Topo-GS: Continuous Volumetric Embedding of High-Dimensional Data via Topological Gaussian Splatting

2026-05-16

AI Summary

This work addresses the limitations of traditional dimensionality reduction methods, which rely on discrete point clouds and often suffer from visual occlusion and artificial discontinuities that obscure the continuous density structure of high-dimensional manifolds. To overcome these issues, the paper introduces 3D Gaussian splatting into dimensionality reduction for the first time, proposing a mesh-free, continuous volumetric representation. The approach replaces photometric loss with local geometric constraints, aligning Gaussian covariances with local tangent spaces and incorporating an As-Rigid-As-Possible prior derived from orthogonal Procrustes analysis. Furthermore, a topology-aware loss is designed to adaptively handle structures of varying intrinsic dimensions—such as 1D trajectories and 2D surfaces—while preserving local topological fidelity. This framework transforms discrete embeddings into continuous representations, rendering projection distortions as explicit geometric deformations.

Abstract

Dimensionality reduction algorithms map high-dimensional data into visualizable 2D or 3D spaces, but traditionally rely on a discrete point-cloud paradigm. This discrete abstraction is susceptible to visual occlusion and artificial discontinuities, often failing to represent the continuous density of the underlying manifold. To address these limitations, we introduce Topo-GS, a framework that repurposes 3D Gaussian Splatting (3DGS) to cast multidimensional projection as a meshless volumetric reconstruction process. Instead of standard photometric losses, Topo-GS is driven by local geometric constraints. By solving orthogonal Procrustes targets, the optimization enforces an As-Rigid-As-Possible prior while explicitly aligning the spatial covariance of each Gaussian to the local tangent space. Recognizing that unrolling data of varying intrinsic dimensionalities requires distinct spatial treatments, we utilize a topology-aware strategy that tailors the loss formulation to preserve either continuous 1D trajectories or cohesive 2D surfaces. Quantitative and visual evaluations demonstrate that Topo-GS successfully transforms discrete scatter plots into continuous volumetric representations, where inherent projection distortions explicitly manifest as observable geometric variations, while preserving local topological fidelity comparable to discrete baselines.

Problem

• dimensionality reduction
• continuous representation
• topological fidelity
• visual occlusion
• manifold density

Innovation

• Topological Gaussian Splatting
• Dimensionality Reduction
• Volumetric Embedding
• Manifold Visualization
• As-Rigid-As-Possible