Kejun Huang

Non-Negative Matrix Factorization Revisited: Uniqueness and Algorithm for Symmetric Decomposition

Non-negative matrix factorization (NMF) has found numerous applications, due to its ability to provide interpretable decompositions. Perhaps surprisingly, existing results regarding its uniqueness properties are rather limited, and there is much room for improvement in terms of algorithms as well. Uniqueness aspects of NMF are revisited here from a geometrical point of view. Both symmetric and asymmetric NMF are considered, the former being tantamount to element-wise non-negative square-root factorization of positive semidefinite matrices. New uniqueness results are derived, e.g., it is shown that a sufficient condition for uniqueness is that the conic hull of the latent factors is a superset of a particular second-order cone. Checking this condition is shown to be NP-complete; yet this and other results offer insights on the role of latent sparsity in this context. On the computational side, a new algorithm for symmetric NMF is proposed, which is very different from existing ones. It alternates between Procrustes rotation and projection onto the non-negative orthant to find a non-negative matrix close to the span of the dominant subspace. Simulation results show promising performance with respect to the state-of-art. Finally, the new algorithm is applied to a clustering problem for co-authorship data, yielding meaningful and interpretable results.

Feasible Point Pursuit and Successive Approximation of Non-Convex QCQPs

Quadratically constrained quadratic programs (QCQPs) have a wide range of applications in signal processing and wireless communications. Non-convex QCQPs are NP-hard in general. Existing approaches relax the non-convexity using semi-definite relaxation (SDR) or linearize the non-convex part and solve the resulting convex problem. However, these techniques are seldom successful in even obtaining a feasible solution when the QCQP matrices are indefinite. In this letter, a new feasible point pursuit successive convex approximation (FPP-SCA) algorithm is proposed for non-convex QCQPs. FPP-SCA linearizes the non-convex parts of the problem as conventional SCA does, but adds slack variables to sustain feasibility, and a penalty to ensure slacks are sparingly used. When FPP-SCA is successful in identifying a feasible point of the non-convex QCQP, convergence to a Karush-Kuhn-Tucker (KKT) point is thereafter ensured. Simulations show the effectiveness of our proposed algorithm in obtaining feasible and near-optimal solutions, significantly outperforming existing approaches.

Consensus-ADMM for General Quadratically Constrained Quadratic Programming

Nonconvex quadratically constrained quadratic programming (QCQP) problems have numerous applications in signal processing, machine learning, and wireless communications, albeit the general QCQP is NP-hard, and several interesting special cases are NP-hard as well. This paper proposes a new algorithm for general QCQP. The problem is first reformulated in consensus optimization form, to which the alternating direction method of multipliers can be applied. The reformulation is done in such a way that each of the subproblems is a QCQP with only one constraint (QCQP-1), which is efficiently solvable irrespective of (non)convexity. The core components are carefully designed to make the overall algorithm more scalable, including efficient methods for solving QCQP-1, memory efficient implementation, parallel/distributed implementation, and smart initialization. The proposed algorithm is then tested in two applications: multicast beamforming and phase retrieval. The results indicate superior performance over prior state-of-the-art methods.

Phase Retrieval from 1D Fourier Measurements: Convexity, Uniqueness, and Algorithms

This paper considers phase retrieval from the magnitude of one-dimensional over-sampled Fourier measurements, a classical problem that has challenged researchers in various fields of science and engineering. We show that an optimal vector in a least-squares sense can be found by solving a convex problem, thus establishing a hidden convexity in Fourier phase retrieval. We then show that the standard semidefinite relaxation approach yields the optimal cost function value (albeit not necessarily an optimal solution). A method is then derived to retrieve an optimal minimum phase solution in polynomial time. Using these results, a new measuring technique is proposed which guarantees uniqueness of the solution, along with an efficient algorithm that can solve large-scale Fourier phase retrieval problems with uniqueness and optimality guarantees.

Blind Separation of Quasi-Stationary Sources: Exploiting Convex Geometry in Covariance Domain

This paper revisits blind source separation of instantaneously mixed quasi-stationary sources (BSS-QSS), motivated by the observation that in certain applications (e.g., speech) there exist time frames during which only one source is active, or locally dominant. Combined with nonnegativity of source powers, this endows the problem with a nice convex geometry that enables elegant and efficient BSS solutions. Local dominance is tantamount to the so-called pure pixel/separability assumption in hyperspectral unmixing/nonnegative matrix factorization, respectively. Building on this link, a very simple algorithm called successive projection algorithm (SPA) is considered for estimating the mixing system in closed form. To complement SPA in the specific BSS-QSS context, an algebraic preprocessing procedure is proposed to suppress short-term source cross-correlation interference. The proposed procedure is simple, effective, and supported by theoretical analysis. Solutions based on volume minimization (VolMin) are also considered. By theoretical analysis, it is shown that VolMin guarantees perfect mixing system identifiability under an assumption more relaxed than (exact) local dominance - which means wider applicability in practice. Exploiting the specific structure of BSS-QSS, a fast VolMin algorithm is proposed for the overdetermined case. Careful simulations using real speech sources showcase the simplicity, efficiency, and accuracy of the proposed algorithms.

