Personalized Convolutional Dictionary Learning of Physiological Time Series

TL;DR

The paper addresses the challenge of modeling physiological time series that exhibit both global, population-level structure and local, individual-specific variations. It introduces Personalized Convolutional Dictionary Learning (PerCDL), which learns a global dictionary $\bm{\Phi}$ and subject-specific personalization parameters $\bm{A}$ to generate personalized atoms $\hat{\bm{\phi}}^{s}_k = f(\bm{\phi}_k, \bm{a}_{k}^{s})$, enabling a mixed-effects-like representation via a time-warping transformation. A modular meta-algorithm alternates between estimating the global dictionary and refining personalization, including a federated-learning variant that scales to distributed data; theoretical guarantees show that the common-atom estimator converges at rate $\mathcal{O}\left( \frac{1}{\rho \sqrt{\sum_{s=1}^S p_s}} \right)$ under mild assumptions, improving over independent approaches. Empirically, PerCDL reliably recovers global motifs and individual-specific variations on synthetic data, gait, and ECG signals, demonstrating robustness to noise and enhanced interpretability for pathology analysis. The work highlights the potential of combining global dictionaries with learned temporal transformations to capture mixed-effects in physiological time series and suggests scalable, privacy-preserving extensions via federated learning.

Abstract

Human physiological signals tend to exhibit both global and local structures: the former are shared across a population, while the latter reflect inter-individual variability. For instance, kinetic measurements of the gait cycle during locomotion present common characteristics, although idiosyncrasies may be observed due to biomechanical disposition or pathology. To better represent datasets with local-global structure, this work extends Convolutional Dictionary Learning (CDL), a popular method for learning interpretable representations, or dictionaries, of time-series data. In particular, we propose Personalized CDL (PerCDL), in which a local dictionary models local information as a personalized spatiotemporal transformation of a global dictionary. The transformation is learnable and can combine operations such as time warping and rotation. Formal computational and statistical guarantees for PerCDL are provided and its effectiveness on synthetic and real human locomotion data is demonstrated.

Figures (16)

• Figure 1: Illustration of PerCDL for the Analysis of Human Locomotion Data. (left) The foot kinematic signals of two individuals ① and ② present a repeated structure called a gait cycle. (middle) PerCDL learns the shared structures and subject-specific variability: the common shape (green) is "personalized" using a transformation function $f$ (parameterized by the personalization parameters $\bm{a}_1$ and $\bm{a}_2$). (right) PerCDL successfully identifies all gait cycles and signal-specific shapes (blue and red), resulting in an accurate reconstruction.
• Figure 2: Time Warping using Eqs. \ref{['eq:transformation_warping']}-\ref{['eq:psi_def']}.
• Figure 3: Convergence Toward the Common Structures. Average distance between the common atoms identified in the synthetic experiment and the ground truth as a function of the dataset's size. Shaded regions represent the $95 \%$ confidence interval.
• Figure 4: Reconstruction Error Under Impulse Noise Contamination. Error bars represent two standard deviations.
• Figure 5: Application of PerCDL to Locomotion Data. (a) Common atom identified by PerCDL (grey) and averaged personalized gait cycle of the healthy population (blue). The shaded region represents two standard deviations. (b) Population-averaged personalized atoms learned by PerCDL in the Orthopedic (brown) and Neurological (green) groups, with two standard deviations (shaded areas). The red-colored curves represent the normalized relative variability in the gait cycle for the current population compared with the Healthy group.

Theorems & Definitions (8)

• Lemma 1
• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 2
• proof