Structural Identifiability in Low-Rank Matrix Factorization

Fritzilas, Epameinondas; Ríos-Solís, Yasmín Á.; Rahmann, Sven

doi:10.1007/978-3-540-69733-6_15

Cited by 2 publications

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“…While the recognition problem for identifiable bipartite graphs is clearly polynomial using bipartite matching algorithms, several natural algorithmic problems concerning identifiable graphs turn out to be NP-complete (see [4,5,7]). In [4], three problems related to finding specific identifiable subgraphs were introduced.…”

Section: Introductionmentioning

confidence: 99%

“…A bipartite graph G = (L, R; E) with at least one edge is said to be identifiable if for every vertex in L, the subgraph of G induced by its non-neighborhood has a matching of cardinality |L| − 1. Identifiable bipartite graphs were studied in several papers [4][5][6][7]; the property arises in the context of low-rank matrix factorization and has applications in data mining, signal processing, and computational biology. For further details on applications of notions and problems discussed in this paper, we refer to [4,5].…”

Section: Introductionmentioning

confidence: 99%

“…Identifiable bipartite graphs were studied in several papers [4][5][6][7]; the property arises in the context of low-rank matrix factorization and has applications in data mining, signal processing, and computational biology. For further details on applications of notions and problems discussed in this paper, we refer to [4,5].…”

Section: Introductionmentioning

confidence: 99%

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On the complexity of the identifiable subgraph problem, revisited

Kratsch

Milanič

2017

Discrete Applied Mathematics

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A bipartite graph G = (L, R; E) with at least one edge is said to be identifiable if for every vertex v ∈ L, the subgraph induced by its non-neighbors has a matching of cardinality |L| − 1. An -subgraph of G is an induced subgraph of G obtained by deleting from it some vertices in L together with all their neighbors. The Identifiable Subgraph problem is the problem of determining whether a given bipartite graph contains an identifiable -subgraph.We show that the Identifiable Subgraph problem is polynomially solvable, along with the version of the problem in which the task is to delete as few vertices from L as possible together with all their neighbors so that the resulting -subgraph is identifiable. We also complement a known APX-hardness result for the complementary problem in which the task is to minimize the number of remaining vertices in L, by showing that two parameterized variants of the problem are W[1]-hard.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the complexity of the identifiable subgraph problem, revisited

Kratsch

Milanič

2017

Discrete Applied Mathematics

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Structural Identifiability in Low-Rank Matrix Factorization

et al. 2009

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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 sourcesensor mixing coefficients. Our task is to monitor the signal sources with the cheapest subset of sensors, while maintaining structural identifiability. Firstly, we show that E. Fritzilas ( ) Faculty 314 Algorithmica (2010) 56: 313-332 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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Structural Identifiability in Low-Rank Matrix Factorization

Cited by 2 publications

References 7 publications

On the complexity of the identifiable subgraph problem, revisited

On the complexity of the identifiable subgraph problem, revisited

Structural Identifiability in Low-Rank Matrix Factorization

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