Recurrent state estimation gets a convergence guarantee: one gain drives the error exponentially into any small neighborhood
Synopsis
Recurrent models for state estimation previously had empirical success but no formal convergence or robustness guarantees. This work treats a recurrent network as a discretization of continuous latent dynamics and requires the latent flow to contract, proving that a smooth injective map embeds the true physical state into latent space; raising a single scalar gain κ drives the estimation error exponentially fast into an arbitrarily small neighborhood of zero, at the cost of decoder capacity growing polynomially in κ. On a real mobile-robot pose-estimation task, the provably contracting model outperforms a vanilla GRU and trains more stably.
Interpretation
Recurrent networks that estimate the state of a dynamical system can now come with formal guarantees on convergence rate and tunability, rather than only post-training empirical behavior. Recurrent models such as LSTMs and GRUs perform strongly on sequence tasks but lack formal guarantees on convergence and robustness; the existing existence result for nonlinear contracting latent dynamics covers only diagonal latent vector fields, whereas practical recurrent architectures mix nonlinearities across latent coordinates. On a real mobile-robot pose-estimation task, the provably contracting model outperforms an unconstrained vanilla GRU and trains more stably across 5 training seeds; on the same task, unconstrained GRUs also converge to contracting dynamics.
The approach lets the latent state evolve under an observation-driven continuous vector field and enforces that the field contracts, so the network exponentially forgets its initial condition. The contraction condition is that the symmetric part of the latent dynamics Jacobian is uniformly negative definite, and it is satisfied by construction through spectral normalization and a fixed dissipation coefficient rather than checked after training. Training uses real navigation data from 4 buildings, 14 floors, 27.4 km and 18 h 12 min, with the validation floor and test building unseen; the trained contracting GRU satisfies the monotonicity condition at every time step along all validation trajectories.
The decoder only has to recover the physical state from the latent state, and its required size has a quantitative relation to the gain κ and the target precision. With an ideal left inverse available, the error decays as κ to the power ℓ times an exponential term, where ℓ is the observability index; when an MLP approximates that inverse, the number of weights grows polynomially in κ to the power ℓ and in 1/ϵ. In a simulated nonlinear oscillator with an unknown parameter, separate observers are trained for κ from 0.5 to 20 and decoder widths from 32 to 4096; at fixed width the error first falls and then rises with κ, consistent with the predicted decoder-Lipschitz cost.
Perspective
The conclusions target systems whose observations and target variables are generated by a deterministic ordinary differential equation, requiring the physical state to be differentially observable, the latent dynamics to contract, and a filter rank condition to hold; the real experiment is pose estimation of a mobile robot relative to its starting position, with lidar localization readings available during training but not at deployment.
The theoretical guarantees are existence results: they show suitable weights exist rather than giving a constructive training algorithm, and enforcing the rank condition during training is non-trivial, with the supporting evidence being empirical. The generative model is deterministic, noise robustness is stated for bounded measurement disturbances with a noise floor that grows with κ to the power ℓ, and a stochastic treatment of error variance is left for future work. Reconstructing only a function of a non-observable state would need a functional observability framework, so far developed only for linear latent dynamics.