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We first present a simple recursive algorithm that generates cyclic rotation Gray codes for stamp foldings and semi-meanders, where consecutive strings differ by a stamp rotation. These are the first known Gray codes for stamp foldings and semi-meanders, and we thus solve an open problem posted by Sawada and Li in [Electron. J. Comb. 19(2), 2012]. We then introduce an iterative algorithm that generates the same rotation Gray codes for stamp foldings and semi-meanders. Both the recursive and iterative algorithms generate stamp foldings and semi-meanders in constant amortized time and $O(n)$-amortized time per string respectively, using a linear amount of memory.

We propose a simple methodology to approximate functions with given asymptotic behavior by specifically constructed terms and an unconstrained deep neural network (DNN). The methodology we describe extends to various asymptotic behaviors and multiple dimensions and is easy to implement. In this work we demonstrate it for linear asymptotic behavior in one-dimensional examples. We apply it to function approximation and regression problems where we measure approximation of only function values (``Vanilla Machine Learning''-VML) or also approximation of function and derivative values (``Differential Machine Learning''-DML) on several examples. We see that enforcing given asymptotic behavior leads to better approximation and faster convergence.

Statistical learning under distribution shift is challenging when neither prior knowledge nor fully accessible data from the target distribution is available. Distributionally robust learning (DRL) aims to control the worst-case statistical performance within an uncertainty set of candidate distributions, but how to properly specify the set remains challenging. To enable distributional robustness without being overly conservative, in this paper, we propose a shape-constrained approach to DRL, which incorporates prior information about the way in which the unknown target distribution differs from its estimate. More specifically, we assume the unknown density ratio between the target distribution and its estimate is isotonic with respect to some partial order. At the population level, we provide a solution to the shape-constrained optimization problem that does not involve the isotonic constraint. At the sample level, we provide consistency results for an empirical estimator of the target in a range of different settings. Empirical studies on both synthetic and real data examples demonstrate the improved accuracy of the proposed shape-constrained approach.

In decision-making, maxitive functions are used for worst-case and best-case evaluations. Maxitivity gives rise to a rich structure that is well-studied in the context of the pointwise order. In this article, we investigate maxitivity with respect to general preorders and provide a representation theorem for such functionals. The results are illustrated for different stochastic orders in the literature, including the usual stochastic order, the increasing convex/concave order, and the dispersive order.

Autonomous morphology, such as inflection class systems and paradigmatic distribution patterns, is widespread and diachronically resilient in natural language. Why this should be so has remained unclear given that autonomous morphology imposes learning costs, offers no clear benefit relative to its absence and could easily be removed by the analogical forces which are constantly reshaping it. Here we propose an explanation for the resilience of autonomous morphology, in terms of a diachronic dynamic of attraction and repulsion between morphomic categories, which emerges spontaneously from a simple paradigm cell filling process. Employing computational evolutionary models, our key innovation is to bring to light the role of `dissociative evidence', i.e., evidence for inflectional distinctiveness which a rational reasoner will have access to during analogical inference. Dissociative evidence creates a repulsion dynamic which prevents morphomic classes from collapsing together entirely, i.e., undergoing complete levelling. As we probe alternative models, we reveal the limits of conditional entropy as a measure for predictability in systems that are undergoing change. Finally, we demonstrate that autonomous morphology, far from being `unnatural' (e.g. \citealt{Aronoff1994}), is rather the natural (emergent) consequence of a natural (rational) process of inference applied to inflectional systems.

The success of over-parameterized neural networks trained to near-zero training error has caused great interest in the phenomenon of benign overfitting, where estimators are statistically consistent even though they interpolate noisy training data. While benign overfitting in fixed dimension has been established for some learning methods, current literature suggests that for regression with typical kernel methods and wide neural networks, benign overfitting requires a high-dimensional setting where the dimension grows with the sample size. In this paper, we show that the smoothness of the estimators, and not the dimension, is the key: benign overfitting is possible if and only if the estimator's derivatives are large enough. We generalize existing inconsistency results to non-interpolating models and more kernels to show that benign overfitting with moderate derivatives is impossible in fixed dimension. Conversely, we show that rate-optimal benign overfitting is possible for regression with a sequence of spiky-smooth kernels with large derivatives. Using neural tangent kernels, we translate our results to wide neural networks. We prove that while infinite-width networks do not overfit benignly with the ReLU activation, this can be fixed by adding small high-frequency fluctuations to the activation function. Our experiments verify that such neural networks, while overfitting, can indeed generalize well even on low-dimensional data sets.

The gradient bounds of generalized barycentric coordinates play an essential role in the $H^1$ norm approximation error estimate of generalized barycentric interpolations. Similarly, the $H^k$ norm, $k>1$, estimate needs upper bounds of high-order derivatives, which are not available in the literature. In this paper, we derive such upper bounds for the Wachspress generalized barycentric coordinates on simple convex $d$-dimensional polytopes, $d\ge 1$. The result can be used to prove optimal convergence for Wachspress-based polytopal finite element approximation of, for example, fourth-order elliptic equations. Another contribution of this paper is to compare various shape-regularity conditions for simple convex polytopes, and to clarify their relations using knowledge from convex geometry.

Randomized trials are considered the gold standard for making informed decisions in medicine, yet they often lack generalizability to the patient populations in clinical practice. Observational studies, on the other hand, cover a broader patient population but are prone to various biases. Thus, before using an observational study for decision-making, it is crucial to benchmark its treatment effect estimates against those derived from a randomized trial. We propose a novel strategy to benchmark observational studies beyond the average treatment effect. First, we design a statistical test for the null hypothesis that the treatment effects estimated from the two studies, conditioned on a set of relevant features, differ up to some tolerance. We then estimate an asymptotically valid lower bound on the maximum bias strength for any subgroup in the observational study. Finally, we validate our benchmarking strategy in a real-world setting and show that it leads to conclusions that align with established medical knowledge.

This paper introduces a new algorithm to improve the accuracy of numerical phase-averaging in oscillatory, multiscale, differential equations. Phase-averaging is a timestepping method which averages a mapped variable to remove highly oscillatory linear terms from the differential equation. This retains the main contribution of fast waves on the low frequencies without explicitly resolving the rapid oscillations. However, this comes at the cost of introducing an averaging error. To offset this, we propose a modified mapping that includes a mean correction term encoding an average measure of the nonlinear interactions. This mapping was introduced in Tao (2019) for weak nonlinearity and relied on classical time-averaging, which leaves only the zero frequencies. Our algorithm instead considers mean corrected phase-averaging when 1) the nonlinearity is not weak but the linear oscillations are fast and 2) finite averaging windows are applied via a smooth kernel, which has the advantage of retaining low frequencies whilst still eliminating the fastest oscillations. In particular, we introduce a local mean correction that combines the concepts of a mean correction and finite averaging; this retains low-frequency components in the mean correction that are removed with classical time-averaging. We show that the new timestepping algorithm reduces phase errors in the mapped variable for the swinging spring ODE in various dynamical configurations. We also show accuracy improvements with a local mean correction compared to standard phase-averaging in the one-dimensional rotating shallow water equations, a useful test case for weather and climate applications.