Gradient flows in uncertainty propagation and filtering of linear Gaussian systems

Halder, Abhishek; Georgiou, Tryphon T.

doi:10.1109/cdc.2017.8264109

Cited by 16 publications

(35 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The purpose of this paper is to pursue a systems-theoretic variational viewpoint for computing ρ(x, t) that breaks away from the "solve PDE as a PDE" philosophy, and instead solves (2) as a gradient descent on the manifold of joint PDFs. This emerging geometric viewpoint for uncertainty propagation and filtering has been reported in our recent work [18], [19], but it remained unclear whether this viewpoint can offer computational benefit over the standard PDE solvers. It is not at all obvious whether and how an infinite-dimensional gradient descent can numerically obviate function approximation or spatial discretization.…”

Section: Introductionmentioning

confidence: 92%

Gradient Flow Algorithms for Density Propagation in Stochastic Systems

Caluya

Halder

2020

IEEE Trans. Automat. Contr.

Self Cite

View full text Add to dashboard Cite

We develop a new computational framework to solve the partial differential equations (PDEs) governing the flow of the joint probability density functions (PDFs) in continuous-time stochastic nonlinear systems. The need for computing the transient joint PDFs subject to prior dynamics arises in uncertainty propagation, nonlinear filtering and stochastic control. Our methodology breaks away from the traditional approach of spatial discretization or function approximation -both of which, in general, suffer from the "curse-of-dimensionality". In the proposed framework, we discretize time but not the state space. We solve infinite dimensional proximal recursions in the manifold of joint PDFs, which in the small time-step limit, is theoretically equivalent to solving the underlying transport PDEs. The resulting computation has the geometric interpretation of gradient flow of certain free energy functional with respect to the Wasserstein metric arising from the theory of optimal mass transport. We show that dualization along with an entropic regularization, leads to a cone-preserving fixed point recursion that is proved to be contractive in Thompson metric. A block co-ordinate iteration scheme is proposed to solve the resulting nonlinear recursions with guaranteed convergence. This approach enables remarkably fast computation for non-parametric transient joint PDF propagation. Numerical examples and various extensions are provided to illustrate the scope and efficacy of the proposed approach.

show abstract

Section: Introductionmentioning

confidence: 92%

Gradient Flow Algorithms for Density Propagation in Stochastic Systems

Caluya

Halder

2020

IEEE Trans. Automat. Contr.

Self Cite

View full text Add to dashboard Cite

show abstract

“…where π(x, y) denotes a joint probability measure supported on M × M, and the geodesic distance d G is given by (13). The infimum in (47) is taken over the set Π 2 (µ, ν), which we define as the set of all joint probability measures having finite second moment that are supported on M×M, with prescribed x-marginal µ, and prescribed y-marginal ν. If µ and ν have respective PDFs ρ x and ρ y , then we can use the notation W G (ρ x , ρ y ) in lieu of W G (µ, ν).…”

Section: A Wasserstein Gradient Flowmentioning

confidence: 99%

“…Just like the proximal viewpoint of finite dimensional gradient descent, (54) can be taken as an alternative definition of gradient descent of F (·) on the manifold P 2 (M) w.r.t. the metric W G ; see e.g., [47]- [50]. The utilitarian value of (54) over (52) is as follows.…”

Section: B Proximal Recursion On P 2 (M)mentioning

confidence: 99%

“…We generate N = 500 samples from the uniform initial joint PDF ρ 0 supported on [0, 1] 2 in the form of the point (59), starting from the uniform initial joint PDF. The joint PDF evolution is computed via (54) with W G given by (47), wherein d G is given by (30). The color (red = high, blue = low) denotes the joint PDF value at a point at that time (see colorbar).…”

Section: Proximal Algorithmmentioning

confidence: 99%

See 1 more Smart Citation

Hopfield Neural Network Flow: A Geometric Viewpoint

Halder

Caluya

Travacca

et al. 2020

IEEE Trans. Neural Netw. Learning Syst.

Self Cite

View full text Add to dashboard Cite

We provide gradient flow interpretations for the continuous-time continuous-state Hopfield neural network (HNN). The ordinary and stochastic differential equations associated with the HNN were introduced in the literature as analog optimizers, and were reported to exhibit good performance in numerical experiments. In this work, we point out that the deterministic HNN can be transcribed into Amari's natural gradient descent, and thereby uncover the explicit relation between the underlying Riemannian metric and the activation functions. By exploiting an equivalence between the natural gradient descent and the mirror descent, we show how the choice of activation function governs the geometry of the HNN dynamics.For the stochastic HNN, we show that the so-called "diffusion machine", while not a gradient flow itself, induces a gradient flow when lifted in the space of probability measures. We characterize this infinite dimensional flow as the gradient descent of certain free energy with respect to a Wasserstein metric that depends on the geodesic distance on the ground manifold. Furthermore, we demonstrate how this gradient flow interpretation can be used for fast computation via recently developed proximal algorithms.

show abstract

“…This allows interpreting the discrete time-stepping as steepest descent of the functional Φ w.r.t. distance d. Proximal operators have also been used in general Hilbert spaces [13], and in the space of probability density functions [4], [5], [11], [12], [14]- [16]. The idea of applying proximal recursion in the space of probability measures appeared first in [14]; see also [15].…”

Section: Main Ideamentioning

confidence: 99%