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Signal reconstruction using determinantal sampling

Ayoub Belhadji, Rémi Bardenet, Pierre Chainais

TL;DR

D determinantal sampling guarantees the so-called instance optimality property for a smaller number of function evaluations than i.i.d. sampling, and mean-square guarantees in $L^2$ norm are proved.

Abstract

We study the approximation of a square-integrable function from a finite number of evaluations on a random set of nodes according to a well-chosen distribution. This is particularly relevant when the function is assumed to belong to a reproducing kernel Hilbert space (RKHS). This work proposes to combine several natural finite-dimensional approximations based two possible probability distributions of nodes. These distributions are related to determinantal point processes, and use the kernel of the RKHS to favor RKHS-adapted regularity in the random design. While previous work on determinantal sampling relied on the RKHS norm, we prove mean-square guarantees in $L^2$ norm. We show that determinantal point processes and mixtures thereof can yield fast convergence rates. Our results also shed light on how the rate changes as more smoothness is assumed, a phenomenon known as superconvergence. Besides, determinantal sampling generalizes i.i.d. sampling from the Christoffel function which is standard in the literature. More importantly, determinantal sampling guarantees the so-called instance optimality property for a smaller number of function evaluations than i.i.d. sampling.

Signal reconstruction using determinantal sampling

TL;DR

D determinantal sampling guarantees the so-called instance optimality property for a smaller number of function evaluations than i.i.d. sampling, and mean-square guarantees in norm are proved.

Abstract

We study the approximation of a square-integrable function from a finite number of evaluations on a random set of nodes according to a well-chosen distribution. This is particularly relevant when the function is assumed to belong to a reproducing kernel Hilbert space (RKHS). This work proposes to combine several natural finite-dimensional approximations based two possible probability distributions of nodes. These distributions are related to determinantal point processes, and use the kernel of the RKHS to favor RKHS-adapted regularity in the random design. While previous work on determinantal sampling relied on the RKHS norm, we prove mean-square guarantees in norm. We show that determinantal point processes and mixtures thereof can yield fast convergence rates. Our results also shed light on how the rate changes as more smoothness is assumed, a phenomenon known as superconvergence. Besides, determinantal sampling generalizes i.i.d. sampling from the Christoffel function which is standard in the literature. More importantly, determinantal sampling guarantees the so-called instance optimality property for a smaller number of function evaluations than i.i.d. sampling.
Paper Structure (36 sections, 12 theorems, 142 equations, 5 figures, 2 tables)

This paper contains 36 sections, 12 theorems, 142 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Sampling and reconstruction in RKHSs
  3. Notations and assumptions
  4. Kernels and RKHSs
  5. Different kernels
  6. Increasing levels of smoothness
  7. Finite-dimensional approximations in RKHSs
  8. Approximations based on mixtures of kernel translates
  9. Approximations living in eigenspaces
  10. Transform-based approximations
  11. The empirical least-squares approximation
  12. Designs for finite-dimensional approximations
  13. Independent samples from the Christoffel function
  14. Determinantal sampling
  15. Existing results on RKHS sampling using DPPs and CVS
  16. ...and 21 more sections

Key Result

Theorem 1

Let $f \in \mathop{\mathrm{\mathcal{F}}}\nolimits$, and for $M\in \mathbb{N}^{*}$, let $f_{M}$ be its projection onto the eigenspace $\mathcal{E}_{M}$ defined by eq:eigen_approximation. Let $N\in \mathbb{N}^{*}$ and $\bm{x}=(x_1, \dots, x_N)$ be distributed according to the DPP in def:density_detsam

Figures (5)

  • Figure 1: A schematic diagram illustrating the relationship between the unit ball $\mathbb{B}_{\mathbb{L}_{2}(\omega)}$ of $\mathbb{L}_{2}(\omega)$, the unit ball $\mathbb{B}_{\mathop{\mathrm{\mathcal{F}}}\nolimits}$ of $\mathop{\mathrm{\mathcal{F}}}\nolimits$, and the image of $\mathbb{B}_{\mathbb{L}_{2}(\omega)}$ by the integration operator $\bm{\Sigma}$.
  • Figure 2: The reconstruction error for $f = e_{m}^{\mathcal{F}}$, when $\mathop{\mathrm{\mathcal{F}}}\nolimits$ is the periodic Sobolev space of order $s \in \{1,2\}$.
  • Figure 3: The reconstruction error for $50$ samples of $f$ from the distribution defined by \ref{['eq:f_random_xi']} when $\mathop{\mathrm{\mathcal{F}}}\nolimits$ is the periodic Sobolev space of order $s =1$.
  • Figure 4: The reconstruction error for $f = e_{m}^{\mathcal{F}}$, when $\mathop{\mathrm{\mathcal{F}}}\nolimits$ is the periodic Sobolev space of order $s \in \{1,2\}$ in the hypersphere $\mathbb{S}^{d-1}$ where $d=3$.
  • Figure 5: The reconstruction error for $f = e_{m}^{\mathcal{F}}$ in the RKHS associated to the Sinc kernel.

Theorems & Definitions (18)

  • Definition 1: A family of projection DPPs
  • Definition 2: continuous volume sampling
  • Theorem 1
  • Corollary 1
  • proof : Proof of \ref{['cor:dpprates']}
  • Corollary 2
  • proof
  • Lemma 1
  • Theorem 2
  • Corollary 3
  • ...and 8 more