Approximating Analytically-Intractable Likelihood Densities with Deterministic Arithmetic for Optimal Particle Filtering

TL;DR

This work tackles the challenge of evaluating likelihoods for analytically-intractable, non-Gaussian, non-linear measurement models in real-time particle filters on resource-constrained devices. It introduces a deterministic arithmetic-on-pdf approach using Uncertainty-extended hardware (UxHw) to compute a predictive-lookahead proxy likelihood marginalized over transition noise within a predictive-lookahead auxiliary PF. Empirical results on the Gordon–Salmond–Smith system show substantial speedups (up to 37.7×) and improved RMSE and robustness compared with Monte Carlo and EKF-like baselines, including dramatically lower false-zero rates for narrow noise. The method enables real-time, reliable likelihood evaluation for complex models on embedded platforms, broadening the applicability of advanced likelihood models in robotics and related fields.

Abstract

Particle filtering algorithms have enabled practical solutions to problems in autonomous robotics (self-driving cars, UAVs, warehouse robots), target tracking, and econometrics, with further applications in speech processing and medicine (patient monitoring). Yet, their inherent weakness at representing the likelihood of the observation (which often leads to particle degeneracy) remains unaddressed for high-frequency and resource-constrained systems. Improvements such as the optimal proposal and auxiliary particle filter mitigate this issue under specific circumstances and with increased computational cost. This work presents a new particle filtering method and its implementation, which enables tunably-approximative representation of arbitrary likelihood densities as program transformations of parametric distributions. Our method leverages a recent computing platform that can perform deterministic computation on probability distribution representations (UxHw) without relying on stochastic methods. For non-Gaussian non-linear systems and with an optimal-auxiliary particle filter, we benchmark the likelihood evaluation error and speed for a total of 294840 evaluation points. For such models, the results show that the UxHw method leads to as much as 37.7x speedup compared to the Monte Carlo alternative. For narrow uniform observation noise, the particle filter falsely assigns zero likelihood as much as 81.89% of the time whereas UxHw achieves 1.52% false-zero rate. The UxHw approach achieves filter RMSE improvement of as much as 18.9% (average 3.3%) over the Monte Carlo alternative.

Figures (6)

• Figure 1: C code for three approaches of the proxy likelihood evaluation: pointwise (Listing \ref{['ListingProxyLyksPointwiseApproach']}), Monte Carlo simulation (Listing \ref{['ListingProxyLyksMonteCarloApproach']}) , and UxHw-based computation (Listing \ref{['ListingProxyLyksUxHwMarginalApproach']}). ➊ On an uncertainty-tracking processor tsoutsouras2022laplace such as the one in Figure \ref{['fig:NativeUncertaintyTrackingHardware']}, the + operator adds the conventional values but also performs addition between the associated distributions for these values. ➋ UxHwDoubleEvaluatePDF is not a C function but rather a function-wrapped UxHw-microarchitecture instruction for evaluating the density.
• Figure 2: a Diagram for the Monte Carlo simulation approach (Listing \ref{['ListingProxyLyksMonteCarloApproach']}). $M$ independent re-executions sample the transition noise and evaluate the analytic observation noise density at $h(f(x, \upsilon), 0)$, where $\upsilon\sim\mu_\upsilon$. The estimate for the proxy likelihood $m^{(i)}_k$ is the average of these evaluations. b Diagram for the UxHw approach (Listing \ref{['ListingProxyLyksUxHwMarginalApproach']}). In a single execution of the code, the UxHw arithmetic logic unit adds the entire transition noise density to the predicted state. Then, it pushes this predictive state distribution , together with the entire observation noise density, through the observation model $h$, computing the predictive observation distribution (Equation \ref{['EqPredictiveLikelihoodIntegralPushforward']}). The $m^{(i)}_k$ estimate is the evaluation of the predictive observation distribution at the measurement.
• Figure 3: (A) Empirical distribution of absolute errors for each method for 7320 instances of the Gordon--Salmond--Smith system with $\upsilon \sim \mathcal{N}(0, 3^2)$ and $\nu \sim \mathcal{U}(-0.25, 0.25)$. A point (X, Y) on the graph corresponds to Y% of errors being smaller than X. UxHw consistently achieves lower errors than the respective equal-speed Monte Carlo. Average (B) RMSE and (C) effective sample size for $\upsilon \sim \mathcal{N}(0, 3^2)$ and $\nu \sim \mathcal{U}(-0.05, 0.05)$ for different particle counts.
• Figure 4: Commercially-available system-on-module UxHw-FPGA-17k for native uncertainty tracking with 45 clock speed, 120 memory, 99 base power, 10$\times$40 size, based on RISC-V RV32IM.
• Figure 5: (a) Percentage of false-zero results for three UxHw and equal-speed Monte Carlo variants, with $\upsilon \sim \mathcal{N}(0, 3^2)$ and $\nu \sim \mathcal{U}(-\frac{s}{2},\frac{s}{2})$. UxHw lowers false-zero events by as much as 80 percentage points ($s=0.05$). (b) Percentage of filter iterations that had to reset the weights due to failure to explain the observation (when $s=0.1$). With UxHw the weight distribution collapses less frequently by as much as 2 percentage points (UxHw 8).