INFORMATION-GEOMETRIC LATENT ENCODING FOR 3D GAUSSIAN SPLATTING

Abstract

3D Gaussian splatting represents a scene as an unordered set of anisotropic Gaussian primitives whose centers, covariances, colors, and opacities jointly determine rendered appearance. This thesis develops the Geometric Latent Encoding Autoencoder (GLEA), a class-agnostic model that maps a variable-cardinality splat set to a 512-dimensional code and reconstructs up to 3,200 Gaussian primitives without encoder skip connections. The central design question is whether primitive comparison should use covariance information directly rather than relying only on learned content features or center proximity. GLEA addresses that question with a gated attention branch derived from the Bhattacharyya coefficient, a closed-form measure of overlap between probability distributions. The coefficient is a positive-semidefinite kernel under the square-root density embedding, and its associated distance agrees locally, to second order, with Fisher–Rao geometry for nearby distributions. The implemented system combines PCA pose alignment, cuboid normalization, multi- resolution hash features, learned grouping, gated geometric and content attention, a hierar- chical intermediate representation, a deterministic latent-only decoder, sparse Bhattacharyya– Chamfer reconstruction, multi-view rendering supervision, and conditional flow matching. On a disjoint ShapeSplat split, evaluation over 128 held-out objects yields mean center Chamfer distance of 1.54 ×10−3, PSNR of 25.57 ±2.64 dB, SSIM of 0.956 ±0.016, and LPIPS of 0.124 ±0.032. Under a matched four-epoch, single-seed ablation budget, removing Bhattacharyya attention lowers mean held-out PSNR by 1.20 dB and worsens every reported external reconstruction metric; removing PCA alignment lowers PSNR by 0.67 dB. These results support covariance-aware attention under the evaluated configuration. A controlled analytic battery independently verifies that the underlying kernel is covariance-sensitive in exactly the way a center-only score is not, matching its closed form to a relative error of 10−12 . Separately, relaxing the strict 512-dimensional bottleneck to decode from the full hierarchical latent raises reconstruction fidelity at the cost of the compact, analyzable code, making the fidelity–interpretability tradeoff explicit. They do not yet establish category-level transfer, unconditional generation, global preservation of Fisher–Rao distances in the learned code, or comparative state-of-the-art performance. Multi-seed replication and matched external baselines remain necessary.