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A Functional Extension of Semi-Structured Networks
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Extends semi-structured networks (SSNs) to function-on-function regression by combining a basis-expanded interpretable functional regression part with an InceptionTime deep network, using a functional post-hoc orthogonalization for identifiability and array computations for scalability.
Problem
Function-on-function regression with functional predictors and outcomes is not expressive enough for all non-linearities and interactions, and naive SSN implementations do not scale to large functional datasets or ensure identifiability of the interpretable part.
Approach
A functional semi-structured network adds a tensor-product-basis functional regression term to a deep network (InceptionTime) residually, uses an extension of post-hoc orthogonalization to make the linear part identifiable, and uses array computation and basis recycling to scale in features and time points.
Key results
- On Fukuchi and Liew data, SSFNN using all features reaches median relative RMSE 0.25 versus 0.30 for boosting (all features).
- On the Carmago dataset, the semi-structured model achieves mean relative MSE difference 0.955 versus 0.923 for the deep model and 0.872 for the structured model.
- Scalability study shows the proposed implementation uses substantially less memory than existing additive model and boosting implementations, which grow superlinearly.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| combine structured functional regression with a deep network rather than use either component alone | SSFNN on Carmago dataset · Carmago dataset (five outcomes) | relative MSE difference: 0.872 -> 0.955; 0.923 -> 0.955 | Helped |
| use SSFNN instead of the additive model on selected features | Fukuchi and Liew datasets · Fukuchi and Liew datasets (selected features) | relative RMSE: 0.36 (0.08) -> 0.30 (0.03) | Helped |
| use SSFNN instead of Boosting with all features | Fukuchi and Liew datasets · Fukuchi and Liew datasets (all features) | relative RMSE: 0.30 (0.10) -> 0.25 (0.04) | Helped |
| replace Deep with SSFNN | SSFNN · EMG from EEG | correlation: 0.171 (0.111) -> 0.201 (0.162) | Mixed |
| add the non-shared deep architecture variant (b) (ablation) | SSFNN · Air quality (NO2) | correlation: 0.880 (0.052) -> 0.897 (0.047) | No clear effect |
| add the deep network to the structured functional model | SSFNN · Hot water consumption | correlation: 0.761 (0.006) -> 0.799 (0.042) | Helped |
| use shared encoding and decoding basis (SSFNN a) rather than non-shared architecture (SSFNN b) (ablation) | SSFNN · Hot water consumption | relative RMSE: 0.893 (0.077) -> 0.887 (0.052) | Helped |
Surprising findings
- Without post-hoc orthogonalization, the deep network absorbs almost all linear structure, so the interpretable weight surfaces are essentially absent until correction.
- On the air quality task, a plain FNN outperforms both the semi-structured variants in correlation, suggesting the task is easy for linear models.
Limitations
- Authors note the model could benefit from sparsity penalties for high-dimensional sensor data, which are not included.
- Non-linear FFR extension is proposed but not implemented; interpretability of such extensions is challenging.
- Deep network architectures in additional experiments were not tuned.
Experimental setup
Datasets: Fukuchi running dataset Liew running dataset Carmago lower limb biomechanics dataset EMG-EEG cognitive affective neuroscience dataset air quality NO2 dataset hot water consumption profiles dataset
Benchmarks: Fukuchi and Liew running datasets Carmago dataset EMG-EEG air quality NO2 hot water consumption
Metrics: relative RMSE absolute RMSE Pearson correlation relative MSE difference (functional R^2) memory consumption
Baselines: additive model FFR boosting FFR FNN (functional MLP) InceptionTime deep-only network non-linear FFR
Builds on: neural additive models semi-structured networks post-hoc orthogonalization function-on-function regression InceptionTime
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