Nested Nonnegative Cone Analysis
This addresses issues in determining component numbers and multi-scale interpretability for researchers working with nonnegative data, though it appears incremental as it builds on NMF.
The paper tackled the problem of analyzing nonnegative data, where traditional methods like PCA/SVD produce non-interpretable results and NMF suffers from non-uniqueness and lack of nested subspaces, by proposing Nested Nonnegative Cone Analysis (NNCA), which uniquely generates nested structures at each rank.
Motivated by the analysis of nonnegative data objects, a novel Nested Nonnegative Cone Analysis (NNCA) approach is proposed to overcome some drawbacks of existing methods. The application of traditional PCA/SVD method to nonnegative data often cause the approximation matrix leave the nonnegative cone, which leads to non-interpretable and sometimes nonsensical results. The nonnegative matrix factorization (NMF) approach overcomes this issue, however the NMF approximation matrices suffer several drawbacks: 1) the factorization may not be unique, 2) the resulting approximation matrix at a specific rank may not be unique, and 3) the subspaces spanned by the approximation matrices at different ranks may not be nested. These drawbacks will cause troubles in determining the number of components and in multi-scale (in ranks) interpretability. The NNCA approach proposed in this paper naturally generates a nested structure, and is shown to be unique at each rank. Simulations are used in this paper to illustrate the drawbacks of the traditional methods, and the usefulness of the NNCA method.