← 研踪 PaperTrailarXiv:2611.16022 · v1
Closed-Form Gaussian Information Projection for Embedded Hypercubes and Beta--Gaussian Mixtures
首次提交 2026-11-12 05:28
版本更新 2026-11-12 05:28
雷达观测 2026-11-13 14:16
摘要
We derive closed-form Gaussian information projections of Beta--Gaussian distributions whose tangential components are supported on affine embedded hypercubes and whose normal components are Gaussian. The optimal Gaussian is uniquely characterized by moment matching, and the corresponding minimum Kullback--Leibler divergence admits an explicit expression depending only on the intrinsic Beta parameters, independently of the embedding geometry, ambient dimension, and normal Gaussian parameters. In the uniform case, this divergence reduces to a linear function of the intrinsic dimension. We further extend this construction to finite, pure embedded cubical complexes by considering mixtures of the componentwise Gaussian projections. Using the joint convexity of the Kullback--Leibler divergence, we establish an explicit upper bound on the approximation error of the resulting Gaussian Mixture. For uniform tangential distributions and volume-based mixture weights, the bound is linear in the intrinsic dimension and independent of the geometry of the complex, its embedding, and the normal Gaussian parameters.
作者备注：11 pages
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v1 · 2026-11-12 05:28 · 最新观测版本
Closed-Form Gaussian Information Projection for Embedded Hypercubes and Beta--Gaussian Mixtures
We derive closed-form Gaussian information projections of Beta--Gaussian distributions whose tangential components are supported on affine embedded hypercubes and whose normal components are Gaussian. The optimal Gaussian is uniquely characterized by moment matching, and the corresponding minimum Kullback--Leibler divergence admits an explicit expression depending only on the intrinsic Beta parameters, independently of the embedding geometry, ambient dimension, and normal Gaussian parameters. In the uniform case, this divergence reduces to a linear function of the intrinsic dimension. We further extend this construction to finite, pure embedded cubical complexes by considering mixtures of the componentwise Gaussian projections. Using the joint convexity of the Kullback--Leibler divergence, we establish an explicit upper bound on the approximation error of the resulting Gaussian Mixture. For uniform tangential distributions and volume-based mixture weights, the bound is linear in the intrinsic dimension and independent of the geometry of the complex, its embedding, and the normal Gaussian parameters.
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