This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors $\Lambda$ and $\Gamma$ between thin categories of relational structures are adjoint if for all structures $\mathbf A$ and $\mathbf B$, we have that $\Lambda(\mathbf A)$ maps homomorphically to $\mathbf B$ if and only if $\mathbf A$ maps homomorphically to $\Gamma(\mathbf B)$. If this is the case, $\Lambda$ is called the left adjoint to $\Gamma$ and $\Gamma$ the right adjoint to $\Lambda$. In 2015, Foniok and Tardif described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr in 1970. We generalise results of Foniok and Tardif to arbitrary relational structures, and coincidently, we also provide more right adjoints on digraphs, and since these constructions are connected to finite duality, we also provide a new construction of duals to trees. Our results are inspired by an application in promise constraint satisfaction -- it has been shown that such functors can be used as efficient reductions between these problems.
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This paper focuses on studying the convergence rate of the density function of the Euler--Maruyama (EM) method, when applied to the overdamped generalized Langevin equation with fractional noise which serves as an important model in many fields. Firstly, we give an improved upper bound estimate for the total variation distance between random variables by their Malliavin--Sobolev norms. Secondly, we establish the existence and smoothness of the density function for both the exact solution and the numerical one. Based on the above results, the convergence rate of the density function of the numerical solution is obtained, which relies on the regularity of the noise and kernel. This convergence result provides a powerful support for numerically capturing the statistical information of the exact solution through the EM method.
The use of propensity score (PS) methods has become ubiquitous in causal inference. At the heart of these methods is the positivity assumption. Violation of the positivity assumption leads to the presence of extreme PS weights when estimating average causal effects of interest, such as the average treatment effect (ATE) or the average treatment effect on the treated (ATT), which renders invalid related statistical inference. To circumvent this issue, trimming or truncating the extreme estimated PSs have been widely used. However, these methods require that we specify a priori a threshold and sometimes an additional smoothing parameter. While there are a number of methods dealing with the lack of positivity when estimating ATE, surprisingly there is no much effort in the same issue for ATT. In this paper, we first review widely used methods, such as trimming and truncation in ATT. We emphasize the underlying intuition behind these methods to better understand their applications and highlight their main limitations. Then, we argue that the current methods simply target estimands that are scaled ATT (and thus move the goalpost to a different target of interest), where we specify the scale and the target populations. We further propose a PS weight-based alternative for the average causal effect on the treated, called overlap weighted average treatment effect on the treated (OWATT). The appeal of our proposed method lies in its ability to obtain similar or even better results than trimming and truncation while relaxing the constraint to choose a priori a threshold (or even specify a smoothing parameter). The performance of the proposed method is illustrated via a series of Monte Carlo simulations and a data analysis on racial disparities in health care expenditures.
We study the use of local consistency methods as reductions between constraint satisfaction problems (CSPs), and promise version thereof, with the aim to classify these reductions in a similar way as the algebraic approach classifies gadget reductions between CSPs. This research is motivated by the requirement of more expressive reductions in the scope of promise CSPs. While gadget reductions are enough to provide all necessary hardness in the scope of (finite domain) non-promise CSP, in promise CSPs a wider class of reductions needs to be used. We provide a general framework of reductions, which we call consistency reductions, that covers most (if not all) reductions recently used for proving NP-hardness of promise CSPs. We prove some basic properties of these reductions, and provide the first steps towards understanding the power of consistency reductions by characterizing a fragment associated to arc-consistency in terms of polymorphisms of the template. In addition to showing hardness, consistency reductions can also be used to provide feasible algorithms by reducing to a fixed tractable (promise) CSP, for example, to solving systems of affine equations. In this direction, among other results, we describe the well-known Sherali-Adams hierarchy for CSP in terms of a consistency reduction to linear programming.
We investigate a class of combinatory algebras, called ribbon combinatory algebras, in which we can interpret both the braided untyped linear lambda calculus and framed oriented tangles. Any reflexive object in a ribbon category gives rise to a ribbon combinatory algebra. Conversely, From a ribbon combinatory algebra, we can construct a ribbon category with a reflexive object, from which the combinatory algebra can be recovered. To show this, and also to give the equational characterisation of ribbon combinatory algebras, we make use of the internal PRO construction developed in Hasegawa's recent work. Interestingly, we can characterise ribbon combinatory algebras in two different ways: as balanced combinatory algebras with a trace combinator, and as balanced combinatory algebras with duality.
Rough set is one of the important methods for rule acquisition and attribute reduction. The current goal of rough set attribute reduction focuses more on minimizing the number of reduced attributes, but ignores the spatial similarity between reduced and decision attributes, which may lead to problems such as increased number of rules and limited generality. In this paper, a rough set attribute reduction algorithm based on spatial optimization is proposed. By introducing the concept of spatial similarity, to find the reduction with the highest spatial similarity, so that the spatial similarity between reduction and decision attributes is higher, and more concise and widespread rules are obtained. In addition, a comparative experiment with the traditional rough set attribute reduction algorithms is designed to prove the effectiveness of the rough set attribute reduction algorithm based on spatial optimization, which has made significant improvements on many datasets.
