Dead Directions: Geometric Singular Learning

• URL: http://arxiv.org/abs/2606.05957v1
• Date: Thu, 04 Jun 2026 09:54:08 GMT
• Title: Dead Directions: Geometric Singular Learning
• Abstract summary: Singular learning theory and information geometry have studied the same parameter spaces in mostly separate vocabularies.<n>We bridge them through one primitive, the dead direction: a unit vector along which the Fisher metric degenerates.<n>A selection rule on smooth fibres translates this rate into Watanabe's single-direction contribution to the real log canonical threshold.<n>We then lift this rate to a deep network: a multi-layer K-FAC factorisation writes each Fisher block as a product of activation- and gradient-side rates.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Singular learning theory and information geometry have studied the same parameter spaces in mostly separate vocabularies: the former computes Bayesian invariants in resolved coordinates, the latter works in original coordinates under a non-degeneracy assumption that overparameterised models routinely violate. We bridge them through one primitive, the dead direction: a unit vector along which the Fisher metric degenerates, equivalently a tangent to the analytic singular set with a definite KL order, set by how fast the KL divergence vanishes. The two readings name the same vector; our central move shows its KL order is recoverable as the decay rate of the directional Fisher curvature approaching the singularity, in original parameter coordinates and without a Hironaka resolution. A selection rule on smooth fibres translates this rate into Watanabe's single-direction contribution to the real log canonical threshold, and we extend the recovery to multi-component crossings, multiplicity $m$, the singular fluctuation $ν$ (universal in the KL order for 1D directions), prior-RLCT shifts, and tempered posteriors. We then lift this rate to a deep network: a multi-layer K-FAC factorisation writes each Fisher block as a product of activation- and gradient-side rates with a duality between them, instantiated at modern-network primitives (residual streams, layer normalisation, attention). A quotient theorem carries the rate to the gauge quotient $Θ/G$ under gradient flow on a $G$-invariant metric; SGD qualifies, standard Adam does not, and we construct a $G$-equivariant Adam-family preconditioner (DDCAdam) that does. The bridge yields a parameter-coordinate handle on singular geometry, closed-form per-architecture predictions, and a trajectory-rate readout of Watanabe's triple $(λ, m, ν)$ from one checkpoint's forward and backward passes, without posterior sampling.

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