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

TL;DR

Topo-GS introduces a novel volumetric embedding method that transforms high-dimensional data into continuous, topology-preserving 3D representations, overcoming limitations of traditional discrete point-cloud approaches.

Contribution

It repurposes 3D Gaussian Splatting with geometric constraints and topology-aware strategies to produce continuous, manifold-aware volumetric embeddings of high-dimensional data.

Findings

Transforms discrete scatter plots into continuous volumetric representations.

Preserves local topological fidelity comparable to discrete baselines.

Effectively captures intrinsic data manifold structures.

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.