  In this paper we present a theoretical analysis to understand sparse
filtering, a recent and effective algorithm for unsupervised learning. The aim
of this research is not to show whether or how well sparse filtering works, but
to understand why and when sparse filtering does work. We provide a thorough
theoretical analysis of sparse filtering and its properties, and further offer
an experimental validation of the main outcomes of our theoretical analysis. We
show that sparse filtering works by explicitly maximizing the entropy of the
learned representation through the maximization of the proxy of sparsity, and
by implicitly preserving mutual information between original and learned
representations through the constraint of preserving a structure of the data,
specifically the structure defined by relations of neighborhoodness under the
cosine distance. Furthermore, we empirically validate our theoretical results
with artificial and real data sets, and we apply our theoretical understanding
to explain the success of sparse filtering on real-world problems. Our work
provides a strong theoretical basis for understanding sparse filtering: it
highlights assumptions and conditions for success behind this feature
distribution learning algorithm, and provides insights for developing new
feature distribution learning algorithms.
