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Optimal Structured Matrix Approximation for Robustness to Incomplete Biosequence Data

Chris Salahub, Jeffrey Uhlmann

TL;DR

The paper addresses the challenge of performing spectral analyses on large $n \times n$ bioinformatics matrices when data are incomplete by introducing a general Frobenius-norm minimization framework that approximates any matrix with a structured one. It proves that the optimum structured matrix $\mathbf{T}_M$ is obtained by replacing each index-set with the mean of the corresponding entries, yielding a space reduction from $O(n^2)$ to $O(|\mathbf{t}|)$ and a residual tied to within-group variance; in the circulant case, the implementation aligns with the DFT. The authors illustrate the approach with simulated and real LD data, showing improved robustness to missing data (fewer negative eigenvalues and smaller eigenvalue-discrepancy) at the cost of bias when data are nearly complete, and discuss extensions such as a structured-plus-diagonal form to preserve positive semidefiniteness. Overall, the method offers a simple, efficient procedure for robust matrix approximation with broad applicability in genomics and beyond.

Abstract

We propose a general method for optimally approximating an arbitrary matrix $\mathbf{M}$ by a structured matrix $\mathbf{T}$ (circulant, Toeplitz/Hankel, etc.) and examine its use for estimating the spectra of genomic linkage disequilibrium matrices. This application is prototypical of a variety of genomic and proteomic problems that demand robustness to incomplete biosequence information. We perform a simulation study and corroborative test of our method using real genomic data from the Mouse Genome Database. The results confirm the predicted utility of the method and provide strong evidence of its potential value to a wide range of bioinformatics applications. Our optimal general matrix approximation method is expected to be of independent interest to an even broader range of applications in applied mathematics and engineering.

Optimal Structured Matrix Approximation for Robustness to Incomplete Biosequence Data

TL;DR

The paper addresses the challenge of performing spectral analyses on large bioinformatics matrices when data are incomplete by introducing a general Frobenius-norm minimization framework that approximates any matrix with a structured one. It proves that the optimum structured matrix is obtained by replacing each index-set with the mean of the corresponding entries, yielding a space reduction from to and a residual tied to within-group variance; in the circulant case, the implementation aligns with the DFT. The authors illustrate the approach with simulated and real LD data, showing improved robustness to missing data (fewer negative eigenvalues and smaller eigenvalue-discrepancy) at the cost of bias when data are nearly complete, and discuss extensions such as a structured-plus-diagonal form to preserve positive semidefiniteness. Overall, the method offers a simple, efficient procedure for robust matrix approximation with broad applicability in genomics and beyond.

Abstract

We propose a general method for optimally approximating an arbitrary matrix by a structured matrix (circulant, Toeplitz/Hankel, etc.) and examine its use for estimating the spectra of genomic linkage disequilibrium matrices. This application is prototypical of a variety of genomic and proteomic problems that demand robustness to incomplete biosequence information. We perform a simulation study and corroborative test of our method using real genomic data from the Mouse Genome Database. The results confirm the predicted utility of the method and provide strong evidence of its potential value to a wide range of bioinformatics applications. Our optimal general matrix approximation method is expected to be of independent interest to an even broader range of applications in applied mathematics and engineering.
Paper Structure (10 sections, 1 theorem, 19 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 10 sections, 1 theorem, 19 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Structured Matrix Approximation
  3. Structured matrices
  4. Optimizing the Frobenius norm
  5. Circulant matrices
  6. Application to Genetic Linkage Disequilibrium
  7. Simulated data
  8. Real data
  9. Conclusion
  10. References

Key Result

Theorem 1

The Frobenius optimal approximating structured matrix $\mathbf{T}$ with index function $f(i,j)$ for $\mathbf{M}$ is given by $\mathbf{T}_M$ with where is the mean of entries in $\mathbf{M}$ over the corresponding index set. Furthermore, $\frac{1}{\sqrt{n}}||{\mathbf{T}_M - \mathbf{M}}||_F$ is the total within-group standard deviation of entries in $\mathbf{M}$ over all index sets.

Figures (2)

  • Figure 1: Paired boxplots of the (a) minimum eigenvalues and (b) sum of squared errors in the ordered eigenvalues for the pairwise LD matrix and the nearest Toeplitz matrix by the proportion of data missing. The nearest Toeplitz, displayed to the right of the line for each pair of boxplots, is more robust to missing data than the pairwise LD matrix, displayed to the left of the line for each pair, but is biased when the data are complete.
  • Figure 2: Paired boxplots of the (a) minimum eigenvalues and (b) sum of squared errors in the ordered eigenvalues for the pairwise LD matrix (to the left of the corresponding line) and the optimal structured matrix (to the right of the corresponding line) by the proportion of data missing. The bias from approximation is more serious in this case than the simulated example: negative eigenvalues are produced for the complete data.

Theorems & Definitions (2)

  • Theorem 1: Means minimize $||{\mathbf{T} - \mathbf{M}}||_F$
  • proof