Fixed-rank matrix factorizations and Riemannian low-rank optimization

Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of fixed-rank matrices. We adopt the geometric framework of optimization on Riemannian quotient manifolds. We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian quotient geometry of the search space in the design of a class of gradient descent and trust-region algorithms. The proposed algorithms generalize our previous results on fixed-rank symmetric positive semidefinite matrices, apply to a broad range of applications, scale to high-dimensional problems, and confer a geometric basis to recent contributions on the learning of fixed-rank non-symmetric matrices. We make connections with existing algorithms in the context of low-rank matrix completion and discuss the usefulness of the proposed framework. Numerical experiments suggest that the proposed algorithms compete with state-of-the-art algorithms and that manifold optimization offers an effective and versatile framework for the design of machine learning algorithms that learn a fixed-rank matrix. © 2013 Springer-Verlag Berlin Heidelberg.

Turbo-smt: Accelerating coupled sparse matrix-tensor factorizations by 200x

How can we correlate the neural activity in the human brain as it responds to typed words, with properties of these terms (like ‘edible’, ‘fits in hand’)? In short, we want to find latent variables, that jointly explain both the brain activity, as well as the behavioral responses. This is one of many settings of the Coupled Matrix-Tensor Factorization (CMTF) problem.

Can we accelerate any CMTF solver, so that it runs within a few minutes instead of tens of hours to a day, while maintaining good accuracy? We introduce Turbo-SMT, a meta-method capable of doing exactly that: it boosts the performance of any CMTF algorithm, by up to 200x, along with an up to 65 fold increase in sparsity, with comparable accuracy to the baseline.

We apply Turbo-SMT to BrainQ, a dataset consisting of a (nouns, brain voxels, human subjects) tensor and a (nouns, properties) matrix, with coupling along the nouns dimension. Turbo-SMT is able to find meaningful latent variables, as well as to predict brain activity with competitive accuracy.

Extended factorizations of exponential functionals of Lévy processes

Pardo, Patie, and Savov derived, under mild conditions, a Wiener-Hopf type factorization for the exponential functional of proper Lévy processes. In this paper, we extend this factorization by relaxing a finite moment assumption as well as by considering the exponential functional for killed Lévy processes. As a by-product, we derive some interesting fine distributional properties enjoyed by a large class of this random variable, such as the absolute continuity of its distribution and the smoothness, boundedness or complete monotonicity of its density. This type of results is then used to derive similar properties for the law of maxima and first passage time of some stable Lévy processes. Thus, for example, we show that for any stable process with $\rho\in(0,\frac{1}{\alpha}-1]$, where $\rho\in[0,1]$ is the positivity parameter and $\alpha$ is the stable index, then the first passage time has a bounded and non-increasing density on $\mathbb{R}_+$. We also generate many instances of integral or power series representations for the law of the exponential functional of Lévy processes with one or two-sided jumps. The proof of our main results requires different devices from the one developed by Pardo, Patie, Savov. It relies in particular on a generalization of a transform recently introduced by Chazal et al together with some extensions to killed Lévy process of Wiener-Hopf techniques. The factorizations developed here also allow for further applications which we only indicate here also allow for further applications which we only indicate here.

The divergence of Stone's factorizations when no parameters are used /

"September 7, 1971."

Cube factorizations of complete graphs

A cube factorization of the complete graph on n vertices, K-n, is a 3-factorization of & in which the components of each factor are cubes. We show that there exists a cube factorization of & if and only if n equivalent to 16 (mod 24), thus providing a new family of uniform 3 -factorizations as well as a partial solution to an open problem posed by Kotzig in 1979. (C) 2004 Wiley Periodicals, Inc.

Factorizations of some Groups of Lie Type of Lie Rank 4

Еленка Генчева, Цанко Генчев В настоящата работа се разглеждат крайни прости групи G , които могат да се представят като произведение на две свои собствени неабелеви прости подгрупи A и B. Всяко такова представяне G = AB е прието да се нарича факторизация на G, а тъй като множителите A и B са избрани да бъдат прости подгрупи на G, то разглежданите факторизации са известни още като прости факторизации на G. Тук се предполага, че G е проста група от лиев тип и лиев ранг 4 над крайно поле GF (q). Ключови думи: крайни прости групи, групи от лиев тип, факторизации на групи.