We consider the nonparametric regression problem when the covariates are located on an unknown smooth compact submanifold of a Euclidean space. Under defining a random geometric graph structure over the covariates we analyze the asymptotic frequentist behaviour of the posterior distribution arising from Bayesian priors designed through random basis expansion in the graph Laplacian eigenbasis. Under Holder smoothness assumption on the regression function and the density of the covariates over the submanifold, we prove that the posterior contraction rates of such methods are minimax optimal (up to logarithmic factors) for any positive smoothness index.
相關內容
The rapid pace of development in quantum computing technology has sparked a proliferation of benchmarks for assessing the performance of quantum computing hardware and software. Good benchmarks empower scientists, engineers, programmers, and users to understand a computing system's power, but bad benchmarks can misdirect research and inhibit progress. In this Perspective, we survey the science of quantum computer benchmarking. We discuss the role of benchmarks and benchmarking, and how good benchmarks can drive and measure progress towards the long-term goal of useful quantum computations, i.e., "quantum utility". We explain how different kinds of benchmark quantify the performance of different parts of a quantum computer, we survey existing benchmarks, critically discuss recent trends in benchmarking, and highlight important open research questions in this field.
The broad class of multivariate unified skew-normal (SUN) distributions has been recently shown to possess important conjugacy properties. When used as priors for the vector of parameters in general probit, tobit, and multinomial probit models, these distributions yield posteriors that still belong to the SUN family. Although such a core result has led to important advancements in Bayesian inference and computation, its applicability beyond likelihoods associated with fully-observed, discretized, or censored realizations from multivariate Gaussian models remains yet unexplored. This article covers such an important gap by proving that the wider family of multivariate unified skew-elliptical (SUE) distributions, which extends SUNs to more general perturbations of elliptical densities, guarantees conjugacy for broader classes of models, beyond those relying on fully-observed, discretized or censored Gaussians. Such a result leverages the closure under linear combinations, conditioning and marginalization of SUE to prove that this family is conjugate to the likelihood induced by general multivariate regression models for fully-observed, censored or dichotomized realizations from skew-elliptical distributions. This advancement enlarges the set of models that enable conjugate Bayesian inference to general formulations arising from elliptical and skew-elliptical families, including the multivariate Student's t and skew-t, among others.
Credences are mental states corresponding to degrees of confidence in propositions. Attribution of credences to Large Language Models (LLMs) is commonplace in the empirical literature on LLM evaluation. Yet the theoretical basis for LLM credence attribution is unclear. We defend three claims. First, our semantic claim is that LLM credence attributions are (at least in general) correctly interpreted literally, as expressing truth-apt beliefs on the part of scientists that purport to describe facts about LLM credences. Second, our metaphysical claim is that the existence of LLM credences is at least plausible, although current evidence is inconclusive. Third, our epistemic claim is that LLM credence attributions made in the empirical literature on LLM evaluation are subject to non-trivial sceptical concerns. It is a distinct possibility that even if LLMs have credences, LLM credence attributions are generally false because the experimental techniques used to assess LLM credences are not truth-tracking.
A common method for estimating the Hessian operator from random samples on a low-dimensional manifold involves locally fitting a quadratic polynomial. Although widely used, it is unclear if this estimator introduces bias, especially in complex manifolds with boundaries and nonuniform sampling. Rigorous theoretical guarantees of its asymptotic behavior have been lacking. We show that, under mild conditions, this estimator asymptotically converges to the Hessian operator, with nonuniform sampling and curvature effects proving negligible, even near boundaries. Our analysis framework simplifies the intensive computations required for direct analysis.
A new decoder for the SIF test problems of the CUTEst collection is described, which produces problem files allowing the computation of values and derivatives of the objective function and constraints of most \cutest\ problems directly within ``native'' Matlab, Python or Julia, without any additional installation or interfacing with MEX files or Fortran programs. When used with Matlab, the new problem files optionally support reduced-precision computations.
We consider zero-sum games in which players move between adjacent states, where in each pair of adjacent states one state dominates the other. The states in our game can represent positional advantages in physical conflict such as high ground or camouflage, or product characteristics that lend an advantage over competing sellers in a duopoly. We study the equilibria of the game as a function of the topological and geometric properties of the underlying graph. Our main result characterizes the expected payoff of both players starting from any initial position, under the assumption that the graph does not contain certain types of small cycles. This characterization leverages the block-cut tree of the graph, a construction that describes the topology of the biconnected components of the graph. We identify three natural types of (on-path) pure equilibria, and characterize when these equilibria exist under the above assumptions. On the geometric side, we show that strongly connected outerplanar graphs with undirected girth at least 4 always support some of these types of on-path pure equilibria. Finally, we show that a data structure describing all pure equilibria can be efficiently computed for these games.
We present a Bayesian method for multivariate changepoint detection that allows for simultaneous inference on the location of a changepoint and the coefficients of a logistic regression model for distinguishing pre-changepoint data from post-changepoint data. In contrast to many methods for multivariate changepoint detection, the proposed method is applicable to data of mixed type and avoids strict assumptions regarding the distribution of the data and the nature of the change. The regression coefficients provide an interpretable description of a potentially complex change. For posterior inference, the model admits a simple Gibbs sampling algorithm based on P\'olya-gamma data augmentation. We establish conditions under which the proposed method is guaranteed to recover the true underlying changepoint. As a testing ground for our method, we consider the problem of detecting topological changes in time series of images. We demonstrate that our proposed method $\mathtt{bclr}$, combined with a topological feature embedding, performs well on both simulated and real image data. The method also successfully recovers the location and nature of changes in more traditional changepoint tasks.
We propose a topological mapping and localization system able to operate on real human colonoscopies, despite significant shape and illumination changes. The map is a graph where each node codes a colon location by a set of real images, while edges represent traversability between nodes. For close-in-time images, where scene changes are minor, place recognition can be successfully managed with the recent transformers-based local feature matching algorithms. However, under long-term changes -- such as different colonoscopies of the same patient -- feature-based matching fails. To address this, we train on real colonoscopies a deep global descriptor achieving high recall with significant changes in the scene. The addition of a Bayesian filter boosts the accuracy of long-term place recognition, enabling relocalization in a previously built map. Our experiments show that ColonMapper is able to autonomously build a map and localize against it in two important use cases: localization within the same colonoscopy or within different colonoscopies of the same patient. Code: //github.com/jmorlana/ColonMapper.
The trustworthiness of Machine Learning (ML) models can be difficult to assess, but is critical in high-risk or ethically sensitive applications. Many models are treated as a `black-box' where the reasoning or criteria for a final decision is opaque to the user. To address this, some existing Explainable AI (XAI) approaches approximate model behaviour using perturbed data. However, such methods have been criticised for ignoring feature dependencies, with explanations being based on potentially unrealistic data. We propose a novel framework, CHILLI, for incorporating data context into XAI by generating contextually aware perturbations, which are faithful to the training data of the base model being explained. This is shown to improve both the soundness and accuracy of the explanations.
We prove the convergence of a damped Newton's method for the nonlinear system resulting from a discretization of the second boundary value problem for the Monge-Ampere equation. The boundary condition is enforced through the use of the notion of asymptotic cone. The differential operator is discretized based on a discrete analogue of the subdifferential.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191