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This paper addresses the challenge of modeling multi-way contingency tables for matched set data with ordinal categories. Although the complete symmetry and marginal homogeneity models are well established, they may not always provide a satisfactory fit to the data. To address this issue, we propose a generalized ordinal quasi-symmetry model that offers increased flexibility when the complete symmetry model fails to capture the underlying structure. We investigate the properties of this new model and provide an information-theoretic interpretation, elucidating its relationship to the ordinal quasi-symmetry model. Moreover, we revisit Agresti's findings and present a new necessary and sufficient condition for the complete symmetry model, proving that the proposed model and the marginal moment equality model are separable hypotheses. The separability of the proposed model and marginal moment equality model is a significant development in the analysis of multi-way contingency tables. It enables researchers to examine the symmetry structure in the data with greater precision, providing a more thorough understanding of the underlying patterns. This powerful framework equips researchers with the necessary tools to explore the complexities of ordinal variable relationships in matched set data, paving the way for new discoveries and insights.

This manuscript derives locally weighted ensemble Kalman methods from the point of view of ensemble-based function approximation. This is done by using pointwise evaluations to build up a local linear or quadratic approximation of a function, tapering off the effect of distant particles via local weighting. This introduces a candidate method (the locally weighted Ensemble Kalman method for inversion) with the motivation of combining some of the strengths of the particle filter (ability to cope with nonlinear maps and non-Gaussian distributions) and the Ensemble Kalman filter (no filter degeneracy).

Most of the recent results in polynomial functional regression have been focused on an in-depth exploration of single-parameter regularization schemes. In contrast, in this study we go beyond that framework by introducing an algorithm for multiple parameter regularization and presenting a theoretically grounded method for dealing with the associated parameters. This method facilitates the aggregation of models with varying regularization parameters. The efficacy of the proposed approach is assessed through evaluations on both synthetic and some real-world medical data, revealing promising results.

This paper presents a regularized recursive identification algorithm with simultaneous on-line estimation of both the model parameters and the algorithms hyperparameters. A new kernel is proposed to facilitate the algorithm development. The performance of this novel scheme is compared with that of the recursive least squares algorithm in simulation.

The property of reversibility is quite meaningful for the classic theoretical computer science model, cellular automata. For the reversibility problem for a CA under null boundary conditions, while linear rules have been studied a lot, the non-linear rules remain unexplored at present. The paper investigates the reversibility problem of general one-dimensional CA on a finite field $\mathbb{Z}_p$, and proposes an approach to optimize the Amoroso's infinite CA surjectivity detection algorithm. This paper proposes algorithms for deciding the reversibility of one-dimensional CA under null boundary conditions. We propose a method to decide the strict reversibility of one-dimensional CA under null boundary conditions. We also provide a bucket chain based algorithm for calculating the reversibility function of one-dimensional CA under null boundary conditions. These decision algorithms work for not only linear rules but also non-linear rules. In addition, it has been confirmed that the reversibility function always has a period, and its periodicity is related to the periodicity of the corresponding bucket chain. Some of our experiment results of reversible CA are presented in the paper, complementing and validating the theoretical aspects, and thereby further supporting the research conclusions of this paper.

This paper analyzes a full discretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. The discretization uses the Euler scheme for temporal discretization and the finite element method for spatial discretization. By deriving a stability estimate of a discrete stochastic convolution and utilizing this stability estimate along with the discrete stochastic maximal $L^p$-regularity estimate, a pathwise uniform convergence rate with the general spatial $ L^q $-norms is derived.

Training data memorization in language models impacts model capability (generalization) and safety (privacy risk). This paper focuses on analyzing prompts' impact on detecting the memorization of 6 masked language model-based named entity recognition models. Specifically, we employ a diverse set of 400 automatically generated prompts, and a pairwise dataset where each pair consists of one person's name from the training set and another name out of the set. A prompt completed with a person's name serves as input for getting the model's confidence in predicting this name. Finally, the prompt performance of detecting model memorization is quantified by the percentage of name pairs for which the model has higher confidence for the name from the training set. We show that the performance of different prompts varies by as much as 16 percentage points on the same model, and prompt engineering further increases the gap. Moreover, our experiments demonstrate that prompt performance is model-dependent but does generalize across different name sets. A comprehensive analysis indicates how prompt performance is influenced by prompt properties, contained tokens, and the model's self-attention weights on the prompt.

We propose an algorithm to construct optimal exact designs (EDs). Most of the work in the optimal regression design literature focuses on the approximate design (AD) paradigm due to its desired properties, including the optimality verification conditions derived by Kiefer (1959, 1974). ADs may have unbalanced weights, and practitioners may have difficulty implementing them with a designated run size $n$. Some EDs are constructed using rounding methods to get an integer number of runs at each support point of an AD, but this approach may not yield optimal results. To construct EDs, one may need to perform new combinatorial constructions for each $n$, and there is no unified approach to construct them. Therefore, we develop a systematic way to construct EDs for any given $n$. Our method can transform ADs into EDs while retaining high statistical efficiency in two steps. The first step involves constructing an AD by utilizing the convex nature of many design criteria. The second step employs a simulated annealing algorithm to search for the ED stochastically. Through several applications, we demonstrate the utility of our method for various design problems. Additionally, we show that the design efficiency approaches unity as the number of design points increases.

Distillation is the task of replacing a complicated machine learning model with a simpler model that approximates the original [BCNM06,HVD15]. Despite many practical applications, basic questions about the extent to which models can be distilled, and the runtime and amount of data needed to distill, remain largely open. To study these questions, we initiate a general theory of distillation, defining PAC-distillation in an analogous way to PAC-learning [Val84]. As applications of this theory: (1) we propose new algorithms to extract the knowledge stored in the trained weights of neural networks -- we show how to efficiently distill neural networks into succinct, explicit decision tree representations when possible by using the ``linear representation hypothesis''; and (2) we prove that distillation can be much cheaper than learning from scratch, and make progress on characterizing its complexity.

Log-linear models are widely used to express the association in multivariate frequency data on contingency tables. The paper focuses on the power analysis for testing the goodness-of-fit hypothesis for this model type. Conventionally, for the power-related sample size calculations a deviation from the null hypothesis (effect size) is specified by means of the chi-square goodness-of-fit index. It is argued that the odds ratio is a more natural measure of effect size, with the advantage of having a data-relevant interpretation. Therefore, a class of log-affine models that are specified by odds ratios whose values deviate from those of the null by a small amount can be chosen as an alternative. Being expressed as sets of constraints on odds ratios, both hypotheses are represented by smooth surfaces in the probability simplex, and thus, the power analysis can be given a geometric interpretation as well. A concept of geometric power is introduced and a Monte-Carlo algorithm for its estimation is proposed. The framework is applied to the power analysis of goodness-of-fit in the context of multinomial sampling. An iterative scaling procedure for generating distributions from a log-affine model is described and its convergence is proved. To illustrate, the geometric power analysis is carried out for data from a clinical study.