Charles Fefferman, Aalok Gangopadhyay, Matti Lassas, Jonathan Marty, Hariharan Narayanan
原文摘要
We study the problem of denoising observations \(Y_i=X_i+Z_i\), where the latent variables \(X_i\) are sampled from a low-dimensional manifold in \(\mathbb{R}^n\) and the noise variables \(Z_i\) are isotropic Gaussian. We propose a convex-relaxation estimator that first reduces dimension by principal component analysis and then projects the observations onto the convex hull of the projected latent manifold. We construct a statistical oracle that estimates its supporting hyperplanes from empirical Gaussian tail probabilities of the noisy sample. Under a lower-mass condition on the latent distribution, we prove finite-sample guarantees for the oracle and derive error bounds for the resulting denoiser. The analysis combines risk bounds for least-squares projection under convex constraints (due to Chatterjee) with entropy bounds for the manifold. We also verify the assumptions of the framework for a Cryo-Electron Microscopy observation model by establishing suitable covering number and Lipschitz estimates for the associated group action and imaging operators.
S David Mis, Trevor Teolis, Ivan Dokmanić, Matti Lassas, Maarten V. de Hoop
原文摘要
Operator learning seeks to approximate function-to-function maps in a way that remains stable when the input discretization changes, a property sometimes called "discretization invariance". We consider a quantitative refinement of this idea, called $\textit{uniform discretization stability}$, which builds on a common representation of input functions and their discretizations as probability measures and provides a Hölder bound relating output changes to the Wasserstein distance between these measures. Our main result is a representation theorem showing all uniformly discretization stable operators can be written exactly as a $\textit{stable Markov kernel network}$---an abstract architecture that is a joint generalization of transformers, Fourier neural operators (FNO), Flowers, and other successful learned operators. Our representation theorem characterizes a design space for discretization-stable operator learning; designing new architectures becomes a question of which parts of a Markov kernel network to parameterize, and how. We further show that the read kernels of these networks can be realized as solutions of regularized variational problems, with particular choices of costs and regularizers recovering the kernels of transformers, FNO, Flowers, their hybrids, and beyond. This variational formulation provides a systematic approach to architecture design, together with sufficient conditions for uniform discretization stability.