How to Avoid the Curse of Dimensionality: Scalability of Particle Filters with and without Importance Weights

IEEE Control Syst. Lett.

Kutschireiter

Pfister

2020

Self Cite

The filtering of a Markov diffusion process on a manifold from counting process observations leads to 'large' changes in the conditional distribution upon an observed event, corresponding to a multiplication of the density by the intensity function of the observation process. If that distribution is represented by unweighted samples or particles, they need to be jointly transformed such that they sample from the modified distribution. In previous work, this transformation has been approximated by a translation of all the particles by a common vector. However, such an operation is ill-defined on a manifold, and on a vector space, a constant gain can lead to a wrong estimate of the uncertainty over the hidden state. Here, taking inspiration from the feedback particle filter (FPF), we derive an asymptotically exact filter (called ppFPF) for point process observations, whose particles evolve according to intrinsic (i.e. parametrization-invariant) dynamics that are composed of the dynamics of the hidden state plus additional control terms. While not sharing the gain-times-error structure of the FPF, the optimal control terms are expressed as solutions to partial differential equations analogous to the weighted Poisson equation for the gain of the FPF. The proposed filter can therefore make use of existing approximation algorithms for solutions of weighted Poisson equations.

Section: Discussionmentioning

confidence: 99%

Asymptotically Exact Unweighted Particle Filter for Manifold-Valued Hidden States and Point Process Observations

IEEE Control Syst. Lett.

Kutschireiter

Pfister

2020

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“…by resampling the particles from the weight distribution and resetting the weights to 1/M . The time scale on which this weight degeneracy happens depends on the number of observable dimensions, in other words it is accelerated as the dimensionality of the system is increased (Surace et al, 2019b). This is a form of the so-called 'curse of dimensionality', a common nuisance in weighted particle filters.…”

Section: Particle Filtering In Continuous Timementioning

confidence: 99%

“…Unweighted particle filters therefore hold the promise of avoiding the curse of dimensionality (see Surace et al, 2019b).…”

Section: Particle Filtering In Continuous Timementioning

confidence: 99%

The Hitchhiker’s guide to nonlinear filtering

Kutschireiter

Journal of Mathematical Psychology

Pfister

2020

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Nonlinear filtering is the problem of online estimation of a dynamic hidden variable from incoming data and has vast applications in different fields, ranging from engineering, machine learning, economic science and natural sciences. We start our review of the theory on nonlinear filtering from the simplest 'filtering' task we can think of, namely static Bayesian inference. From there we continue our journey through discrete-time models, which is usually encountered in machine learning, and generalize to and further emphasize continuous-time filtering theory. The idea of changing the probability measure connects and elucidates several aspects of the theory, such as the parallels between the discrete-and continuous-time problems and between different observation models. Furthermore, it gives insight into the construction of particle filtering algorithms. This tutorial is targeted at scientists and engineers and should serve as an introduction to the main ideas of nonlinear filtering, and as a segway to more advanced and specialized literature.

“…The issue has become even more severe in the era of big data where the dimensionality of the processes is very large. The main approach to approximate the conditional distribution, sequential Monte-Carlo or particle filtering (see [2] for a survey and pointers to the literature), is known to exhibit a curse of dimensionality as the number of dimensions of the observations grows [3]- [6]. The problem can be traced down to the use of importance weights and their increasing degeneracy as time progresses (see [6] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Gauge Freedom within the Class of Linear Feedback Particle Filters

Abedi

2019 IEEE 58th Conference on Decision and Control (CDC)

2019

Self Cite

Feedback particle filters (FPFs) are Monte-Carlo approximations of the solution of the filtering problem in continuous time. The samples or particles evolve according to a feedback control law in order to track the posterior distribution. However, it is known that by itself, the requirement to track the posterior does not lead to a unique algorithm. Given a particle filter, another one can be constructed by applying a time-dependent transformation of the particles that keeps the posterior distribution invariant. Here, we characterize this gauge freedom within the class of FPFs for the linear-Gaussian filtering problem, and thereby extend previously known parametrized families of linear FPFs.