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 [67], 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 [65], head pose estimation in images [16], sound source separation and localization [66], and virtually-supervised acoustic space learning (see Section 7.4.4). 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.

Phase Estimation in Multichannel Mixtures

Participants : Antoine Deleforge, Yann Traonmilin.

Main collaboration: Angélique Drémeau (ENSTA Bretagne and Lab-STICC, Brest)

The problem of estimating source signals given an observed multichannel mixture is fundamentally ill-posed when the mixing matrix is unknown or when the number of sources is larger that the number of microphones. Hence, prior information on the desired source signals must be incorporated in order to tackle it. An important line of research in audio source separation over the past decade consists in using a model of the source signals' magnitudes in the short-time Fourier domain [9]. Such models can be inferred through, e.g., non-negative matrix factorization [9] or deep neural networks [91]. Magnitudes estimates are often interpreted as instantaneous variances of Gaussian-process source signals, and are combined with Wiener filtering for source separation. In [26], we introduced a shift of this paradigm by considering the Phase Unmixing problem: how can one recover the instantaneous phases of complex mixed source signals when their magnitudes and mixing matrix are known? This problem was showed to be NP-hard, and three approaches were proposed to tackle it: a heuristic method, an alternate minimization method, and a convex relaxation into a semi-definite program. The last two approaches were showed to outperform the oracle multichannel Wiener filter in under-determined informed source separation tasks. The latter yielded best results, including the potential for exact source separation in under-determined settings. In [27] we applied this framework to the classical problem of phase retrieval with a novel multivariate Von Mises prior on phases. We showed that enforcing this prior yielded more accurate estimates than state-of-the art phase retrieval methods.

Audio Inpainting and Denoising

Participants : Rémi Gribonval, Nancy Bertin, Clément Gaultier.

Main collaborations: Srdan Kitic (Technicolor R&I France, 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 (definition of the audio inpainting problem as a general framework for many audio processing tasks, application to the audio declipping or desaturation problem, formulation as a sparse recovery problem [50]), we proposed over the last two years a series of algorithms leveraging the competitive cosparse approach, which offers a very appealing trade-off between reconstruction performance and computational time [75], [78] [6]. The work on cosparse audio declipping which was awarded the Conexant best paper award at the LVA/ICA 2015 conference [78] resulted in a software release in 2016.

In 2017, this work was extended towards advanced (co)sparse decompositions, including several forms of structured sparsity in the time-frequency domain and across channels, and towards their application to the denoising task, in addition to the previously introduced declipping task, which we continued to improve. In particular, we investigated the incorporation of the so-called “social” structure constraint [79] into problems regularized by a cosparse prior [28], [41], and exhibited a common framework allowing to tackle both denoising and declipping in a unified fashion [40]. A new algorithm for joint declipping of multichannel audio was also derived (one submitted conference publication.)