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Proximal splitting algorithms for monotone inclusions (and convex optimization problems) in Hilbert spaces share the common feature to guarantee for the generated sequences in general weak convergence to a solution. In order to achieve…

Integration is indispensable, not only in mathematics, but also in a wide range of other fields. A deep learning method has recently been developed and shown to be capable of integrating mathematical functions that could not previously be…

In this paper, we analyze the mirror descent algorithm for non-smooth optimization problems in which the objective function is relatively strongly convex, without relying on the standard Lipschitz continuity assumption commonly used in the…

A Radial Basis Function Generated Finite-Differences (RBF-FD) inspired technique for evaluating definite integrals over bounded volumes that have smooth boundaries in three dimensions is described. A key aspect of this approach is that it…

This paper studies first-order algorithms for solving fully composite optimization problems over convex and compact sets. We leverage the structure of the objective by handling its differentiable and non-differentiable components…

We investigate robust optimization problems defined for maximizing convex functions. For finite uncertainty set, we develop a geometric branch-and-bound algorithmic approach to solve this problem. The geometric branch-and-bound algorithm…

For many applications in signal processing and machine learning, we are tasked with minimizing a large sum of convex functions subject to a large number of convex constraints. In this paper, we devise a new random projection method (RPM) to…

Low rank matrix approximation is a popular topic in machine learning. In this paper, we propose a new algorithm for this topic by minimizing the least-squares estimation over the Riemannian manifold of fixed-rank matrices. The algorithm is…

We provide a unifying framework where artificial neural networks and their architectures can be formally described as particular cases of a general mathematical construction--machines of finite depth. Unlike neural networks, machines have a…

Zeroth-order methods have become important tools for solving problems where we have access only to function evaluations. However, the zeroth-order methods only using gradient approximations are $n$ times slower than classical first-order…

Extending the idea of Even and Lehrer [3], we discuss a general approach to integration based on a given decomposition system equipped with a weighting function, and a decomposition of the integrated function. We distinguish two type of…

By restricting the iterate on a nonlinear manifold, the recently proposed Riemannian optimization methods prove to be both efficient and effective in low rank tensor completion problems. However, existing methods fail to exploit the easily…

For nonconvex and nonsmooth restoration models, the lower bound theory reveals their good edge recovery ability, and related analysis can help to design convergent algorithms. Existing such discussions are focused on isotropic…

An integral equation is a way to encapsulate the relationships between a function and its integrals. We develop a systematic way of describing Volterra integral equations -- specifically an algorithm that reduces any separable Volterra…

In this paper, we propose and analyse a family of generalised stochastic composite mirror descent algorithms. With adaptive step sizes, the proposed algorithms converge without requiring prior knowledge of the problem. Combined with an…

Optimization problems, arise in many practical applications, from the view points of both theory and numerical methods. Especially, significant improvement in deep learning training came from the Quasi-Newton methods. Quasi-Newton search…

The Douglas--Rachford algorithm is a classic splitting method for finding a zero of the sum of two maximal monotone operators. It has also been applied to settings that involve one weakly and one strongly monotone operator. In this work, we…

Numerical integration is a foundational technique in scientific computing and is at the core of many computer vision applications. Among these applications, neural volume rendering has recently been proposed as a new paradigm for view…

This article is written with the hope to draw attention to a method that uses integral transforms to find exact values for a large class of convergent series (and, in particular, series of rational terms). We apply the method to some series…

The downward closure of a word language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of any language is regular. While the downward closure appears to be a powerful…