A Functional Extension of Semi-Structured Networks

TL;DR

This work proposes a functional SSN method that retains the advantageous properties of classical functional regression approaches while also improving scalability, and demonstrates that this approach accurately recovers underlying signals, enhances predictive performance, and performs favorably compared to competing methods.

Abstract

Semi-structured networks (SSNs) merge the structures familiar from additive models with deep neural networks, allowing the modeling of interpretable partial feature effects while capturing higher-order non-linearities at the same time. A significant challenge in this integration is maintaining the interpretability of the additive model component. Inspired by large-scale biomechanics datasets, this paper explores extending SSNs to functional data. Existing methods in functional data analysis are promising but often not expressive enough to account for all interactions and non-linearities and do not scale well to large datasets. Although the SSN approach presents a compelling potential solution, its adaptation to functional data remains complex. In this work, we propose a functional SSN method that retains the advantageous properties of classical functional regression approaches while also improving scalability. Our numerical experiments demonstrate that this approach accurately recovers underlying signals, enhances predictive performance, and performs favorably compared to competing methods.

Figures (8)

• Figure 1: Exemplary weight surface (center), feature signal (bottom), and the resulting response signal (left) when integrating $\int x(s)w(s,t) ds$.
• Figure 2: Different semi-structured architectures.
• Figure 3: Estimated weight surfaces of the one functional shank gyroscope predictor in $\lambda^+$ for the different joints (columns), before and after correcting with the orthogonalization (rows). The color of each surface corresponds to the weight a predictor sensor signal at time points $s$ (x-axis) is estimated to have on the $t$th time point (y-axis) of the outcome (cf. Fig. \ref{['fig:ex']}). Without correction (upper row), the interpretable model part is not only incorrectly estimated but no effect at all is attributed to it. After correction, distinct patterns for some of the time point combinations and joints are visible, e.g., suggesting that an increased gyroscope value for early time points $s$ has a negative influence (dark purple color) on the first half of the hip adduction moment (bottom row, second plot from the left).
• Figure 4: Simulation study results.
• Figure 5: Comparison of performance improvements (larger is better) in mean squared error (MSE) for different joints (outcomes).