  This study presents PRISM, a probabilistic simplex component analysis
approach to identifying the vertices of a data-circumscribing simplex from
data. The problem has a rich variety of applications, the most notable being
hyperspectral unmixing in remote sensing and non-negative matrix factorization
in machine learning. PRISM uses a simple probabilistic model, namely, uniform
simplex data distribution and additive Gaussian noise, and it carries out
inference by maximum likelihood. The inference model is sound in the sense that
the vertices are provably identifiable under some assumptions, and it suggests
that PRISM can be effective in combating noise when the number of data points
is large. PRISM has strong, but hidden, relationships with simplex volume
minimization, a powerful geometric approach for the same problem. We study
these fundamental aspects, and we also consider algorithmic schemes based on
importance sampling and variational inference. In particular, the variational
inference scheme is shown to resemble a matrix factorization problem with a
special regularizer, which draws an interesting connection to the matrix
factorization approach. Numerical results are provided to demonstrate the
potential of PRISM.
