A Bayesian approach to geometric subspace estimation

Srivastava, Anuj

doi:10.1109/78.839985

Cited by 46 publications

(47 citation statements)

References 28 publications

(57 reference statements)

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“…Moreover, is strictly negative definite. This follows from (12) and the fact that is orthogonal. Indeed where .…”

Section: Brownian Distributionsmentioning

confidence: 95%

“…The resulting numerical scheme is common in the literature, for example, [12], [23]. In the present situation, this can be particularly helpful.…”

Section: Application: Polarized Light In a Dispersive Fibermentioning

confidence: 99%

“…Precisely, matrix elements of span the corresponding eigenspace of . A direct calculation shows that this is equivalent to (12) where is the identity matrix and . Note that is skew-Hermitian.…”

Section: B Characteristic Functionsmentioning

confidence: 99%

“…This is due to nonuniqueness of the intrinsic mean and to the fact that the extrinsic mean is generally considerably easier to compute. In [11], Monte Carlo simulation is used to obtain extrinsic means in several classical compact Lie groups and quotient manifolds, which are of particular importance to signal processing-see [12] and [13]. In this paper, Section IV applies the extrinsic mean to parametric estimation of Brownian distributions.…”

Section: Introductionmentioning

confidence: 99%

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Extrinsic Mean of Brownian Distributions on Compact Lie Groups

Said

Manton

2012

IEEE Trans. Inform. Theory

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Abstract-This paper studies Brownian distributions on compact Lie groups. These are defined as the marginal distributions of Brownian processes and are intended as a natural extension of the well-known normal distributions to compact Lie groups. It is shown that this definition preserves key properties of normal distributions. In particular, Brownian distributions transform in a nice way under group operations and satisfy an extension of the central limit theorem. Brownian distributions on a compact Lie group belong to one of two parametric families and -and a positive-definite symmetric matrix. In particular, the parameter appears as a location parameter. An approach based on the extrinsic mean for estimation of the parameters and is studied in detail. It is shown that is the unique extrinsic mean for a Brownian distribution or . Resulting estimates are proved to be consistent and asymptotically normal. While they may also be used to simultaneously estimate and , it is seen this requires that be embedded into a higher dimensional matrix Lie group. Going beyond Brownian distributions, it is shown the extrinsic mean can be used to recover the location parameter for a wider class of distributions arising more generally from Lévy processes. The compact Lie group structure places limitations on the analogy between normal distributions and Brownian distributions. This is illustrated by the study of multivariate Brownian distributions. These are introduced as Brownian distributions on some product group-e.g., . This paper describes their covariance structure and considers its transformation under group operations.Index Terms-Brownian motion, central limit theorem, compact Lie group, extrinsic and intrinsic mean, noncommutative harmonic analysis, Lévy process.

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“…Moreover, is strictly negative definite. This follows from (12) and the fact that is orthogonal. Indeed where .…”

Section: Brownian Distributionsmentioning

confidence: 95%

“…The resulting numerical scheme is common in the literature, for example, [12], [23]. In the present situation, this can be particularly helpful.…”

Section: Application: Polarized Light In a Dispersive Fibermentioning

confidence: 99%

Section: B Characteristic Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Extrinsic Mean of Brownian Distributions on Compact Lie Groups

Said

Manton

2012

IEEE Trans. Inform. Theory

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“…However, this surface is not amenable to minimization due to numerous local minima and areas of small or zero gradient. Numerous randomized techniques for minimizing a surface with multiple minima have been developed, such as simulated annealing [11], [12] or by using a jump diffusion technique [13], [14]. With these techniques, the parameter set is updated by a step increment.…”

Section: Introductionmentioning

confidence: 99%

Template matching based object recognition with unknown geometric parameters

Dufour

Miller²,

Galatsanos³

2002

IEEE Trans. on Image Process.

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Abstract-In this paper, we examine the problem of locating an object in an image when size and rotation are unknown. Previous work has shown that with known geometric parameters, an image restoration method can be useful by estimating a delta function at the object location. When the geometric parameters are unknown, this method becomes impractical because the likelihood surface to be minimized across size and rotation has numerous local minima and areas of zero gradient. In this paper, we propose a new approach where a smooth approximation of the template is used to minimize a well-behaved likelihood surface. A coarse-to-fine approximation of the original template using a diffusion-like equation is used to create a library of templates. Using this library, we can successively perform minimizations which are locally well-behaved. As detail is added to the template, the likelihood surface gains local minima, but previous estimates place us within a wellbehaved "bowl" around the global minimum, leading to an accurate estimate. Numerical experiments are shown which verify the value of this approach for a wide range of values of the geometric parameters.

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Statistical Modeling on Nonlinear Manifolds

Srivastava¹,

Klassen²

2016

Functional and Shape Data Analysis

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A Bayesian approach to geometric subspace estimation

Cited by 46 publications

References 28 publications

Extrinsic Mean of Brownian Distributions on Compact Lie Groups

Extrinsic Mean of Brownian Distributions on Compact Lie Groups

Template matching based object recognition with unknown geometric parameters

Statistical Modeling on Nonlinear Manifolds

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