Section: New Results

Emerging activities on Nonlinear Inverse Problems

Compressive sensing, compressive learning, audio inpainting, phase estimation

Locally-Linear Inverse Regression

Participant : Antoine Deleforge.

Main collaborations: Florence Forbes (MISTIS Inria project-team, Grenoble), Emeline Perthame (HUB team, Institut Pasteur, Paris), Vincent Drouard, Radu Horaud, Sileye Ba and Georgios Evangelidis (PERCEPTION Inria project-team, Grenoble)

A general problem in machine learning and statistics is that of high- to low-dimensional mapping. In other words, given two spaces ${ℝ}^D$ and ${ℝ}^L$ with $D\gg L$, how to find a relation between these two spaces such that given a new observation vector $y\in {ℝ}^D$ its associated vector $x\in {ℝ}^L$ can be estimated? In regression, a set of training pairs ${\left\{(y_n,x_n)\right\}}_{n=1}^N$ is used to learn the relation. In dimensionality reduction, only vectors ${\left\{y_n\right\}}_{n=1}^N$ are observed, and an intrinsic low-dimensional representation ${\left\{x_n\right\}}_{n=1}^N$ is sought. In [73], we introduced a probabilistic framework unifying both tasks referred to as Gaussian Locally Linear Mapping (GLLiM). The key idea is to learn an easier other-way-around locally-linear relationship from $x$ to $y$ using a joint Gaussian Mixture model on $x$ and $y$. This mapping is then easily reversed via Bayes' inversion. This framework was notably applied to hyperspectral imaging of Mars [71], head pose estimation in images [79], sound source separation and localization [72], and virtually-supervised acoustic space learning (see Section 7.6.1). This year, in [19], we introduced the Student Locally Linear Mapping (SLLiM) framework. The use of heavy-tailed Student's t-distributions instead of Gaussian ones leads to more robustness and better regression performance on several datasets.

Audio Inpainting and Denoising

Participants : Rémi Gribonval, Nancy Bertin, Clément Gaultier.

Main collaborations: Srdan Kitic (Orange, Rennes)

Inpainting is a particular kind of inverse problems that has been extensively addressed in the recent years in the field of image processing. Building upon our previous pioneering contributions [54]), we proposed over the last three years a series of algorithms leveraging the competitive cosparse approach, which offers a very appealing trade-off between reconstruction performance and computational time [100], [102] [6]. The work on cosparse audio declipping which was awarded the Conexant best paper award at the LVA/ICA 2015 conference [102] resulted in a software release in 2016. In 2017, this work was extended towards advanced (co)sparse decompositions, including several forms of structured sparsityand towards their application to the denoising task.In particular, we investigated the incorporation of the so-called “social” structure constraint [103] into problems regularized by a cosparse prior [84], [85], and exhibited a common framework allowing to tackle both denoising and declipping in a unified fashion [82].

In 2018, a new algorithm for joint declipping of multichannel audio was derived and published [29]. Extensive experimental benchmarks were conducted, questioning the previous state-of-the-art habits in degradation levels (usually moderate to inaudible) and evaluation (small datasets, SNR-based performance criteria) and setting up new standards for the task (large and diverse datasets, severe saturation, perceptual quality evaluation) as well as guidelines for the choice of the best variant (sparse or cosparse, with or without structural time-frequency constraints...) depending on the data and operational conditions. These new results will be included in an ongoing journal paper, to be submitted in 2019.