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The Restricted Isometry Property of Subsampled Fourier Matrices
Ishay Haviv, Oded Regev
TL;DR
The paper addresses how many measurements are needed for restricted isometry in settings where measurement matrices may be sampled from a Fourier transform. It analyzes random row sampling and establishes a high-probability restricted-isometry result with an improved row bound.
Problem
Compressed sensing seeks measurement matrices with few rows, while many applications use randomly sampled unitary rows such as Fourier rows rather than arbitrary matrices.
Source: §1 · pp. 1–2Method
The paper analyzes matrices formed by uniformly and independently sampling and scaling rows of a unitary matrix.
Source: §1.1 · p. 2, §4.1 · p. 14Results
q = O(log^2(1/ε) · ε^-2 · k · log^2(k/ε) · log N) rows suffice for a high-probability restricted-isometry result.
Source: §3.1 · p. 9, §4.1 · p. 14Takeaways & Limitations
Subsampled Fourier matrices combine restricted-isometry guarantees with fast matrix-vector multiplication in compressed sensing.
Source: §1 · p. 2, §1.1 · p. 2The improved-analysis theorem is stated for sufficiently small ε and η.
Source: §4 · p. 10
Abstract
from arXiv · showhide
A matrix $A \in \mathbb{C}^{q \times N}$ satisfies the restricted isometry property of order $k$ with constant $\varepsilon$ if it preserves the $\ell_2$ norm of all $k$-sparse vectors up to a factor of $1\pm \varepsilon$. We prove that a matrix $A$ obtained by randomly sampling $q = O(k \cdot \log^2 k \cdot \log N)$ rows from an $N \times N$ Fourier matrix satisfies the restricted isometry property of order $k$ with a fixed $\varepsilon$ with high probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math., 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).
1 Introduction
The introduction frames restricted isometry as a condition relevant to sparse recovery and presents improved row bounds for randomly sampled Fourier matrices. It also outlines a multiscale-net proof and places the result among prior bounds and fast-multiplication approaches.
1 Introduction
- The restricted isometry property is motivated by sparse recovery: under the stated condition, reconstruction is equivalent to minimizing the ℓ1 norm among vectors matching the measurements.The introduction notes that this minimization can be formulated as a linear program and solved efficiently.Source: §1 · p. 1
- Compressed sensing seeks measurement matrices with few rows, and subsampled Fourier matrices retain fast matrix-vector multiplication for efficient compression and reconstruction.The introduction cites MRI, ultrasound imaging, and radio telescopes as settings where measurements may come from sampled unitary-matrix rows.Source: §1 · pp. 1–2
1.1 Our Contribution
- For fixed ε, q = O(k · log^2 k · log N) sampled rows yield the restricted isometry property of order k with constant ε with high probability.The rows are sampled uniformly and independently and scaled by N/q.Source: §1.1 · p. 2
- Unlike Rudelson and Vershynin and their improvements, the proof applies chaining directly with elementary arguments rather than Gaussian processes, symmetrization, or Dudley’s inequality.The authors describe the approach as a careful union bound over progressively finer nets.Source: §1.1 · p. 3
1.2 Related Literature
- The related-literature discussion compares the result with a tweaked Fourier construction, which bunches rows with random signs, and says the original matrices attain the same or slightly fewer rows.The tweaked construction is attributed to Nelson, Price, and Wootters.Source: §1.2 · p. 3
1.3 Proof Overview
- The proof overview normalizes sparse vectors so their transformed squared norm is at least 1/k, which sets the allowable additive approximation error.It then uses a sequence of nets for different scales; larger sets at lower-value coordinates need less accurate approximation, balancing the row-count requirements.Source: §1.3 · p. 4
2 Preliminaries
The preliminaries define the restricted isometry property and introduce notation used in the analysis. They also state concentration bounds and corollaries for bounded independent random variables.
Notation
- The notation x ≈ε,α y means x lies between (1 − ε)y − α and (1 + ε)y + α, while ∥M∥∞ denotes the largest absolute entry of M.Source: §2 · p. 5
The Restricted Isometry Property
- The restricted isometry property requires squared norms of every k-sparse vector and its image under A to agree within factors 1 − ε and 1 + ε.Source: §2 · p. 5
Chernoff-Hoeffding Bounds
- The preliminaries state a Chernoff-Hoeffding bound and derive corollaries for bounded independent variables, including complex-valued variables.The complex-valued corollary is obtained by applying a real-valued bound to real and imaginary parts.Source: §2 · pp. 5–6
3 The Simpler Analysis
The section proves a high-probability norm-preservation bound for random samples from a unitary matrix, then derives a restricted isometry guarantee for sparse vectors.
3 The Simpler Analysis
- q = O(ε^-3 · η^-1 · log N · log²(1/η)) uniform independent samples give the theorem’s high-probability guarantee for every vector.The failure probability is 2^-Ω(ε^-2 · log N · log(1/η)).Source: §3 · p. 6
- The proof bounds the sizes of finite vector families, applies concentration to their members, and uses union bounds across vectors and levels.Approximation vectors are constructed to match Mx on all but a γ fraction of coordinates in both [N] and the sample Q.Source: §3 · p. 7
3.1 The Restricted Isometry Property
- q = O(ε^-4 · k · log²(k/ε) · log N) sampled rows yield order-k restricted isometry with constant ε with probability 1 − 2^-Ω(ε^-2 · log N · log(k/ε)).The theorem concerns unitary matrices satisfying the stated entry-size condition.Source: §3.1 · pp. 9–10
4 The Improved Analysis
The improved analysis sharpens the ε dependence and supplies the sampling bound used to establish a restricted-isometry guarantee for randomly sampled rows of a unitary matrix.
4 The Improved Analysis
- Theorem 4.1 improves the ε dependence over Theorem 3.1, using q = O(log^2(1/ε) ε^-1η^-1 log N·log^2(1/η)) for its uniform-in-x guarantee with probability 1 − 2^-Ω(log N·log(1/η)).Source: §4 · p. 10
- The proof partitions coordinates by the first threshold index i met by g(i), then constructs h(i,m) and ∆(i,m) vectors supported within each set Ci.The collections Di,m comprise the resulting ∆(i,m) vectors.Source: §4 · p. 10
- The proof bounds the effect of bad coordinates on the expectation difference by 60(t + r)γ, which equals η by the choice of γ.Source: §4 · pp. 10, 11, 13
4.1 The Restricted Isometry Property
- q = O(log^2(1/ε) ε^-2 k log^2(k/ε) log N) rows suffice for the restricted isometry property of order k with constant ε, with probability 1 − 2^-Ω(log N·log(k/ε)).The sampled rows are multiplied by N/q.Source: §4.1 · p. 14