Volume Growth and General Rate Quantization on Grassmann Manifolds

Dai, Wei; Rider, Brian; Liu, Youjian

doi:10.1109/glocom.2007.277

Cited by 11 publications

(15 citation statements)

References 13 publications

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“…3 The assumption is motivated by the fact that the observed column does not provide much information when q i ≤ r. Suppose that q i ≤ r for some i ∈ [n]. A randomly generated U from the uniform distribution on U m,r will have full column rank and gives f i (U Ωi ) = 0 with probability one [11].…”

Section: Problem Formulation and Preliminariesmentioning

confidence: 99%

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Low rank matrix completion: A smoothed l<inf>0</inf>-search

Zhou

Zhao

Dai

2012

2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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This paper focuses on algorithmic development for the low-rank matrix completion problem. It has been shown that in the 0-search for low-rank matrix completion, the singular points in the objective function are the major reasons for failures. While different methods have been proposed to handle singular points, this paper rigorously analyzes them to show that there is a need for further improvement. To address the singularity issue, we propose a new objective function that is continuous everywhere. The new objective function is a good approximation of the original objective function in the sense that in the limit, the lower level sets of the new objective function are the closure of those of the original objective function. We formulate the matrix completion problem as the minimization of the new objective function and design a quasiNewton method to solve it. Simulations demonstrate that the new method achieves excellent numerical performance.

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Section: Problem Formulation and Preliminariesmentioning

confidence: 99%

“…Note that although the above results are asymptotic, they provide a good approximation for finite m, r, q i . See [11] for the detailed discussion of the convergence rate and a numerical comparison between the empirical distribution F i (t) and the asymptotic distribution.…”

Section: A a Choice Of Parameter ρ I Smentioning

confidence: 99%

Low rank matrix completion: A smoothed l<inf>0</inf>-search

Zhou

Zhao

Dai

2012

2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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“…where V 2k (1) = vol S 2k−1 is given in (29). The two different conventions are related by vol V C n,p = 2…”

Section: A Spherical Volumes and Hyperspherical Capsmentioning

confidence: 99%

“…where V dim (r) is according to (29) with the dimension dim of the manifold in Table I. Intuitively, in a small neighborhood the manifold looks like a Euclidean space and can be approximated by the tangent space.…”

Section: A Spherical Volumes and Hyperspherical Capsmentioning

confidence: 99%

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Density of Spherically Embedded Stiefel and Grassmann Codes

Pitaval

Wei

Tirkkonen

et al. 2018

IEEE Trans. Inform. Theory

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The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. A high density indicates that a code performs well when used as a uniform point-wise discretization of an ambient space. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal distance arising from an Euclidean embedding, including the unitary group as a special case. The choice of distance enables the treatment of the manifolds as subspaces of Euclidean hyperspheres. In this geometry, the densest packings are not necessarily equivalent to maximum-minimum-distance codes. Computing a code's density follows from computing: i) the normalized volume of a metric ball and ii) the kissing radius, the radius of the largest balls one can pack around the codewords without overlapping. First, the normalized volume of a metric ball is evaluated by asymptotic approximations. The volume of a small ball can be wellapproximated by the volume of a locally-equivalent tangential ball. In order to properly normalize this approximation, the precise volumes of the manifolds induced by their spherical embedding are computed. For larger balls, a hyperspherical cap approximation is used, which is justified by a volume comparison theorem showing that the normalized volume of a ball in the Stiefel or Grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity. Then, bounds on the kissing radius are derived alongside corresponding bounds on the density. Unlike spherical codes or codes in flat spaces, the kissing radius of Grassmann or Stiefel codes cannot be exactly determined from its minimum distance. It is nonetheless possible to derive bounds on density as functions of the minimum distance. Stiefel and Grassmann codes have larger density than their image spherical codes when dimensions tend to infinity. Finally, the bounds on density lead to refinements of the standard Hamming bounds for Stiefel and Grassmann codes.

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Algebraic and Geometric Coding Theory

Chirikjian

2011

Stochastic Models, Information Theory, and Lie Groups, Volume 2

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Volume Growth and General Rate Quantization on Grassmann Manifolds

Cited by 11 publications

References 13 publications

Low rank matrix completion: A smoothed l<inf>0</inf>-search

Low rank matrix completion: A smoothed l<inf>0</inf>-search

Density of Spherically Embedded Stiefel and Grassmann Codes

Algebraic and Geometric Coding Theory

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