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Sparse Recovery by Non-convex Optimization -- Instance Optimality

Rayan Saab, Ozgur Yilmaz

arXiv:0809.0745v2cs.ITmath.IT

TL;DR

Sparse recovery is challenging because the measurement system is underdetermined and, when consistent, admits infinitely many solutions, so a decoder must select the desired solution. The paper analyzes ℓp-minimization decoders for 0 < p < 1, extending deterministic ℓ1 guarantees and the probabilistic argument for random measurement matrices. For 0 < p < 1, the paper establishes robust, stable (2,p) instance optimality under sufficient conditions and (2,2) instance optimality in probability for appropriate random matrices.

  • Problem

    Sparse recovery is challenging because the measurement system is underdetermined and, when consistent, admits infinitely many solutions, so a decoder must select the desired solution.

    Source: §1 · p. 2

  • Method

    The paper analyzes ℓp-minimization decoders for 0 < p < 1, extending deterministic ℓ1 guarantees and the probabilistic argument for random measurement matrices.

    Source: §1.3 · p. 7, §2.3 · p. 11

  • Results

    For 0 < p < 1, the paper establishes robust, stable (2,p) instance optimality under sufficient conditions and (2,2) instance optimality in probability for appropriate random matrices.

    Source: §1.3 · p. 7, §2.1 · p. 8

  • Takeaways & Limitations

    The results extend ℓ1 recovery guarantees to ℓp decoders, including deterministic noise robustness and probabilistic instance optimality under the stated matrix conditions and distributions.

    Source: §1.3 · p. 7, §2.1 · p. 8

    The modified algorithms discussed for ℓp minimization have only been shown to converge to local minima, though the results also hold for feasible points with objective no larger than the target’s.

    Source: §1.2.1 · p. 7

Abstract

from arXiv · show

In this note, we address the theoretical properties of $\Delta_p$, a class of compressed sensing decoders that rely on $\ell^p$ minimization with 0<p<1 to recover estimates of sparse and compressible signals from incomplete and inaccurate measurements. In particular, we extend the results of Candes, Romberg and Tao, and Wojtaszczyk regarding the decoder $\Delta_1$, based on $\ell^1$ minimization, to $\Delta_p$ with 0<p<1. Our results are two-fold. First, we show that under certain sufficient conditions that are weaker than the analogous sufficient conditions for $\Delta_1$ the decoders $\Delta_p$ are robust to noise and stable in the sense that they are (2,p) instance optimal for a large class of encoders. Second, we extend the results of Wojtaszczyk to show that, like $\Delta_1$, the decoders $\Delta_p$ are (2,2) instance optimal in probability provided the measurement matrix is drawn from an appropriate distribution.

1 Introduction

Sparse recovery must select a useful solution from infinitely many solutions to an underdetermined measurement system; this paper extends ℓ1 recovery guarantees to non-convex ℓp decoders. It addresses recovery under noise and signal compressibility, including deterministic stability and robustness and probabilistic instance optimality.

1 Introduction

  • An underdetermined measurement system can have infinitely many solutions, so sparse recovery must select the desired solution using a sparsity measure.In the noise-free case, a decoder ∆0 recovers all sufficiently sparse signals under certain conditions on A.Source: §1 · p. 2

1.1 Decoding by ℓ1 minimization

  • For ℓ1 recovery, error bounds involve σS(x)ℓ1/S, while replacing it uniformly with σS(x)ℓ2 for a fixed matrix requires M > cN.That measurement scale conflicts with recovery from significantly fewer measurements than the ambient dimension.Source: §1.1 · pp. 4–5

1.2 Decoding by ℓp minimization

  • ℓp decoders with 0 < p < 1 approximate the combinatorial ℓ0 decoder through non-convex optimization, and the paper asks whether they inherit ℓ1 guarantees under weaker conditions.Earlier work reported less restrictive sufficient conditions for perfect recovery, while the paper studies robustness, stability, and instance optimality.Source: §1.2 · p. 6

