Data is often available in matrix form, in which columns are samples, and
processing of such data often entails finding an approximate factorisation of
the matrix in two factors. The first factor yields recurring patterns
characteristic of the data. The second factor describes in which proportions
each data sample is made of these patterns. Latent factor estimation (LFE)
is the problem of finding such a factorisation, usually under given constraints.
LFE appears under other domain-specific names such as
dictionary learning, low-rank approximation, factor analysis or
latent semantic analysis. It is used for tasks such as dimensionality
reduction, unmixing, soft clustering, coding or matrix completion in very diverse fields.

In this project, we propose to explore three new paradigms that push the
frontiers of traditional LFE. The first objective is to break beyond the
ubiquitous Gaussian assumption, a practical choice that too rarely complies
with the nature and geometry of the data. Estimation in non-Gaussian models
is more difficult, but recent work in audio and text processing has shown
that it pays off in practice. Second, in traditional settings the data
matrix is often a collection of features computed from raw data.
These features are computed with generic off-the-shelf transforms
that loosely preprocess the data, setting a limit to performance.
We propose to explore a new paradigm in which an optimal low-rank inducing
transform is learnt together with the factors in a single step.
Thirdly, we propose to reconsider the dominant deterministic approach to
LFE and explore a novel statistical estimation paradigm,
based on the marginal likelihood, with enhanced capabilities.
The new methodology is applied to real-world problems in audio signal
processing (speech enhancement, music remastering),
remote sensing (Earth observation, cosmic object discovery) and data mining
(multimodal information retrieval, user recommendation).

Latent factor estimation (LFE)

General principle of LFE

LFE for dimensionality reduction (e.g., coding, low-dimensional embedding)

LFE for matrix completion (e.g., collaborative filtering, inpainting)

LFE for unmixing (e.g., source separation, latent topic discovery)