Structured nonnegative matrix factorization with applications to hidden Markov realization and clustering |
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Authors: | Bart Vanluyten Jan C. Willems Bart De Moor |
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Affiliation: | K.U.Leuven, ESAT/SCD(SISTA), Kasteelpark Arenberg 10, B-3001 Heverlee-Leuven, Belgium |
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Abstract: | In this paper, we study the structured nonnegative matrix factorization problem: given a square, nonnegative matrix P, decompose it as P=VAV? with V and A nonnegative matrices and with the dimension of A as small as possible. We propose an iterative approach that minimizes the Kullback-Leibler divergence between P and VAV? subject to the nonnegativity constraints on A and V with the dimension of A given. The approximate structured decomposition P?VAV? is closely related to the approximate symmetric decomposition P?VV?. It is shown that the approach for finding an approximate structured decomposition can be adapted to solve the symmetric decomposition problem approximately. Finally, we apply the nonnegative decomposition VAV? to the hidden Markov realization problem and to the clustering of data vectors based on their distance matrix. |
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Keywords: | Nonnegative matrix factorization Kullback-Leibler divergence Multiplicative update formulas cp-Rank |
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