Computing Principal Components Dynamically

TL;DR

This paper introduces efficient closed-form solutions for updating principal components dynamically when points are added or removed, significantly reducing computation time for applications like bounding box calculations in computer graphics.

Contribution

The paper provides novel closed-form formulas for updating principal components in real-time, applicable to both discrete and continuous point sets without extra data structures.

Findings

Update principal components in O(m) time for fixed dimension

Achieved faster dynamic PCA updates compared to recomputation

Validated methods through implementation in 3D bounding box applications

Abstract

In this paper we present closed-form solutions for efficiently updating the principal components of a set of $n$ points, when $m$ points are added or deleted from the point set. For both operations performed on a discrete point set in $\mathbb{R}^d$, we can compute the new principal components in $O(m)$ time for fixed $d$. This is a significant improvement over the commonly used approach of recomputing the principal components from scratch, which takes $O(n+m)$ time. An important application of the above result is the dynamical computation of bounding boxes based on principal component analysis. PCA bounding boxes are very often used in many fields, among others in computer graphics for collision detection and fast rendering. We have implemented and evaluated few algorithms for computing dynamically PCA bounding boxes in $\mathbb{R}^3$. In addition, we present closed-form solutions for computing dynamically principal components of continuous point sets in $\mathbb{R}^2$ and $\mathbb{R}^3$. In both cases, discrete and continuous, to compute the new principal components, no additional data structures or storage are needed.