Asynchronous Gibbs sampling has been recently shown to be fast-mixing and an accurate method for estimating probabilities of events on a small number of variables of a graphical model satisfying Dobrushin's condition [DSOR16].

This allows us to bound each term in the sum using Lemma 1. Therefore, we conclude that 64 C ˆ C . For the last equation Eq. (27) is due to The proof of the first inequality Eq. (30) can be found in [ Eq. (31) can be obtained via induction. We prove this lemma via induction. Hence we finish our proof.

However, fitting surfaces explicitly from raw point clouds suffers from overfitting or underfitting issues caused by inappropriatepolynomial orders andoutliers, which significantly limits theperformance of existing methods.

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With proper prior knowledge of a task, we can choose a suitable coordination level.

Mean-field characterizations of first-order iterative algorithms -- including Approximate Message Passing (AMP), stochastic and proximal gradient descent, and Langevin diffusions -- have enabled a precise understanding of learning dynamics in many statistical applications. For algorithms whose non-linearities have a coordinate-separable form, it is known that such characterizations enjoy a degree of universality with respect to the underlying data distribution. However, mean-field characterizations of non-separable algorithm dynamics have largely remained restricted to i.i.d. Gaussian or rotationally-invariant data. In this work, we initiate a study of universality for non-separable AMP algorithms. We identify a general condition for AMP with polynomial non-linearities, in terms of a Bounded Composition Property (BCP) for their representing tensors, to admit a state evolution that holds universally for matrices with non-Gaussian entries. We then formalize a condition of BCP-approximability for Lipschitz AMP algorithms to enjoy a similar universal guarantee. We demonstrate that many common classes of non-separable non-linearities are BCP-approximable, including local denoisers, spectral denoisers for generic signals, and compositions of separable functions with generic linear maps, implying the universality of state evolution for AMP algorithms employing these non-linearities.

In this paper, we prove that every multivariate polynomial with even degree can be decomposed into a sum of convex and concave polynomials. Motivated by this property, we exploit the concave-convex procedure to perform inference on continuous Markov random fields with polynomial potentials. In particular, we show that the concave-convex decomposition of polynomials can be expressed as a sum-of-squares optimization, which can be efficiently solved via semidefinite programing. We demonstrate the effectiveness of our approach in the context of 3D reconstruction, shape from shading and image denoising, and show that our method significantly outperforms existing techniques in terms of efficiency as well as quality of the retrieved solution.

We examine two in context learning (ICL) tasks with mathematical functions in several train and test settings for transformer models. Our study generalizes work on linear functions by showing that small transformers, even models with attention layers only, can approximate arbitrary polynomial functions and hence continuous functions under certain conditions. Our models also can approximate previously unseen classes of polynomial functions, as well as the zeros of complex functions. Our models perform far better on this task than LLMs like GPT4 and involve complex reasoning when provided with suitable training data and methods. Our models also have important limitations; they fail to generalize outside of training distributions and so don't learn class forms of functions. We explain why this is so.