A Primer on Stochastic Differential Geometry for Signal Processing

Manton, Jonathan H.

doi:10.1109/jstsp.2013.2264798

Cited by 22 publications

(14 citation statements)

References 66 publications

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“…A stochastic process X is a collection of random variables X(t) indexed by a parameter t ∈ T often taken to be time. An archetypal process is the Wiener process, described from first principles in [43], among other introductory material. A sample path is a realisation t → X(t) of X, also known as a randomly chosen signal or waveform.…”

Section: The Rkhs Associated With a Stochastic Processmentioning

confidence: 99%

A Primer on Reproducing Kernel Hilbert Spaces

Manton

Amblard

2015

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Reproducing kernel Hilbert spaces are elucidated without assuming prior familiarity with Hilbert spaces. Compared with extant pedagogic material, greater care is placed on motivating the definition of reproducing kernel Hilbert spaces and explaining when and why these spaces are efficacious. The novel viewpoint is that reproducing kernel Hilbert space theory studies extrinsic geometry, associating with each geometric configuration a canonical overdetermined coordinate system. This coordinate system varies continuously with changing geometric configurations, making it well-suited for studying problems whose solutions also vary continuously with changing geometry. This primer can also serve as an introduction to infinite-dimensional linear algebra because reproducing kernel Hilbert spaces have more properties in common with Euclidean spaces than do more general Hilbert spaces.In several contexts, RKHS methods have been described as providing a unified framework [77,32,46,59]; although a subclass of problems was solved earlier by other techniques, a RKHS approach was found to be more elegant, have broader applicability, or offer new insight for obtaining actual solutions, either in closed form or numerically. Parzen denote a straight line in R 2 passing through the origin and intersecting the horizontal axis at an angle of θ radians; it is a one-dimensional subspace of R 2 . Fix an arbitrary point p = (p 1 , p 2 ) ∈ R 2 and define f (θ) to be the point on L(θ) closest to p with respect to the Euclidean metric. It can be shown thatVisualising f (θ) as the projection of p onto L(θ) shows that f (θ) depends continuously on the orientation of the line. While (1.2) veri- Pointwise Coordinates and Canonical CoordinatesFinite-dimensional RKHSs space also be finite dimensional? For convenience, we adopt the latter interpretation. While this interpretation is not standard, it is consistent with our emphasis on the embedding space playing a significant role in RKHS theory. Precisely, we say a RKHS is finite dimensional if the set X in §4 has finite cardinality. The Kernel of an Inner Product SubspaceCentral to RKHS theory is the following question. Let V ⊂ R n be endowed with an inner product. How can this configuration be described efficiently? The configuration involves three aspects: the vector space V , the orientation of V in R n , and the inner product on V . Importantly, the inner product is not defined on the whole of R n , unless V = R n .(An alternative viewpoint that will emerge later is that RKHS theory studies possibly degenerate inner products on R n , where V represents the largest subspace on which the inner product is not degenerate.) One way of describing the configuration is by writing down a basis {v 1 , · · · , v r } ⊂ V ⊂ R n for V and the corresponding Gram matrix G whose ijth element is G ij = v i , v j . Alternatively, the configuration is completely described by giving an orthonormal basis {u 1 , · · · , u r } ⊂ V ⊂ R n ; the Gram matrix associated with an orthonormal basis is the identity matrix and...

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Section: The Rkhs Associated With a Stochastic Processmentioning

confidence: 99%

A Primer on Reproducing Kernel Hilbert Spaces

Manton

Amblard

2015

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“…Therefore, to define "Brownian motion on S", we define some diffusion (X t ) t≥0 that takes values in R 2 , for which the process (r(X t )) t≥0 "looks locally" like a Brownian motion (and lies on S). See [42] for more intuition here. Our goal, therefore, is to define a diffusion on Euclidean space, which, when mapped onto a manifold through r, becomes the Langevin diffusion described in (24) by the above procedure.…”

Section: Diffusions On Manifoldsmentioning

confidence: 99%

“…We turn first to (B t ) t≥0 , which we use to denote Brownian motion on a manifold. Intuitively, we may think of a construction based on embedded manifolds, by settingB 0 = p ∈ M , and for each increment sampling some random vector in the tangent space T p M , and then moving along the manifold in the prescribed direction for an infinitesimal period of time before re-sampling another velocity vector from the next tangent space [42]. In fact, we can define such a construction using Stratonovich calculus and show that the infinitesimal generator can be written using only local coordinates [28].…”

Section: Diffusions On Manifoldsmentioning

confidence: 99%

Information-Geometric Markov Chain Monte Carlo Methods Using Diffusions

Livingstone

Girolami

2014

Entropy

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Recent work incorporating geometric ideas in Markov chain Monte Carlo is reviewed in order to highlight these advances and their possible application in a range of domains beyond statistics. A full exposition of Markov chains and their use in Monte Carlo simulation for statistical inference and molecular dynamics is provided, with particular emphasis on methods based on Langevin diffusions. After this, geometric concepts in Markov chain Monte Carlo are introduced. A full derivation of the Langevin diffusion on a Riemannian manifold is given, together with a discussion of the appropriate Riemannian metric choice for different problems. A survey of applications is provided, and some open questions are discussed.

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“…Symmetric positive-definite matrices have wide usage in many fields of information science, such as stability analysis of signal processing, linear stationary systems, optimal control strategies and imaging analysis [1][2][3]. Its importance is beyond words [4,5].…”

Section: Introductionmentioning

confidence: 99%

Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing

Luo

Zhang

Cao

et al. 2021

Entropy

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The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence (AI) and other branches of computer science. In this paper, by involving the Wasserstein metric on SPD(n), we obtain computationally feasible expressions for some geometric quantities, including geodesics, exponential maps, the Riemannian connection, Jacobi fields and curvatures, particularly the scalar curvature. Furthermore, we discuss the behavior of geodesics and prove that the manifold is globally geodesic convex. Finally, we design algorithms for point cloud denoising and edge detecting of a polluted image based on the Wasserstein curvature on SPD(n). The experimental results show the efficiency and robustness of our curvature-based methods.

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A Primer on Stochastic Differential Geometry for Signal Processing

Cited by 22 publications

References 66 publications

A Primer on Reproducing Kernel Hilbert Spaces

A Primer on Reproducing Kernel Hilbert Spaces

Information-Geometric Markov Chain Monte Carlo Methods Using Diffusions

Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing

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