How I work
I take well-established objects from classical mathematics — the Kac–Rice formula, Ornstein’s d̄-distance, Doob’s h-transform, Heckman’s selection model — and instantiate them as practical tools and diagnostics for modern machine learning.
All of it runs on a single RTX 3080 in a home workstation. The constraint is deliberate: it forces questions sharp enough to be answered with careful experimental design rather than scale.
Research
The Kac–Rice formula, Minkowski functionals, and Rice's formula, made differentiable — mesh-free losses, sparsity budgets, and topology control for neural fields, PINNs, and event-driven networks.
What controls when capabilities appear? Causal experiments on the grokking delay, representational priors, critical windows, and state tracking in linear RNNs.
Ergodic-theoretic tools — Ornstein's d̄-distance, transfer-operator spectra, point-process likelihoods — for evaluating surrogates of chaotic systems and auditing latent reasoning.
Econometric selection models (Heckman 1979) brought to deep learning: calibrated uncertainty under selection on unobservables, and survivor bias in AutoML.
Doob-conditioned consolidation that turns analog device noise into a memory resource — validated on real BrainScaleS-2 neuromorphic silicon.
Positional information relative to learned semantic anchors, and what linear RNNs actually compute underneath their parameterization.
Selected publications
Controlled experiments showing the grokking delay is causally the time to form task-structured representations. Injecting the right structure through a contrastive representational prior collapses the delay — up to 22× faster generalization — while structure-agnostic interventions do not.
When are two neural networks the same computation? A process-level equivalence metric from ergodic theory that separates what representation-geometry tools (CKA, SVCCA) and linearized-dynamics tools (DSA) conflate — such as a model versus its pruned, distilled, or noise-injected twin.
Casts per-synapse memory consolidation as a Doob h-transform: condition each weight's stochastic dynamics on never crossing a memory barrier, and intrinsic device noise becomes a consolidation resource instead of an accuracy tax. Confirmed on real BrainScaleS-2 analog neuromorphic hardware, with +15.6 points retention.
Training data is routinely collected by a selection process the model never sees — loans observed only when granted, outcomes only when a test was ordered. When selection acts on unobservables, importance weighting and covariate-shift corrections fail; adapting Heckman's econometric selection model to deep networks restores calibrated uncertainty.
Coordinate-MLP neural fields exhibit spectral bias: low frequencies fit quickly, high frequencies slowly or never. The Kac–Rice level-crossing density gives a mesh-free, FFT-free auxiliary loss that acts directly on scattered data in the spatial domain — where frequency-domain remedies require grids.
Two chaotic systems can share an invariant measure yet compute different dynamics — a failure Wasserstein-style evaluation cannot see. Ornstein's d̄-distance from ergodic isomorphism theory catches it, validated on Lorenz-63 and Kuramoto–Sivashinsky surrogates.
In the pipeline
A single architectural term — the additive input injection — is the causal culprit behind why expressive linear RNNs fail to length-generalize on hard state tracking. Delete it and the same architecture learns the exact automaton, generalizing perfectly at 16× training length (S5 accuracy 0.20 → 1.00). Zeroing the term at inference restores exact generalization: the model had learned the automaton underneath all along. 202 pre-registered runs; every number regenerates byte-for-byte.
github.com/gunnerhowe/automaton-underneathThe capstone of the grokking program: forecasting when a capability will emerge — per seed, before it happens, with calibrated intervals, certified false-alarm rates, and a blind pre-registered gate.
An AI-safety study formalizing whether a reward hack's exploit set is finite (closable by patching) or open-ended — and predicting which, directly from the task specification and verifier code.
Contact
Open to collaboration, review, and conversations about any of the work above — especially replication attempts and adversarial reads. The fastest way to reach me is email.