Optimal shift-invariant spaces from uniform measurements
Rohan Joy, Radha Ramakrishnan
TL;DR
The paper develops a fiber-map framework to identify optimal finitely generated shift-invariant subspaces from uniform, noisy measurements of multiple signals. It proves existence of optimal FSISs under three common subspace classes: unrestricted FSISs, FSISs with $\mathbb{Z}/n_0$ extra invariance, and translation-invariant FSISs, and provides explicit constructions where possible. The methodology recasts data-fitting as a fiber-wise projection problem on a mixed space $\mathcal{R}_\lambda$ and uses range-function theory to derive per-fiber decompositions and optimal subspaces. It also shows that Paley–Wiener-type finite-dimensional approximations yield implementable optimization and outlines a spectral-selection procedure to maximize measurement-energy capture. These results offer a principled, data-driven route to model-order selection for signal approximation using shift-invariant frameworks.
Abstract
Let $m$ be a positive integer and $\mathcal{C}$ be a collection of closed subspaces in $L^2(\mathbb{R})$. Given the measurements $\mathcal{F}_Y=\left\lbrace \left\lbrace y_k^1 \right\rbrace_{k\in \mathbb{Z}},\ldots, \left\lbrace y_k^m \right\rbrace_{k\in \mathbb{Z}} \right\rbrace \subset \ell^2(\mathbb{Z})$ of unknown functions $\mathcal{F}=\left\{f_1, \ldots,f_m \right\} \subset L^2( \mathbb{R})$, in this paper we study the problem of finding an optimal space $S$ in $\mathcal{C}$ that is ``closest" to the measurements $\mathcal{F}_Y$ of $\mathcal{F}$. Since the class of finitely generated shift-invariant spaces (FSISs) is popularly used for modelling signals, we assume $\mathcal{C}$ consists of FSISs. We will be considering three cases. In the first case, $\mathcal{C}$ consists of FSISs without any assumption on extra invariance. In the second case, we assume $\mathcal{C}$ consists of extra invariant FSISs, and in the third case, we assume $\mathcal{C}$ has translation-invariant FSISs. In all three cases, we prove the existence of an optimal space.