提出一种无需采样的确定性高斯化方法，实现可组合的潜在变量表示。
The Wristband Gaussian Loss: Deterministic, Composable Latents via a Sphere-Interval Decomposition
- 将点嵌入映射到球面与区间的乘积空间，通过方向与概率变换半径实现高斯化。
- 在2D和10D上优于现有方法，尤其对径向-角度耦合数据表现最佳。
- 适合需要独立因子生成与可逆建模的生成模型研究者。
我们提出腕带高斯损失（Wristband Gaussian Loss），一种无需采样、无KL项或迭代传输的确定性批损失函数，用于高斯化点嵌入。每个 $x \in \mathbb{R}^d$ 被映射为方向 $u = x/\|x\|$ 和经过卡方分布累积分布函数变换的半径 $t = F_{χ^2_d}(\|x\|^2)$，位于腕带 $S^{d-1} \times [0,1]$ 上。我们证明（并在 Lean~4 中机器验证）当 $d \ge 2$ 时，若源分布为 $\mathcal{N}(0, I_d)$，则前推映射等于 $σ_{d-1} \otimes \mathrm{Unif}[0,1]$，且诺伊曼反射腕带排斥能唯一最小化于均匀目标。该反射核目标以两种方式计算：$O(N^2 d)$ 的最近三图像配对截断，以及 $O(N d K)$ 的谱诺伊曼路径（连接球谐与余弦基），梯度经验匹配。一维沃尔德斯坦径向项与矩惩罚作为有限样本加速器，共享最优解；蒙特卡洛零校准使各组件合成单一标准化统计量。在轴向均匀的 X 基准测试中，腕带在2D表现优异，在10D得分最高；在更难的径向-角度-耦合伪造数据（径向与角度边际正确但依赖）上，10D与128D得分均领先。结合可学习键欧氏注意力与精确可逆流，所提确定性高斯自编码器提供独立因子的反事实采样接口，及依赖因子的上下文/残差构造。
原文摘要 · Abstract (English)
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.
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