1.2.1 Algorithmic Issues

  • The modified algorithms discussed for ℓp recovery have only been shown to converge to local minima, although the paper’s results also hold for feasible points with ℓp objective no larger than that of the target.The optimization problem has many local minimizers.Source: §1.2.1 · p. 7

1.3 Paper Outline

  • The paper extends stability and robustness guarantees to ℓp minimization, with sufficient matrix conditions that are weaker for smaller p than for p = 1.It also extends probabilistic (2,2) instance optimality to ℓp decoders for appropriate random measurement matrices.Source: §1.3 · p. 7
  • The paper’s outline covers deterministic stability and robustness, probabilistic (2,2) instance optimality, numerical results, and proofs.The probabilistic result uses the same dimension range as the ℓ1 result, M > cS log(N), for an appropriate measurement-matrix distribution.Source: §1.3 · p. 7

2 Main Results

The paper extends sparse recovery guarantees for ℓ1 minimization to ℓp decoders, establishing deterministic stability and robustness as well as instance optimality in probability under appropriate conditions. It also relates the sparsity levels recoverable by Δ1 and Δp.

2.1 Sparse recovery with ∆p: stability and robustness

  • The sufficient conditions are weaker for smaller p than for p=1, though the compared conditions (16) and (17) are not uniformly ordered across m.Condition (16) is weaker near m=2, while (17) is weaker as m grows.Source: §2.1 · pp. 8–9, Figure 1 · p. 10
  • For 0 < p ≤ 1, if A satisfies P(k,S,p), then (A, Delta_p) is (2,p) instance optimal of order S with constant C1/p.Source: §2.1 · p. 9

2.2 The relationship between S1 and Sp

  • Proposition 2.10 relates the sparsity levels S1 and Sp: Δ1 recovers all S1-sparse vectors, while Δp recovers all Sp-sparse vectors.The supplied passage does not specify the relation’s formula.Source: §2.2 · p. 11

2.3 Instance optimality in probability and ∆p

  • The probability result generalizes LQ1 to LQp; after establishing LQp for the random matrices, the proof adapts Wojtaszczyk’s argument for the non-convex ℓp-quasinorm.The approach modifies the earlier proof to account for non-convexity when 0<p<1.Source: §2.3 · pp. 11–12
  • For Gaussian matrices with N≥M[log(M)]^2 and S≤c1M/log(N/M), the paper establishes (2,2) instance optimality in probability for Δp.The statement also holds for matrices whose columns are independently drawn uniformly from the sphere.Source: §2.3 · pp. 11, 13
  • The Δp decoder requires no knowledge of the noise level, which is useful when noise estimates are unavailable.The paper identifies this as a practical advantage over Theorem 2.1.Source: §2.3 · p. 14

3 Numerical Experiments

The experiments examine recovery guarantees and reconstruction error for ℓp decoders across compressible signals and noisy observations. They report roughly linear error growth, advantages for smaller p in some settings, and empirical success beyond the stated sufficient conditions.

3 Numerical Experiments

  • For Gaussian A∈R^100×300, Figure 2 compares theoretical recovery regions with empirical recovery rates for different (S,p) pairs.The theoretical regions rely on estimated RIP constants being correct, although the estimates are only lower bounds.Source: §3 · p. 14, Figure 2 · p. 15
  • Empirical recovery succeeds over a wider range of scenarios than the sufficient conditions predict, possibly because those conditions are only sufficient or recovery is not uniform.The authors also interpret the practical results as possible high-probability success on either the signal or the measurement matrix.Source: §3 · pp. 14–15
  • Reconstruction error grows roughly linearly with compressibility error and noise, while p<1 lowers error for highly compressible signals or low noise.For p near one, sparse noiseless signals still had nonzero average error in the experiment.Source: §3 · p. 16, Figure 3 · p. 16
  • At q=0.4, p<0.6 yields a 5 dB gain over p=1; at q=0.6 the advantage is about 2 dB, and at q=0.9 performance is nearly identical.The tested signals have power-law-decaying sorted coefficients; the results indicate p≈q gives the lowest reconstruction errors.Source: §3 · p. 16, Figure 4 · p. 17

