained optimization. It is well known that the problem can be transformed into a nonlinearly constrained optimization problem with one extra variable, where the objective and constraint functions are continuously differentiable. This equivalent problem has special properties which are ignored if solved by a general-purpose constrained optimization method. The algorithm we present exploits the special structure of the equivalent problem. A direction of search is obtained at each iteration of the algorithm by solving an equality-constrained quadratic programming problem, related to one a projected Lagrangian method might use to solve the equivalent constrained optimization problem. Special Lagrange multiplier estimates are used to form an approximation to the Hessian of the Lagrangian function, which appears in the quadratic program. Analytical Hessians, finite differencing or quasi-Newton updating may be used in the approximation of this matrix. The resulting direction of search is guaranteed to be a descent direction for the minimax objective function. Under mild conditions the algorithms are locally quadratically convergent if analytical Hessians are used. We propose a trust region algorithm for solving nonconvex smooth optimization problems. For any @math ∈¯?(0,?), the algorithm requires at most @math O(∈-3 2) iterations, function evaluations, and derivative evaluations to drive the norm of the gradient of the objective function below any @math ∈?(0,∈¯]. This improves upon the @math O(∈-2) bound known to hold for some other trust region algorithms and matches the @math O(∈-3 2) bound for the recently proposed Adaptive Regularisation framework using Cubics, also known as the arc algorithm. Our algorithm, entitled trace, follows a trust region framework, but employs modified step acceptance criteria and a novel trust region update mechanism that allow the algorithm to achieve such a worst-case global complexity bound. Importantly, we prove that our algorithm also attains global and fast local convergence guarantees under similar assumptions as for other trust region algorithms. We also prove a worst-case upper bound on the number of iterations, function evaluations, and derivative evaluations that the algorithm requires to obtain an approximate second-order stationary point. Given @math and @math we consider a family of functionals, the @math -frame potentials FP @math , defined on the set of all collections of @math unit-norm vectors in @math . For the special case @math and @math , both the minima and the minimizers of these potentials have been thoroughly investigated. In this paper, we investigate the minimizers of the functionals FP @math , by first establishing some general properties of their minima. Thereafter, we focus on the special case @math , for which, surprisingly, not much is known. One of our main results establishes the unique minimizer for big enough @math . Moreover, this minimizer is universal in the sense that it minimizes a large range of energy functions that includes the @math -frame potential. We conclude the paper by reporting some numerical experiments for the case @math , @math , @math . These experiments lead to some conjectures that we pose. Fitting a function by using linear combinations of a large number @math of simple' components is one of the most fruitful ideas in statistical learning. This idea lies at the core of a variety of methods, from two-layer neural networks to kernel regression, to boosting. In general, the resulting risk minimization problem is non-convex and is solved by gradient descent or its variants. Unfortunately, little is known about global convergence properties of these approaches. Here we consider the problem of learning a concave function @math on a compact convex domain @math , using linear combinations of bump-like' components (neurons). The parameters to be fitted are the centers of @math bumps, and the resulting empirical risk minimization problem is highly non-convex. We prove that, in the limit in which the number of neurons diverges, the evolution of gradient descent converges to a Wasserstein gradient flow in the space of probability distributions over @math . Further, when the bump width @math tends to @math , this gradient flow has a limit which is a viscous porous medium equation. Remarkably, the cost function optimized by this gradient flow exhibits a special property known as displacement convexity, which implies exponential convergence rates for @math , @math . Surprisingly, this asymptotic theory appears to capture well the behavior for moderate values of @math . Explaining this phenomenon, and understanding the dependence on @math in a quantitative manner remains an outstanding challenge. Convex optimization is an essential tool for modern data analysis, as it provides a framework to formulate and solve many problems in machine learning and data mining. However, general convex optimization solvers do not scale well, and scalable solvers are often specialized to only work on a narrow class of problems. Therefore, there is a need for simple, scalable algorithms that can solve many common optimization problems. In this paper, we introduce the network lasso, a generalization of the group lasso to a network setting that allows for simultaneous clustering and optimization on graphs. We develop an algorithm based on the Alternating Direction Method of Multipliers (ADMM) to solve this problem in a distributed and scalable manner, which allows for guaranteed global convergence even on large graphs. We also examine a non-convex extension of this approach. We then demonstrate that many types of problems can be expressed in our framework. We focus on three in particular --- binary classification, predicting housing prices, and event detection in time series data --- comparing the network lasso to baseline approaches and showing that it is both a fast and accurate method of solving large optimization problems. We develop several efficient algorithms for the classical problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input @math matrix @math , this problem asks to find diagonal (scaling) matrices @math and @math (if they exist), so that @math @math -approximates a doubly stochastic, or more generally a matrix with prescribed row and column sums. We address the general scaling problem as well as some important special cases. In particular, if @math has @math nonzero entries, and if there exist @math and @math with polynomially large entries such that @math is doubly stochastic, then we can solve the problem in total complexity @math . This greatly improves on the best known previous results, which were either @math or @math . Our algorithms are based on tailor-made first and second order techniques, combined with other recent advances in continuous optimization, which may be of independent interest for solving similar problems. Convex and submodular functions play an important role in many applications, and in particular in combinatorial optimization. Here we study two special cases: convexity in one dimension and submodularity in two dimensions. The latter type of functions are equivalent to the well-known Monge matrices. A matrix @math is called a Monge matrix if for every @math and @math we have @math . If inequality holds in the opposite direction, then V is an inverse Monge matrix (supermodular function). Many problems, such as the traveling salesperson problem and various transportation problems, can be solved more efficiently if the input is a Monge matrix. In this work we present testing