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Certificate for Orthogonal Equivalence of Real Polynomials by Polynomial-Weighted Principal Component Analysis

Martin Helmer, David Hong, Hoon Hong

TL;DR

The paper addresses certifying orthogonal equivalence of real polynomials under the orthogonal group $O(n)$ by introducing Polynomial-Weighted Principal Component Analysis (PW-PCA). PW-PCA homogenizes polynomials and defines polynomial-weighted variance and covariance to produce principal axes via the leading eigenstructure of a submatrix of the polynomial-weighted covariance, yielding an efficiently computable PW-PCA $(\lambda,V)$. A key theorem connects PW-PCA to orthogonal equivalence, showing that if $f$ and $g$ are orthogonally equivalent, then $\hat{f}=V_f\bullet f$ and $\hat{g}=V_g\bullet g$ are equivalent up to a finite signflip, enabling a practical certificate $R=V_f\mathrm{diag}(\sigma)V_g^T$. Empirical results demonstrate substantial speedups over baseline algebraic methods and illustrate the method’s scalability, while identifying bottlenecks in canonical-form computations and signflip search. The approach provides a scalable, numerically stable certificate mechanism with potential applicability to invariant-theory problems and polynomial orbit tests.

Abstract

Suppose that $f(x) \in \mathbb{R}[x_1,\dots, x_n]$ and $g(x) \in \mathbb{R}[x_1,\dots, x_n]$ are two real polynomials of degree $d$ in $n$ variables. If the polynomials $f$ and $g$ are the same up to orthogonal symmetry a natural question is then what element of the orthogonal group induces the orthogonal symmetry; i.e. to find the element $R\in O(n)$ such that $f(Rx)=g(x)$. One may directly solve this problem by constructing a nonlinear system of equations induced by the relation $f(Rx)=g(x)$ along with the identities of the orthogonal group however this approach becomes quite computationally expensive for larger values of $n$ and $d$. To give an alternative and significantly more scalable solution to this problem, we introduce the concept of Polynomial-Weighted Principal Component Analysis (PW-PCA). We in particular show how PW-PCA can be effectively computed and how these techniques can be used to obtain a certificate of orthogonal equivalence, that is we find the $R\in O(n)$ such that $f(Rx)=g(x)$.

Certificate for Orthogonal Equivalence of Real Polynomials by Polynomial-Weighted Principal Component Analysis

TL;DR

The paper addresses certifying orthogonal equivalence of real polynomials under the orthogonal group by introducing Polynomial-Weighted Principal Component Analysis (PW-PCA). PW-PCA homogenizes polynomials and defines polynomial-weighted variance and covariance to produce principal axes via the leading eigenstructure of a submatrix of the polynomial-weighted covariance, yielding an efficiently computable PW-PCA . A key theorem connects PW-PCA to orthogonal equivalence, showing that if and are orthogonally equivalent, then and are equivalent up to a finite signflip, enabling a practical certificate . Empirical results demonstrate substantial speedups over baseline algebraic methods and illustrate the method’s scalability, while identifying bottlenecks in canonical-form computations and signflip search. The approach provides a scalable, numerically stable certificate mechanism with potential applicability to invariant-theory problems and polynomial orbit tests.

Abstract

Suppose that and are two real polynomials of degree in variables. If the polynomials and are the same up to orthogonal symmetry a natural question is then what element of the orthogonal group induces the orthogonal symmetry; i.e. to find the element such that . One may directly solve this problem by constructing a nonlinear system of equations induced by the relation along with the identities of the orthogonal group however this approach becomes quite computationally expensive for larger values of and . To give an alternative and significantly more scalable solution to this problem, we introduce the concept of Polynomial-Weighted Principal Component Analysis (PW-PCA). We in particular show how PW-PCA can be effectively computed and how these techniques can be used to obtain a certificate of orthogonal equivalence, that is we find the such that .
Paper Structure (14 sections, 9 theorems, 48 equations, 3 algorithms)

This paper contains 14 sections, 9 theorems, 48 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem
  3. Polynomial-Weighted Principal Component Analysis
  4. A brief review of Principal Component Analysis
  5. Definition of a polynomial-weighted variance and covariance
  6. Main definition: Polynomial-Weighted Principal Component Analysis
  7. Algorithm for Polynomial-Weighted Principal Component Analysis
  8. Certificate of Orthogonal Equivalence via PW-PCA
  9. Performance of the proposed algorithm
  10. Setup for the experiments
  11. Comparison with the baseline method
  12. Details on the performance of the proposed algorithm
  13. Acknowledgments:
  14. Proof of \ref{['prp:genericity']}

Key Result

Proposition 9

For any $f \in \mathbb{R}[x_1,\dots,x_n]$, the following are equivalent: where the eigendecomposition here sorts the eigenvalues in non-increasing order, i.e., $\lambda_1 \geq \cdots \geq \lambda_n$.

Theorems & Definitions (31)

  • Definition 1: Orthogonal equivalence
  • Example 3: Simple running example
  • Remark 4
  • Remark 5: Relationship to symmetric tensors and the Procrustes problem
  • Definition 6: Polynomial-weighted variance of a homogeneous polynomial
  • Definition 7: Polynomial-weighted covariance of a homogeneous polynomial
  • Definition 8: Polynomial-Weighted Principal Component Analysis
  • Proposition 9: PW-PCA via eigendecomposition
  • proof : Proof of \ref{['thm:pwpca:cov']}
  • Theorem 11: Expression for the polynomial-weighted covariance
  • ...and 21 more