Dead-direction signatures
Dead-Direction Signatures: A Cheap Spectral Reading of Singular Complexity · arXiv:2606.211581
Where the theory paper proves the order is readable in principle, this paper builds the cheap instruments. Three observables, each a single pass: $\sigma_{\min}$ of the activations detects a dead direction from the forward side; $\lambda^{+}_{\min}(G)$ detects and localises it from the gradient side; and the active volume $\log\det^{+}(G)$ counts how many are dying together. The hero result is the rank-multiplicative volume identity, the volume slope tracking the planted rank-deficit across $r = 1, 2, 3, 4$, a discriminating test that no single-eigenvalue monitor can run. Around it sit the structural correlation between the two sides (down to $\rho = +0.75$ even where Adam breaks the trajectory rate), the ground-truth anchor on closed-form cells, and a width sweep where the cheap reads separate models that a fixed-budget posterior sampler reads as identical.
The relationship to the established SGLD estimator is stated carefully and matters: the signatures are directional, layer-local, and three to four orders of magnitude cheaper per checkpoint; the sampler stays canonical for the integrated posterior readings the signatures do not provide.
Where it is explained here
- The two sides and their lock: reading a dead direction from both sides#7
- The counting identity: counting dead directions#8
- The forward-side floor at depth: a floor that survives depth#9
- The price of each read: the cost of reading a dead direction#11
The signature demo
The volume identity, on the measured runs:
Everything the later papers measure at scale runs on reads this paper validated first.
- Tejas Pradeep Shirodkar and P. J. Narayanan, Dead-Direction Signatures: A Cheap Spectral Reading of Singular Complexity, arXiv:2606.21158 (2026). ↩︎