Strong NP-Hardness of Rank-2 Nonnegative Matrix Factorization
Arnaud Vandaele
Abstract
Nonnegative matrix factorization (NMF) approximates a nonnegative matrix by the product of two nonnegative matrices with a prescribed inner dimension~r. The existence of an exact NMF can be decided in polynomial time for every fixed~r, and a best rank-1 NMF in the Frobenius norm can be obtained from a truncated singular value decomposition. We prove that computing a best rank-2 NMF in the Frobenius norm is strongly NP-hard. Our proof uses a polynomial-time reduction from Clique that produces symmetric positive-definite, completely positive matrices with polynomially bounded integer entries. The same reduction also establishes strong NP-hardness of rank-2 symmetric NMF.
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