Primal-Dual Algorithms for Non-negative Matrix Factorization with the Kullback-Leibler Divergence

Non-negative matrix factorization (NMF) approximates a given matrix as a product of two non-negative matrices. Multiplicative algorithms deliver reliable results, but they show slow convergence for high-dimensional data and may be stuck away from local minima. Gradient descent methods have better behavior, but only apply to smooth losses such as the least-squares loss. In this article, we propose a first-order primal-dual algorithm for non-negative decomposition problems (where one factor is fixed) with the KL divergence, based on the Chambolle-Pock algorithm. All required computations may be obtained in closed form and we provide an efficient heuristic way to select step-sizes. By using alternating optimization, our algorithm readily extends to NMF and, on synthetic examples, face recognition or music source separation datasets, it is either faster than existing algorithms, or leads to improved local optima, or both

Acquiring and processing verb argument structure : distributional learning in a miniature language

Adult knowledge of a language involves correctly balancing lexically-based and more language-general patterns. For example, verb argument structures may sometimes readily generalize to new verbs, yet with particular verbs may resist generalization. From the perspective of acquisition, this creates significant learnability problems, with some researchers claiming a crucial role for verb semantics in the determination of when generalization may and may not occur. Similarly, there has been debate regarding how verb-specific and more generalized constraints interact in sentence processing and on the role of semantics in this process. The current work explores these issues using artificial language learning. In three experiments using languages without semantic cues to verb distribution, we demonstrate that learners can acquire both verb-specific and verb-general patterns, based on distributional information in the linguistic input regarding each of the verbs as well as across the language as a whole. As with natural languages, these factors are shown to affect production, judgments and real-time processing. We demonstrate that learners apply a rational procedure in determining their usage of these different input statistics and conclude by suggesting that a Bayesian perspective on statistical learning may be an appropriate framework for capturing our findings

Reliability approach to rotating-component design

A probabilistic methodology for designing rotating mechanical components using reliability to relate stress to strength is explained. The experimental test machines and data obtained for steel to verify this methodology are described. A sample mechanical rotating component design problem is solved by comparing a deterministic design method with the new design-by reliability approach. The new method shows that a smaller size and weight can be obtained for specified rotating shaft life and reliability, and uses the statistical distortion-energy theory with statistical fatigue diagrams for optimum shaft design. Statistical methods are presented for (1) determining strength distributions for steel experimentally, (2) determining a failure theory for stress variations in a rotating shaft subjected to reversed bending and steady torque, and (3) relating strength to stress by reliability

Fast ADMM Algorithm for Distributed Optimization with Adaptive Penalty

We propose new methods to speed up convergence of the Alternating Direction Method of Multipliers (ADMM), a common optimization tool in the context of large scale and distributed learning. The proposed method accelerates the speed of convergence by automatically deciding the constraint penalty needed for parameter consensus in each iteration. In addition, we also propose an extension of the method that adaptively determines the maximum number of iterations to update the penalty. We show that this approach effectively leads to an adaptive, dynamic network topology underlying the distributed optimization. The utility of the new penalty update schemes is demonstrated on both synthetic and real data, including a computer vision application of distributed structure from motion.Comment: 8 pages manuscript, 2 pages appendix, 5 figure

Hybrid PDE solver for data-driven problems and modern branching

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for nonlinear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European Journal of Applied Mathematics (EJAM

Alternation-Trading Proofs, Linear Programming, and Lower Bounds

A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, Mod6-SAT, Majority-of-Majority-SAT, and Tautologies, to name a few. The proofs of these lower bounds follow a certain proof-by-contradiction strategy that we call alternation-trading. An important open problem is to determine how powerful such proofs can possibly be. We propose a methodology for studying these proofs that makes them amenable to both formal analysis and automated theorem proving. We prove that the search for better lower bounds can often be turned into a problem of solving a large series of linear programming instances. Implementing a small-scale theorem prover based on this result, we extract new human-readable time lower bounds for several problems. This framework can also be used to prove concrete limitations on the current techniques.Comment: To appear in STACS 2010, 12 page