4 Proofs

The proofs adapt ℓ1 recovery arguments to the non-convex ℓp setting and establish probabilistic guarantees for random matrices. Their arguments use sparse-vector decompositions, geometric bounds, and RIP-related properties.

4.1 Proof of Proposition 2.10

  • For Proposition 2.10, the proof uses monotonicity of the RIP constants and an increasing scalar map to choose integer parameters satisfying the needed relation.That choice establishes the condition required by Corollary 2.5.Source: §4.1 · pp. 17–18

4.2 Proof of Theorem 2.1

  • The proof of Theorem 2.1 adapts the Candès et al. ℓ1 argument to the non-convex ℓp quasinorm and decomposes the error using its largest coefficients and successive support blocks.The resulting error bound requires a positive constant condition on the matrix RIP constants.Source: §4.2 · pp. 18, 20
  • For Gaussian matrices, the proof combines RIP and LQp properties holding with high probability to obtain instance-optimality results for ∆p.The argument first establishes an (2,p) guarantee over a suitable sparsity range, then uses the properties jointly on high-probability events.Source: §4.5 · pp. 24–25

5 Appendix: Proof of Lemma 4.2

The appendix supplies geometric details for Lemma 4.2, introducing notation for p-convex bodies and proving a supporting bound through properties of the gauge functional.

5 Appendix: Proof of Lemma 4.2

  • The appendix defines the gauge functional and the distance δ_K from a p-convex body to its convex hull.It also introduces a type constant for bodies and states the associated inequality for the gauge.Source: §5 · pp. 25–26
  • The proof uses p-convexity to relate the gauge of a sum to the gauges of its two terms.The argument normalizes the vectors by their gauge values and applies the defining p-convexity property.Source: §5 · p. 26
  • The appendix establishes that the constant T2 for the Euclidean unit ball equals 1.The proof bounds the quantity from above and below by 1.Source: §5 · pp. 26–27

Proof of Lemma 4.2

The proof develops a general p-convex-body argument for Lemma 4.2, using d1 and a bound parameterized by q. It then specializes to q = 2 to recover Lemma 4.2 as a corollary.

Proof of Lemma 4.2

  • For 0 < p < 1 and q in (1, 2], the proof considers a p-convex body K and a symmetric body B containing conv(K), with d1(K, B) and Tq(B) setting the bound.The proof begins by taking collections of points in K and choosing signs to control their signed sum using Tq(B).Source: Proof of Lemma 4.2 · pp. 27–28
  • The proof derives bounds for α at successive scales, then combines them across scales to bound α at any k ≤ m.The argument uses sub-additivity of the K gauge and the inclusion (1/d)B ⊂ K.Source: Proof of Lemma 4.2 · p. 28
  • Choosing k to minimize the bound leads to a continuous critical point k*, which the proof adjusts to an integer using ⌈k*⌉.The proof solves f′(k*) = 0 and uses f(k* + 1) as a bound when k* is not an integer.Source: Proof of Lemma 4.2 · p. 28
  • The argument extends the bound from α at dyadic indices to δK = sup_r αr by changing between index and logarithmic scales.This scale conversion is justified by replacing 2m and 2k with m and k, then with log2 m and log2 k.Source: Proof of Lemma 4.2 · pp. 28–29
  • The proof combines the resulting bounds with the d1 inclusions, then takes q = 2 and T = 1 to obtain Lemma 4.2 as a corollary.The q = 2 specialization uses T = T2(B2^n) = 1.Source: Proof of Lemma 4.2 · p. 29
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