WHERE PRIOR LEARNING CAN AND CAN'T WORK IN UNSUPERVISED INVERSE PROBLEMS

Abstract

Linear inverse problems consist in recovering a signal from its noisy observation in a lower dimensional space. Many popular resolution methods rely on data-driven algorithms that learn a prior from pairs of signals and observations to overcome the loss of information. However, these approaches are difficult, if not impossible, to adapt to unsupervised contexts -where no ground truth data are available -due to the need for learning from clean signals. This paper studies situations that do or do not allow learning a prior in unsupervised inverse problems. First, we focus on dictionary learning and point out that recovering the dictionary is unfeasible without constraints when the signal is observed through only one measurement operator. It can, however, be learned with multiple operators, given that they are diverse enough to span the whole signal space. Then, we study methods where weak priors are made available either through optimization constraints or deep learning architectures. We empirically emphasize that they perform better than hand-crafted priors only if they are adapted to the inverse problem.

1. INTRODUCTION

Linear inverse problems are ubiquitous in observational science such as imaging (Ribes & Schmitt, 2008) , neurosciences (Gramfort et al., 2012) or astrophysics (Starck, 2016) . They consist in reconstructing signals X ∈ R n×N from remote and noisy measurements Y ∈ R m×N which are obtained as a linear transformation A ∈ R m×n of X, corrupted with noise B ∈ R m×N : Y = AX + B. As the dimension m of Y is usually much smaller than the dimension n of X, these problems are ill-posed, and several solutions could lead to a given set of observations. The uncertainty of the measurements, which can be noisy, increases the number of potential solutions. Therefore, practitioners rely on prior knowledge of the data to select a plausible solution among all possible ones. On the one hand, hand-crafted priors relying on sparsity in a basis produce satisfying results on specific data, such as wavelets in imaging or Gaborlets in audio (Mallat, 2008) . However, the complexity and variability of the signals often make ad hoc priors inadequate. On the other hand, the prior can be learned from ground truth data when available. For instance, frameworks based on Plug-and-Play (Brifman et al., 2016) and Deep Learning (Chan et al., 2016; Romano et al., 2017; Rick Chang et al., 2017) propose to integrate a pre-trained denoiser in an iterative algorithm to solve the problem. Supervised methods leveraging sparsity also allow to summarize the structure of the signal (Elad, 2010). In particular, dictionary learning (Olshausen & Field, 1997; Aharon et al., 2006; Mairal et al., 2009) is efficient on pattern learning tasks such as blood cell detection or MEG signals analysis (Yellin et al., 2017; Dupré la Tour et al., 2018) . Nevertheless, these methods require clean data, sometimes available in audio and imaging but not in fields like neuroimaging or astrophysics. While data-driven methods have been extensively studied in the context of supervised inverse problems, recent works have focused on unsupervised scenarios and provided new algorithms to learn from corrupted data only (Lehtinen et al., 2018; Bora et al., 2018; Liu et al., 2020) . Chen et al. (2021) and Tachella et al. (2022) demonstrate that a necessary condition to learn extensive priors from degraded signals is either to measure them with multiple operators which span the whole space, or to introduce weak prior knowledge such as group structures and equivariance in the model when only one operator is available. Other works based on Deep Learning have leveraged successful architectures to recover images without access to any ground truth data. In particular, Deep Image Prior shows that CNNs contain enough prior information to recover an image in several 1

