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We consider minimization of a smooth nonconvex function with inexact oracle access to gradient and Hessian (without assuming access to the function value) to achieve approximate second-order optimality. A novel feature of our method is that…

In quantum mechanics, a norm squared wave function can be interpreted as the probability density that describes the likelihood of a particle to be measured in a given position or momentum. This statistical property is at the core of the…

Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples.…

We investigate the convergence properties of a class of iterative algorithms designed to minimize a potentially non-smooth and noisy objective function, which may be algebraically intractable and whose values may be obtained as the output…

Several optimization schemes have been known for convex optimization problems. However, numerical algorithms for solving nonconvex optimization problems are still underdeveloped. A progress to go beyond convexity was made by considering the…

This paper is concerned with function reconstruction from samples. The sampling points used in several approaches are (1) structured points connected with fast algorithms or (2) unstructured points coming from, e.g., an initial random draw…

We present a systematic study of the reconstruction of a non-negative function via maximum entropy approach utilizing the information contained in a finite number of moments of the function. For testing the efficacy of the approach, we…

In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. In several applications the…

Efficiency criteria for conformal prediction, such as \emph{observed fuzziness} (i.e., the sum of p-values associated with false labels), are commonly used to \emph{evaluate} the performance of given conformal predictors. Here, we…

The Bouncy Particle Sampler is a novel rejection-free non-reversible sampler for differentiable probability distributions over continuous variables. We generalize the algorithm to piecewise differentiable distributions and apply it to…

We derive an exact and efficient Bayesian regression algorithm for piecewise constant functions of unknown segment number, boundary location, and levels. It works for any noise and segment level prior, e.g. Cauchy which can handle outliers.…

Gradient boosting is a prediction method that iteratively combines weak learners to produce a complex and accurate model. From an optimization point of view, the learning procedure of gradient boosting mimics a gradient descent on a…

The optimization on the structure of process of information management under uncertain environment has attracted lots of attention from researchers around the world. Nevertheless, how to obtain accurate and rational evaluation from…

We propose a new probabilistic method for unsupervised recovery of corrupted data. Given a large ensemble of degraded samples, our method recovers accurate posteriors of clean values, allowing the exploration of the manifold of possible…

This paper discusses a class of uncertain optimization problems, in which unknown parameters are modeled by fuzzy intervals. The membership functions of the fuzzy intervals are interpreted as possibility distributions for the values of the…

Lucky imaging is a high-resolution astronomical image recovery technique with two classic implementation algorithms, i.e. image selecting, shifting and adding in image space and data selecting and image synthesizing in Fourier space. This…

This paper describes Picky, a probabilistic agenda-based chart parsing algorithm which uses a technique called {\em probabilistic prediction} to predict which grammar rules are likely to lead to an acceptable parse of the input. Using a…

Orthogonal matching pursuit (OMP) is a greedy algorithm popularly being used for the recovery of sparse signals. In this paper, we study the performance of OMP for support recovery of sparse signal under noise. Our analysis shows that under…

Maximizing high-dimensional, non-convex functions through noisy observations is a notoriously hard problem, but one that arises in many applications. In this paper, we tackle this challenge by modeling the unknown function as a sample from…

In this paper we show that the negative sample distance covariance function is a quasi-concave set function of samples of random variables that are not statistically independent. We use these properties to propose greedy algorithms to…