Variational approaches for signal and image processing
Quand ? |
Le 18/11/2022, de 10:00 à 17:00 |
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Participants |
Barbara Pascal / Audrey Repetti / Julian Tachella / Titouan Vayer |
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Location: Salle 101, ENSL
Program:
Learning Maximally Monotone Operators for Image Recovery
14h-15h: Titouan Vayer (INRIA & LIP, ENSL)
Towards Compressive Recovery of Sparse Precision Matrices
15h-16h: Julian Tachella (CNRS & LP, ENSL)
Imaging with Equivariant Deep Learning
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Abstract Kadambari Sai Kiran : This work focuses on learning product graphs from multi-domain data. We assume that the product graph is formed by the Cartesian product of two smaller graphs, which we refer to as graph factors. We pose the product graph learning problem as the problem of estimating the graph factor Laplacian matrices. To capture local interactions in data, we seek sparse graph factors and assume a smoothness model for data. We propose an efficient iterative solver for learning sparse product graphs from data. We then extend this solver to infer multi-component graph factors with applications to product graph clustering by imposing rank constraints on the graph Laplacian matrices. Although working with smaller graph factors is computationally more attractive, not all graphs readily admit an exact Cartesian product factorization. To this end, we propose efficient algorithms to approximate a graph by a nearest Cartesian product of two smaller graphs. The efficacy of the developed framework is demonstrated using several numerical experiments on synthetic and real data.
Abstract Barbara Pascal : Monitoring the Covid19 pandemic constitutes a critical societal stake that received considerable research efforts. Raw infection counts are not informative enough about the pandemic spread dynamics, and one has to recourse to more advanced epidemiological indicators, the most popular being the reproduction number, defined in Cori's model as the average number of secondary cases caused by an infected individual. Though, the quality of Covid19 data, consisting in daily new infection counts reported by public health authorities, is low (due, e.g., to pseudo-seasonalities, irrelevant or missing counts), making robust estimation of daily reproduction numbers very challenging.
Abstract Julian Tachella: From early image processing to modern computational imaging, successful models and algorithms have relied on a fundamental property of natural signals: symmetry. Here symmetry refers to the invariance property of signal sets to transformations such as translation, rotation or scaling. Symmetry can also be incorporated into deep neural networks in the form of equivariance, allowing for more data-efficient learning. While there has been important advances in the design of end-to-end equivariant networks for image classification in recent years, computational imaging introduces unique challenges for equivariant network solutions since we typically only observe the image through some noisy ill-conditioned forward operator that itself may not be equivariant. In this talk, I will present the emerging field of equivariant imaging and show how it can provide improved generalization and new imaging opportunities. Along the way I will show the interplay between the acquisition physics and group actions and links to iterative reconstruction, blind compressed sensing and self-supervised learning.
Abstract Titouan Vayer: We consider the problem of learning a graph modeling the statistical relations of the $d$ variables of a dataset with $n$ samples. Standard approaches amount to searching for a precision matrix $\boldsymbol\Theta$ representative of a Gaussian graphical model that adequately explains the data. However, most maximum likelihood based estimators usually require to store the $d^{2}$ values of the empirical covariance matrix, which is often too costly in a high-dimensional setting. In this work we adopt a ‘‘compressive'' viewpoint and look for estimating a sparse $\boldsymbol\Theta$ from a \emph{sketch} of the data, \textit{i.e.} a low-dimensional vector of size $m << d^{2}$ carefully designed from the data using nonlinear random features. Under certain assumptions on the spectrum of $\boldsymbol\Theta$, we show that it is possible to estimate it robustly from a sketch of size $m=\mathcal{O}\left((d+2k)\ln(d)\right)$ where $k$ is the maximal number of edges of the underlying graph. We also show that our estimator has an error that decreases in $\mathcal{O}(n^{-1/2})$. These guarantees are inspired from the compressed sensing theory and involve restricted isometric properties and instance optimal decoders. Our estimator requires solving a non-convex inverse problem and we investigate the possibility of achieving practical recovery by a variant of the Davis-Yin three operator splitting algorithm. We compare our approach with ‘‘Graphical LASSO'' type estimators on synthetic datasets. Finally, we discuss in a last part the limitations and perspectives of this work, which partially answers some questions but also opens many others.