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Compressed Sensing: A Discrete Optimization Approach

Dimitris Bertsimas, Nicholas A. G. Johnson

TL;DR

This work introduces a discrete, sparsity-promoting reformulation of compressed sensing by adding an ${\ell_2^2}$ regularization to the ${\ell_0}$ objective and recasting the problem as a mixed-integer second-order cone program. It develops a cascade of convex relaxations, including a second-order cone (SOC) relaxation equivalent to Basis Pursuit Denoising under mild parameter conditions, and a stronger semidefinite (PSD/SOS) relaxation; a custom branch-and-bound algorithm then computes certifiably optimal solutions. Theoretical results establish when the SOC relaxation aligns with established CS methods and quantify bounds for small regularization, while empirical results demonstrate substantial sparsity gains and improved reconstruction accuracy on synthetic data, real ECG signals, and multi-label classification tasks, albeit with significantly higher computational cost. The findings highlight the potential of integrating exact integer optimization into compressed sensing to achieve sparser, more accurate solutions and reveal practical trade-offs between solution quality and runtime. Overall, the paper provides a rigorous framework and practical evidence that discrete optimization can outperform traditional CS benchmarks in sparsity and accuracy when runtime is not the primary constraint.

Abstract

We study the Compressed Sensing (CS) problem, which is the problem of finding the most sparse vector that satisfies a set of linear measurements up to some numerical tolerance. We introduce an $\ell_2$ regularized formulation of CS which we reformulate as a mixed integer second order cone program. We derive a second order cone relaxation of this problem and show that under mild conditions on the regularization parameter, the resulting relaxation is equivalent to the well studied basis pursuit denoising problem. We present a semidefinite relaxation that strengthens the second order cone relaxation and develop a custom branch-and-bound algorithm that leverages our second order cone relaxation to solve small-scale instances of CS to certifiable optimality. When compared against solutions produced by three state of the art benchmark methods on synthetic data, our numerical results show that our approach produces solutions that are on average $6.22\%$ more sparse. When compared only against the experiment-wise best performing benchmark method on synthetic data, our approach produces solutions that are on average $3.10\%$ more sparse. On real world ECG data, for a given $\ell_2$ reconstruction error our approach produces solutions that are on average $9.95\%$ more sparse than benchmark methods ($3.88\%$ more sparse if only compared against the best performing benchmark), while for a given sparsity level our approach produces solutions that have on average $10.77\%$ lower reconstruction error than benchmark methods ($1.42\%$ lower error if only compared against the best performing benchmark). When used as a component of a multi-label classification algorithm, our approach achieves greater classification accuracy than benchmark compressed sensing methods. This improved accuracy comes at the cost of an increase in computation time by several orders of magnitude.

Compressed Sensing: A Discrete Optimization Approach

TL;DR

This work introduces a discrete, sparsity-promoting reformulation of compressed sensing by adding an regularization to the objective and recasting the problem as a mixed-integer second-order cone program. It develops a cascade of convex relaxations, including a second-order cone (SOC) relaxation equivalent to Basis Pursuit Denoising under mild parameter conditions, and a stronger semidefinite (PSD/SOS) relaxation; a custom branch-and-bound algorithm then computes certifiably optimal solutions. Theoretical results establish when the SOC relaxation aligns with established CS methods and quantify bounds for small regularization, while empirical results demonstrate substantial sparsity gains and improved reconstruction accuracy on synthetic data, real ECG signals, and multi-label classification tasks, albeit with significantly higher computational cost. The findings highlight the potential of integrating exact integer optimization into compressed sensing to achieve sparser, more accurate solutions and reveal practical trade-offs between solution quality and runtime. Overall, the paper provides a rigorous framework and practical evidence that discrete optimization can outperform traditional CS benchmarks in sparsity and accuracy when runtime is not the primary constraint.

