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Fast solution of Sylvester-structured systems for spatial source separation of the Cosmic Microwave Background

Dung Pham, Kirk M. Soodhalter, Simon Wilson

TL;DR

Two distinct approaches are described: one that uses the method of conjugate gradients directly to the Kronecker-structured problem and another that reformulates the system as a Sylvester matrix equation.

Abstract

Implementation of many statistical methods for large, multivariate data sets requires one to solve a linear system that, depending on the method, is of the dimension of the number of observations or each individual data vector. This is often the limiting factor in scaling the method with data size and complexity. In this paper we illustrate the use of Krylov subspace methods to address this issue in a statistical solution to a source separation problem in cosmology where the data size is prohibitively large for direct solution of the required system. Two distinct approaches, adapted from techniques in the literature, are described: one that uses the method of conjugate gradients directly to the Kronecker-structured problem and another that reformulates the system as a Sylvester matrix equation. We show that both approaches produce an accurate solution within an acceptable computation time and with practical memory requirements for the data size that is currently available.

Fast solution of Sylvester-structured systems for spatial source separation of the Cosmic Microwave Background

TL;DR

Two distinct approaches are described: one that uses the method of conjugate gradients directly to the Kronecker-structured problem and another that reformulates the system as a Sylvester matrix equation.

Abstract

Implementation of many statistical methods for large, multivariate data sets requires one to solve a linear system that, depending on the method, is of the dimension of the number of observations or each individual data vector. This is often the limiting factor in scaling the method with data size and complexity. In this paper we illustrate the use of Krylov subspace methods to address this issue in a statistical solution to a source separation problem in cosmology where the data size is prohibitively large for direct solution of the required system. Two distinct approaches, adapted from techniques in the literature, are described: one that uses the method of conjugate gradients directly to the Kronecker-structured problem and another that reformulates the system as a Sylvester matrix equation. We show that both approaches produce an accurate solution within an acceptable computation time and with practical memory requirements for the data size that is currently available.
Paper Structure (21 sections, 2 theorems, 43 equations, 5 figures, 4 tables)

This paper contains 21 sections, 2 theorems, 43 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Motivating Example
  3. Source separation model for all-sky maps
  4. Source separation solution
  5. Solving the Linear System
  6. Efficient matrix-vector products
  7. Solution of \ref{['eqn.Axb']} using matrix-free iterative methods
  8. Solution of \ref{['eqn.Axb']} using a Sylvester interpretation
  9. An existing technique for the treating sparse-dense Sylvester equations
  10. Application of block Krylov approach for large-scale problems
  11. The block Lanczos process with respect to $\left(\cdot, \cdot\right)_{\mathbf{N}}$
  12. Projection methods for Sylvester equations
  13. Evaluation and Application
  14. Accuracy
  15. Time and memory usage
  16. ...and 6 more sections

Key Result

lemma 1

The coefficient matrix $\mathbf{N}^{-1}\mathbf{D}^{2}$ is symmetric with respect to $\left(\cdot, \cdot\right)_{\mathbf{N}}$.

Figures (5)

  • Figure 1: Solution accuracy. Scatter plot of components of the true source vector $\mathcal{S}$ versus the solution vector (left) and source vector versus the error (right).
  • Figure 2: Running time (in seconds) by HEALPix level (left) and data size (right).
  • Figure 3: A comparison of total memory usage by HEALPix level (left) and data size (right) for Algorithm \ref{['alg.CGnoprecond']} (CG) versus Algorithm \ref{['alg:Sylvester']} (Lanczos-Sylvester -- i.e., Sylv.) for both synthetic data and actual Planck satellite data.
  • Figure 4: Code profiling. The functions that consume the most resources as a function of problem size. The top row is percentage of total computation time and the bottom row is percentage of maximum memory use. Table \ref{['tab:profile_defs']} provides more information on the functions. Sylv. refers to Algorithm \ref{['alg:Sylvester']}.
  • Figure 5: HEALPix partitions. Clockwise from top left: the base partition into 12 pixels, then level 1, 2 and 3 (48, 192 and 768 pixels). Taken from https://healpix.sourceforge.io/.

Theorems & Definitions (4)

  • lemma 1
  • lemma 2
  • remark 1
  • remark 2