High-dimensional ellipsoids converge to Gaussian spaces

Kazukawa, Daisuke; Shioya, Takashi

doi:10.48550/arxiv.2003.05105

Cited by 2 publications

(3 citation statements)

References 11 publications

(12 reference statements)

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“…In this section, we use the cost model (37). For the exact search, carried out on the whole feature vector of dimensionality N , there is no T list, so τ = 0 and the cost is (I + N)S(D, N).…”

Section: Computational Costmentioning

confidence: 99%

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A meta-indexing method for fast probably approximately correct nearest neighbor searches

Santini

2022

Multimed Tools Appl

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In this paper we present an indexing method for probably approximately correct nearest neighbor queries in high dimensional spaces capable of improving the performance of any index whose performance degrades with the increased dimensionality of the query space. The basic idea of the method is quite simple: we use SVD to concentrate the variance of the inter-element distance in a lower dimensional space, Ξ. We do a nearest neighbor query in this space and then we “peek” forward from the nearest neighbor by gathering all the elements whose distance from the query is less than $d_{\Xi }(1+\zeta \sigma _{\Xi }^{2})$ d Ξ ( 1 + ζ σ Ξ 2 ) , where dΞ is the distance from the nearest neighbor in Ξ, $\sigma _{\Xi }^{2}$ σ Ξ 2 is the variance of the data in Ξ, and ζ a parameter. All the data thus collected form a tentative set T, in which we do a scan using the complete feature space to find the point closest to the query. The advantages of the method are that (1) it can be built on top of virtually any indexing method and (2) we can build a model of the distribution of the error precise enough to allow designing a compromise between error and speed. We show the improvement that we can obtain using data from the SUN data base.

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Section: Computational Costmentioning

confidence: 99%

“…Let n be a family of mm-spaces indexed by an integer parameter n. The spaces form a Lévy family if for all > 0, lim n→∞ α n ( ) = 0 [37].…”

Section: Introductionmentioning

confidence: 99%

A meta-indexing method for fast probably approximately correct nearest neighbor searches

Santini

2022

Multimed Tools Appl

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“…The study of the concentration and the weak topologies has been growing in recent years (see [2,[4][5][6][10][11][12][13][14][15]). In particular, we have obtained in [6,14,15] some nontrivial examples of weak convergent sequences, for example, spheres with the restriction of Euclidean norm, solid ellipses, (projective) Stiefel manifolds with the Frobenius norm, whose dimensions are unbounded. However, in all these examples, the distance function comes from the Euclidean distance.…”

Section: Introductionmentioning

confidence: 99%

Convergence of metric transformed spaces

Kazukawa¹

2020

Preprint

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We consider the metric transformation of metric measure spaces/pyramids. We clarify the conditions to obtain the convergence of the sequence of transformed spaces from that of the original sequence, and, conversely, to obtain the convergence of the original sequence from that of the transformed sequence, respectively. As an application, we prove that spheres and projective spaces with standard Riemannian distance converge to a Gaussian space and the Hopf quotient of a Gaussian space, respectively, as the dimension diverges to infinity.

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High-dimensional ellipsoids converge to Gaussian spaces

Cited by 2 publications

References 11 publications

A meta-indexing method for fast probably approximately correct nearest neighbor searches

A meta-indexing method for fast probably approximately correct nearest neighbor searches

Convergence of metric transformed spaces

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