Minh-Hai Nguyen

Minh-Hai Nguyen, Edouard Pauwels, Pierre Weiss

The Maximum A Posteriori (MAP) estimation is a widely used framework in blind deconvolution to recover sharp images from blurred observations. The estimated image and blur filter are defined as the maximizer of the posterior distribution. However, when paired with sparsity-promoting image priors, MAP estimation has been shown to favors blurry solutions, limiting its effectiveness. In this paper, we revisit this result using diffusion-based priors, a class of models that capture realistic image distributions. Through an empirical examination of the prior's likelihood landscape, we uncover two key properties: first, blurry images tend to have higher likelihoods; second, the landscape contains numerous local minimizers that correspond to natural images. Building on these insights, we provide a theoretical analysis of the blind deblurring posterior. This reveals that the MAP estimator tends to produce sharp filters (close to the Dirac delta function) and blurry solutions. However local minimizers of the posterior, which can be obtained with gradient descent, correspond to realistic, natural images, effectively solving the blind deconvolution problem. Our findings suggest that overcoming MAP's limitations requires good local initialization to local minima in the posterior landscape. We validate our analysis with numerical experiments, demonstrating the practical implications of our insights for designing improved priors and optimization techniques.

Graham E. Rowlands, Minh-Hai Nguyen, Guilhem J. Ribeill et al.

The rapidity and low power consumption of superconducting electronics makes them an ideal substrate for physical reservoir computing, which commandeers the computational power inherent to the evolution of a dynamical system for the purposes of performing machine learning tasks. We focus on a subset of superconducting circuits that exhibit soliton-like dynamics in simple transmission line geometries. With numerical simulations we demonstrate the effectiveness of these circuits in performing higher-order parity calculations and channel equalization at rates approaching 100 Gb/s. The availability of a proven superconducting logic scheme considerably simplifies the path to a fully integrated reservoir computing platform and makes superconducting reservoirs an enticing substrate for high rate signal processing applications.

Wendson A. S. Barbosa, Aaron Griffith, Graham E. Rowlands et al.

We demonstrate that matching the symmetry properties of a reservoir computer (RC) to the data being processed dramatically increases its processing power. We apply our method to the parity task, a challenging benchmark problem that highlights inversion and permutation symmetries, and to a chaotic system inference task that presents an inversion symmetry rule. For the parity task, our symmetry-aware RC obtains zero error using an exponentially reduced neural network and training data, greatly speeding up the time to result and outperforming hand crafted artificial neural networks. When both symmetries are respected, we find that the network size $N$ necessary to obtain zero error for 50 different RC instances scales linearly with the parity-order $n$. Moreover, some symmetry-aware RC instances perform a zero error classification with only $N=1$ for $n\leq7$. Furthermore, we show that a symmetry-aware RC only needs a training data set with size on the order of $(n+n/2)$ to obtain such performance, an exponential reduction in comparison to a regular RC which requires a training data set with size on the order of $n2^n$ to contain all $2^n$ possible $n-$bit-long sequences. For the inference task, we show that a symmetry-aware RC presents a normalized root-mean-square error three orders-of-magnitude smaller than regular RCs. For both tasks, our RC approach respects the symmetries by adjusting only the input and the output layers, and not by problem-based modifications to the neural network. We anticipate that generalizations of our procedure can be applied in information processing for problems with known symmetries.