---
title: "Dead Directions: Geometric Singular Learning"
source: "arXiv:2606.05957v1"
authors: "Tejas Pradeep Shirodkar"
affiliation: "IIIT Hyderabad"
year: 2026
category: "cs.LG, stat.ML"
published: "2026-06-04"
pages: 139
---

# Dead Directions: Geometric Singular Learning

**Author**: Tejas Pradeep Shirodkar (IIIT Hyderabad)
**arXiv**: 2606.05957v1 [cs.LG, stat.ML]
**Published**: 2026-06-04 | 139 pages

## Abstract

Bridges singular learning theory and information geometry through one primitive: the **dead direction** — a unit vector where the Fisher metric degenerates, with KL order recoverable from directional Fisher curvature decay rate in original coordinates (no Hironaka resolution). Lifts to deep networks via K-FAC factorization, constructs DDCAdam optimizer, and enables readout of Watanabe's triple (lambda, m, nu) from a single checkpoint.

## Key Concepts
- [[dead-direction|Dead Direction]] — core primitive bridging SLT and info geometry
- [[singular-learning-theory|Singular Learning Theory]] — Watanabe's framework
- [[information-geometry|Information Geometry]] — Amari's framework
- [[fisher-information-metric|Fisher Information Metric]] — the geometry object
- [[real-log-canonical-threshold|RLCT (lambda)]] — Watanabe's Bayesian invariant
- [[kl-order|KL Order]] — the bridge invariant
- [[watanabe-triple|Watanabe's Triple]] — (lambda, m, nu)
- [[ddcadam|DDCAdam]] — G-equivariant Adam-family preconditioner