The symmetric Nonnegative Matrix Factorization (NMF), a special but important class of the general NMF, has found numerous applications in data analysis such as various clustering tasks. Unfortunately, designing fast algorithms for the symmetric NMF is not as easy as for its nonsymmetric counterpart, since the latter admits the splitting property that allows state-of-the-art alternating-type algorithms. To overcome this issue, we first split the decision variable and transform the symmetric NMF to a penalized nonsymmetric one, paving the way for designing efficient alternating-type algorithms. We then show that solving the penalized nonsymmetric reformulation returns a solution to the original symmetric NMF. Moreover, we design a family of alternating-type algorithms and show that they all admit strong convergence guarantee: the generated sequence of iterates is convergent and converges at least sublinearly to a critical point of the original symmetric NMF. Finally, we conduct experiments on both synthetic data and real image clustering to support our theoretical results and demonstrate the performance of the alternating-type algorithms.

A promising trend in deep learning replaces traditional feedforward networks with implicit networks. Unlike traditional networks, implicit networks solve a fixed point equation to compute inferences. Solving for the fixed point varies in complexity, depending on provided data and an error tolerance. Importantly, implicit networks may be trained with fixed memory costs in stark contrast to feedforward networks, whose memory requirements scale linearly with depth. However, there is no free lunch --- backpropagation through implicit networks often requires solving a costly Jacobian-based equation arising from the implicit function theorem. We propose Jacobian-Free Backpropagation (JFB), a fixed-memory approach that circumvents the need to solve Jacobian-based equations. JFB makes implicit networks faster to train and significantly easier to implement, without sacrificing test accuracy. Our experiments show implicit networks trained with JFB are competitive with feedforward networks and prior implicit networks given the same number of parameters.

Tensor optimization is crucial to massive machine learning and signal processing tasks. In this paper, we consider tensor optimization with a convex and well-conditioned objective function and reformulate it into a nonconvex optimization using the Burer-Monteiro type parameterization. We analyze the local convergence of applying vanilla gradient descent to the factored formulation and establish a local regularity condition under mild assumptions. We also provide a linear convergence analysis of the gradient descent algorithm started in a neighborhood of the true tensor factors. Complementary to the local analysis, this work also characterizes the global geometry of the best rank-one tensor approximation problem and demonstrates that for orthogonally decomposable tensors the problem has no spurious local minima and all saddle points are strict except for the one at zero which is a third-order saddle point.

This paper considers general rank-constrained optimization problems that minimize a general objective function${f}( {X})$over the set of rectangular${n}\times {m}$matrices that have rank at most r. To tackle the rank constraint and also to reduce the computational burden, we factorize$ {X}$into$ {U} {V} ^{\mathrm {T}}$where$ {U}$and$ {V}$are${n}\times {r}$and${m}\times {r}$matrices, respectively, and then optimize over the small matrices$ {U}$and$ {V}$. We characterize the global optimization geometry of the nonconvex factored problem and show that the corresponding objective function satisfies the robust strict saddle property as long as the original objective function f satisfies restricted strong convexity and smoothness properties, ensuring global convergence of many local search algorithms (such as noisy gradient descent) in polynomial time for solving the factored problem. We also provide a comprehensive analysis for the optimization geometry of a matrix factorization problem where we aim to find${n}\times {r}$and${m}\times {r}$matrices$ {U}$and$ {V}$such that$ {U} {V} ^{\mathrm {T}}$approximates a given matrix$ {X}^\star $. Aside from the robust strict saddle property, we show that the objective function of the matrix factorization problem has no spurious local minima and obeys the strict saddle property not only for the exact-parameterization case where$\mathrm {rank}( {X}^\star) = {r}$, but also for the over-parameterization case where$\mathrm {rank}( {X}^\star) < {r}$and the under-parameterization case where$\mathrm {rank}( {X}^\star) > {r}$. These geometric properties imply that a number of iterative optimization algorithms (such as gradient descent) converge to a global solution with random initialization.

Low-rank matrix recovery is a fundamental problem in signal processing and machine learning. A recent very popular approach to recovering a low-rank matrix X is to factorize it as a product of two smaller matrices, i.e., X = UVT, and then optimize over U, V instead of X. Despite the resulting non-convexity, recent results have shown that many factorized objective functions actually have benign global geometry-with no spurious local minima and satisfying the so-called strict saddle property-ensuring convergence to a global minimum for many local-search algorithms. Such results hold whenever the original objective function is restricted strongly convex and smooth. However, most of these results actually consider a modified cost function that includes a balancing regularizer. While useful for deriving theory, this balancing regularizer does not appear to be necessary in practice. In this work, we close this theory-practice gap by proving that the unaltered factorized non-convex problem, without the balancing regularizer, also has similar benign global geometry. Moreover, we also extend our theoretical results to the field of distributed optimization.

A variety of unconstrained nonconvex optimization problems have been shown to have benign geometric landscapes that satisfy the strict saddle property and have no spurious local minima. We present a general result relating the geometry of an unconstrained centralized problem to its equality-constrained distributed extension. It follows that many global consensus problems inherit the benign geometry of their original centralized counterpart. Taking advantage of this fact, we demonstrate the favorable performance of the Gradient ADMM algorithm on a distributed low-rank matrix approximation problem.

This work studies the second-order convergence for both standard alternating minimization and proximal alternating minimization. We show that under mild assumptions on the (nonconvex) objective function, both algorithms avoid strict saddles almost surely from random initialization. Together with known first-order convergence results, this implies both algorithms converge to a second-order stationary point. This solves an open problem for the second-order convergence of alternating minimization algorithms that have been widely used in practice to solve large-scale nonconvex problems due to their simple implementation, fast convergence, and superb empirical performance.

