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Structural Identifiability in Low-Rank Matrix Factorization

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Abstract

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.

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Authors and Affiliations

  1. Faculty of Technology, Bielefeld University, Bielefeld, Germany

    Epameinondas Fritzilas

  2. Faculty of Mathematics, Natural Sciences and Information Technologies, University of Primorska, Koper, Slovenia

    Martin Milanič

  3. Computer Science 11, Technische Universität Dortmund, Dortmund, Germany

    Sven Rahmann

  4. Graduate Program of Systems Engineering, UANL, Monterrey, Mexico

    Yasmin A. Rios-Solis

Authors
  1. Epameinondas Fritzilas
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  2. Martin Milanič
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  3. Sven Rahmann
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  4. Yasmin A. Rios-Solis
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Correspondence to Epameinondas Fritzilas.

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Fritzilas, E., Milanič, M., Rahmann, S. et al. Structural Identifiability in Low-Rank Matrix Factorization. Algorithmica 56, 313–332 (2010). https://doi.org/10.1007/s00453-009-9331-2

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  • DOI: https://doi.org/10.1007/s00453-009-9331-2

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