🤖 AI Summary
This work proposes a deterministic Gaussianization method that eschews sampling, KL divergence, and iterative transport. By decomposing input vectors into direction and radius—where the radius is transformed via the cumulative distribution function—the approach maps data to a spherical–interval product space (a “wristband”) and employs an energy minimization mechanism based on the Neumann reflection kernel to achieve Gaussianization. Key contributions include the first deterministic Gaussianization loss, spectral regularization combining spherical harmonics with cosine Mercer modes, and formally verified theoretical guarantees in Lean 4. The method integrates a 1D Wasserstein radial term, moment penalties, invertible flows, and learnable key attention, while constructing Gaussian reference batches via Hungarian recursive averaging. It achieves state-of-the-art Gaussianization performance on complex, non-independent radial–angular joint distributions in both 10D and 128D settings and supports counterfactual generation with either independent or dependent factors.
📝 Abstract
We present the Wristband Gaussian Loss, a deterministic batch loss for Gaussianizing point embeddings without sampling, KL terms, or iterative transport. Each $x \in \mathbb{R}^d$ is mapped to a direction $u = x/\|x\|$ and a CDF-transformed radius $t = F_{χ^2_d}(\|x\|^2)$ on the wristband $S^{d-1} \times [0,1]$. We prove (and machine-verify in Lean~4) that for $d \ge 2$ the pushforward wristband map equals $σ_{d-1} \otimes \mathrm{Unif}[0,1]$ iff the source is $\mathcal{N}(0, I_d)$, 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.