Related papers: Matrix approximations of operators
While many problems in machine learning focus on learning mappings between finite-dimensional spaces, scientific applications require approximating mappings between function spaces, i.e., operators. We study the problem of learning…
For finite-dimensional operator systems $\mathcal{S}_{\mathsf T}$, ${\mathsf T} \in B({\mathcal H})^d$, we show that the local lifting property and $1$-exactness of $\mathcal{S}_{\mathsf T}$ may be characterized by measurements of the…
The problem of approximating the discrete spectra of families of self-adjoint operators that are merely strongly continuous is addressed. It is well-known that the spectrum need not vary continuously (as a set) under strong perturbations.…
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…
We describe a procedure based on the iteration of an initial function by an appropriated operator, acting on continuous functions, in order to get a fixed point. This fixed point will be a calibrated subaction for the doubling map on the…
Operator learning based on neural operators has emerged as a promising paradigm for the data-driven approximation of operators, mapping between infinite-dimensional Banach spaces. Despite significant empirical progress, our theoretical…
We present a method to approximate functionals $\text{Tr} \, f(A)$ of very high-dimensional hermitian matrices $A$ represented as Matrix Product Operators (MPOs). Our method is based on a reformulation of a block Lanczos algorithm in tensor…
Condensates of active particles such as cells form almost-crystalline lattices which play a central role in many biological systems. Typically, their properties have been determined merely by analogy to the rather trivial one-dimensional…
In this paper, we study matrix representations of truncated Toeplitz operators with respect to orthonormal bases which are invariant under a canonical conjugation map. In particular, we determine necessary and sufficient conditions for when…
This investigation seeks to establish the practicality of numerical frame approximations. Specifically, it develops a new method to approximate the inverse frame operator and analyzes its convergence properties. It is established that…
The mathematical models used to represent physical phenomena are generally known to be imperfect representations of reality. Model inadequacies arise for numerous reasons, such as incomplete knowledge of the phenomena or computational…
Starting from the matrix elements of the nucleon-nucleon interaction in momentum space we present a method to derive an operator representation with a minimal set of operators that is required to provide an optimal description of the…
The integration operators (*) $({\mathcal J}^+\,g)(x) = \int_a^x g(t) \, dt$ and (**) $({\mathcal J}^-\,g)(x) = \int_x^b g(t) \, dt$ defined on an interval $(a,b) \subseteq {\mathbf R}$ yield new identities for indefinite convolutions,…
The condition number of a diagonally scaled matrix, for appropriately chosen scaling matrices, is often less than that of the original. Equilibration scales a matrix so that the scaled matrix's row and column norms are equal. Scaling can be…
We consider discretizations of the hyper-singular integral operator on closed surfaces and show that the inverses of the corresponding system matrices can be approximated by blockwise low-rank matrices at an exponential rate in the block…
We consider the problem of matrix completion on an $n \times m$ matrix. We introduce the problem of Interpretable Matrix Completion that aims to provide meaningful insights for the low-rank matrix using side information. We show that the…
Recent years have seen rapid advances in the data-driven analysis of dynamical systems based on Koopman operator theory and related approaches. On the other hand, low-rank tensor product approximations -- in particular the tensor train (TT)…
We consider a minimal realization of a rational matrix functions. We perturb the polynomial part and one of the constant matrices from the realization part. We derive explicit computable expressions of backward errors of approximate…
The effect of matrix perturbations on the polar decomposition has been studied by several authors and various results are known. However, for operators between infinite-dimensional spaces the problem has not been considered so far. Here, we…
Quantum algorithms for diverse problems, including search and optimization problems, require the implementation of a reflection operator over a target state. Commonly, such reflections are approximately implemented using phase estimation.…