Topological Singularity Detection at Multiple Scales
- URL: http://arxiv.org/abs/2210.00069v4
- Date: Wed, 14 Jun 2023 21:18:38 GMT
- Title: Topological Singularity Detection at Multiple Scales
- Authors: Julius von Rohrscheidt and Bastian Rieck
- Abstract summary: Real-world data exhibits distinct non-manifold structures that can lead to erroneous findings.
We develop a framework that quantifies the local intrinsic dimension, and yields a Euclidicity score for assessing the'manifoldness' of a point along multiple scales.
Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.
- Score: 11.396560798899413
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: The manifold hypothesis, which assumes that data lies on or close to an
unknown manifold of low intrinsic dimension, is a staple of modern machine
learning research. However, recent work has shown that real-world data exhibits
distinct non-manifold structures, i.e. singularities, that can lead to
erroneous findings. Detecting such singularities is therefore crucial as a
precursor to interpolation and inference tasks. We address this issue by
developing a topological framework that (i) quantifies the local intrinsic
dimension, and (ii) yields a Euclidicity score for assessing the 'manifoldness'
of a point along multiple scales. Our approach identifies singularities of
complex spaces, while also capturing singular structures and local geometric
complexity in image data.
Related papers
- Decoder ensembling for learned latent geometries [15.484595752241122]
We show how to easily compute geodesics on the associated expected manifold.
We find this simple and reliable, thereby coming one step closer to easy-to-use latent geometries.
arXiv Detail & Related papers (2024-08-14T12:35:41Z) - Adversarial Estimation of Topological Dimension with Harmonic Score Maps [7.34158170612151]
We show that it is possible to retrieve the topological dimension of the manifold learned by the score map.
We then introduce a novel method to measure the learned manifold's topological dimension using adversarial attacks.
arXiv Detail & Related papers (2023-12-11T22:29:54Z) - Higher-order topological kernels via quantum computation [68.8204255655161]
Topological data analysis (TDA) has emerged as a powerful tool for extracting meaningful insights from complex data.
We propose a quantum approach to defining Betti kernels, which is based on constructing Betti curves with increasing order.
arXiv Detail & Related papers (2023-07-14T14:48:52Z) - Towards a mathematical understanding of learning from few examples with
nonlinear feature maps [68.8204255655161]
We consider the problem of data classification where the training set consists of just a few data points.
We reveal key relationships between the geometry of an AI model's feature space, the structure of the underlying data distributions, and the model's generalisation capabilities.
arXiv Detail & Related papers (2022-11-07T14:52:58Z) - Statistical exploration of the Manifold Hypothesis [10.389701595098922]
The Manifold Hypothesis asserts that nominally high-dimensional data are in fact concentrated near a low-dimensional manifold, embedded in high-dimensional space.
We show that rich and sometimes intricate manifold structure in data can emerge from a generic and remarkably simple statistical model.
We derive procedures to discover and interpret the geometry of high-dimensional data, and explore hypotheses about the data generating mechanism.
arXiv Detail & Related papers (2022-08-24T17:00:16Z) - Intrinsic dimension estimation for discrete metrics [65.5438227932088]
In this letter we introduce an algorithm to infer the intrinsic dimension (ID) of datasets embedded in discrete spaces.
We demonstrate its accuracy on benchmark datasets, and we apply it to analyze a metagenomic dataset for species fingerprinting.
This suggests that evolutive pressure acts on a low-dimensional manifold despite the high-dimensionality of sequences' space.
arXiv Detail & Related papers (2022-07-20T06:38:36Z) - A geometric framework for outlier detection in high-dimensional data [0.0]
Outlier or anomaly detection is an important task in data analysis.
We provide a framework that exploits the metric structure of a data set.
We show that exploiting this structure significantly improves the detection of outlying observations in high-dimensional data.
arXiv Detail & Related papers (2022-07-01T12:07:51Z) - Intrinsic Dimension Estimation [92.87600241234344]
We introduce a new estimator of the intrinsic dimension and provide finite sample, non-asymptotic guarantees.
We then apply our techniques to get new sample complexity bounds for Generative Adversarial Networks (GANs) depending on the intrinsic dimension of the data.
arXiv Detail & Related papers (2021-06-08T00:05:39Z) - Quadric hypersurface intersection for manifold learning in feature space [52.83976795260532]
manifold learning technique suitable for moderately high dimension and large datasets.
The technique is learned from the training data in the form of an intersection of quadric hypersurfaces.
At test time, this manifold can be used to introduce an outlier score for arbitrary new points.
arXiv Detail & Related papers (2021-02-11T18:52:08Z) - Manifold Learning via Manifold Deflation [105.7418091051558]
dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data.
Many popular methods can fail dramatically, even on simple two-dimensional Manifolds.
This paper presents an embedding method for a novel, incremental tangent space estimator that incorporates global structure as coordinates.
Empirically, we show our algorithm recovers novel and interesting embeddings on real-world and synthetic datasets.
arXiv Detail & Related papers (2020-07-07T10:04:28Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.