Fast Rank-1 NMF for Missing Data with KL Divergence
This work addresses a specific computational bottleneck in matrix factorization for data with missing values, offering an incremental improvement in speed for domain-specific applications.
The paper tackles the problem of fast non-negative matrix factorization for missing data by proposing A1GM, a non-gradient method that minimizes KL divergence, achieving competitive reconstruction errors with greater efficiency than gradient methods.
We propose a fast non-gradient-based method of rank-1 non-negative matrix factorization (NMF) for missing data, called A1GM, that minimizes the KL divergence from an input matrix to the reconstructed rank-1 matrix. Our method is based on our new finding of an analytical closed-formula of the best rank-1 non-negative multiple matrix factorization (NMMF), a variety of NMF. NMMF is known to exactly solve NMF for missing data if positions of missing values satisfy a certain condition, and A1GM transforms a given matrix so that the analytical solution to NMMF can be applied. We empirically show that A1GM is more efficient than a gradient method with competitive reconstruction errors.