nSimplex Zen: A Novel Dimensionality Reduction for Euclidean and Hilbert Spaces

• URL: http://arxiv.org/abs/2302.11508v2
• Date: Tue, 13 Feb 2024 11:33:18 GMT
• Title: nSimplex Zen: A Novel Dimensionality Reduction for Euclidean and Hilbert Spaces
• Authors: Richard Connor, Lucia Vadicamo
• Abstract summary: Dimensionality reduction techniques map values from a high dimensional space to one with a lower dimension. We introduce nSimplex Zen, a novel topological method of reducing dimensionality.
• Score: 0.5076419064097734
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Dimensionality reduction techniques map values from a high dimensional space to one with a lower dimension. The result is a space which requires less physical memory and has a faster distance calculation. These techniques are widely used where required properties of the reduced-dimension space give an acceptable accuracy with respect to the original space. Many such transforms have been described. They have been classified in two main groups: linear and topological. Linear methods such as Principal Component Analysis (PCA) and Random Projection (RP) define matrix-based transforms into a lower dimension of Euclidean space. Topological methods such as Multidimensional Scaling (MDS) attempt to preserve higher-level aspects such as the nearest-neighbour relation, and some may be applied to non-Euclidean spaces. Here, we introduce nSimplex Zen, a novel topological method of reducing dimensionality. Like MDS, it relies only upon pairwise distances measured in the original space. The use of distances, rather than coordinates, allows the technique to be applied to both Euclidean and other Hilbert spaces, including those governed by Cosine, Jensen-Shannon and Quadratic Form distances. We show that in almost all cases, due to geometric properties of high-dimensional spaces, our new technique gives better properties than others, especially with reduction to very low dimensions.

Related papers

• Krylov Cubic Regularized Newton: A Subspace Second-Order Method with Dimension-Free Convergence Rate [83.3933097134767]
  We introduce a novel subspace cubic regularized Newton method that achieves a dimension-independent global convergence rate of $Oleft(frac1mk+frac1k2right)$. Our method converges faster than existing random subspace methods, especially for high-dimensional problems.
  arXiv  Detail & Related papers  (2024-01-05T20:24:18Z)
• Splitting the local Hilbert space: MPS-based approach to large local dimensions [0.0]
  A large, or even infinite, local Hilbert space dimension poses a significant computational challenge for simulating quantum systems. We present a matrix product state (MPS)-based method for simulating one-dimensional quantum systems with a large local Hilbert space dimension.
  arXiv  Detail & Related papers  (2023-07-29T17:41:27Z)
• Principal Component Analysis in Space Forms [7.822210329345704]
  We study Principal Component Analysis (PCA) in space forms. Finding the optimal low-dimensional affine subspace for given points in a space form amounts to dimensionality reduction. We propose proper cost functions that enjoy two properties: (1) their optimal affine subspace is the solution to an eigenequation, and (2) optimal affine subspaces of different dimensions form a nested set.
  arXiv  Detail & Related papers  (2023-01-06T23:48:37Z)
• Dimensional reduction of the Dirac equation in arbitrary spatial dimensions [0.0]
  We show that the Dirac equation reduces to either a single Dirac equation or two decoupled Dirac equations. We construct and discuss an explicit hierarchy of representations in which this procedure becomes manifest and can easily be iterated.
  arXiv  Detail & Related papers  (2022-12-22T18:50:05Z)
• Manifold Hypothesis in Data Analysis: Double Geometrically-Probabilistic Approach to Manifold Dimension Estimation [92.81218653234669]
  We present new approach to manifold hypothesis checking and underlying manifold dimension estimation. Our geometrical method is a modification for sparse data of a well-known box-counting algorithm for Minkowski dimension calculation. Experiments on real datasets show that the suggested approach based on two methods combination is powerful and effective.
  arXiv  Detail & Related papers  (2021-07-08T15:35:54Z)
• Improving Metric Dimensionality Reduction with Distributed Topology [68.8204255655161]
  DIPOLE is a dimensionality-reduction post-processing step that corrects an initial embedding by minimizing a loss functional with both a local, metric term and a global, topological term. We observe that DIPOLE outperforms popular methods like UMAP, t-SNE, and Isomap on a number of popular datasets.
  arXiv  Detail & Related papers  (2021-06-14T17:19:44Z)
• A Local Similarity-Preserving Framework for Nonlinear Dimensionality Reduction with Neural Networks [56.068488417457935]
  We propose a novel local nonlinear approach named Vec2vec for general purpose dimensionality reduction. To train the neural network, we build the neighborhood similarity graph of a matrix and define the context of data points. Experiments of data classification and clustering on eight real datasets show that Vec2vec is better than several classical dimensionality reduction methods in the statistical hypothesis test.
  arXiv  Detail & Related papers  (2021-03-10T23:10:47Z)
• Linear Classifiers in Mixed Constant Curvature Spaces [40.82908295137667]
  We address the problem of linear classification in a product space form -- a mix of Euclidean, spherical, and hyperbolic spaces. We prove that linear classifiers in $d$-dimensional constant curvature spaces can shatter exactly $d+1$ points. We describe a novel perceptron classification algorithm, and establish rigorous convergence results.
  arXiv  Detail & Related papers  (2021-02-19T23:29:03Z)
• Multi-point dimensionality reduction to improve projection layout reliability [77.34726150561087]
  In ordinary Dimensionality Reduction (DR), each data instance in an m-dimensional space (original space) is mapped to one point in a d-dimensional space (visual space) Our solution, named Red Gray Plus, is built upon and extends a combination of ordinary DR and graph drawing techniques.
  arXiv  Detail & Related papers  (2021-01-15T17:17:02Z)
• 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)
• ABID: Angle Based Intrinsic Dimensionality [0.0]
  The intrinsic dimensionality refers to the true'' dimensionality of the data, as opposed to the dimensionality of the data representation. Most popular methods for estimating the local intrinsic dimensionality are based on distances. We derive the theoretical distribution of angles and use this to construct an estimator for intrinsic dimensionality.
  arXiv  Detail & Related papers  (2020-06-23T10:19:34Z)

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.