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Robust Filtering and Smoothing with Gaussian Processes
Marc Peter Deisenroth, Ryan Turner, Marco F. Huber, Uwe D. Hanebeck, Carl Edward Rasmussen
TL;DR
Filtering and smoothing estimate latent states in nonlinear systems, where unknown transition and measurement functions may be modeled as Gaussian processes. The paper derives GP-RTSS, an analytic smoother that accounts for GP model uncertainty, and reports greater robustness to nonlinearities than state-of-the-art smoothers.
Problem
In GP dynamic systems, robust Bayesian state estimation must handle uncertainty in unknown transition and measurement functions.
Source: B. Gaussian RTS Smoothing · p. 2, §III · p. 3Method
GP-RTSS analytically computes an RTS smoothing approximation for GP dynamics while accounting for model uncertainty.
Source: §V · p. 6Results
GP-RTSS is reported to be more robust to nonlinearities than state-of-the-art smoothers.
Source: §V · p. 6Takeaways & Limitations
The smoother computes its Gaussian approximation without function linearization or sigma/cubature-point density representations, while integrating GP-induced model uncertainty.
Source: §V · p. 7GP-RTSS is computationally more involved than URTSS, EKS, and CKS, though this does not necessarily make it slower.
Source: §V · p. 7
Abstract
from arXiv · showhide
We propose a principled algorithm for robust Bayesian filtering and smoothing in nonlinear stochastic dynamic systems when both the transition function and the measurement function are described by non-parametric Gaussian process (GP) models. GPs are gaining increasing importance in signal processing, machine learning, robotics, and control for representing unknown system functions by posterior probability distributions. This modern way of "system identification" is more robust than finding point estimates of a parametric function representation. In this article, we present a principled algorithm for robust analytic smoothing in GP dynamic systems, which are increasingly used in robotics and control. Our numerical evaluations demonstrate the robustness of the proposed approach in situations where other state-of-the-art Gaussian filters and smoothers can fail.
I. INTRODUCTION
The paper introduces GP-RTSS, a Gaussian RTS smoother for GP dynamic systems that computes a closed-form approximation without linearization or small sampling approximations. It situates this method within Bayesian state estimation and Gaussian RTS smoothing for nonlinear systems.
I. INTRODUCTION
- Filtering and smoothing estimate latent-state posteriors from noisy measurement histories, with established linear-system solutions and many nonlinear approximations.Applications include navigation, control engineering, robotics, and machine learning.Source: §I · p. 1
- GP-RTSS computes a Gaussian approximation to the smoothing distribution in closed form, without linearization or small sampling approximations.It is derived for dynamic systems whose functions are modeled by Gaussian processes.Source: §I · p. 1
B. Gaussian RTS Smoothing
- Gaussian RTS smoothers differ in how they compute the means and covariances needed for smoothing, and GP-RTSS derives these quantities analytically for GP dynamics.The paper also describes GP-URTSS as following the URTSS derivation, which it does not detail.Source: B. Gaussian RTS Smoothing · p. 2
II. GAUSSIAN PROCESSES
The paper presents Gaussian processes as distributions over unknown functions, with predictions made from a posterior learned using noisy training data. Its default kernel assumptions are a zero prior mean and a squared-exponential covariance function.
II. GAUSSIAN PROCESSES
- In GP regression, noisy observations are generated from an unknown function, and the posterior predicts its values at arbitrary inputs.The model treats the function as random and assumes independent Gaussian observation noise.Source: §II · p. 2
- A GP is specified by a mean function and a covariance function, also called a kernel, which gives covariances between function values.The covariance function is defined using expectation over the function.Source: §II · p. 2
- The default prior uses a zero mean and a squared-exponential covariance function with added noise covariance; this assumes a smooth, stationary latent function.The paper motivates smoothness and stationarity as easier to justify than a fixed parametric form.Source: §II · p. 2
A. Expressiveness of the Model
The paper relates its zero-mean, squared-exponential GP prior to a universal function approximator built from Gaussian-shaped basis functions with random weights. It notes that other covariance functions encode different assumptions.
