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Multiscale estimates for the condition number of non-harmonic Fourier matrices

Weilin Li

Abstract

This paper studies the extreme singular values of non-harmonic Fourier matrices. Such a matrix of size $m\times s$ can be written as $Φ=[ e^{-2πi j x_k}]_{j=0,1,\dots,m-1, k=1,2,\dots,s}$ for some set $\mathcal{X}=\{x_k\}_{k=1}^s$. Its condition number controls the stability of inversion, which is of great importance to super-resolution and nonuniform Fourier transforms. Under the assumption $m\geq 6s$ and without any restrictions on $\mathcal{X}$, the main theorems provide explicit lower bounds for the smallest singular value $σ_s(Φ)$ in terms of distances between elements in $\mathcal{X}$. More specifically, distances exceeding an appropriate scale $τ$ have modest influence on $σ_s(Φ)$, while the product of distances that are less than $τ$ dominates the behavior of $σ_s(Φ)$. These estimates reveal how the multiscale structure of $\mathcal{X}$ affects the condition number of Fourier matrices. Theoretical and numerical comparisons indicate that the main theorems significantly improve upon classical bounds and recover the same rate for special cases but with relaxed assumptions.

Multiscale estimates for the condition number of non-harmonic Fourier matrices

Abstract

This paper studies the extreme singular values of non-harmonic Fourier matrices. Such a matrix of size can be written as for some set . Its condition number controls the stability of inversion, which is of great importance to super-resolution and nonuniform Fourier transforms. Under the assumption and without any restrictions on , the main theorems provide explicit lower bounds for the smallest singular value in terms of distances between elements in . More specifically, distances exceeding an appropriate scale have modest influence on , while the product of distances that are less than dominates the behavior of . These estimates reveal how the multiscale structure of affects the condition number of Fourier matrices. Theoretical and numerical comparisons indicate that the main theorems significantly improve upon classical bounds and recover the same rate for special cases but with relaxed assumptions.
Paper Structure (25 sections, 10 theorems, 112 equations, 8 figures)

This paper contains 25 sections, 10 theorems, 112 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. A motivational multiscale example
  4. Main results
  5. Comparison with prior art
  6. Comparison to classical estimates
  7. Comparison to clumps
  8. Other related work
  9. Numerical simulations and examples
  10. Setup and definitions
  11. The motivational example revisited
  12. Another multiscale example
  13. Sparse spike train
  14. Colliding clumps
  15. Proof strategy
  16. ...and 10 more sections

Key Result

Theorem 1

Let $m,s\in\mathbb{N}_+$ such that $s\geq 2$ and $m\geq 6s$. Suppose $\mathcal{X}=\{x_k\}_{k=1}^s\subseteq\mathbb{T}$ and pick any $\tau$ such that $\mathcal{X}$ satisfies the $(m,\tau)$ density criterion. For each $k=1,2,\dots,s$, define and the subsets Then we have and in particular,

Figures (8)

  • Figure 1: Left: Plot of $\mathcal{X}$ defined in \ref{['eq:Xmotivation']}. Right: Plot of $\sigma_s(\Phi(m,\mathcal{X}))$ and two different lower bounds as functions of $m$ for $\mathcal{X}$ defined in \ref{['eq:Xmotivation']}.
  • Figure 2: An example of the three sets $\mathcal{I}_k$, $\mathcal{J}_k$ and $\mathcal{G}_k$ defined in \ref{['thm:main']}.
  • Figure 3: Plot of $\nu(\tau,\mathcal{X})$ as a function of $\tau\in (0,\frac{1}{2}]$.
  • Figure 4: For $\mathcal{X}$ defined in \ref{['eq:multiplescales']}, plot of $\sigma_s(\Phi(m,\mathcal{X}_\varepsilon))$, main theorem, and Gautschi-Bazán bound, as a function of $\varepsilon$, with $m=400$ on the left and $m=100$ on the right.
  • Figure 5: Left: Plot of the local sparsity $\nu(\tau,\mathcal{X}_{s,\varepsilon})$ as a function of $\tau$ for $\varepsilon=\frac{1}{2}$. Right: For $\mathcal{X}_{s,\varepsilon}$ defined in \ref{['eq:spiketrain']}, $m=200$, and $\varepsilon=0.9,0.7,0.5$, the graphs of $\sigma_s(\Phi(m,\mathcal{X}_{s,\varepsilon}))$, upper bound \ref{['eq:barnettupper']} with $C=500$, and lower bound \ref{['eq:main1']} are shown in solid, dashed circular markers, and dashed plus sign markers, respectively. The slopes of the upper bounds are $-\frac{\pi (1-\varepsilon)s}{2}$. The slopes of the lower bounds are $-1.2729$, $-1.5259$, and $-1.8625$, found by best linear fit.
  • ...and 3 more figures

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Theorem 1
  • Theorem 2
  • Definition 2.3
  • Corollary 1
  • Theorem 3
  • Definition 5.1
  • Lemma 5.2
  • proof
  • ...and 14 more