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GT-PCA: Effective and Interpretable Dimensionality Reduction with General Transform-Invariant Principal Component Analysis

Florian Heinrichs

TL;DR

GT-PCA introduces an interpretable extension of PCA that yields components invariant to problem-dependent transformations by solving a transform-invariant energy maximization. A neural network GT-PC-layer approximates these components, enabling sequential estimation of multiple GT-PCs in settings such as time shifts and image rotations. Empirical results show GT-PCA outperforms PCA, KPCA, autoencoders, and VAEs on synthetic and real data when nontrivial transformations are present, while retaining interpretability similar to functional PCA. This approach offers a principled, diagnostically meaningful dimensionality reduction tool with potential applications across EEG analysis, computer vision, and functional time series.

Abstract

Data analysis often requires methods that are invariant with respect to specific transformations, such as rotations in case of images or shifts in case of images and time series. While principal component analysis (PCA) is a widely-used dimension reduction technique, it lacks robustness with respect to these transformations. Modern alternatives, such as autoencoders, can be invariant with respect to specific transformations but are generally not interpretable. We introduce General Transform-Invariant Principal Component Analysis (GT-PCA) as an effective and interpretable alternative to PCA and autoencoders. We propose a neural network that efficiently estimates the components and show that GT-PCA significantly outperforms alternative methods in experiments based on synthetic and real data.

GT-PCA: Effective and Interpretable Dimensionality Reduction with General Transform-Invariant Principal Component Analysis

TL;DR

GT-PCA introduces an interpretable extension of PCA that yields components invariant to problem-dependent transformations by solving a transform-invariant energy maximization. A neural network GT-PC-layer approximates these components, enabling sequential estimation of multiple GT-PCs in settings such as time shifts and image rotations. Empirical results show GT-PCA outperforms PCA, KPCA, autoencoders, and VAEs on synthetic and real data when nontrivial transformations are present, while retaining interpretability similar to functional PCA. This approach offers a principled, diagnostically meaningful dimensionality reduction tool with potential applications across EEG analysis, computer vision, and functional time series.

Abstract

Data analysis often requires methods that are invariant with respect to specific transformations, such as rotations in case of images or shifts in case of images and time series. While principal component analysis (PCA) is a widely-used dimension reduction technique, it lacks robustness with respect to these transformations. Modern alternatives, such as autoencoders, can be invariant with respect to specific transformations but are generally not interpretable. We introduce General Transform-Invariant Principal Component Analysis (GT-PCA) as an effective and interpretable alternative to PCA and autoencoders. We propose a neural network that efficiently estimates the components and show that GT-PCA significantly outperforms alternative methods in experiments based on synthetic and real data.
Paper Structure (17 sections, 9 equations, 11 figures, 8 tables)

This paper contains 17 sections, 9 equations, 11 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Methodology
  4. General Transform-Invariant Principal Components Analysis
  5. Time-dependent Data
  6. Images
  7. Neural Network-based Approximation of GT-PCs
  8. Experimental Results
  9. Synthetic Data Experiments
  10. Projection
  11. Classification & Outlier Detection
  12. Real Data Experiments
  13. Projection
  14. Classification & Outlier Detection
  15. Conclusion
  16. ...and 2 more sections

Figures (11)

  • Figure 1: Exemplary transformed MNIST data. Left: rotated digits. Right: shifted digits.
  • Figure 2: Top: First 3 GT-PCs of generated spikes with window length $\tfrac{1}{2}$. Bottom: Generated signals of length $1$ with spikes at different times, different widths and amplitudes and their reconstructions based on 1 (dashed), 2 (dashdot) and 3 (dotted) components.
  • Figure 3: Top: First 3 GT-PCs of generated spikes with window length $1$. Bottom: Generated signals of length $\tfrac{1}{2}$ with spikes at different times, different widths and amplitudes and their reconstructions based on 1 (dashed), 2 (dashdot) and 3 (dotted) components.
  • Figure 4: Top left: First four GT-PCs of rotated digits. Top right: Reconstruction of rotated digits based on first four GT-PCs. Bottom left: First four GT-PCs of shifted digits. Bottom right: Reconstruction of shifted digits based on first four GT-PCs.
  • Figure 5: Neural network architecture for the approximation of general-transform invariant principal components.
  • ...and 6 more figures