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We here elaborate on a quantitative argument to support the validity of the Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The analysis is structured as follows. First, three distinct fixed points are found for the…
Algorithmic stability is a classical approach to understanding and analysis of the generalization error of learning algorithms. A notable weakness of most stability-based generalization bounds is that they hold only in expectation.…
We propose a machine-learning algorithm for Bayesian inverse problems in the function-space regime based on one-step generative transport. Building on the Mean Flows, we learn a fully conditional amortized sampler with a neural-operator…
Regression Discontinuity Design (RDD) is a popular framework for estimating a causal effect in settings where treatment is assigned if an observed covariate exceeds a fixed threshold. We consider estimation and inference in the common…
The model averaging problem is to average multiple models to achieve a prediction accuracy not much worse than that of the best single model in terms of mean squared error. It is known that if the models are misspecified, model averaging is…
Stochastic iterative methods are useful in a variety of large-scale numerical linear algebraic, machine learning, and statistical problems, in part due to their low-memory footprint. They are frequently used in a variety of applications,…
Boltzmann machines are undirected graphical models with two-state stochastic variables, in which the logarithms of the clique potentials are quadratic functions of the node states. They have been widely studied in the neural computing…
This paper proposes the asymmetric linear double autoregression, which jointly models the conditional mean and conditional heteroscedasticity characterized by asymmetric effects. A sufficient condition is established for the existence of a…
Prompt learning has emerged as an effective technique for fine-tuning large-scale foundation models for downstream tasks. However, conventional prompt learning methods are prone to overfitting and can struggle with out-of-distribution…
We present a randomized parallel algorithm that computes the greatest common divisor of two integers of n bits in length with probability 1-o(1) that takes O(n loglog n / log n) expected time using n^{6+\epsilon} processors on the EREW PRAM…
We propose a high-order stochastic-statistical moment closure model for efficient ensemble prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. The statistical moment…
In this paper we study the problem of convergence and generalization error bound of stochastic momentum for deep learning from the perspective of regularization. To do so, we first interpret momentum as solving an $\ell_2$-regularized…
This article presents a numerical illustration of a recently proposed strongly polynomial-time algorithm for the general linear programming (LP) problem. Each iteration of the proposed algorithm consists of two Gauss-Jordan pivoting…
This paper presents constant-time and near-constant-time distributed algorithms for a variety of problems in the congested clique model. We show how to compute a 3-ruling set in expected $O(\log \log \log n)$ rounds and using this, we…
Restricted Boltzmann Machines (RBMs) are general unsupervised learning devices to ascertain generative models of data distributions. RBMs are often trained using the Contrastive Divergence learning algorithm (CD), an approximation to the…
Clustering mixtures of Gaussian distributions is a fundamental and challenging problem that is ubiquitous in various high-dimensional data processing tasks. While state-of-the-art work on learning Gaussian mixture models has focused…
We investigate the behavior of the Generalized Likelihood Ratio Test (GLRT) (Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153-193]) for time varying coefficient models where the regressors and errors are non-stationary time series and can…
The mixing time of a graph is an important metric, which is not only useful in analyzing connectivity and expansion properties of the network, but also serves as a key parameter in designing efficient algorithms. We present an efficient…
We consider the problem of estimating the number of distinct elements in a large data set (or, equivalently, the support size of the distribution induced by the data set) from a random sample of its elements. The problem occurs in many…
In this paper, we develop and analyze an integral fixed-time sliding mode control method for a scenario in which the system model is only partially known, utilizing Gaussian processes. We present two theorems on fixed-time convergence. The…