Related papers: Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embedding

Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embedding

URL: http://arxiv.org/abs/2503.18010v2
Date: Sat, 29 Mar 2025 10:33:09 GMT
Title: Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embedding
Authors: Thomas Dagès, Simon Weber, Ya-Wei Eileen Lin, Ronen Talmon, Daniel Cremers, Michael Lindenbaum, Alfred M. Bruckstein, Ron Kimmel,
Abstract summary: Dimensionality reduction aims to simplify complex data by reducing its feature dimensionality while preserving essential patterns, with core applications in data analysis and visualisation.<n>To preserve the underlying data structure, multi-dimensional scaling (MDS) methods focus on preserving pairwise dissimilarities, such as distances.
Score: 41.601022263772535
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
Abstract: Dimensionality reduction is a fundamental task that aims to simplify complex data by reducing its feature dimensionality while preserving essential patterns, with core applications in data analysis and visualisation. To preserve the underlying data structure, multi-dimensional scaling (MDS) methods focus on preserving pairwise dissimilarities, such as distances. They optimise the embedding to have pairwise distances as close as possible to the data dissimilarities. However, the current standard is limited to embedding data in Riemannian manifolds. Motivated by the lack of asymmetry in the Riemannian metric of the embedding space, this paper extends the MDS problem to a natural asymmetric generalisation of Riemannian manifolds called Finsler manifolds. Inspired by Euclidean space, we define a canonical Finsler space for embedding asymmetric data. Due to its simplicity with respect to geodesics, data representation in this space is both intuitive and simple to analyse. We demonstrate that our generalisation benefits from the same theoretical convergence guarantees. We reveal the effectiveness of our Finsler embedding across various types of non-symmetric data, highlighting its value in applications such as data visualisation, dimensionality reduction, directed graph embedding, and link prediction.

Related papers

Learning symmetries in datasets [0.0]
We investigate how symmetries present in datasets affect the structure of the latent space learned by Variational Autoencoders (VAEs) We show that when symmetries or approximate symmetries are present, the VAE self-organizes its latent space, effectively compressing the data along a reduced number of latent variables. Our results highlight the potential of unsupervised generative models to expose underlying structures in data and offer a novel approach to symmetry discovery without explicit supervision.
arXiv Detail & Related papers (2025-04-07T15:17:41Z)
Riemann$^2$: Learning Riemannian Submanifolds from Riemannian Data [12.424539896723603]
Latent variable models are powerful tools for learning low-dimensional manifold from high-dimensional data.<n>This paper generalizes previous work and allows us to handle complex tasks in various domains, including robot motion synthesis and analysis of brain connectomes.
arXiv Detail & Related papers (2025-03-07T16:08:53Z)
Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasets [11.105392318582677]
We propose a principled approach for aligning and jointly embedding a pair of datasets with theoretical guarantees. Our approach leverages the leading singular vectors of the EOT plan matrix between two datasets to extract their shared underlying structure. We show that in a high-dimensional regime, the EOT plan recovers the shared manifold structure by approximating a kernel function evaluated at the locations of the latent variables.
arXiv Detail & Related papers (2024-07-01T18:48:55Z)
Pulling back symmetric Riemannian geometry for data analysis [0.0]
Ideal data analysis tools for data sets should account for non-linear geometry. A rich mathematical structure to account for non-linear geometries has been shown to be able to capture the data geometry. Many standard data analysis tools initially developed for data in Euclidean space can be generalised efficiently to data on a symmetric Riemannian manifold.
arXiv Detail & Related papers (2024-03-11T10:59:55Z)
Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein [56.62376364594194]
Unsupervised learning aims to capture the underlying structure of potentially large and high-dimensional datasets. In this work, we revisit these approaches under the lens of optimal transport and exhibit relationships with the Gromov-Wasserstein problem. This unveils a new general framework, called distributional reduction, that recovers DR and clustering as special cases and allows addressing them jointly within a single optimization problem.
arXiv Detail & Related papers (2024-02-03T19:00:19Z)
Scaling Riemannian Diffusion Models [68.52820280448991]
We show that our method enables us to scale to high dimensional tasks on nontrivial manifold. We model QCD densities on $SU(n)$ lattices and contrastively learned embeddings on high dimensional hyperspheres.
arXiv Detail & Related papers (2023-10-30T21:27:53Z)
Oracle-Preserving Latent Flows [58.720142291102135]
We develop a methodology for the simultaneous discovery of multiple nontrivial continuous symmetries across an entire labelled dataset. The symmetry transformations and the corresponding generators are modeled with fully connected neural networks trained with a specially constructed loss function. The two new elements in this work are the use of a reduced-dimensionality latent space and the generalization to transformations invariant with respect to high-dimensional oracles.
arXiv Detail & Related papers (2023-02-02T00:13:32Z)
Laplacian-based Cluster-Contractive t-SNE for High Dimensional Data Visualization [20.43471678277403]
We propose LaptSNE, a new graph-based dimensionality reduction method based on t-SNE. Specifically, LaptSNE leverages the eigenvalue information of the graph Laplacian to shrink the potential clusters in the low-dimensional embedding. We show how to calculate the gradient analytically, which may be of broad interest when considering optimization with Laplacian-composited objective.
arXiv Detail & Related papers (2022-07-25T14:10:24Z)
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)
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)
Learning Flat Latent Manifolds with VAEs [16.725880610265378]
We propose an extension to the framework of variational auto-encoders, where the Euclidean metric is a proxy for the similarity between data points. We replace the compact prior typically used in variational auto-encoders with a recently presented, more expressive hierarchical one. We evaluate our method on a range of data-sets, including a video-tracking benchmark.
arXiv Detail & Related papers (2020-02-12T09:54:52Z)

This list is automatically generated from the titles and abstracts of the papers in this site.