The Wristband Gaussian Loss: Deterministic, Composable Latents via a Sphere-Interval Decomposition

arXiv:2605.08749v1 Announce Type: new Abstract: We present the Wristband Gaussian Loss, a deterministic batch loss for Gaussianizing point embeddings without sampling, KL terms, or iterative transport. Each x in R^d is mapped to a direction u = x/|x| and a CDF-transformed radius t = F_{hi^2_d}(|x|^2) on the wristband S^{d-1} imes [0,1]. We prove (and machine-verify in Lean~4) that for d ge 2 the pushforward wristband map equals sigma_{d-1} otimes Unif[0,1] iff the source is N(0, I_d), and that the Neumann-reflected wristband repulsion energy is uniquely minimized at the uniform target. We compute this reflected-kernel objective in two ways: a nearest three-image pairwise truncation at O(N^2 d), and a spectral Neumann path joining angular and radial Mercer modes (spherical-harmonic and cosine) at O(N d K), with empirically matched gradients. A 1D Wasserstein radial term and a moment penalty serve as finite-sample accelerators with the same optimum, and Monte-Carlo null calibration turns the components into a single standardized statistic. We evaluate direct point-cloud Gaussianization with a calibrated barycentric W_2 score: a deterministic Gaussian reference batch is built by recursive Hungarian averaging, with each method reported as a z-score against same-size Gaussian batches. On the axis-uniform X benchmark, Wristband is competitive in 2D and gives the best 10D score. On a harder radial--angular-copula impostor whose Gaussian radial and angular marginals are correct but dependent, Wristband gives the best 10D and 128D scores. Coupled with learnable-key Euclidean attention and exact invertible flows, the resulting Deterministic Gaussian Autoencoder delivers a Gaussian-latent interface for counterfactual sampling with independent factors and a context/residual construction for dependent factors.

Source: arXiv cs.LG | 2026-05-12