Chen Li

Yuxin Dong, Tieliang Gong, Shujian Yu et al.

The matrix-based Rényi's entropy allows us to directly quantify information measures from given data, without explicit estimation of the underlying probability distribution. This intriguing property makes it widely applied in statistical inference and machine learning tasks. However, this information theoretical quantity is not robust against noise in the data, and is computationally prohibitive in large-scale applications. To address these issues, we propose a novel measure of information, termed low-rank matrix-based Rényi's entropy, based on low-rank representations of infinitely divisible kernel matrices. The proposed entropy functional inherits the specialty of of the original definition to directly quantify information from data, but enjoys additional advantages including robustness and effective calculation. Specifically, our low-rank variant is more sensitive to informative perturbations induced by changes in underlying distributions, while being insensitive to uninformative ones caused by noises. Moreover, low-rank Rényi's entropy can be efficiently approximated by random projection and Lanczos iteration techniques, reducing the overall complexity from $\mathcal{O}(n^3)$ to $\mathcal{O}(n^2 s)$ or even $\mathcal{O}(ns^2)$, where $n$ is the number of data samples and $s \ll n$. We conduct large-scale experiments to evaluate the effectiveness of this new information measure, demonstrating superior results compared to matrix-based Rényi's entropy in terms of both performance and computational efficiency.

Bingxue Wu, Jiao Wei, Chen Li et al.

The iterative weighted shrinkage-thresholding algorithm (IWSTA) has shown superiority to the classic unweighted iterative shrinkage-thresholding algorithm (ISTA) for solving linear inverse problems, which address the attributes differently. This paper proposes a new entropy regularized IWSTA (ERIWSTA) that adds an entropy regularizer to the cost function to measure the uncertainty of the weights to stimulate attributes to participate in problem solving. Then, the weights are solved with a Lagrange multiplier method to obtain a simple iterative update. The weights can be explained as the probability of the contribution of an attribute to the problem solution. Experimental results on CT image restoration show that the proposed method has better performance in terms of convergence speed and restoration accuracy than the existing methods.

Chen Li, Matthew Dunlop, Georg Stadler

We consider Bayesian inverse problems wherein the unknown state is assumed to be a function with discontinuous structure a priori. A class of prior distributions based on the output of neural networks with heavy-tailed weights is introduced, motivated by existing results concerning the infinite-width limit of such networks. We show theoretically that samples from such priors have desirable discontinuous-like properties even when the network width is finite, making them appropriate for edge-preserving inversion. Numerically we consider deconvolution problems defined on one- and two-dimensional spatial domains to illustrate the effectiveness of these priors; MAP estimation, dimension-robust MCMC sampling and ensemble-based approximations are utilized to probe the posterior distribution. The accuracy of point estimates is shown to exceed those obtained from non-heavy tailed priors, and uncertainty estimates are shown to provide more useful qualitative information.