Abstract

We study the Compressed Sensing (CS) problem, which is the problem of finding the most sparse vector that satisfies a set of linear measurements up to some numerical tolerance. We introduce an regularized formulation of CS which we reformulate as a mixed integer second order cone program. We derive a second order cone relaxation of this problem and show that under mild conditions on the regularization parameter, the resulting relaxation is equivalent to the well studied basis pursuit denoising problem. We present a semidefinite relaxation that strengthens the second order cone relaxation and develop a custom branch-and-bound algorithm that leverages our second order cone relaxation to solve small-scale instances of CS to certifiable optimality. When compared against solutions produced by three state of the art benchmark methods on synthetic data, our numerical results show that our approach produces solutions that are on average more sparse. When compared only against the experiment-wise best performing benchmark method on synthetic data, our approach produces solutions that are on average more sparse. On real world ECG data, for a given reconstruction error our approach produces solutions that are on average more sparse than benchmark methods ( more sparse if only compared against the best performing benchmark), while for a given sparsity level our approach produces solutions that have on average lower reconstruction error than benchmark methods ( lower error if only compared against the best performing benchmark). When used as a component of a multi-label classification algorithm, our approach achieves greater classification accuracy than benchmark compressed sensing methods. This improved accuracy comes at the cost of an increase in computation time by several orders of magnitude.
Paper Structure (41 sections, 6 theorems, 59 equations, 8 figures, 10 tables, 2 algorithms)

This paper contains 41 sections, 6 theorems, 59 equations, 8 figures, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions and Structure
  3. Notation:
  4. Literature Review
  5. Basis Pursuit Denoising
  6. Iterative Reweighted L1
  7. Orthogonal Matching Pursuit
  8. Formulation Properties
  9. An Exact Reformulation and Convex Relaxations
  10. A Second Order Cone Relaxation
  11. A Positive Semidefinite Cone Relaxation
  12. Branch-and-Bound
  13. Subproblems
  14. Subproblem Lower Bound
  15. Subproblem Upper Bound
  16. ...and 26 more sections

Key Result

Theorem 1

There exists a finite value $\gamma_0 < \infty$ such that for all $\Bar{\gamma} \geq \gamma_0$, there exists a vector $\bm{x}^\star$ such that $\bm{x}^\star$ is an optimal solution of opt:main_problem and also an optimal solution of opt:our_formulation where we set $\gamma = \Bar{\gamma}$. Letting $

Figures (8)

  • Figure 1: Sparsity (top left), accuracy (top right), true positive rate (bottom left) and true negative rate (bottom right) versus $n$ with $k=10$, and $\alpha=0.2$. Averaged over $100$ trials for each parameter configuration.
  • Figure 2: Sparsity (top left), accuracy (top right), true positive rate (bottom left) and true negative rate (bottom right) versus $k$ with $N=200$ and $\alpha=0.2$. Averaged over $100$ trials for each parameter configuration.
  • Figure 3: Sparsity (top left), accuracy (top right), true positive rate (bottom left) and true negative rate (bottom right) versus $\alpha$ with $n=200$ and $k=10$. Averaged over $100$ trials for each parameter configuration.
  • Figure 4: Problem \ref{['opt:our_formulation']} lower bound (left) produced by Problem \ref{['opt:MISOC_W_relax']} (SOC) and Problem \ref{['opt:poly_relax']} (SOS) with $d=1$. Percent improvement of Problem \ref{['opt:our_formulation']} lower bound of compared to (right). $n=25, m=100$ and $k=10$.
  • Figure 5: Problem \ref{['opt:our_formulation']} lower bound (left) produced by Problem \ref{['opt:MISOC_W_relax']} (SOC) and Problem \ref{['opt:poly_relax']} (SOS) with $d=1$. Percent improvement of Problem \ref{['opt:our_formulation']} lower bound of compared to (right). $n=50, m=25$ and $k=10$.
  • ...and 3 more figures

Theorems & Definitions (15)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Remark 2
  • ...and 5 more