The Emergence of Topological Dimension Theory

Crilly, Tony

doi:10.1016/b978-044482375-5/50002-x

Cited by 17 publications

(12 citation statements)

References 13 publications

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“…At a theoretical level, this brings up the question, what is dimension in the context of statistical learning? It took mathematicians roughly half a century, to isolate the correct notion of dimension of a topological space and obtain the basic results of the theory (roughly, 1873-1921, see [38]). Will it take the same amount of time to put forward a satisfactory theory of intrinsic dimension of data in the context of statistical learning, in order to explain away the curse of dimensionality?…”

Section: Discussionmentioning

confidence: 99%

Is thek-NN classifier in high dimensions affected by the curse of dimensionality?

Pestov

2013

Computers & Mathematics with Applications

107

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There is an increasing body of evidence suggesting that exact nearest neighbour search in high-dimensional spaces is affected by the curse of dimensionality at a fundamental level. Does it necessarily mean that the same is true for k nearest neighbours based learning algorithms such as the k-NN classifier? We analyse this question at a number of levels and show that the answer is different at each of them. As our first main observation, we show the consistency of a k approximate nearest neighbour classifier. However, the performance of the classifier in very high dimensions is provably unstable. As our second main observation, we point out that the existing model for statistical learning is oblivious of dimension of the domain and so every learning problem admits a universally consistent deterministic reduction to the one-dimensional case by means of a Borel isomorphism.

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Section: Discussionmentioning

confidence: 99%

Is thek-NN classifier in high dimensions affected by the curse of dimensionality?

Pestov

2013

Computers & Mathematics with Applications

107

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“…Arthur Cayley in 1878 helped to spread the problem by putting in first in print and giving it the status of an open problem. Even Cayley, who worked in graph theory did not use the language of graphs yet to describe the problem [19]. Alfred Kempe published a proof in 1879 which turned out to be incomplete.…”

Section: Discussionmentioning

confidence: 99%

Coloring graphs using topology

Knill

2014

Preprint

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Higher dimensional graphs can be used to color 2dimensional geometric graphs G. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colorable with 4 colors. The later goal is achieved if all interior edge degrees are even. Using a refinement process which cuts the interior along surfaces S we can adapt the degrees along the boundary S. More efficient is a self-cobordism of G with itself with a host graph discretizing the product of G with an interval. It follows from the fact that Euler curvature is zero everywhere for three dimensional geometric graphs, that the odd degree edge set O is a cycle and so a boundary if H is simply connected. A reduction to minimal coloring would imply the four color theorem. The method is expected to give a reason "why 4 colors suffice" and suggests that every two dimensional geometric graph of arbitrary degree and orientation can be colored by 5 colors: since the projective plane can not be a boundary of a 3-dimensional graph and because for higher genus surfaces, the interior H is not simply connected, we need in general to embed a surface into a 4-dimensional simply connected graph in order to color it. This explains the appearance of the chromatic number 5 for higher degree or non-orientable situations, a number we believe to be the upper limit. For every surface type, we construct examples with chromatic number 3, 4 or 5, where the construction of surfaces with chromatic number 5 is based on a method of Fisk. We have implemented and illustrated all the topological aspects described in this paper on a computer. So far we still need human guidance or simulated annealing to do the refinements in the higher dimensional host graph.

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“…For the reverse step, when removing a point, we might have to modify the topology first and take a rougher topology. For example, lets look at the line graph L 4 with four vertices equipped with the topology generated by B = {(1, 2, 3), (2,3,4)}. This topological graph has the nerve K 2 .…”

Section: Proof Ofmentioning

confidence: 99%

“…This topological graph has the nerve K 2 . Removing the vertex 2 would produce C = {(1, 3), (3,4)} which has a disconnected nerve graph. But if we take the rougher topology B = {(1, 2, 3, 4)} then this becomes {(1, 3, 4)} after removing the vertex and the homotopy reduction is continuous.…”

Section: Proof Ofmentioning

confidence: 99%

A notion of graph homeomorphism

Knill

2014

Preprint

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We introduce a notion of graph homeomorphisms which uses the concept of dimension and homotopy for graphs. It preserves the dimension of a subbasis, cohomology and Euler characteristic. Connectivity and homotopy look as in classical topology. The Brouwer-Lefshetz fixed point leads to the following discretiszation of the Kakutani fixed point theorem: any graph homeomorphism T with nonzero Lefschetz number has a nontrivial invariant open set which is fixed by T .

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The Emergence of Topological Dimension Theory

Cited by 17 publications

References 13 publications

Is thek-NN classifier in high dimensions affected by the curse of dimensionality?

Is thek-NN classifier in high dimensions affected by the curse of dimensionality?

Coloring graphs using topology

A notion of graph homeomorphism

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