On the nonexistence of bilipschitz parameterizations and geometric problems about $A_\infty$-weights

Semmes, Stephen

doi:10.4171/rmi/201

Cited by 145 publications

(151 citation statements)

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“…Note that the difference between the original and the snowflake version is not bounded for any 0 < γ < 1. Assouad conjectured that the embedding also applies to the original source metric (corresponding to γ = 1), but that was disproved by Semmes [16]. As a further step, Gottlieb and Krauthgamer [17] extended Assouad's Theorem by showing that if the source metric is also Euclidean (beyond having constant doubling dimension), then the embedding of the snowflake version into constant dimension can be done with 1+ϵ distortion with arbitrary ϵ >0.…”

Section: International Journal Of Machine Learning and Computing Volmentioning

confidence: 87%

Optimizing Massive Distance Computations in Pattern Recognition

Faragó¹

2014

IJMLC

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Abstract-It is a common task in pattern recognition to evaluate the similarity of large data objects. These are often represented by high dimensional vectors. A frequently used mathematical model for evaluating their similarity is to view them as points (vectors) in a high dimensional space, and compute their distances from each other. The "distance," however, can be defined in a very complicated way, it may be much more complex than the well known Euclidean distance. Therefore, the algorithmic bottleneck often becomes the number of distance computations that need to be carried out. We consider the case when we have to compute all the distances between n objects, where n is large. Without any shortcuts it takes n(n − 1)/2 = O(n 2 ) distance computations. In those applications where the distances are complicated, being defined by sophisticated algorithms (such as in speech and image recognition), a quadratically growing number of distance computations becomes a severe bottleneck. We prove the following general result that can help eliminating the bottleneck: for a large and general class of distances it is possible to obtain a very close approximation of each of the O(n 2 ) pairwise distances of n objects by doing only a linear number distance computations, which is optimal with respect to the order of magnitude. Moreover, the approximation factor can be made arbitrarily close to 1, making the approximation error negligible. The needed side computations to achieve this reduction can also be done in polynomial time.

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Section: International Journal Of Machine Learning and Computing Volmentioning

confidence: 87%

Optimizing Massive Distance Computations in Pattern Recognition

Faragó¹

2014

IJMLC

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“…Theorem 1.5 admits no converse, that is, there exist isotropically doubling weights that are not comparable to Df for any δ-monotone mapping f . This follows from the results of Semmes [18] and Laakso [16], who constructed compact ULNC sets A ⊂ R n such that for some p > 0 the weight w(x) := dist(x, A) p is not comparable to Df for any quasiconformal mapping f : R n → R n . By Proposition 3.6 the weight w is isotropic doubling.…”

Section: Proof Of Propositionmentioning

confidence: 97%

“…We also prove that the distance functions of certain fractal subsets of R n give rise to isotropic doubling weights (Proposition 3.6). By virtue of this result, the sets constructed by Semmes [18] and Laakso [16] provide examples of isotropic doubling weights that are not comparable to Df for any δ-monotone, or even quasiconformal, mapping f : R n → R n .…”

Section: Introductionmentioning

confidence: 99%

“…By Proposition 3.6 the weight w is isotropic doubling. The example of Semmes is a form of the Antoine's necklace, and requires n ≥ 3 (see the proof of Theorem 1.10 in [18]). The example of Laakso is two-dimensional [16,Theorem 1.7].…”

Section: Proof Of Propositionmentioning

confidence: 99%

“…x ∈ R n (1.3) 1 12C 3 I n−1 µ(x) ≤ Df µ (x) ≤ πI n−1 µ(x), where C is the doubling constant of µ. Theorem 1.1 expands the class of weights that are known to be comparable to Jacobians of quasiconformal mappings. The quasiconformal Jacobian problem posed by David and Semmes in [9] asks for a characterization of all such weights (see also [4,6,7,16,18]). The authors of [6] point out that such a characterization would give a good idea of which metric spaces are bi-Lipschitz equivalent to R n .…”

Section: Introductionmentioning

confidence: 99%

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Doubling measures, monotonicity, and quasiconformality

Kovalev

Maldonado

2007

Math. Z.

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Abstract. We construct quasiconformal mappings in Euclidean spaces by integration of a discontinuous kernel against doubling measures with suitable decay. The differentials of mappings that arise in this way satisfy an isotropic form of the doubling condition. We prove that this isotropic doubling condition is satisfied by the distance functions of certain fractal sets. Finally, we construct an isotropic doubling measure that is not absolutely continuous with respect to the Lebesgue measure.

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The coarea formula for real‐valued Lipschitz maps on stratified groups

Magnani¹

2005

Mathematische Nachrichten

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We establish a coarea formula for real-valued Lipschitz maps on stratified groups when the domain is endowed with a homogeneous distance and level sets are measured by the Q − 1 dimensional spherical Hausdorff measure. The number Q is the Hausdorff dimension of the group with respect to its Carnot-Carathéodory distance. We construct a Lipschitz function on the Heisenberg group which is not approximately differentiable on a set of positive measure, provided that the Euclidean notion of differentiability is adopted. The coarea formula for stratified groups also applies to this function, where the Euclidean one clearly fails. This phenomenon shows that the coarea formula holds for the natural class of Lipschitz functions which arises from the geometry of the group and that this class may be strictly larger than the usual one.

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On the nonexistence of bilipschitz parameterizations and geometric problems about $A_\infty$-weights

Cited by 145 publications

References 0 publications

Optimizing Massive Distance Computations in Pattern Recognition

Optimizing Massive Distance Computations in Pattern Recognition

Doubling measures, monotonicity, and quasiconformality

The coarea formula for real‐valued Lipschitz maps on stratified groups

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