We propose a new asymptotic expansion method for nonlinear filtering, based on a small parameter in the system noise. The conditional expectation is expanded as a power series in the noise level, with each coefficient computed by solving a system of ordinary differential equations. This approach mitigates the trade-off between computational efficiency and accuracy inherent in existing methods such as Gaussian approximations and particle filters. Moreover, by incorporating an Edgeworth-type expansion, our method captures complex features of the conditional distribution, such as multimodality, with significantly lower computational cost than conventional filtering algorithms.

This paper is concerned with the problem of nonlinear filtering, i.e., computing the conditional distribution of the state of a stochastic dynamical system given a history of noisy partial observations. Conventional sequential importance resampling (SIR) particle filters suffer from fundamental limitations, in scenarios involving degenerate likelihoods or high-dimensional states, due to the weight degeneracy issue. In this paper, we explore an alternative method, which is based on estimating the Brenier optimal transport (OT) map from the current prior distribution of the state to the posterior distribution at the next time step. Unlike SIR particle filters, the OT formulation does not require the analytical form of the likelihood. Moreover, it allows us to harness the approximation power of neural networks to model complex and multi-modal distributions and employ stochastic optimization algorithms to enhance scalability. Extensive numerical experiments are presented that compare the OT method to the SIR particle filter and the ensemble Kalman filter, evaluating the performance in terms of sample efficiency, high-dimensional scalability, and the ability to capture complex and multi-modal distributions.

One attractive instance of measure transport for Bayesian Multi-agent systems are commonplace in today's technological inference is through the approximation of the Knothe-landscape, and many problems that were once cast in Rosenblatt (KR) rearrangement [13], [14]. This transformation a centralized setting, have been recast in a distributed manner can be easily approximated given only samples of [1]. With the introduction of multiple agents, various considerations a distribution and has been applied for various higherdimensional must be made due to information flow, changes and nonlinear filtering problems [15], [16].

Nonlinear (cid:12)ltering can solve very complex problems, but typically involve very time consuming calculations. Here we show that for (cid:12)lters that are constructed as a RBF network with Gaussian basis functions, a decomposition into linear (cid:12)lters exists, which can be computed e(cid:14)ciently in the frequency domain, yielding dramatic improvement in speed. We present an application of this idea to image processing. In electron micrograph images of photoreceptor terminals of the fruit (cid:13)y, Drosophila, synaptic vesicles containing neurotransmitter should be detected and labeled automatically. We use hand labels, provided by human experts, to learn a RBF (cid:12)lter using Support Vector Regression with Gaussian kernels.

Nonlinear filtering can solve very complex problems, but typically involve very time consuming calculations. Here we show that for filters that are constructed as a RBF network with Gaussian basis functions, a decomposition into linear filters exists, which can be computed efficiently in the frequency domain, yielding dramatic improvement in speed. We present an application of this idea to image processing. In electron micrograph images of photoreceptor terminals of the fruit fly, Drosophila, synaptic vesicles containing neurotransmitter should be detected and labeled automatically. We use hand labels, provided by human experts, to learn a RBF filter using Support Vector Regression with Gaussian kernels. We will show that the resulting nonlinear filter solves the task to a degree of accuracy, which is close to what can be achieved by human experts. This allows the very time consuming task of data evaluation to be done efficiently.

The most simple filter is linear mapping.