On Coverings of Ellipsoids in Euclidean Spaces

Dumer, Ilya; Pinsker, M. S.; Prelov, V. V.

doi:10.1109/tit.2004.834759

Cited by 13 publications

(28 citation statements)

References 14 publications

(25 reference statements)

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“…In particular, it fails on a ball B n ρ of any given radius ρ > 1. The following asymptotic bound of [8] removes this drawback. (8) holds for the ellipsoids E n a provided that log a n log ρ…”

Section: Prior and Present Resultsmentioning

confidence: 99%

“…Replacing an original ellipsoid with direct products of the balls was first used in [8]. Present design differs in the following aspects.…”

Section: Discussionmentioning

confidence: 99%

“…Present design differs in the following aspects. Firstly, exponentially declining steps are now used instead of the uniform quantization of [8]. Secondly, different approximation grids are applied to different positions.…”

Section: Discussionmentioning

confidence: 99%

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Covering an ellipsoid with equal balls

Dumer

2006

Journal of Combinatorial Theory, Series A

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Section: Prior and Present Resultsmentioning

confidence: 99%

“…Replacing an original ellipsoid with direct products of the balls was first used in [8]. Present design differs in the following aspects.…”

Section: Discussionmentioning

confidence: 99%

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Covering an ellipsoid with equal balls

Dumer

2006

Journal of Combinatorial Theory, Series A

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“…Theorem 9 is proved in [20]. The main idea of the proof is rather close to that of the upper bound in the Hamming space.…”

Section: Asymptotic Upper Boundmentioning

confidence: 90%

“…Some results on optimal coverings of other sets can be found in [18] and [19]. Below we briefly describe some new results obtained in the recent paper [20] for optimal coverings of ellipsoids in Euclidean spaces.…”

Section: Covering Of Ellipsoids In Euclidean Spacesmentioning

confidence: 96%

On the Thinnest Coverings of Spheres and Ellipsoids with Balls in Hamming and Euclidean Spaces

Dumer¹,

Pinsker²,

Prelov³

2006

Lecture Notes in Computer Science

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Abstract. In this paper, we present some new results on the thinnest coverings that can be obtained in Hamming or Euclidean spaces if spheres and ellipsoids are covered with balls of some radius ε. In particular, we tighten the bounds currently known for the ε-entropy of Hamming spheres of an arbitrary radius r. New bounds for the ε-entropy of Hamming balls are also derived. If both parameters ε and r are linear in dimension n, then the upper bounds exceed the lower ones by an additive term of order log n. We also present the uniform bounds valid for all values of ε and r.In the second part of the paper, new sufficient conditions are obtained, which allow one to verify the validity of the asymptotic formula for the size of an ellipsoid in a Hamming space. Finally, we survey recent results concerning coverings of ellipsoids in Hamming and Euclidean spaces.

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Signal reconstruction in the presence of finite‐rate measurements: finite‐horizon control applications

Sarma

Dahleh

2009

Intl J Robust & Nonlinear

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SUMMARYIn this paper, we study finite-length signal reconstruction over a finite-rate noiseless channel. We allow the class of signals to belong to a bounded ellipsoid and derive a universal lower bound on a worstcase reconstruction error. We then compute upper bounds on the error that arise from different coding schemes and under different causality assumptions. When the encoder and decoder are noncausal, we derive an upper bound that either achieves the universal lower bound or is comparable to it. When the decoder and encoder are both causal operators, we show that within a very broad class of causal coding schemes, memoryless coding prevails as optimal, imposing a hard limitation on reconstruction.Finally, we map our general reconstruction problem into two important control problems in which the plant and controller are local to each other, but are together driven by a remote reference signal that is transmitted through a finite-rate noiseless channel. The first problem is to minimize a finite-horizon weighted tracking error between the remote system output and a reference command. The second * Correspondence to: sree@mit.edu problem is to navigate the state of the remote system from a nonzero initial condition to as close to the origin as possible in finite-time. Our analysis enables us to quantify the tradeoff between time horizon and performance accuracy which is not well-studied in the area of control with limited information as most works address infinite-horizon control objectives (eg. stability, disturbance rejection).

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On Coverings of Ellipsoids in Euclidean Spaces

Cited by 13 publications

References 14 publications

Covering an ellipsoid with equal balls

Covering an ellipsoid with equal balls

On the Thinnest Coverings of Spheres and Ellipsoids with Balls in Hamming and Euclidean Spaces

Signal reconstruction in the presence of finite‐rate measurements: finite‐horizon control applications

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