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Quotient Manifold Optimization for Spectral Compressed Sensing

Wenlong Wang, Wen Huang, Zai Yang

TL;DR

This paper addresses reconstructing spectral-sparse signals from partial samples by formulating spectral compressed sensing as optimization over equivalence classes on a quotient manifold of low-rank PSD Hankel-Toeplitz matrices. It establishes a precise geometric framework with a Riemannian metric, horizontal/vertical spaces, and a retraction, enabling a Hankel-Toeplitz Riemannian Conjugate Gradient Descent (HT-RCGD) algorithm. The method leverages FFT-based convolutions to achieve $O(KN\log N)$ per-iteration cost (plus $O(K^2N)$ terms), and it includes a convergence guarantee for accumulation points. Numerical experiments show HT-RCGD outperforms state-of-the-art techniques in convergence speed and accuracy, validating its effectiveness for scalable spectral super-resolution in radar and wireless channels. Overall, the work provides a principled, geometry-aware, computationally efficient solution for spectral compressed sensing with exact structural modeling.

Abstract

Spectral compressed sensing involves reconstructing a spectral-sparse signal from a subset of uniformly spaced samples, with applications in radar imaging and wireless channel estimation. By fully exploiting the signal structures, this problem is formulated as a rank-constrained semidefinite program subject to Hankel-Toeplitz structural constraints in our previous work. To further enhance computational efficiency, this paper proposes a quotient-manifold-based optimization framework that leverages the underlying Riemannian geometry in a matrix factorization space. Specifically, we establish an equivalence between spectral-sparse signals and matrix equivalence classes under the action of the real orthogonal group, where each class member corresponds to a rank-constrained positive-semidefinite Hankel-Toeplitz structured matrix. The associated quotient manifold geometry--including the Riemannian metric, horizontal space, retraction, and vector transport--is rigorously derived. Based on these results, we develop a Riemannian conjugate gradient descent algorithm, where each iteration is efficiently implemented using fast Fourier transforms (FFTs) by exploiting the Hankel and Toeplitz structures. Extensive numerical experiments demonstrate the superior performance of the proposed algorithm in both computational speed and accuracy compared to state-of-the-art methods.

Quotient Manifold Optimization for Spectral Compressed Sensing

TL;DR

This paper addresses reconstructing spectral-sparse signals from partial samples by formulating spectral compressed sensing as optimization over equivalence classes on a quotient manifold of low-rank PSD Hankel-Toeplitz matrices. It establishes a precise geometric framework with a Riemannian metric, horizontal/vertical spaces, and a retraction, enabling a Hankel-Toeplitz Riemannian Conjugate Gradient Descent (HT-RCGD) algorithm. The method leverages FFT-based convolutions to achieve per-iteration cost (plus terms), and it includes a convergence guarantee for accumulation points. Numerical experiments show HT-RCGD outperforms state-of-the-art techniques in convergence speed and accuracy, validating its effectiveness for scalable spectral super-resolution in radar and wireless channels. Overall, the work provides a principled, geometry-aware, computationally efficient solution for spectral compressed sensing with exact structural modeling.

Abstract

Spectral compressed sensing involves reconstructing a spectral-sparse signal from a subset of uniformly spaced samples, with applications in radar imaging and wireless channel estimation. By fully exploiting the signal structures, this problem is formulated as a rank-constrained semidefinite program subject to Hankel-Toeplitz structural constraints in our previous work. To further enhance computational efficiency, this paper proposes a quotient-manifold-based optimization framework that leverages the underlying Riemannian geometry in a matrix factorization space. Specifically, we establish an equivalence between spectral-sparse signals and matrix equivalence classes under the action of the real orthogonal group, where each class member corresponds to a rank-constrained positive-semidefinite Hankel-Toeplitz structured matrix. The associated quotient manifold geometry--including the Riemannian metric, horizontal space, retraction, and vector transport--is rigorously derived. Based on these results, we develop a Riemannian conjugate gradient descent algorithm, where each iteration is efficiently implemented using fast Fourier transforms (FFTs) by exploiting the Hankel and Toeplitz structures. Extensive numerical experiments demonstrate the superior performance of the proposed algorithm in both computational speed and accuracy compared to state-of-the-art methods.

Paper Structure

This paper contains 16 sections, 6 theorems, 55 equations, 6 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Notations and organization
  4. Preliminaries
  5. Optimization Problem on Quotient Manifold
  6. Problem formulation
  7. Riemannian geometry of
  8. Riemanian Optimization Algorithm
  9. Objective function
  10. Riemannian conjugate gradient method
  11. Fast implementation and computational complexity
  12. Computing the update direction
  13. Computing the step size
  14. Convergence
  15. Numerical Simulations
  16. ...and 1 more sections

Key Result

Theorem 1

Assume $K<p$. For any spectral-sparse signal $\boldsymbol{x}\in\mathcal{S}_0$ with frequency vector $\boldsymbol{f}$ and cofficient vector $\boldsymbol{s}$, there exists a unique equivalence class $\left[\boldsymbol{Z}\right]=\left\{\boldsymbol{Z}\boldsymbol{O}\in\mathbb{C}^{p\times K}_*:\;\boldsymb

Figures (6)

  • Figure 1: NMSE vs iteration.
  • Figure 2: Phase transition behaviors for frequencies with minimum separation. White means complete success and black means complete failure.
  • Figure 3: Comparison of iteration numbers for varying $M$ and $K$ under frequency separation constraint.
  • Figure 4: Phase transition behavior for frequencies without minimum separation. White means complete success and black means complete failure.
  • Figure 5: Comparison of iteration numbers for varying $M$ and $K$ without frequency separation.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Proof 1
  • Theorem 2
  • Proof 2
  • Proposition 1
  • Proof 3
  • Proposition 2
  • Proof 4
  • Proposition 3
  • Proof 5
  • ...and 2 more