Structural Identifiability in Low-Rank Matrix Factorization

In many signal processing and data mining applications, we need to approximate a given matrix Y with a low-rank product Y ≈ AX . Both matrices A and X are to be determined, but we assume that from the specifics of the application we have an important piece of a-priori knowledge: A must have zeros at...

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Vydáno v:Algorithmica Ročník 56; číslo 3; s. 313 - 332
Hlavní autoři: Fritzilas, Epameinondas, Milanič, Martin, Rahmann, Sven, Rios-Solis, Yasmin A.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer-Verlag 01.03.2010
Témata:
Algorithm Analysis and Problem Complexity
Algorithms
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Mathematics of Computing
Theory of Computation
Integer programming
Low-rank matrix factorization
Bipartite matching
Approximation algorithm
Computational complexity
Structural rank
ISSN:0178-4617, 1432-0541
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Popis
Shrnutí:In many signal processing and data mining applications, we need to approximate a given matrix Y with a low-rank product Y ≈ AX . Both matrices A and X are to be determined, but we assume that from the specifics of the application we have an important piece of a-priori knowledge: A must have zeros at certain positions. In general, different AX factorizations approximate a given Y equally well, so a fundamental question is whether the known zero pattern of A contributes to the uniqueness of the factorization. Using the notion of structural rank, we present a combinatorial characterization of uniqueness up to diagonal scaling (subject to a mild non-degeneracy condition on the factors), called structural identifiability of the model. Next, we define an optimization problem that arises in the need for efficient experimental design. In this context, Y contains sensor measurements over several time samples, X contains source signals over time samples and A contains the source-sensor mixing coefficients. Our task is to monitor the signal sources with the cheapest subset of sensors, while maintaining structural identifiability. Firstly, we show that this problem is NP-hard. Secondly, we present a mixed integer linear program for its exact solution together with two practical incremental approaches. We also propose a greedy approximation algorithm. Finally, we perform computational experiments on simulated problem instances of various sizes.
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-009-9331-2