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Recovering sparse DFT from missing signals via interior point method on GPU

Wei Kuang, Alexis Montoison, Vishwas Rao, François Pacaud, Mihai Anitescu

TL;DR

The paper addresses recovering a sparse Discrete Fourier Transform from noisy, incomplete data by formulating an $\ell_1$-regularized inverse problem in the real domain and solving it with a GPU-accelerated primal-dual interior-point method. A matrix-free approach leverages FFT/IFFT as operators, paired with a specialized Krylov solver and a tailored preconditioner to solve the KKT systems efficiently at very large scales. The authors provide theoretical bounds on the preconditioned system’s conditioning and demonstrate practical scalability on datasets with up to $10^8$ variables, including real 3D crystallography data, achieving substantial speedups over traditional interpolation-based methods. The work yields a robust, scalable framework for sparse DFT recovery applicable to crystallography, with open-source Julia implementations and clear paths for extension to broader optimization problems.

Abstract

We propose a method to recover the sparse discrete Fourier transform (DFT) of a signal that is both noisy and potentially incomplete, with missing values. The problem is formulated as a penalized least-squares minimization based on the inverse discrete Fourier transform (IDFT) with an $\ell_1$-penalty term, reformulated to be solvable using a primal-dual interior point method (IPM). Although Krylov methods are not typically used to solve Karush-Kuhn-Tucker (KKT) systems arising in IPMs due to their ill-conditioning, we employ a tailored preconditioner and establish new asymptotic bounds on the condition number of preconditioned KKT matrices. Thanks to this dedicated preconditioner -- and the fact that FFT and IFFT operate as linear operators without requiring explicit matrix materialization -- KKT systems can be solved efficiently at large scales in a matrix-free manner. Numerical results from a Julia implementation leveraging GPU-accelerated interior point methods, Krylov methods, and FFT toolkits demonstrate the scalability of our approach on problems with hundreds of millions of variables, inclusive of real data obtained from the diffuse scattering from a slightly disordered Molybdenum Vanadium Dioxide crystal.

Recovering sparse DFT from missing signals via interior point method on GPU

TL;DR

The paper addresses recovering a sparse Discrete Fourier Transform from noisy, incomplete data by formulating an -regularized inverse problem in the real domain and solving it with a GPU-accelerated primal-dual interior-point method. A matrix-free approach leverages FFT/IFFT as operators, paired with a specialized Krylov solver and a tailored preconditioner to solve the KKT systems efficiently at very large scales. The authors provide theoretical bounds on the preconditioned system’s conditioning and demonstrate practical scalability on datasets with up to variables, including real 3D crystallography data, achieving substantial speedups over traditional interpolation-based methods. The work yields a robust, scalable framework for sparse DFT recovery applicable to crystallography, with open-source Julia implementations and clear paths for extension to broader optimization problems.

Abstract

We propose a method to recover the sparse discrete Fourier transform (DFT) of a signal that is both noisy and potentially incomplete, with missing values. The problem is formulated as a penalized least-squares minimization based on the inverse discrete Fourier transform (IDFT) with an -penalty term, reformulated to be solvable using a primal-dual interior point method (IPM). Although Krylov methods are not typically used to solve Karush-Kuhn-Tucker (KKT) systems arising in IPMs due to their ill-conditioning, we employ a tailored preconditioner and establish new asymptotic bounds on the condition number of preconditioned KKT matrices. Thanks to this dedicated preconditioner -- and the fact that FFT and IFFT operate as linear operators without requiring explicit matrix materialization -- KKT systems can be solved efficiently at large scales in a matrix-free manner. Numerical results from a Julia implementation leveraging GPU-accelerated interior point methods, Krylov methods, and FFT toolkits demonstrate the scalability of our approach on problems with hundreds of millions of variables, inclusive of real data obtained from the diffuse scattering from a slightly disordered Molybdenum Vanadium Dioxide crystal.

Paper Structure

This paper contains 16 sections, 8 theorems, 67 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notations
  3. Problem formulation
  4. Matrix-free operators
  5. Karush-Kuhn-Tucker stationary conditions
  6. Primal-dual interior-point method
  7. KKT linear systems
  8. Augmented KKT system
  9. Condensed KKT system
  10. Krylov methods and preconditioners
  11. Applications
  12. Synthetic Dataset
  13. Recovering the 3D-$\Delta$PDF in crystals
  14. Implementation and numerical experiments
  15. Conclusions and perspectives
  16. ...and 1 more sections

Key Result

Proposition 3.4

\newlabelprop:wellposedness0 The problem eq:slackproblem satisfies LICQ. If SCS hold and the active submatrix $N_{\mathcal{A}}$ is full row rank, then the problem eq:slackproblem also satisfies SOSC.

Figures (3)

  • Figure 1: \newlabelfig:VO20 Left panel shows the ordinary punch-and-fill, right panel shows the result obtained through $\ell_1$ constraint minimization approach.
  • Figure 1: Number of Krylov iterations per IPM iteration for a problem with 104 million variables.
  • Figure 2: Scalability of CPU and GPU performance across different problem sizes.

Theorems & Definitions (18)

  • Proposition 3.4
  • Proof 1
  • Definition 3.5: Centrality conditions
  • Proposition 3.6
  • Proof 2
  • Corollary 3.7
  • Proposition 3.8
  • Proof 3
  • Proposition 3.9
  • Proof 4
  • ...and 8 more