Inverse Problems

Chairs: Xiuyuan Cheng (Duke) and Yimin Zhong (Duke)

Time: August 19th, 10:20am-11:10am ET, 16:20-17:10 CET, 22:20-23:10 GMT+8

  • Spectral Geometric Matrix Completion, Amit Boyarski (Technion); Sanketh Vedula (Technion), Alex Bronstein (Technion)

Paper Highlight, by Rachel Ward

Consider a matrix that is composed of a linear combination of the lowest harmonic vectors of some product graph. This is a structured low-rank model, but one which arises naturally in recommender systems, where similar users tend to give similar ratings, and similar items should have similar ratings. This paper proposes a novel approach to matrix completion, within this model framework: (1) Explicit spectral regularization via the Dirichlet energy, and (2) multi-resolution of the spectral loss. The multiresolution of the spectral loss is motivated by applications arising in shape correspondence. The approach can also be viewed as a generalization of a recent matrix completion method called deep matrix factorization inspired by overparameterized neural networks. Empirical results show that the above approach achieves state-of-art performance compared to other recent methods, particularly in the setting of data scarcity, where the number of measurements is small compared to the target rank of the matrix. This work is a significant contribution to matrix completion, in the form of introducing a simple geometric model and optimization method which is natural for recommender systems and which can practically outperform overparameterized deep nonlinear networks.

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  • Solving Bayesian Inverse Problems via Variational Autoencoders, Hwan Goh (Oden Institute of Computational Sciences and Engineering), Sheroze Sheriffdeen (Oden Institute); Jonathan Wittmer (Oden Institute of Computational Sciences and Engineering); Tan Bui-Thanh (Oden Institute of Computational Sciences and Engineering)

Paper Highlight, by Chiara Cammarota

Generally the power of machine learning develops in two main directions: discriminative modeling aims to learn a predictor given the observations, in generative modelling the more general problem of correctly sampling new real-like observations from the entire joint distribution and simulating real-world data generation processes is attacked. Variational Autoencoders (VAE) represent one of the most successful realizations of the second challenge. In “Solving Bayesian Inverse Problems via Variational Autoencoders” the authors propose an interesting perspective shift on VEAs by re-adapting them to a full-fledged modelling reconstruction with application to uncertainty quantification in scientific inverse problems. Uncertainty quantification variational autoencoder (UQ-VAE) results as a flexible modelling framework which takes full advantage, already during the training procedure, both from data and model structure inputs. To show its efficacy and robustness, the implementation of the UQ-VAE framework is proposed in a simple setting to the two dimensional steady state heat conduction problem and the results are compared with those obtained within the Laplace Approximation.

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  • Interpretable and Learnable Super-Resolution Time-Frequency Representation, Randall Balestriero (Rice University), Herve Glotin (); Richard Baraniuk (Rice University)

Paper Highlight, by Dennis Elbrachter

The paper introduces a method of obtaining super-resolved quadratic time-frequency representations via Gaussian filtering of the Wigner-Ville transform. It is both interpretable as well as computationally feasible, achieving state-of-the-art results on various datasets. I particularly enjoyed the clean presentation of formal results augmented by helpful explanations of the intuitions behind them.

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  • Phase Retrieval with Holography and Untrained Priors: Tackling the Challenges of Low-Photon Nanoscale Imaging, Hannah Lawrence (Flatiron Institute); David Barmherzig (); Henry Li (Yale); Michael Eickenberg (UC Berkeley); Marylou GabriĆ© (NYU / Flatiron Institute)

Paper Highlight, by Reinhard Heckel

The paper introduces a novel dataset-free deep learning framework for holographic phase retrieval. It shows, in a realistic simulation setups, that un-trained neural network enable to regularize holographic phase retrieval. It thus shows that non-linear inverse problems can be regularized with neural networks without any training, thereby making an important contribution in the intersection of machine learning and inverse problems.

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  • Multilevel Stein variational gradient descent with applications to Bayesian inverse problems, Terrence Alsup (New York University), Luca Venturi (Courant Institute of Mathematical Sciences); Benjamin Peherstorfer (Courant Institute of Mathematical Sciences),

Paper Highlight, by Joan Bruna

This paper presents a multilevel variant of Stein variational gradient descent, a canonical method to sample from a target distribution. The key idea of the method developed by Alstrup, Venturi and Peherstorfer is to obtain a sequence of increasingly complex measures converging to the true target distribution, and then using the previous levels as preconditioners for sampling the next one. The authors provide a nice theoretical analysis, establishing advantages of the multiscale method over the standard one, complemented with extensive numerical experiments on realistic scenarios, demonstrating speedups up to an order of magnitude.

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