Near-Convex Archetypal Analysis

Nonnegative matrix factorization (NMF) is a widely used linear dimensionality reduction technique for nonnegative data. NMF requires that each data point is approximated by a convex combination of basis elements. Archetypal analysis (AA), also referred to as convex NMF, is a well-known NMF variant imposing that the basis elements are themselves convex combinations of the data points. AA has the advantage to be more interpretable than NMF because the basis elements are directly constructed from the data points. However, it usually suffers from a high data fitting error because the basis elements are constrained to be contained in the convex cone of the data points. In this letter, we introduce near-convex archetypal analysis (NCAA) which combines the advantages of both AA and NMF. As for AA, the basis vectors are required to be linear combinations of the data points and hence are easily interpretable. As for NMF, the additional flexibility in choosing the basis elements allows NCAA to have a low data fitting error. We show that NCAA compares favorably with a state-of-the-art minimum-volume NMF method on synthetic datasets and on a real-world hyperspectral image.