This paper investigates the numerical resolution of the Beurling-LASSO (BLASSO), a convex optimization framework that promotes sparsity in the space of measures. We consider its application to the estimation of Gaussian mixture models (GMMs) with an unknown number of components and unknown diagonal covariance matrices. Our approach combines the Conic Particle Gradient Descent (CPGD) principle with Riemannian gradient descent, to account for the underlying Fisher-Rao geometry of Gaussian distributions. Our contributions are twofold. First, we provide theoretical guarantees for the convergence of our algorithm. In particular, we establish exponential local convergence under a non-degeneracy condition on the solution and relate this assumption to a separation condition on the underlying statistical target. Second, we address practical implementation aspects of CPGD and present numerical experiments illustrating its performance. On the test cases considered, these experiments suggest that CPGD is more robust to overspecification of the number of components than the EM algorithm. We also investigate the impact of component separation on recovery accuracy.
Active subspaces identify low-dimensional linear structure in high-dimensional parameter-to-output maps by estimating the dominant eigenspace of a gradient covariance operator. In practice this covariance is replaced by a Monte Carlo estimator built from a limited number of gradient evaluations. Classical analyses base...
Fabio Nobile, Matteo Raviola, R. Tempone· 0 citations
We consider nonparametric tangent vector field regression on a Riemannian manifold without boundary. Because responses at different points lie in different tangent spaces, the proposed kernel estimator first parallel transports nearby responses to the target tangent space and then forms a volume-corrected local average...
As an alternative to the standard geometric analyses, we give an exact, information-theoretic analysis of stochastic gradient descent (SGD) and its variants. We show that a preconditioned SGD step is the posterior-mean update of a Gaussian Bayes model, and that its one-step regret splits into an intrinsic-time cost and...
This work proposes PRISM-ZO, a projection-robust framework that samples low-dimensional random tangent subspaces and combines symmetric finite differences with median-of-means or Huber aggregation and establishes the unbiasedness of the correctly rescaled projected direction in expectation over the random subspace.
Yin-Pu Ma, Cunlin Li, Shiyue Zhang· Journal of King Saud Univers...· 0 citations
We develop a Gaussian comparison theory for posterior inference in non-Gaussian high-dimensional models. The framework allows for model misspecification and does not require the posterior distribution itself to be approximately Gaussian. Working directly with likelihood processes on separable function spaces, we establ...
A unified framework for scalable estimation of tensor covariances based on a Kronecker-structured sparse inverse Cholesky (KSIC) projection, proving that the KSIC estimator gainfully exploits cross-mode information and is robust to data scarcity.
Wentao Zhan, Matthias Katzfuss· 0 citations
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