Abstract :
[en] Matrix factorizations (MF) are standard techniques in linear algebra
that approximate a data matrix as the product of two smaller
matrices whose inner dimension is called the rank of the factorization.
A general assumption in MFs is the nonnegativity of the input
matrix which led to the development of nonnegative matrix factorization
(NMF) models, where the factors are constrained to be nonnegative
as well, motivated by several real-world applications. NMF
allows one to express the columns of the input matrix, which generally
represent data points, as linear combinations of a small number
of latent features.
In the first part of this thesis, we introduce near-convex archetypal
analysis (NCAA), a flexible extension of archetypal analysis,
a well-known NMF variant, which has an interesting geometric
interpretation and performs competitively with minimum-volume
(minVol) NMF.
In the second part of this thesis, we study deep MF, that is, the
extension of MF to several layers, inspired by the recent advances in
deep learning. We conduct a thorough literature review of multilayer
and deep MF, focusing on the models, the choice of the parameters,
the applications, and the theoretical aspects. We also illustrate the
abilities of deep MF to extract hierarchical features within complex
data sets on three showcase examples, namely the extraction of facial
features, hyperspectral unmixing (HU) and recommender systems.
We then introduce two new loss functions for deep MF together
with a general optimization framework. We show their efficiency to
tackle sparse and minVol deep MF on both synthetic and real-world
data. These loss functions alleviate the drawbacks of the current approaches
both in a theoretical and experimental point of view. We
also design a new greedy initialization algorithm for deep MF and
extend symmetric NMF to the deep case. We apply this latter successfully
to the extraction of overlapping communities of symptoms
in psychiatric networks, with promising clinical interpretation.
Finally, we sketch perspectives of future works, including the
study of the identifiability of deep MF, the investigation of the
connexions between deep MF and deep neural networks, and the
exploration of new deep MF models.
