Related papers: The foundations of spectral computations via the S…
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
Classical approach of solvability problem has shed much light on what we can solve and what we cannot solve mathematically. Starting with quadratic equation, we know that we can solve it by the quadratic formula which uses square root.…
Spectral clustering is a popular algorithm that clusters points using the eigenvalues and eigenvectors of Laplacian matrices derived from the data. For years, spectral clustering has been working mysteriously. This paper explains spectral…
This is a survey on the use of low-degree polynomials to predict and explain the apparent statistical-computational tradeoffs in a variety of average-case computational problems. In a nutshell, this framework measures the complexity of a…
Inverse spectral problem for a self-adjoint differential operator, which is the sum of the operator of the third derivative on a finite interval and of the operator of multiplication by a real function (potential), is solved. Closed system…
Our goal is to provide simple and practical algorithms in higher-order Fourier analysis which are based on spectral decompositions of operators. We propose a general framework for such algorithms and provide a detailed analysis of the…
We show that (i) any constrained polynomial optimization problem (POP) has an equivalent formulation on a variety contained in an Euclidean sphere and (ii) the resulting semidefinite relaxations in the moment-SOS hierarchy have the constant…
We study the problem of detecting zeros of continuous functions that are known only up to an error bound, extending the earlier theoretical work with explicit algorithms and experiments with an implementation. More formally, the robustness…
Solving ill-posed inverse problems necessitates effective regularization strategies to stabilize the inversion process against measurement noise. While classical methods like Tikhonov regularization require heuristic parameter tuning, and…
This paper investigates stochastic invariance for control systems through probabilistic controlled invariant sets (PCISs). As a natural complement to robust controlled invariant sets~(RCISs), we propose finite- and infinite-horizon PCISs,…
Despite the recent progress of automated program verification techniques, fully automated verification of programs manipulating recursive data structures remains a challenge. We introduce solvable tuple patterns (STPs) and conjunctive STPs…
Stochastic Optimal Control (SOC) problems arise in systems influenced by uncertainty, such as autonomous robots or financial models. Traditional methods like dynamic programming are often intractable for high-dimensional, nonlinear systems…
Motivated by inverse problems with a single passive measurement, we introduce and analyze a new class of inverse spectral problems on closed Riemannian manifolds. Specifically, we establish two general uniqueness results for the recovery of…
We derive, similar to Lau and Riha, a matrix formulation of a general best approximation theorem of Singer for the special case of spectral approximations of a given matrix from a given subspace. Using our matrix formulation we describe the…
A consistent functional calculus approach to the spectral theorem for strongly commuting normal operators on Hilbert spaces is presented. In contrast to the common approaches using projection-valued measures or multiplication operators,…
The spectral theory on the $S$-spectrum was born out of the need to give quaternionic quantum mechanics (formulated by Birkhoff and von Neumann) a precise mathematical foundation. Then it turned out that this theory has important…
The variation of spectral subspaces for linear self-adjoint operators under an additive bounded semidefinite perturbation is considered. A variant of the Davis-Kahan $ \sin2\Theta $ theorem from [SIAM J. Numer. Anal. 7 (1970), 1--46]…
This paper investigates spectral properties of certain classes of positive operators originated from different matrices appeared in linear complementarity problem. These positive operators play a crucial role in various areas of mathematics…
Conditional independence (CI) is central to causal inference, feature selection, and graphical modeling, yet it is untestable in many settings without additional assumptions. Existing CI tests often rely on restrictive structural…
Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been…