  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.
