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

arXiv: https://arxiv.org/abs/2605.08749 · HTML: https://mlsanity.com/paper/2605.08749/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 22 references that are in the map.
- First posted on arXiv 9 May 2026.

## Builds on (most cited first, 11 of 11)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [VICReg: Variance-Invariance-Covariance Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2105.04906/) (2021)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Sparse Gaussian Processes with Spherical Harmonic Features](https://mlsanity.com/paper/2006.16649/) (2020)
- [Invertible Generative Modeling using Linear Rational Splines](https://mlsanity.com/paper/2001.05168/) (2020)
- [Auto-Compressing Networks](https://mlsanity.com/paper/2506.09714/) (2025)
- [Radial-VCReg: More Informative Representation Learning Through Radial Gaussianization](https://mlsanity.com/paper/2602.14272/) (2026)

## Built on by (most cited first, 1 of 1)

- [Shared Gaussianization: What Gaussian Regularizers Certify About Contrastive Learning, and What They Miss](https://mlsanity.com/paper/2610.10299/) (2026)

## Abstract (arXiv)

We present the Wristband Gaussian Loss, a deterministic batch loss for Gaussianizing point embeddings without sampling, KL terms, or iterative transport. Each x ∈ ℝd is mapped to a direction u = x/|x| and a CDF-transformed radius t = Fchi2d(|x|2) on the wristband Sd-1 × [0,1]. We prove (and machine-verify in Lean~4) that for d ≥ 2 the pushforward wristband map equals σd-1 otimes Unif[0,1] iff the source is N(0, Id), 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(N2 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 W2 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.