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.
We study Gaussian approximation of Dirichlet--Gaussian probability density functions on embedded simplices. We derive the unique information projection onto the family of nondegenerate ambient Gaussian probability density functions, obtaining closed-form expressions for the optimal Gaussian and the minimum divergence for arbitrary positive Dirichlet parameters. The minimum divergence depends only on the intrinsic simplex dimension and Dirichlet parameters, is invariant up to an affine isomorphism, independent of the ambient dimension, and independent of the normal covariance. Furthermore, by joint convexity of Kullback--Leibler divergence, we establish explicit intrinsic dimension-dependent sharp upper bounds for finite Gaussian Mixture with shared component weights. For uniform probability density functions, these bounds reduce to dimension-dependent constants. The results provide closed-form Gaussian Mixture constructions and rigorous approximation guarantees for probability density functions associated with embedded simplicial complexes. A numerical example on a circular curve illustrates the theoretical results and their connections to geometric Gaussian Mixture representations.