A. Expressiveness of the Model
- In the infinite-basis limit, the model represents a function using Gaussian-shaped basis functions and Gaussian random weights.The construction uses a white-noise process convolved with a Gaussian kernel.Source: A. Expressiveness of the Model · p. 2
- The resulting universal function approximator has the same zero prior mean and squared-exponential covariance as the GP model in one input dimension.The paper derives the mean and covariance by averaging over the random weights.Source: A. Expressiveness of the Model · p. 2
- The considered GP prior implicitly assumes latent functions representable by this universal approximator, while other covariance functions encode different model assumptions.Source: A. Expressiveness of the Model · p. 3
B. Training via Evidence Maximization
The paper describes evidence maximization for training GP models and then develops analytic moment-based smoothing for GP dynamic systems. The smoother accounts for transition-model uncertainty and computes the joint-state approximation without sampling, linearization, or numerical integration.
B. Training via Evidence Maximization
- Evidence maximization estimates GP covariance and noise hyperparameters by optimizing the data likelihood, trading off data fit with function complexity.The paper then uses trained GPs for the state-transition and measurement mappings.Source: B. Training via Evidence Maximization · p. 3
III. ROBUST SMOOTHING IN GAUSSIAN PROCESS DYNAMIC SYSTEMS
- The smoothing method extends analytic moment-based filtering to nonlinear GP models without linearization or sigma/cubature-point representations.It computes means and covariances for the adjacent-state joint distribution needed by the RTS smoother.Source: §III · p. 3
- The transition function is distributed according to a GP, so robust filtering and smoothing account for model uncertainty by Bayesian averaging.The derivation includes the predicted-state marginal and its cross-covariance with the previous state.Source: §III · p. 3
A. Marginal Distribution
The marginal predicted mean averages over system noise, the previous-state distribution, and uncertainty in the transition function. With a squared-exponential covariance function, the required integral is analytic; other tractable kernel choices are also identified.
1) Marginal Mean:
- The marginal predicted mean integrates over system noise, the previous state, and uncertainty in the transition function.The previous state is treated as a Gaussian filter distribution and as a test input to the GP regression model.Source: 1) Marginal Mean · p. 3
- The predicted mean is the expectation of the GP posterior mean function over the previous-state filter distribution.The posterior mean function is expressed as a finite sum over squared-exponential kernels centered at the GP training inputs.Source: 1) Marginal Mean · p. 3
- The squared-exponential covariance integral is analytic, and tractable alternatives include kernels combining squared exponentials, trigonometric functions, and polynomials.The mean expression uses GP training inputs, targets, kernel matrices, and dimension-specific hyperparameters.Source: 1) Marginal Mean · p. 3
2) Marginal Covariance Matrix:
The marginal covariance is computed using the law of total covariance, combining covariance of conditional expectations with expected conditional covariance. Under the stated diagonal-noise and GP training assumptions, the expected-covariance term contributes only to diagonal entries.
2) Marginal Covariance Matrix:
- The law of total covariance decomposes the marginal covariance into covariance of conditional expectations and expected conditional covariance.The system noise is assumed to have mean zero in the derivation.Source: 2) Marginal Covariance Matrix · p. 4
- Because the noise covariance is diagonal and GP target dimensions do not covary at deterministic inputs, the expected-covariance term contributes only to diagonal entries.The cited derivation gives this term as zero when target dimensions differ.Source: 2) Marginal Covariance Matrix · p. 4
- The marginal covariance combines the covariance-of-expectations term with the expected-covariance term on the diagonal, while retaining the former elsewhere.The resulting entries determine the marginal covariance in the predicted Gaussian distribution.Source: 2) Marginal Covariance Matrix · p. 4
B. Cross-Covariance
The section derives the predicted mean while explicitly averaging over transition-function uncertainty, then computes the cross-covariance needed to complete the joint Gaussian approximation. These joint moments provide the components used for analytic smoothing.
B. Cross-Covariance
- The cross-covariance between the previous state and next state is computed from the system equation using integrals expressed through squared-exponential kernels.The derivation writes the GP mean as a finite kernel sum and evaluates the integral analytically for the squared-exponential covariance function.Source: B. Cross-Covariance · p. 4
- The mean calculation averages out transition-function uncertainty and evaluates the GP posterior mean function over the previous-state distribution.The law of total expectation is used, and the GP mean function maps the previous state to the next state.Source: B. Cross-Covariance · p. 4
- The cross-covariance completes the joint Gaussian approximation, whose moments and filter step supply the components for analytic smoothing.The cited passage identifies the mean and covariance from the preceding equations as the necessary components.Source: B. Cross-Covariance · p. 5
IV. SIMULATIONS
The simulations section examines robustness of state-of-the-art nonlinear filters and evaluates the performances of the corresponding smoothers.
