2025/12/11 by Nicholas J. Daras, Daras, Nicholas J.
Computer Science · Mathematics · #30C30 #68P99 #FOS: Mathematics #General Mathematics (math.GM) #Morphological variations and asymmetry #Stochastic Gradient Optimization Techniques #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2512.11916
openalex publication_date 2025/12/11 · openalex created_date 2025/12/17 · openalex updated_date 2026/07/28
We give two low-complexity algorithms, one for dimensionality reduction and one for dimensionality increase, which are applicable to any dataset, regardless of whether the set has an intrinsic dimension or not. The corresponding methods introduce chains of compositions of conformal homeomorphisms that transform any data set \mathbbX in a Euclidean space ℝD+1 into an isopleth dataset \mathbbY within a Euclidean space ℝ^\mathfrakD+1 of arbitrarily smaller or of arbitrarily larger dimension \mathfrakD+1 and preserve all angles, in the sense that all angles formed between points in the original dataset \mathbbX are equal to the angles formed between the images of these points in the new dataset \mathbbY. Because they preserve angles, the two methods also preserve shapes locally, although, in general, the overall sizes and shapes are distorted away from a center point.