Online Infinite-Dimensional Regression: Learning Linear Operators
We consider the problem of learning linear operators under squared loss between two infinite-dimensional Hilbert spaces in the online setting. We show that the class of linear operators with uniformly bounded $p$-Schatten norm is online learnable for any $p \in [1, \infty)$. On the other hand, we prove an impossibility result by showing that the class of uniformly bounded linear operators with respect to the operator norm is \textit{not} online learnable. Moreover, we show a separation between sequential uniform convergence and online learnability by identifying a class of bounded linear operators that is online learnable but uniform convergence does not hold. Finally, we prove that the impossibility result and the separation between uniform convergence and learnability also hold in the batch setting.
Code (0)
등록된 구현이 없습니다.
Tasks
regressionSimilar Papers 제목 키워드 기반
Universal Approximation of Nonlinear Operators and Their Derivatives
Establishing Universal Approximation Theorems (UATs) for nonlinear operators and their derivatives is a foundational open problem in Operator Learning (OL) and raises delicate questions in Nonlinear Functional Analysis. …
Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem
We consider the problem of learning a linear operator $\theta$ between two Hilbert spaces from empirical observations, which we interpret as least squares regression in infinite dimensions. We show that this goal can be …
regressionRigorous data-driven computation of spectral properties of Koopman operators for dynamical systems
Koopman operators are infinite-dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continu…
Learning Schatten--von Neumann Operators
We study the learnability of a class of compact operators known as Schatten--von Neumann operators. These operators between infinite-dimensional function spaces play a central role in a variety of applications in learnin…
Learning TheoryResidual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems
We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems (BIPs) with models governed by nonlinear parame…