IV. SIMULATIONS
- The simulations analyze robustness of state-of-the-art nonlinear filters.The filter analysis is assigned to Sec. IV-A.Source: §IV · p. 5
- The simulations also analyze the performances of the corresponding smoothers.The smoother analysis is assigned to Sec. IV-B.Source: §IV · p. 5
A. Filter Robustness
The section evaluates one-step filter robustness on a modified nonlinear system with a sinusoidal measurement function. GP-ADF is reported as the most robust filter, with linearization and small-sample approximation errors affecting competing methods.
A. Filter Robustness
- The modified system replaces a quadratic measurement function with a sinusoidal one, which can produce arbitrarily many modes for a broad input distribution.The transition function also omits a purely time-dependent term to allow learning stationary transition dynamics.Source: A. Filter Robustness · p. 5
- The evaluation compares one filter step against a Gibbs-filter ground truth, making individual algorithms easier to assess than over longer trajectories.It tests EKF, UKF, CKF, SIR PF, GP-UKF, and GP-ADF.Source: A. Filter Robustness · p. 5
- GP-ADF is the most robust filter and statistically significantly outperforms all filters except the Gibbs-filter and SIR PF.The comparison averages 1,000 test runs and 100 randomly sampled start states per run.Source: A. Filter Robustness · p. 5
- Linearization errors explain the EKF’s poor performance, while degeneracy in small-sample density approximations affects UKF, GP-UKF, and CKF.The CKF uses fewer cubature points than the UKF and is statistically even less robust.Source: A. Filter Robustness · p. 5
B. Smoother Robustness
The pendulum experiment tests filters and smoothers over six seconds, including models learned from limited data. GP-RTSS is the only smoother reported to consistently reduce NLL relative to its corresponding filter, and remains robust at smaller signal-to-noise ratios.
B. Smoother Robustness
- Figure 1 contrasts shaded true predictive distributions with solid Gaussian UT predictions for the transition and measurement mappings.Source: Figure 1 · p. 6
- The experiment tracks a pendulum with a scalar measurement depending only on angle, so reconstructing its full state relies on angle–angular-velocity cross-correlation.The pendulum is driven by randomly sampled torque under zero-order-hold control, with measurements every 0.2 seconds.Source: B. Smoother Robustness · pp. 5–6
- For each trajectory, GP models GPf and GPg are learned from randomly generated data using either 250 or 20 data points.Source: B. Smoother Robustness · p. 6
- GP-RTSS is the only method that consistently reduces NLL relative to its corresponding filtering algorithm, and GP-ADF/GP-RTSS outperform EKF/EKS, UKF/URTSS, and CKF/CKS with 20 training points.The NLL results average 1,000 runs over a six-second tracking horizon.Source: B. Smoother Robustness · p. 6
- At even smaller signal-to-noise ratios, GP-RTSS remains robust while the other smoothers remain unstable.Source: B. Smoother Robustness · p. 6
V. DISCUSSION AND CONCLUSION
GP-RTSS is an analytic Rauch–Tung–Striebel smoother for GP dynamic systems that computes a Gaussian approximation to the smoothing distribution in closed form. The discussion attributes its robustness to avoiding linearization and density approximations while accounting for GP model uncertainty.
V. DISCUSSION AND CONCLUSION
- GP-RTSS is more robust to nonlinearities than state-of-the-art smoothers by avoiding linearization and density approximations that can produce incoherent filtering and smoothing estimates.It also represents uncertainty about system and measurement functions at locations not encountered during data collection.Source: §V · p. 6
- After offline GP training, GP-RTSS filtering costs O(T(E^3 + n^2(D^3 + E^3))), versus O(T(D^3 + E^3)) for URTSS and EKS, and scales linearly with T.Here n is the GP training-set size, and D and E are the state and measurement dimensions.Source: §V · p. 6
- Explicit equations for the integrals make numerical integration unnecessary, while grid-based approximations scale poorly with increasing dimension.The discussion identifies this dimensional scaling issue as already appearing in the histogram filter.Source: §V · p. 7
- In the pendulum example, repeated ODE-solver calls made EKS, CKS, and URTSS twice as slow as GP-RTSS with 250 training points.Although GP-RTSS is computationally more involved, it does not require those function evaluations after training.Source: §V · p. 7
- To the authors’ best knowledge, GP-RTSS is the most principled GP smoother because its computations are analytic and exactly integrate GP-induced model uncertainty.The method avoids function linearization and sigma- or cubature-point representations of densities.Source: §V · p. 7