Parallel Algorithms for Constrained Tensor Factorization via the Alternating Direction Method of Multipliers

Tensor factorization has proven useful in a wide range of applications, from sensor array processing to communications, speech and audio signal processing, and machine learning. With few recent exceptions, all tensor factorization algorithms were originally developed for centralized, in-memory computation on a single machine; and the few that break away from this mold do not easily incorporate practically important constraints, such as nonnegativity. A new constrained tensor factorization framework is proposed in this paper, building upon the Alternating Direction method of Multipliers (ADMoM). It is shown that this simplifies computations, bypassing the need to solve constrained optimization problems in each iteration; and it naturally leads to distributed algorithms suitable for parallel implementation on regular high-performance computing (e.g., mesh) architectures. This opens the door for many emerging big data-enabled applications. The methodology is exemplified using nonnegativity as a baseline constraint, but the proposed framework can more-or-less readily incorporate many other types of constraints. Numerical experiments are very encouraging, indicating that the ADMoM-based nonnegative tensor factorization (NTF) has high potential as an alternative to state-of-the-art approaches.Comment: Submitted to the IEEE Transactions on Signal Processin

Non-negative Matrix Factorization: A Survey

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A convex model for non-negative matrix factorization and dimensionality reduction on physical space

A collaborative convex framework for factoring a data matrix $X$ into a non-negative product $AS$, with a sparse coefficient matrix $S$, is proposed. We restrict the columns of the dictionary matrix $A$ to coincide with certain columns of the data matrix $X$, thereby guaranteeing a physically meaningful dictionary and dimensionality reduction. We use $l_{1,\infty}$ regularization to select the dictionary from the data and show this leads to an exact convex relaxation of $l_0$ in the case of distinct noise free data. We also show how to relax the restriction-to-$X$ constraint by initializing an alternating minimization approach with the solution of the convex model, obtaining a dictionary close to but not necessarily in $X$. We focus on applications of the proposed framework to hyperspectral endmember and abundances identification and also show an application to blind source separation of NMR data.Comment: 14 pages, 9 figures. EE and JX were supported by NSF grants {DMS-0911277}, {PRISM-0948247}, MM by the German Academic Exchange Service (DAAD), SO and MM by NSF grants {DMS-0835863}, {DMS-0914561}, {DMS-0914856} and ONR grant {N00014-08-1119}, and GS was supported by NSF, NGA, ONR, ARO, DARPA, and {NSSEFF.