Related papers: Algorithmic Information Bounds for Distances and O…
We consider the problem of minimizing a non-convex function over a smooth manifold $\mathcal{M}$. We propose a novel algorithm, the Orthogonal Directions Constrained Gradient Method (ODCGM) which only requires computing a projection onto a…
We study the problem of reconstructing the latent geometry of a $d$-dimensional Riemannian manifold from a random geometric graph. While recent works have made significant progress in manifold recovery from random geometric graphs, and more…
While Kolmogorov complexity is the accepted absolute measure of information content in an individual finite object, a similarly absolute notion is needed for the information distance between two individual objects, for example, two…
In this paper, we propose a fast and convergent algorithm to solve unassigned distance geometry problems (uDGP). Technically, we construct a novel quadratic measurement model by leveraging $\ell_0$-norm instead of $\ell_1$-norm in the…
Resource-constrained systems are prevalent in communications. Such a system is composed of many components but only some of them can be allocated with resources such as time slots. According to the amount of information about the system,…
We propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator…
We investigate the relationship between algorithmic fractal dimensions and the classical local fractal dimensions of outer measures in Euclidean spaces. We introduce global and local optimality conditions for lower semicomputable outer…
We investigate error bounds for numerical solutions of divergence structure linear elliptic PDEs on compact manifolds without boundary. Our focus is on a class of monotone finite difference approximations, which provide a strong form of…
This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We…
We propose a machine learning enhanced algorithm for solving the optimal landing problem. Using Pontryagin's minimum principle, we derive a two-point boundary value problem for the landing problem. The proposed algorithm uses deep learning…
In this paper we develop proximal methods for statistical learning. Proximal point algorithms are useful in statistics and machine learning for obtaining optimization solutions for composite functions. Our approach exploits closed-form…
Error bounds have been studied for more than seventy years, beginning with the seminal result of Hoffman (1952) [{\it J. Res. Natl. Bur. Standards}, 49 (1952), 263--265], which establishes an upper bound for the distance from an arbitrary…
Autonomous exploration in mobile robotics often involves a trade-off between two objectives: maximizing environmental coverage and minimizing the total path length. In the widely used information gain paradigm, exploration is guided by the…
We study projections onto non-degenerate one-dimensional families of lines and planes in $\mathbb{R}^{3}$. Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most…
We investigate the information complexity of mixed-integer convex optimization under different types of oracles. We establish new lower bounds for the standard first-order oracle, improving upon the previous best known lower bound. This…
This paper addresses the question of when projections of a high-dimensional random vector are approximately Gaussian. This problem has been studied previously in the context of high-dimensional data analysis, where the focus is on…
We consider statistics for stochastic evolution equations in Hilbert space with emphasis on stochastic partial differential equations (SPDEs). We observe a solution process under additional measurement errors and want to estimate a real or…
We study finite horizon reachable set estimation for unknown discrete-time dynamical systems using only sampled state trajectories. Rather than treating scenario optimization as a black-box tool, we show how it can be tailored to reachable…
Many classical and modern machine learning algorithms require solving optimization tasks under orthogonality constraints. Solving these tasks with feasible methods requires a gradient descent update followed by a retraction operation on the…
For conforming finite element approximations of the Laplacian eigenfunctions, a fully computable guaranteed error bound in the $L^2$ norm sense is proposed. The bound is based on the a priori error estimate for the Galerkin projection of…