# Dimension Detection with Local Homology

@article{Dey2014DimensionDW, title={Dimension Detection with Local Homology}, author={Tamal K. Dey and Fengtao Fan and Yusu Wang}, journal={ArXiv}, year={2014}, volume={abs/1405.3534} }

Detecting the dimension of a hidden manifold from a point sample has become an important problem in the current data-driven era. Indeed, estimating the shape dimension is often the first step in studying the processes or phenomena associated to the data. Among the many dimension detection algorithms proposed in various fields, a few can provide theoretical guarantee on the correctness of the estimated dimension. However, the correctness usually requires certain regularity of the input: the…

## 10 Citations

### Another look at recovering local homology from samples of stratified sets

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### Another look at recovering local homology from samples of stratified sets

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### Multi-scale local shape analysis and feature selection in machine learning applications

- Computer Science2015 International Joint Conference on Neural Networks (IJCNN)
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We introduce a method called multi-scale local shape analysis for extracting features that describe the local structure of points within a dataset. The method uses both geometric and topological…

### Exploring persistent local homology in topological data analysis

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### The Localized Union-of-Balls Bifiltration

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- 2023

We propose an extension of the classical union-of-balls filtration of persistent homology: fixing a point $q$, we focus our attention to a ball centered at $q$ whose radius is controlled by a second…

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A topological framework is developed that identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data, and yields a Euclidicity score for assessing the ‘manifoldness’ of a point along multiple scales.

### Towards Stratified Space Learning: Linearly Embedded Graphs

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An algorithm is presented that learns the abstract structure of an embedded graph and models the specific embedding from a point cloud sampled from it and proves the correctness of the identified abstract structure under assumptions on the embedding.

## References

SHOWING 1-10 OF 32 REFERENCES

### Towards persistence-based reconstruction in euclidean spaces

- Mathematics, Computer ScienceSCG '08
- 2008

A novel approach that stands in-between classical reconstruction and topological estimation, and whose complexity scales up with the intrinsic dimension of the data is introduced.

### Provable dimension detection using principal component analysis

- Computer ScienceInt. J. Comput. Geom. Appl.
- 2005

The experimental results validate the effectiveness of the approach in computing the dimension and both the algorithm and its analysis can be generalized to the noisy case, in which outliers and a small perturbation of the samples are allowed.

### Local homology transfer and stratification learning

- Mathematics, Computer ScienceSODA
- 2012

This paper uses methods derived from kernel and cokernel persistent homology to cluster the data points into different strata, and provides a probabilistic guarantee for the clustering for the point sample setting.

### Finding the Homology of Submanifolds with High Confidence from Random Samples

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 2008

This work considers the case where data are drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space and shows how to “learn” the homology of the sub manifold with high confidence.

### Manifold-Adaptive Dimension Estimation

- Computer ScienceICML '07
- 2007

This paper proposes an algorithm for dimension estimation and studies its finite-sample behaviour and shows that the rate of convergence of the resulting estimate is independent of the dimension of the input space and hence the algorithm is “manifold-adaptive”.

### Dimension detection via slivers

- MathematicsSODA
- 2009

A novel approach to estimate the dimension m of an unknown manifold M⊂ Rd with positive reach from a set of point samples P ⊂ M by analyzing the shape of simplices formed by point samples, which proves that the dimension can be correctly output in O(kd|P|1+1/k) time with probability greater than 1-2-k.

### Approximating Local Homology from Samples

- MathematicsSODA
- 2014

It is shown that the persistence diagrams used for estimating local homology, can be approximated using families of Vietoris-Rips complexes, whose simple constructions are robust in any dimension.

### Inferring Local Homology from Sampled Stratified Spaces

- Mathematics48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
- 2007

This work uses the vineyard of the distance function restricted to a 1-parameter family of neighborhoods of a point to assess the local homology of the stratified space at that point and proves the correctness of this assessment under the assumption of a sufficiently dense sample.

### Shape Dimension and Approximation from Samples

- Computer ScienceSODA '02
- 2002

A Voronoi-based dimension detection algorithm is presented that assigns a dimension to a sample point which is the topological dimension of the manifold it belongs to and an algorithm to approximate shapes of arbitrary dimension from their samples is presented.

### Multiscale Geometric Methods for Estimating Intrinsic Dimension

- Mathematics
- 2010

We present a novel approach for estimating the intrinsic dimension of certain point clouds: we assume that the points are sampled from a manifold M of dimension k, with k << D, and corrupted by…