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Joint Approximate Partial Diagonalization of Large Matrices

Abd-Krim Seghouane, Yousef Saad

TL;DR

It is argued that when the matrices are of large dimension, then the natural generalization of this problem is to seek an orthonormal basis of a certain subspace that is a near eigenspace for all the matrices in the set.

Abstract

Given a set of $p$ symmetric (real) matrices, the Orthogonal Joint Diagonalization (OJD) problem consists of finding an orthonormal basis in which the representation of each of these $p$ matrices is as close as possible to a diagonal matrix. We argue that when the matrices are of large dimension, then the natural generalization of this problem is to seek an orthonormal basis of a certain subspace that is a near eigenspace for all the matrices in the set. We refer to this as the problem of ``partial joint diagonalization of matrices.'' The approach proposed first finds this approximate common near eigenspace and then proceeds to a joint diagonalization of the restrictions of the input matrices in this subspace. A few solution methods for this problem are proposed and illustrations of its potential applications are provided.

Joint Approximate Partial Diagonalization of Large Matrices

TL;DR

It is argued that when the matrices are of large dimension, then the natural generalization of this problem is to seek an orthonormal basis of a certain subspace that is a near eigenspace for all the matrices in the set.

Abstract

Given a set of symmetric (real) matrices, the Orthogonal Joint Diagonalization (OJD) problem consists of finding an orthonormal basis in which the representation of each of these matrices is as close as possible to a diagonal matrix. We argue that when the matrices are of large dimension, then the natural generalization of this problem is to seek an orthonormal basis of a certain subspace that is a near eigenspace for all the matrices in the set. We refer to this as the problem of ``partial joint diagonalization of matrices.'' The approach proposed first finds this approximate common near eigenspace and then proceeds to a joint diagonalization of the restrictions of the input matrices in this subspace. A few solution methods for this problem are proposed and illustrations of its potential applications are provided.
Paper Structure (17 sections, 9 theorems, 62 equations, 4 figures, 3 algorithms)

This paper contains 17 sections, 9 theorems, 62 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction and problem statement
  2. Approximate Joint Partial Diagonalization
  3. Joint eigenspaces
  4. Joint partial diagonalization
  5. General approach
  6. General results
  7. Problem decoupling
  8. The Grassmannian perspective
  9. Comparison with the global optimum
  10. Computing an optimal $Q$
  11. A subspace iteration procedure
  12. Gradient-based Algorithms
  13. Simulation results
  14. Small diagonalizable data sets
  15. Small approximately diagonalizable data sets
  16. ...and 2 more sections

Key Result

theorem 1

For a fixed orthogonal matrix $Q$, define $D_{Q,i} = Q^T C_i Q$. Then the objective function (eq:obj) is minimized when $D_i = D_{Q,i}$. In addition, if we define the residual matrix as $R_i = C_i Q - Q D_{Q,i}$ then $R_i = (I-Q Q^T) C_i Q$ and in particular $R_i \perp_F Q$ and $\|R_i \|_F = \| (I-

Figures (4)

  • Figure 1: Simulation results for the small diagonalizable data sets. The figures present the boxplots of the measure $E$ generated over 100 realization for Cardoso2 in the left figure and the proposed method in the right figure.
  • Figure 2: Simulation results for the small diagonalizable data set of size $N=10$. The figures present the boxplots of the measure $E$ generated over 100 realization with the proposed method with dimension $k=3,4,5,6$ and $7$.
  • Figure 3: Simulation results for the small approximately diagonalizable data sets. The figures present the boxplots of the measure $E$ generated over 100 realization for Cardoso2 in the left figure and the proposed method in the right figure.
  • Figure 4: Resting state fMRI results. Left: tICA McKeown, right: proposed approach

Theorems & Definitions (17)

  • theorem 1
  • proof
  • proposition 1: Error decoupling
  • proof
  • theorem 2
  • proof
  • lemma 1
  • proof
  • proposition 2
  • proof
  • ...and 7 more