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Approximating Latent Manifolds in Neural Networks via Vanishing Ideals
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Proposes truncating a pretrained network at an intermediate layer and classifying with polynomial generators of class-manifold vanishing ideals followed by one polynomial layer, yielding shallower, faster models with sharper generalization bounds.
Problem
Deep networks learn latent representations that approximate data manifolds, but these latent manifolds are not explicitly characterized, and deep models are costly; the paper asks whether vanishing ideals can describe latent class manifolds to yield compact classifiers.
Approach
Truncate a pretrained network at an intermediate layer, approximate each class's latent manifold with polynomial generators of its vanishing ideal (computational algebra), and map the resulting features to linearly separable space via a single polynomial layer.
Key results
- The truncated models have significantly fewer layers than pretrained baselines while maintaining comparable accuracy, per the abstract.
- The approach achieves higher throughput and uses fewer parameters than baselines, per the abstract.
- Spectral complexity analysis yields generalization bounds that can in principle be tighter than standard deep networks.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace the pretrained baseline with the proposed truncated network and polynomial layer | Proposed vanishing-ideal model · Numerical experiments | accuracy | No clear effect |
| Replace the pretrained baseline with the proposed truncated network and polynomial layer | Proposed vanishing-ideal model · Numerical experiments | throughput | Helped |
| Replace the pretrained baseline with the proposed truncated network and polynomial layer | Proposed vanishing-ideal model · Numerical experiments | number of parameters | Helped |
Experimental setup
Metrics: accuracy throughput parameter count
Baselines: pretrained baseline networks
Builds on: manifold hypothesis vanishing ideal approximate vanishing ideal algorithms
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