algorithms for the above properties. A testing algorithm for a predetermined property @math is given query access to an unknown function f and a distance parameter @math . The algorithm should accept f with high probability if it has the property @math and reject it with high probability if more than an @math -fraction of the function values should be modified so that f obtains the property. Our algorithm for testing whether a 1-dimensional function @math is convex (concave) has query complexity and running time of @math . Our algorithm for testing whether an n1 × n2 matrix V is a Monge (inverse Monge) matrix has query complexity and running time of @math . We present a method for solving the shortest transshipmen problem - also known as uncapacitated minimum cost flow - up to a multiplicative error of @math in undirected graphs with polynomially bounded integer edge weights using a tailored gradient descent algorithm. An important special case of the transshipment problem is the single-source shortest paths (SSSP) problem. Our gradient descent algorithm takes @math iterations, and in each iteration it needs to solve the transshipment problem up to a multiplicative error of @math , where @math is the number of nodes. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. As a consequence, we improve prior work by obtaining the following results: (1) Broadcast congest model: @math -approximate SSSP using @math rounds, where @math is the (hop) diameter of the network. (2) Broadcast congested clique model: @math -approximate transshipment and SSSP using @math rounds. (3) Multipass streaming model: @math -approximate transshipment and SSSP using @math space and @math passes. The previously fastest algorithms for these models leverage sparse hop sets. We bypass the hop set construction; computing a spanner is sufficient with our method. The above bounds assume non-negative integer edge weights that are polynomially bounded in @math ; for general non-negative weights, running times scale with the logarithm of the maximum ratio between non-zero weights. In case of asymmetric costs, running times scale with the maximum ratio between the costs of both directions over all edges. We study the performance of a family of randomized parallel coordinate descent methods for minimizing the sum of a nonsmooth and separable convex functions. The problem class includes as a special case L1-regularized L1 regression and the minimization of the exponential loss ("AdaBoost problem"). We assume the input data defining the loss function is contained in a sparse @math matrix @math with at most @math nonzeros in each row. Our methods need @math iterations to find an approximate solution with high probability, where @math is the number of processors and @math for the fastest variant. The notation hides dependence on quantities such as the required accuracy and confidence levels and the distance of the starting iterate from an optimal point. Since @math is a decreasing function of @math , the method needs fewer iterations when more processors are used. Certain variants of our algorithms perform on average only @math arithmetic operations during a single iteration per processor and, because @math decreases when @math does, fewer iterations are needed for sparser problems. We study inherent structural properties of a strongly NP-hard problem of scheduling @math jobs with release times and due dates on a single machine to minimize the number of late jobs. Our study leads to two polynomial-time algorithms. The first algorithm with the time complexity @math solves the problem if during its execution no job with some special property occurs. The second algorithm solves the version of the problem when all jobs have the same length. The time complexity of the latter algorithm is @math , which is an improvement over the earlier known algorithm with the time complexity @math . This paper introduces a parallel and distributed algorithm for solving the following minimization problem with linear constraints: @math minimizef1(x1)+ź+fN(xN)subject toA1x1+ź+ANxN=c,x1źX1,ź,xNźXN,where @math Nź2, @math fi are convex functions, @math Ai are matrices, and @math Xi are feasible sets for variable @math xi. Our algorithm extends the alternating direction method of multipliers (ADMM) and decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This paper shows that the classic ADMM can be extended to the N-block Jacobi fashion and preserve convergence in the following two cases: (i) matrices @math Ai are mutually near-orthogonal and have full column-rank, or (ii) proximal terms are added to the N subproblems (but without any assumption on matrices @math Ai). In the latter case, certain proximal terms can let the subproblem be solved in more flexible and efficient ways. We show that @math źxk+1-xkźM2 converges at a rate of o(1 k) where M is a symmetric positive semi-definte matrix. Since the parameters used in the convergence analysis are conservative, we introduce a strategy for automatically tuning the parameters to substantially accelerate our algorithm in practice. We implemented our algorithm (for the case ii above) on Amazon EC2 and tested it on basis pursuit problems with >300 GB of distributed data. This is the first time that successfully solving a compressive sensing problem of such a large scale is reported. We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with @math constraint matrices, each of dimension @math , rank @math , and sparsity @math . The first algorithm assumes an input model where one is given access to entries of the matrices at unit cost. We show that it has run time @math , where @math is the error. This gives an optimal dependence in terms of @math and quadratic improvement over previous quantum algorithms when @math . The second algorithm assumes a fully quantum input model in which the matrices are given as quantum states. We show that its run time is @math , with @math an upper bound on the trace-norm of all input matrices. In particular the complexity depends only poly-logarithmically in @math and polynomially in @math . We apply the second SDP solver to the problem of learning a good description of a quantum state with respect to a set of measurements: Given @math measurements and copies of an unknown state @math , we show we can find in time @math a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as @math on the @math measurements, up to error @math . The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians with a poly-logarithmic dependence on its dimension, which could be of independent interest. We propose a new randomized coordinate descent method for minimizing the sum of convex functions each of which depends on a small number of coordinates only. Our method (APPROX) is simultaneously Accelerated, Parallel, and PROXimal; this is the first time such a method is proposed. In the special case when the number of processors is equal to the number of coordinates, the method converges at the rate @math , where @math is the iteration counter, @math is a data-weighted average degree of separability of the loss function, @math is the average of Lipschitz constants associated with the coordinates and individual functions in the sum, and @math is the distance of the initial point from the minimizer. We show that the method can be implemented without the need to perform full-dimensional vector operations, which is the major bottleneck of accelerated coordinate descent. The fact that the method depends on the average degree of separability, and not on the maximum degree...