Covering a Ball with Smaller Equal Balls in ℝ n
TLDR
An explicit upper bound of the minimal number νT,n of balls of radius 1/2 which form a covering of a ball of radius T >1/2 in ℝn, n is given.Abstract:
We give an explicit upper bound of the minimal number νT,n of balls of radius 1/2 which form a covering of a ball of radius T > 1/2 in ℝn, n \geq 2 The asymptotic estimates of νT,n we deduce when n is large are improved further by recent results of Boroczky, Jr and Wintsche on the asymptotic estimates of the minimal numberof equal balls of ℝn covering the sphere Sn-1 The optimality of the asymptotic estimates is discussedread more
Citations
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Journal ArticleDOI
Packing and Covering. By C. A. Rogers. Pp. viii, 111. 30s. 1964. (Cambridge)
TL;DR: The existence of reasonably dense lattice coverings and reasonably economical lattice covers has been studied in this paper, where the authors show that simplices cannot be very dense and coverings with spheres cannot have very economical coverings.
Proceedings Article
Stochastic Convex Optimization.
TL;DR: Stochastic convex optimization is studied, and it is shown that the key ingredient is strong convexity and regularization, which is only a sufficient, but not necessary, condition for meaningful non-trivial learnability.
Journal ArticleDOI
Fixed-Length Lossy Compression in the Finite Blocklength Regime
Victoria Kostina,Sergio Verdu +1 more
TL;DR: For stationary memoryless sources with separable distortion, the minimum rate achievable is shown to be closely approximated by the standard Gaussian complementary cumulative distribution function.
Proceedings ArticleDOI
Fast construction of nets in low dimensional metrics, and their applications
Sariel Har-Peled,Manor Mendel +1 more
TL;DR: A near linear time algorithm for constructing hierarchical nets in finite metric spaces with constant doubling dimension is presented and this data-structure is applied to obtain improved algorithms for the following problems: Approximate nearest neighbor search, well-separated pair decomposition, spanner construction, compact representation scheme, doubling measure, and computation of the Lipschitz constant of a function.
Journal ArticleDOI
ByRDiE: Byzantine-Resilient Distributed Coordinate Descent for Decentralized Learning
Zhixiong Yang,Waheed U. Bajwa +1 more
TL;DR: This paper focuses on the problem of Byzantine failures, which are the hardest to safeguard against in distributed algorithms, and develops and analyzes an algorithm termed Byzantine-resilient distributed coordinate descent that enables distributed learning in the presence of Byzantine fails.
References
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Journal ArticleDOI
Packing and covering
TL;DR: The existence of reasonably dense lattice coverings and reasonably economical lattice covers has been studied in this article, where the authors show that simplices cannot be very dense and coverings with spheres cannot have very economical coverings.
Journal ArticleDOI
The Number of Circles Covering a Set
TL;DR: In this paper, it is shown that the best way to cover a given area with circles of a given radius E is to place the centers of the circles on an equilateral triangle network, i.e., to circumscribe (inscribe) the circles about the hexagons of a regular hexagon network or honeycomb.
Journal ArticleDOI
Packing and Covering. By C. A. Rogers. Pp. viii, 111. 30s. 1964. (Cambridge)
TL;DR: The existence of reasonably dense lattice coverings and reasonably economical lattice covers has been studied in this paper, where the authors show that simplices cannot be very dense and coverings with spheres cannot have very economical coverings.
Journal ArticleDOI
Borsuk's problem and the chromatic numbers of some metric spaces
TL;DR: In this article, a detailed survey is given of various results pertaining to Borsuk's problem on partitions of an arbitrary bounded d-dimensional set of non-zero diameter into parts of smaller diameter, and the problem of finding chromatic numbers of some metric spaces.
Book ChapterDOI
Packing and covering
TL;DR: In this age of modern era, the use of internet must be maximized to get the on-line packing and covering book, as the world window, as many people suggest.