Volume of geodesic balls in the complex Stiefel manifold

Krishnamachari, Rajesh Tembarai; Varanasi, Mahesh K.

doi:10.1109/allerton.2008.4797653

Cited by 10 publications

(8 citation statements)

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“…For channel quantizations in precoded multi-antenna systems, the characterization of rate-distortion tradeoff is directly related to volume calculations in the manifold of interest [6,7]. Estimating fundamental coding bounds such as Gilbert-Varshamov and Hamming bounds relies on the volume of the corresponding metric ball [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

“…Despite the need to accurately estimate the volume of metric balls in the unitary group, results in this direction are rather limited. Volume estimates have been derived in [9][10][11] when the radius of metric ball is small. In this paper, we study the volume of a metric ball, valid for any radius, in the unitary group with chordal distance.…”

Section: Introductionmentioning

confidence: 99%

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From Random Matrix Theory to Coding Theory: Volume of a Metric Ball in Unitary Group

Wei

Pitaval

Corander

et al. 2017

IEEE Trans. Inform. Theory

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Volume estimates of metric balls in manifolds find diverse applications in information and coding theory. In this paper, some new results for the volume of a metric ball in unitary group are derived via various tools from random matrix theory. The first result is an integral representation of the exact volume, which involves a Toeplitz determinant of Bessel functions. The connection to matrix-variate hypergeometric functions and Szegő's strong limit theorem lead independently from the finite size formula to an asymptotic one. The convergence of the limiting formula is exceptionally fast due to an underlying mock-Gaussian behavior. The proposed volume estimate enables simple but accurate analytical evaluation of coding-theoretic bounds of unitary codes. In particular, the Gilbert-Varshamov lower bound and the Hamming upper bound on cardinality as well as the resulting bounds on code rate and minimum distance are derived. Moreover, bounds on the scaling law of code rate are found. Lastly, a closed-form bound on diversity sum relevant to unitary space-time codes is obtained, which was only computed numerically in literature. Index TermsCoding-theoretic bounds, random matrix theory, unitary group, volume of metric balls. L. Wei and J. Corander are with the

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

From Random Matrix Theory to Coding Theory: Volume of a Metric Ball in Unitary Group

Wei

Pitaval

Corander

et al. 2017

IEEE Trans. Inform. Theory

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“…However this series of papers - [4], [5], and the current paperdoes open up the interesting possibility of conducting a numerical perturbation analysis of information rate of a MIMO system as a function of feedback rate.…”

Section: Discussion Of the Above Theoremmentioning

confidence: 95%

“…Similar approaches have been used in [1], [4] and [5] for the Grassmann and the real/complex Stiefel manifolds, respectively.…”

Section: Bounds On Distortion-rate Tradeoffmentioning

confidence: 95%

“…We extended the analysis to the more general case of the Stiefel manifold V n,k , the set of all n × k semi-orthogonal (or semi-unitary) matrices, by characterizing the distortion-rate tradeoff D * (K) of a source invariantly distributed over it -for real matrices in [4] and for complex matrices in [5]. In this paper, we consider the more general case of arbitrary covariance matrix feedback by relating it to quantization on P F (n), F = R or C, and by providing asymptotically tight bounds for D * (K) for uniform sources over them.…”

Section: Introductionmentioning

confidence: 99%

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Distortion-rate tradeoff of a source uniformly distributed over positive semi-definite matrices

Krishnamachari

Varanasi

2008

2008 42nd Asilomar Conference on Signals, Systems and Computers

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We study the distortion-rate tradeoff D * (K) of a source uniformly distributed over a manifold comprising of positive semi-definite matrices -over both real and complex fields -under a trace constraint. This models the impact exercised by limited-rate feedback on the information transfer of optimal 'input covariance matrices' to a transmitter from a channel-aware receiver in a multiple-input mulitple-output (MIMO) system. Utilizing the volume of a geodesic ball in the manifold, D * (K) is bounded between asymptotically tight lower and upper limits obtained via sphere-packing and random coding arguments, respectively.

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Volume of ball and Hamming-type bounds for Stiefel manifold with Euclidean distance

Pitaval

Tirkkonen

2012

2012 Conference Record of the Forty Sixth Asilomar Conference on Signals, Systems and Computers (ASILOMAR)

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We compute the volume of the Stiefel manifold induced by the canonical embedding of the manifold as a surface in a Euclidean hypersphere and taking the corresponding Euclidean/chordal distance. Exploiting a power series expansion of the volume element, the volume of a small metric ball under the chordal distance is evaluated. Evaluating the volume of a metric ball is critical to derive Hamming-type bounds. Using a spherical embedding argument, we provide results generalizing previously known bounds on codes in the Grassmann manifold and the unitary group.

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Volume of geodesic balls in the complex Stiefel manifold

Cited by 10 publications

References 10 publications

From Random Matrix Theory to Coding Theory: Volume of a Metric Ball in Unitary Group

From Random Matrix Theory to Coding Theory: Volume of a Metric Ball in Unitary Group

Distortion-rate tradeoff of a source uniformly distributed over positive semi-definite matrices

Volume of ball and Hamming-type bounds for Stiefel manifold with Euclidean distance

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