# Fast approximation of orthogonal matrices and application to PCA

## TL;DR

This paper introduces a fast approximation method for orthogonal matrices using extended Givens transformations, enabling efficient PCA computations with a good balance of accuracy and speed.

## Contribution

It proposes a novel approximation technique for orthogonal matrices based on extended Givens transformations and an efficient greedy algorithm, improving computational speed for spectral methods.

## Key findings

- The method achieves a good trade-off between accuracy and computational speed.
- It effectively approximates orthogonal matrices for spectral methods like PCA.
- Experimental results demonstrate improved efficiency in PCA applications.

## Abstract

We study the problem of approximating orthogonal matrices so that their application is numerically fast and yet accurate. We find an approximation by solving an optimization problem over a set of structured matrices, that we call extended orthogonal Givens transformations, including Givens rotations as a special case. We propose an efficient greedy algorithm to solve such a problem and show that it strikes a balance between approximation accuracy and speed of computation. The approach is relevant to spectral methods and we illustrate its application to PCA.