The Geometry of Projection Heads: Conditioning, Invariance, and Collapse
We develop a geometric theory of projection heads in self-supervised learning by modeling the head as a trainable Riemannian metric on the backbone representation manifold. We show that linear heads perform implicit subspace whitening, while nonlinear heads adapt local metrics to satisfy the specific topological constraints of the loss, with head depth empirically dictating this capacity. Analyzing dimensional collapse, we prove that smooth nonlinear heads natively induce negative eigenvalues in the Hessian at collapsed equilibria, making them unstable. We empirically validate this by continuously tracking the optimization geometry during training, which reveals that smooth activations li...
| Authors | Faris Chaudhry |
|---|---|
| Primary category | cs.LG |
| Published | 16 May 2026 |
| Last updated at source | 16 May 2026 |
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