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.

翻译：降维算法将高维数据映射到可视化的二维或三维空间，但传统上依赖离散点云范式。这种离散抽象易受视觉遮挡和人为不连续性影响，常常无法表示底层流形的连续密度。为解决这些限制，我们提出Topo-GS框架，该框架将三维高斯溅射（3DGS）重新用于将多维投影转化为无网格体积重建过程。与标准光度损失不同，Topo-GS由局部几何约束驱动。通过求解正交普鲁克问题，优化过程在明确对齐每个高斯空间协方差与局部切空间的同时，强制执行“尽可能刚性”先验。认识到展开不同固有维度的数据需要不同的空间处理方式，我们采用拓扑感知策略，定制损失函数以保留连续的1D轨迹或内聚的2D表面。定量与视觉评估表明，Topo-GS成功将离散散点图转化为连续体积表示，其中固有的投影畸变显式表现为可观测的几何变化，同时保留了与离散基线相当的整体拓扑保真度。