Structural Identifiability in Low-Rank Matrix Factorization

• Published in: Algorithmica Vol. 56; no. 3; pp. 313 - 332
• Main Authors: Fritzilas, Epameinondas, Milanič, Martin, Rahmann, Sven, Rios-Solis, Yasmin A.
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.03.2010
• Subjects: 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
• DOI: 10.1007/s00453-009-9331-2

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.