[Paper Review] Relevant sampling in finitely generated shift-invariant spaces

Hartmut Führ, Jun Xian|arXiv (Cornell University)|Oct 17, 2014

Mathematical Analysis and Transform Methods16 references3 citations

TL;DR

This paper establishes a probabilistic sampling framework for finitely generated shift-invariant spaces in $L^2(\mathbb{R}^n)$, showing that $O(R^n \log(R^n / \alpha'))$ independent uniform samples within a cube $C_R$ yield stable sampling for functions concentrated in $C_R$, under mild conditions on the generators. The key contribution is a high-probability sampling estimate with explicit bounds on constants derived from generator properties, extending Bass and Gr"ochenig's relevant sampling theory to general shift-invariant spaces with minimal assumptions.

ABSTRACT

We consider random sampling in finitely generated shift-invariant spaces $V(Φ) \subset { m L}^2(\mathbb{R}^n)$ generated by a vector $Φ= (φ_1,\ldots,φ_r) \in { m L}^2(\mathbb{R}^n)^r$. Following the approach introduced by Bass and Gröchenig, we consider certain relatively compact subsets $V_{R,δ}(Φ)$ of such a space, defined in terms of a concentration inequality with respect to a cube with side lengths $R$. Under very mild assumptions on the generators, we show that for $R$ sufficiently large, taking $O(R^n log(R^{n^2/α'}))$ many random samples (taken independently uniformly distributed within $C_R$) yields a sampling set for $V_{R,δ}(Φ)$ with high probability. Here $α' \le n$ is a suitable constant.We give explicit estimates of all involved constants in terms of the generators $φ_1, \ldots, φ_r$.

Motivation & Objective

• To develop a relevant sampling theory for finitely generated shift-invariant spaces in $L^2(\mathbb{R}^n)$, where sampling is restricted to a compact cube $C_R$ to address the lack of uniform distribution on $\mathbb{R}^n$.
• To extend Bass and Gr"ochenig's relevant sampling framework from bandlimited and trigonometric polynomial spaces to general finitely generated shift-invariant spaces with minimal assumptions on the generators.
• To derive explicit, quantitative bounds on the number of random samples required to achieve stable sampling with high probability, in terms of the generator functions' properties.
• To provide a complete probabilistic sampling estimate with explicit constants for the lower and upper sampling bounds, valid for functions in $V_R(\Phi)$, the subset of functions with energy mostly concentrated in $C_R$.

Proposed method

• Define the relevant space $V_{R,\delta}(\Phi)$ as functions in the shift-invariant space $V(\Phi)$ whose $L^2$-norm is at least $(1-\delta)$ concentrated on the cube $C_R = [-R/2, R/2]^n$.
• Use the frame and dual frame structure of the generators to represent functions via the projection operator $P_\Phi = \sum_{m,i} (T_m \phi_i) \otimes (T_m \hat{\phi}_i)$, ensuring reconstruction via inner products.
• Introduce the localization operator $Q_R P_\Phi$, whose eigenvalues $\lambda_j$ and eigenfunctions $\psi_j$ are used to decompose the space into finite-dimensional subspaces $P_N$.
• Apply matrix Bernstein inequalities to control the deviation of the empirical sampling operator from its expectation, by analyzing the random matrix $T_j = (\delta_{x_j} \otimes \delta_{x_j}) \circ Q_R P_\Phi$.
• Use covering index estimates and tail bounds for the covering number $N_0$ of random point sets to control the probability of poor sampling configurations.
• Combine norm estimates for projections onto finite-dimensional subspaces with probabilistic concentration inequalities to derive a high-probability sampling estimate with explicit constants.

Experimental results

Research questions

• RQ1 What is the minimal number of independent uniform random samples within a compact cube $C_R$ required to stably sample functions in $V_{R,\delta}(\Phi)$ with high probability?
• RQ2 How do the sampling bounds depend explicitly on the generator functions $\phi_1, \dots, \phi_r$ and their properties such as decay and frame bounds?
• RQ3 Can the relevant sampling framework of Bass and Gr"ochenig be extended to general finitely generated shift-invariant spaces with only mild assumptions on the generators?
• RQ4 What is the precise dependence of the sampling probability on the concentration parameter $\delta$, the dimension $n$, and the cube size $R$?
• RQ5 How can the constants in the sampling inequalities be explicitly estimated in terms of the generator functions' $L^2$-norms, frame bounds, and decay rates?

Key findings

• For $R$ sufficiently large, $O(R^n \log(R^n / \alpha'))$ independent uniform samples in $C_R$ yield a stable sampling set for $V_{R,\delta}(\Phi)$ with high probability, where $\alpha' \leq n$ is a constant derived from the generator decay.
• The lower sampling bound is $A \|f\|_2^2$ with $A = \frac{s}{R^n} \left( \frac{1}{2} - \delta - \nu - 12\delta C_2 \right)$, which is strictly positive under the stated conditions on $\delta$ and $\nu$, ensuring stability.
• The upper sampling bound is $s C_1^2 \|f\|_2^2$, where $C_1 = \sup_x \|v_x\|_2$ is the uniform bound on the reproducing kernel, ensuring no over-sampling.
• The probability of failure is bounded by $1 - \text{exp}(-\Omega(s/R^n))$, with explicit dependence on $R$, $s$, $C_1$, $C_2$, and $\delta$, showing high-probability sampling for large $s$.
• The constants in the sampling bounds are explicitly estimated in terms of the Bessel constants $C_0, \hat{C}_0$, the reproducing kernel bound $C_1$, the Plancherel-Polya constant $C_2$, and the decay constant $C_3$ of the generators.
• The results hold under mild assumptions: Bessel bounds, bounded point evaluation, Plancherel-Polya inequality, and polynomial decay of generators outside $C_R$, which are satisfied by compactly supported generators.