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Time series analysis with Gumbel dynamics

NeurIPS 2026 · Time series › Causal and generative time series · approach: Dynamical system reconstruction

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Proposes the Gumbel Dynamical Model (GDM), a differentiable switching dynamical model that relaxes discrete states with Gumbel-Softmax and trains via amortized black-box variational inference for interpretable time series states.

Problem

Switching dynamical systems have discrete states that struggle with smooth, variable-speed transitions, stochastic mixtures of overlapping states, and spurious rapid switching on real data.

Approach

Replace discrete states with Gumbel-Softmax relaxed states in a recurrent switching linear dynamical system, making the model fully differentiable and enabling amortized variational inference with standard gradient descent, so states become soft mixtures of dynamics primitives.

Key results

• On soft sticky NASCAR, GDM reaches 0.70 ± 0.03 inferred state accuracy vs 0.32–0.43 for baselines (rSLDS RO best baseline at 0.43 ± 0.09).
• On standard NASCAR, rSLDS (recurrent only) scores highest at 0.96 ± 0.06 while GDM gets 0.88 ± 0.10.
• On F1 telemetry, GDM's state accuracy is consistently higher than rSLDS across state dimensions K, with rapid gains at small K; on CalMS21 GDM gives significantly higher inferred state accuracy on all test trials.

Surprising findings

• On standard NASCAR, the recurrent-only rSLDS outperforms GDM in inferred state accuracy, which the authors attribute to GDM's stochastic treatment of observations.

Limitations

• Performance on CalMS21 could be improved with nonlinear recurrent functions, which were not the main evaluated setting
• Characterization of the impact of Gumbel parameters (temperature, scale) on performance is left open
• Standard NASCAR case where rSLDS is best shows the stochastic observation assumption is not always advantageous

Experimental setup

Datasets: synthetic NASCAR F1 telemetry via FastF1 (Shanghai International Circuit, Suzuka Circuit) CalMS21

Benchmarks: synthetic NASCAR F1 telemetry CalMS21

Metrics: R^2 (train/test) inferred state accuracy (k-NN mapping)

Baselines: SLDS (sticky) rSLDS (sticky) rSLDS (recurrent only) p-dLDS rSLDS

Builds on: Gumbel-Softmax trick switching linear dynamical systems recurrent SLDS probabilistic decomposed LDS black-box variational inference

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