We study the problem of constructing an estimator of the average treatment effect (ATE) that exhibits doubly-robust asymptotic linearity (DRAL). This is a stronger requirement than doubly-robust consistency. A DRAL estimator can yield asymptotically valid Wald-type confidence intervals even when the propensity score or the outcome model is inconsistently estimated. On the contrary, the celebrated doubly-robust, augmented-IPW (AIPW) estimator generally requires consistent estimation of both nuisance functions for standard root-n inference. We make three main contributions. First, we propose a new hybrid class of distributions that consists of the structure-agnostic class introduced in Balakrishnan et al (2023) with additional smoothness constraints. While DRAL is generally not possible in the pure structure-agnostic class, we show that it can be attained in the new hybrid one. Second, we calculate minimax lower bounds for estimating the ATE in the new class, as well as in the pure structure-agnostic one. Third, building upon the literature on doubly-robust inference (van der Laan, 2014, Benkeser et al, 2017, Dukes et al 2021), we propose a new estimator of the ATE that enjoys DRAL. Under certain conditions, we show that its rate of convergence in the new class can be much faster than that achieved by the AIPW estimator and, in particular, matches the minimax lower bound rate, thereby establishing its optimality. Finally, we clarify the connection between DRAL estimators and those based on higher-order influence functions (Robins et al, 2017) and complement our theoretical findings with simulations.
Speech perception involves storing and integrating sequentially presented items. Recent work in cognitive neuroscience has identified temporal and contextual characteristics in humans' neural encoding of speech that may facilitate this temporal processing. In this study, we simulated similar analyses with representations extracted from a computational model that was trained on unlabelled speech with the learning objective of predicting upcoming acoustics. Our simulations revealed temporal dynamics similar to those in brain signals, implying that these properties can arise without linguistic knowledge. Another property shared between brains and the model is that the encoding patterns of phonemes support some degree of cross-context generalization. However, we found evidence that the effectiveness of these generalizations depends on the specific contexts, which suggests that this analysis alone is insufficient to support the presence of context-invariant encoding.
We extend the scope of a recently introduced dependence coefficient between a scalar response $Y$ and a multivariate covariate $X$ to the case where $X$ takes values in a general metric space. Particular attention is paid to the case where $X$ is a curve. While on the population level, this extension is straight forward, the asymptotic behavior of the estimator we consider is delicate. It crucially depends on the nearest neighbor structure of the infinite-dimensional covariate sample, where deterministic bounds on the degrees of the nearest neighbor graphs available in multivariate settings do no longer exist. The main contribution of this paper is to give some insight into this matter and to advise a way how to overcome the problem for our purposes. As an important application of our results, we consider an independence test.
This paper develops an in-depth treatment concerning the problem of approximating the Gaussian smoothing and Gaussian derivative computations in scale-space theory for application on discrete data. With close connections to previous axiomatic treatments of continuous and discrete scale-space theory, we consider three main ways discretizing these scale-space operations in terms of explicit discrete convolutions, based on either (i) sampling the Gaussian kernels and the Gaussian derivative kernels, (ii) locally integrating the Gaussian kernels and the Gaussian derivative kernels over each pixel support region and (iii) basing the scale-space analysis on the discrete analogue of the Gaussian kernel, and then computing derivative approximations by applying small-support central difference operators to the spatially smoothed image data. We study the properties of these three main discretization methods both theoretically and experimentally, and characterize their performance by quantitative measures, including the results they give rise to with respect to the task of scale selection, investigated for four different use cases, and with emphasis on the behaviour at fine scales. The results show that the sampled Gaussian kernels and derivatives as well as the integrated Gaussian kernels and derivatives perform very poorly at very fine scales. At very fine scales, the discrete analogue of the Gaussian kernel with its corresponding discrete derivative approximations performs substantially better. The sampled Gaussian kernel and the sampled Gaussian derivatives do, on the other hand, lead to numerically very good approximations of the corresponding continuous results, when the scale parameter is sufficiently large, in the experiments presented in the paper, when the scale parameter is greater than a value of about 1, in units of the grid spacing.
We study stochastic approximation procedures for approximately solving a $d$-dimensional linear fixed point equation based on observing a trajectory of length $n$ from an ergodic Markov chain. We first exhibit a non-asymptotic bound of the order $t_{\mathrm{mix}} \tfrac{d}{n}$ on the squared error of the last iterate of a standard scheme, where $t_{\mathrm{mix}}$ is a mixing time. We then prove a non-asymptotic instance-dependent bound on a suitably averaged sequence of iterates, with a leading term that matches the local asymptotic minimax limit, including sharp dependence on the parameters $(d, t_{\mathrm{mix}})$ in the higher order terms. We complement these upper bounds with a non-asymptotic minimax lower bound that establishes the instance-optimality of the averaged SA estimator. We derive corollaries of these results for policy evaluation with Markov noise -- covering the TD($\lambda$) family of algorithms for all $\lambda \in [0, 1)$ -- and linear autoregressive models. Our instance-dependent characterizations open the door to the design of fine-grained model selection procedures for hyperparameter tuning (e.g., choosing the value of $\lambda$ when running the TD($\lambda$) algorithm).
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