Low-rank matrix factorization is a problem of broad importance, owing to the ubiquity of low-rank models in machine learning contexts. In spite of its non- convexity, this problem has a well-behaved geometric landscape, permitting local search algorithms such as gradient descent to converge to global minimizers. In this paper, we study low-rank matrix factorization in the distributed setting, where local variables at each node encode parts of the overall matrix factors, and consensus is encouraged among certain such variables. We identify conditions under which this new problem also has a well-behaved geometric landscape, and we propose an extension of distributed gradient descent (DGD) to solve this problem. The favorable landscape allows us to prove convergence to global optimality with exact consensus, a stronger result than what is provided by off-the-shelf DGD theory.

Principal Component Analysis (PCA) is one of the most broadly used methods to analyze high-dimensional data. However, most existing studies on PCA aim to minimize the reconstruction error measured by the Euclidean distance, although in some fields, such as text analysis in information retrieval, analysis using the angle distance is known to be more effective. In this paper, we propose a novel PCA formulation by adding a constraint on the factors to unify the Euclidean distance and the angle distance. Because the objective and constraints are nonconvex, the optimization problem is difficult to solve in general. To tackle the optimization problem, we propose an alternating linearized minimization method with guaranteed convergence and provable convergence rate. Experiments on synthetic data and real-world data sets have validated the effectiveness of our new method and demonstrated its advantages over state-of-art competing methods.

BACKGROUND: Gene selection is one of the critical steps in the course of the classification of microarray data. Since particle swarm optimization has no complicated evolutionary operators and fewer parameters need to be adjusted, it has been used increasingly as an effective technique for gene selection. Since particle swarm optimization is apt to converge to local minima which lead to premature convergence, some particle swarm optimization based gene selection methods may select non-optimal genes with high probability. To select predictive genes with low redundancy as well as not filtering out key genes is still a challenge. RESULTS: To obtain predictive genes with lower redundancy as well as overcome the deficiencies of traditional particle swarm optimization based gene selection methods, a hybrid gene selection method based on gene scoring strategy and improved particle swarm optimization is proposed in this paper. To select the genes highly related to out samples' classes, a gene scoring strategy based on randomization and extreme learning machine is proposed to filter much irrelevant genes. With the third-level gene pool established by multiple filter strategy, an improved particle swarm optimization is proposed to perform gene selection. In the improved particle swarm optimization, to decrease the likelihood of the premature of the swarm the Metropolis criterion of simulated annealing algorithm is introduced to update the particles, and the half of the swarm are reinitialized when the swarm is trapped into local minima. CONCLUSIONS: Combining the gene scoring strategy with the improved particle swarm optimization, the new method could select functional gene subsets which are significantly sensitive to the samples' classes. With the few discriminative genes selected by the proposed method, extreme learning machine and support vector machine classifiers achieve much high prediction accuracy on several public microarray data, which in turn verifies the efficiency and effectiveness of the proposed gene selection method.

Symmetric nonnegative matrix factorization (NMF)---a special but important class of the general NMF---is demonstrated to be useful for data analysis and in particular for various clustering tasks. Unfortunately, designing fast algorithms for Symmetric NMF is not as easy as for the nonsymmetric counterpart, the latter admitting the splitting property that allows efficient alternating-type algorithms. To overcome this issue, we transfer the symmetric NMF to a nonsymmetric one, then we can adopt the idea from the state-of-the-art algorithms for nonsymmetric NMF to design fast algorithms solving symmetric NMF. We rigorously establish that solving nonsymmetric reformulation returns a solution for symmetric NMF and then apply fast alternating based algorithms for the corresponding reformulated problem. Furthermore, we show these fast algorithms admit strong convergence guarantee in the sense that the generated sequence is convergent at least at a sublinear rate and it converges globally to a critical point of the symmetric NMF. We conduct experiments on both synthetic data and image clustering to support our result.

Based on the maximum likelihood estimation principle, we derive a collaborative estimation framework that fuses several different estimators and yields a better estimate. Applying it to compressive sensing (CS), we propose a collaborative CS (CCS) scheme consisting of a bank of $K$ CS systems that share the same sensing matrix but have different sparsifying dictionaries. This CCS system is expected to yield better performance than each individual CS system, while requiring the same time as that needed for each individual CS system when a parallel computing strategy is used. We then provide an approach to designing optimal CCS systems by utilizing a measure that involves both the sensing matrix and dictionaries and hence allows us to simultaneously optimize the sensing matrix and all the $K$ dictionaries. An alternating minimization-based algorithm is derived for solving the corresponding optimal design problem. With a rigorous convergence analysis, we show that the proposed algorithm is convergent. Experiments are carried out to confirm the theoretical results and show that the proposed CCS system yields significant improvements over the existing CS systems in terms of the signal recovery accuracy.

Frequency estimation is a classical problem in signal processing, with applications ranging from sensor array processing to wireless communications and structural health monitoring. Modern algorithms based on atomic norm minimization can cope with missing data but incur a high computational cost. To recover missing data from an ensemble of frequency-sparse signals, we propose a computationally efficient low-rank tensor completion algorithm that exploits the fact that each signal in the ensemble can be associated with a Toeplitz matrix. We name our algorithm JAZZ in the spirit of the classical MUSIC algorithm for frequency estimation and in tribute to the random, improvisational nature of jazz music.

In this paper, we propose a new method to optimize the sensing matrix and the overcomplete dictionary simultaneously in a block-sparse system. This method mainly includes two parts: the optimization of the sensing matrix for a given dictionary and the optimization of the overcomplete dictionary with a block structure for a predefined sensing matrix. Simulation results show that our novel method can significantly improve the dictionary recovery ability and lower the representation error compared with other dictionary learning methods in block-sparse systems.