Approximate Nonlinear Filtering by Projection on Exponential Manifolds of Densities

Brigo, Damiano; Hanzon, Bernard; Gland, François Le

doi:10.2307/3318714

Cited by 94 publications

(163 citation statements)

References 27 publications

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“…In particular, while the d H metric leads to the Fisher Information and to an equivalence between the projection filter and assumed density filters (ADFs) when using exponential families, see [18], the d D metric for simple mixture families is equivalent to a Galerkin method, as we show now following the second named author preprint [16]. For applications of Galerkin methods to nonlinear filtering, we refer for example to [9,24,27,41].…”

Section: Equivalence With Assumed Density Filters and Galerkin Methodsmentioning

confidence: 99%

“…A similar algorithm is described in [17,18] for projection using the Hellinger metric onto an exponential family. We refer to this as the HE projection filter.…”

Section: Comparison With the Hellinger Exponential (He) Projection Almentioning

confidence: 99%

“…As we shall see, the choice of the "best" Hilbert space structure is determined by the manifold one wishes to consider-for manifolds associated with exponential families of distributions the Hellinger metric leads to the simplest equations, whereas the direct L 2 metric works well with mixture distributions. It was proven in [18] that the projection filter in Hellinger metric on exponential families is equivalent to the classical assumed density filters. In this paper, we show that the projection filter for basic mixture manifolds in L 2 metric is equivalent to a Galerkin method.…”

Section: Introductionmentioning

confidence: 99%

“…We will call this scheme the L 2 normal mixture projection filter or simply the L2NM projection filter. The stochastic ODE for the Hellinger metric was considered in [17,19] and [18]. In particular, a precise numerical scheme is given in [19] for finding solutions by projecting onto an exponential family of distributions.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear filtering via stochastic PDE projection on mixture manifolds in $$L^2$$ L 2 direct metric

Armstrong

Brigo

2015

Math. Control Signals Syst.

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We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a new projection method for the optimal filter infinite-dimensional stochastic partial differential equation (SPDE), based on the direct L 2 metric and on a family of normal mixtures. This results in a new finite-dimensional approximate filter based on the differential geometric approach to statistics. We compare this new filter to earlier projection methods based on the Hellinger distance/Fisher metric and exponential families, and compare the L 2 mixture projection filter with a particle method with the same number of parameters, using the Levy metric. We discuss differences between projecting the SPDE for the normalized density, known as Kushner-Stratonovich equation, and the SPDE for the unnormalized density known as Zakai equation. We prove that for a simple choice of the mixture manifold the L 2 mixture projection filter coincides with a Galerkin method, whereas for more general mixture manifolds the equivalence does not hold and the L 2 mixture filter is more general. We study particular systems that may illustrate the advantages of this new filter over other algorithms when comparing outputs with the optimal filter. We finally consider a specific software design that is suited for a numerically efficient implementation of this filter and provide numerical examples. We leverage an algebraic ring structure by proving that in presence of a given structure in the system coefficients the key integrations needed to implement the new filter equations can be executed offline. B Damiano Brigo

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Section: Equivalence With Assumed Density Filters and Galerkin Methodsmentioning

confidence: 99%

“…A similar algorithm is described in [17,18] for projection using the Hellinger metric onto an exponential family. We refer to this as the HE projection filter.…”

Section: Comparison With the Hellinger Exponential (He) Projection Almentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonlinear filtering via stochastic PDE projection on mixture manifolds in $$L^2$$ L 2 direct metric

Armstrong

Brigo

2015

Math. Control Signals Syst.

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“…However, aside from the fact that the method only performs well for nearly linear systems, it is not clear how it can be applied to quantum models 1 . A much more flexible approximation for nonlinear filtering equations was proposed by Brigo, Hanzon and LeGland [10][11][12], based on the differential geometric methods of information geometry [13]. In this scheme we fix a finite-dimensional family of densities that are assumed to be good approximations to the information state.…”

Section: Introductionmentioning

confidence: 99%

Quantum projection filter for a highly nonlinear model in cavity QED

Handel¹,

Mabuchi²

2005

J. Opt. B: Quantum Semiclass. Opt.

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Both in classical and quantum stochastic control theory a major role is played by the filtering equation, which recursively updates the information state of the system under observation. Unfortunately, the theory is plagued by infinite dimensionality of the information state which severely limits its practical applicability, except in a few select cases (e.g. the linear Gaussian case). One solution proposed in classical filtering theory is that of the projection filter. In this scheme, the filter is constrained to evolve in a finite-dimensional family of densities through orthogonal projection on the tangent space with respect to the Fisher metric. Here we apply this approach to the simple but highly nonlinear quantum model of optical phase bistability of a strongly coupled two-level atom in an optical cavity. We observe near-optimal performance of the quantum projection filter, demonstrating the utility of such an approach.

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Variational Gaussian Approximation of the Kushner Optimal Filter

Lambert

Bonnabel

Bach

2023

Lecture Notes in Computer Science

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In estimation theory, the Kushner equation provides the evolution of the probability density of the state of a dynamical system given continuous-time observations. Building upon our recent work, we propose a new way to approximate the solution of the Kushner equation through tractable variational Gaussian approximations of two proximal losses associated with the propagation and Bayesian update of the probability density. The first is a proximal loss based on the Wasserstein metric and the second is a proximal loss based on the Fisher metric. The solution to this last proximal loss is given by implicit updates on the mean and covariance that we proposed earlier. These two variational updates can be fused and shown to satisfy a set of stochastic differential equations on the Gaussian's mean and covariance matrix. This Gaussian flow is consistent with the Kalman-Bucy and Riccati flows in the linear case and generalize them in the nonlinear one.

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Approximate Nonlinear Filtering by Projection on Exponential Manifolds of Densities

Cited by 94 publications

References 27 publications

Nonlinear filtering via stochastic PDE projection on mixture manifolds in $$L^2$$ L 2 direct metric

Nonlinear filtering via stochastic PDE projection on mixture manifolds in $$L^2$$ L 2 direct metric

Quantum projection filter for a highly nonlinear model in cavity QED

Variational Gaussian Approximation of the Kushner Optimal Filter

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