Strong NP-Hardness of Rank-2 Nonnegative Matrix Factorization

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 $\text{textsc}Clique$ 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.