Phase Retrieval Using Feasible Point Pursuit: Algorithms and Cramér–Rao Bound

Reconstructing a signal from squared linear (rank-1 quadratic) measurements is a challenging problem with important applications in optics and imaging, where it is known as phase retrieval. This paper proposes two new phase retrieval algorithms based on nonconvex quadratically constrained quadratic programming) formulations, and a recently proposed approximation technique dubbed feasible point pursuit (FPP). The first is designed for uniformly distributed bounded measurement errors, such as those arising from high-rate quantization (B-FPP). The second is designed for Gaussian measurement errors, using a least-squares criterion (LS-FPP). Their performance is measured against state-of-the-art algorithms and the Cramér-Rao bound (CRB), which is also derived here. Simulations show that LS-FPP outperforms the existing schemes and operates close to the CRB. Compact CRB expressions, properties, and insights are obtained by explicitly computing the CRB in various special cases-including when the signal of interest admits a sparse parametrization, using harmonic retrieval as an example.

Joint Tensor Factorization and Outlying Slab Suppression With Applications

We consider factoring low-rank tensors in the presence of outlying slabs. This problem is important in practice, because data collected in many real-world applications, such as speech, fluorescence, and some social network data, fit this paradigm. Prior work tackles this problem by iteratively selecting a fixed number of slabs and fitting, a procedure which may not converge. We formulate this problem from a group-sparsity promoting point of view, and propose an alternating optimization framework to handle the corresponding ℓp (0 <; p ≤ 1) minimization-based low-rank tensor factorization problem. The proposed algorithm features a similar per-iteration complexity as the plain trilinear alternating least squares (TALS) algorithm. Convergence of the proposed algorithm is also easy to analyze under the framework of alternating optimization and its variants. In addition, regularization and constraints can be easily incorporated to make use of a priori information on the latent loading factors. Simulations and real data experiments on blind speech separation, fluorescence data analysis, and social network mining are used to showcase the effectiveness of the proposed algorithm.

A Flexible and Efficient Algorithmic Framework for Constrained Matrix and Tensor Factorization

We propose a general algorithmic framework for constrained matrix and tensor factorization, which is widely used in signal processing and machine learning. The new framework is a hybrid between alternating optimization (AO) and the alternating direction method of multipliers (ADMM): each matrix factor is updated in turn, using ADMM, hence the name AO-ADMM. This combination can naturally accommodate a great variety of constraints on the factor matrices, and almost all possible loss measures for the fitting. Computation caching and warm start strategies are used to ensure that each update is evaluated efficiently, while the outer AO framework exploits recent developments in block coordinate descent (BCD)-type methods which help ensure that every limit point is a stationary point, as well as faster and more robust convergence in practice. Three special cases are studied in detail: non-negative matrix/tensor factorization, constrained matrix/tensor completion, and dictionary learning. Extensive simulations and experiments with real data are used to showcase the effectiveness and broad applicability of the proposed framework.

Putting Nonnegative Matrix Factorization to the Test: A tutorial derivation of pertinent Cramer?Rao bounds and performance benchmarking

Nonnegative matrix factorization (NMF) is a useful tool in a broad range of applications, from signal separation to computer vision and machine learning. NMF is a hard (NP-hard) computational problem for which various approximate solutions have been developed over the years. Given the widespread interest in NMF and its applications, it is perhaps surprising that the pertinent Cramer-Rao lower bound (CRLB) on the accuracy of the nonnegative latent factor estimates has not been worked out in the literature. In hindsight, one reason may be that the required computations are more subtle than usual: the problem involves constraints and ambiguities that must be dealt with, and the Fisher information matrix is always singular. We provide a concise tutorial derivation of the CRLB for both symmetric NMF and asymmetric NMF, using the latest CRLB tools, which should be of broad interest for analogous derivations in related factor analysis problems. We illustrate the behavior of these bounds with respect to model parameters and put some of the best NMF algorithms to the test against one another and the CRLB. The results help illuminate what can be expected from the current state of art in NMF algorithms, and they are reassuring in that the gap to optimality is small in relatively sparse and low rank scenarios.

Translation Invariant Word Embeddings

This work focuses on the task of finding latent vector representations of the words in a corpus. In particular, we address the issue of what to do when there are multiple languages in the corpus. Prior work has, among other techniques, used canonical correlation analysis to project pre-trained vectors in two languages into a common space. We propose a simple and scalable method that is inspired by the notion that the learned vector representations should be invariant to translation between languages. We show empirically that our method outperforms prior work on multilingual tasks, matches the performance of prior work on monolingual tasks, and scales linearly with the size of the input data (and thus the number of languages being embedded).