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Learning with Category-Equivariant Representations for Human Activity Recognition

Yoshihiro Maruyama

TL;DR

This work addresses robustness in human activity recognition by modeling sensor data variability with a Category-Equivariant framework that combines a time-gain group $M=C_T\times\Lambda$ and a sensor-hierarchy poset $P$ into a product category. Data and model are formalized as functors, and category-equivariant representations are natural transformations that commute with both group actions and hierarchical morphisms. A concrete HAR realization uses RMS gain normalization, axis-to-magnitude pooling, and low-frequency real FFT magnitudes, yielding a compact, interpretable feature map that is invariant to time shifts and gain while respecting sensor hierarchies. Experiments on UCI HAR show substantial out-of-distribution robustness (≈3.6× baseline) under time/gain/pose perturbations, with theoretical guarantees: naturality on generators suffices for full equivariance and invariant blocks provide exact robustness under group perturbations. The approach offers data efficiency, robustness, and interpretability, and suggests broader applicability to multimodal sensing and hierarchical, non-invertible symmetry settings.

Abstract

Human activity recognition is challenging because sensor signals shift with context, motion, and environment; effective models must therefore remain stable as the world around them changes. We introduce a categorical symmetry-aware learning framework that captures how signals vary over time, scale, and sensor hierarchy. We build these factors into the structure of feature representations, yielding models that automatically preserve the relationships between sensors and remain stable under realistic distortions such as time shifts, amplitude drift, and device orientation changes. On the UCI Human Activity Recognition benchmark, this categorical symmetry-driven design improves out-of-distribution accuracy by approx. 46 percentage points (approx. 3.6x over the baseline), demonstrating that abstract symmetry principles can translate into concrete performance gains in everyday sensing tasks via category-equivariant representation theory.

Learning with Category-Equivariant Representations for Human Activity Recognition

TL;DR

This work addresses robustness in human activity recognition by modeling sensor data variability with a Category-Equivariant framework that combines a time-gain group and a sensor-hierarchy poset into a product category. Data and model are formalized as functors, and category-equivariant representations are natural transformations that commute with both group actions and hierarchical morphisms. A concrete HAR realization uses RMS gain normalization, axis-to-magnitude pooling, and low-frequency real FFT magnitudes, yielding a compact, interpretable feature map that is invariant to time shifts and gain while respecting sensor hierarchies. Experiments on UCI HAR show substantial out-of-distribution robustness (≈3.6× baseline) under time/gain/pose perturbations, with theoretical guarantees: naturality on generators suffices for full equivariance and invariant blocks provide exact robustness under group perturbations. The approach offers data efficiency, robustness, and interpretability, and suggests broader applicability to multimodal sensing and hierarchical, non-invertible symmetry settings.

Abstract

Human activity recognition is challenging because sensor signals shift with context, motion, and environment; effective models must therefore remain stable as the world around them changes. We introduce a categorical symmetry-aware learning framework that captures how signals vary over time, scale, and sensor hierarchy. We build these factors into the structure of feature representations, yielding models that automatically preserve the relationships between sensors and remain stable under realistic distortions such as time shifts, amplitude drift, and device orientation changes. On the UCI Human Activity Recognition benchmark, this categorical symmetry-driven design improves out-of-distribution accuracy by approx. 46 percentage points (approx. 3.6x over the baseline), demonstrating that abstract symmetry principles can translate into concrete performance gains in everyday sensing tasks via category-equivariant representation theory.

Paper Structure

This paper contains 38 sections, 9 theorems, 47 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Categorical Symmetry and Category-Equivariant Representations
  3. Problem setting.
  4. Product category of symmetries
  5. Group $M$ (time and gain).
  6. Poset $P$ (sensor hierarchy).
  7. Product category.
  8. Data and model functors; category-equivariant representations
  9. Concrete realization on HAR (Group × Poset)
  10. Spaces.
  11. Group actions on $X$ and $Y$.
  12. Representation maps.
  13. Transport maps on $Y$.
  14. Group invariance and poset naturality
  15. Remarks for HAR
  16. ...and 23 more sections

Key Result

proposition 1

We let $u_s: s{:}\mathrm{axes}\to s{:}\mathrm{mag}$ be the axis$\to$magnitude arrow in the poset, and let $X(u_s)$ denote its action on signals. For any $(t,\lambda)\in C_T\times\Lambda$, any $x_s\in X(s{:}\mathrm{axes})$, and any $x\in X(\mathrm{TOTAL})$, and the amplitude scalar satisfies $a_s((t,\lambda)\cdot x_s)=\lambda_s\,a_s(x_s).$

Theorems & Definitions (20)

  • definition 1: Data and model functors
  • definition 2: Category-equivariant representation
  • proposition 1: Group invariance
  • proof
  • proposition 2: Poset naturality
  • proof
  • lemma 1: M-action natural in $P$
  • proof
  • lemma 2: Generators determine naturality
  • proof
  • ...and 10 more