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Mechanistic Interpretability of RNNs emulating Hidden Markov Models
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Trains vanilla RNNs with a Sinkhorn loss and Gumbel-Softmax outputs to emulate HMM emission statistics, then reverse-engineers them to find noise-sustained orbital dynamics, kick neurons, and self-induced stochastic resonance.
Problem
How RNNs can generate discrete, stochastic, HMM-like latent dynamics given that RNNs typically model deterministic, input-driven tasks with continuous state spaces.
Approach
Fit vanilla RNNs directly to HMM emission sequences using a Sinkhorn divergence loss with Gumbel-Softmax sampling, then reverse-engineer them: stochastic input sustains closed orbits with slow cluster regions linked by fast kick-zone transitions, driven by kick-neurons and noise-integrating populations in a self-induced stochastic resonance regime.
Key results
- Trained RNNs (|h|=150, d=100) replicate HMM emission statistics across linear-chain, fully-connected and cyclic HMMs, with transition-matrix deviations approaching near-perfect correspondence at d>=100 for linear-chain models.
- Without input, trained RNNs converge to a single fixed point; under Gaussian input noise they show closed orbital dynamics whose radius scales linearly with input variance.
- Ablating kick-neurons or their noise drive traps trajectories in clusters, while enhancement (mu=2) causes overshoot; the mechanism is interpreted as self-induced stochastic resonance.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase input-noise dimensionality | Vanilla RNN trained on linear-chain HMMs · Linear-chain HMMs | Sinkhorn-aligned sequence distance | Helped |
| Increase input-noise dimensionality | Vanilla RNN trained on linear-chain HMMs · Linear-chain HMMs | transition-matrix deviation | Helped |
| Increase hidden size beyond 150 | Vanilla RNN · HMM emulation | emission-statistics replication | No clear effect |
| Remove stochastic input | Trained vanilla RNN · HMM emulation | orbital dynamics | Hurt |
| Silence the kick-neurons (ablation) | Trained vanilla RNN · Two-state HMM | state transitions | Hurt |
| Ablate input noise to the corresponding noise-integrating population (ablation) | Trained vanilla RNN · Two-state HMM | state transitions | Hurt |
| Double kick-neuron activity | Trained vanilla RNN · Two-state HMM | trajectory transition behavior | Hurt |
| Amplify noise-drive from the corresponding noise-integrating population | Trained vanilla RNN · Two-state HMM | trajectory transition behavior | Hurt |
Surprising findings
- Trained RNNs converge to a single fixed point without input rather than forming one attractor per HMM state.
- Input noise, rather than being a nuisance, is used as the computational signal driving state transitions.
Limitations
- Only vanilla ReLU RNNs trained on small synthetic HMMs; no validation on real neural or behavioral data.
- Convergence becomes less consistent as the number of HMM states grows, requiring shorter sequences.
- Second-order perturbation analysis uses an approximation because ReLU has no second derivative.
Experimental setup
Datasets: synthetic sequences sampled from linear-chain, fully-connected, and cyclic HMMs
Benchmarks: synthetic HMM emission statistics
Metrics: Sinkhorn-aligned Euclidean distance transition matrix squared differences observation frequencies observation volatility
Builds on: Sinkhorn divergence Gumbel-Softmax Adam
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