We study vantage-point trees constructed using an independent sample from the uniform distribution on a fixed convex body $K$ in $(\mathbb{R}^d,\|\cdot\|)$, where $\|\cdot\|$ is an arbitrary homogeneous norm on $\mathbb{R}^d$. We prove that a sequence of sets, associated with the left boundary of a vantage-point tree, forms a recurrent Harris chain on the space of convex bodies in $(\mathbb{R}^d,\|\cdot\|)$. The limiting object is a ball polyhedron, that is, an a.s.~finite intersection of closed balls in $(\mathbb{R}^d,\|\cdot\|)$ of possibly different radii. As a consequence, we derive a limit theorem for the length of the leftmost path of a vantage-point tree.
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Neural operators (NO) are discretization invariant deep learning methods with functional output and can approximate any continuous operator. NO have demonstrated the superiority of solving partial differential equations (PDEs) over other deep learning methods. However, the spatial domain of its input function needs to be identical to its output, which limits its applicability. For instance, the widely used Fourier neural operator (FNO) fails to approximate the operator that maps the boundary condition to the PDE solution. To address this issue, we propose a novel framework called resolution-invariant deep operator (RDO) that decouples the spatial domain of the input and output. RDO is motivated by the Deep operator network (DeepONet) and it does not require retraining the network when the input/output is changed compared with DeepONet. RDO takes functional input and its output is also functional so that it keeps the resolution invariant property of NO. It can also resolve PDEs with complex geometries whereas NO fail. Various numerical experiments demonstrate the advantage of our method over DeepONet and FNO.
We present a monolithic finite element formulation for (nonlinear) fluid-structure interaction in Eulerian coordinates. For the discretization we employ an unfitted finite element method based on inf-sup stable finite elements. So-called ghost penalty terms are used to guarantee the robustness of the approach independently of the way the interface cuts the finite element mesh. The resulting system is solved in a monolithic fashion using Newton's method. Our developments are tested on a numerical example with fixed interface.
We prove that, for each fixed genus g, the portion of semigroups belonging to infinite chains in the semigroup tree approaches 0 as the genus grows to infinite. This problem has been open since 2009.
Using validated numerical methods, interval arithmetic and Taylor models, we propose a certified predictor-corrector loop for tracking zeros of polynomial systems with a parameter. We provide a Rust implementation which shows tremendous improvement over existing software for certified path tracking.
In this paper we study the Cayley graph $\mathrm{Cay}(S_n,T)$ of the symmetric group $S_n$ generated by a set of transpositions $T$. We show that for $n\geq 5$ the Cayley graph is normal. As a corollary, we show that its automorphism group is a direct product of $S_n$ and the automorphism group of the transposition graph associated to $T$. This provides an affirmative answer to a conjecture raised by Ganesan in arXiv:1703.08109, showing that $\mathrm{Cay}(S_n,T)$ is normal if and only if the transposition graph is not $C_4$ or $K_n$.
Among semiparametric regression models, partially linear additive models provide a useful tool to include additive nonparametric components as well as a parametric component, when explaining the relationship between the response and a set of explanatory variables. This paper concerns such models under sparsity assumptions for the covariates included in the linear component. Sparse covariates are frequent in regression problems where the task of variable selection is usually of interest. As in other settings, outliers either in the residuals or in the covariates involved in the linear component have a harmful effect. To simultaneously achieve model selection for the parametric component of the model and resistance to outliers, we combine preliminary robust estimators of the additive component, robust linear $MM-$regression estimators with a penalty such as SCAD on the coefficients in the parametric part. Under mild assumptions, consistency results and rates of convergence for the proposed estimators are derived. A Monte Carlo study is carried out to compare, under different models and contamination schemes, the performance of the robust proposal with its classical counterpart. The obtained results show the advantage of using the robust approach. Through the analysis of a real data set, we also illustrate the benefits of the proposed procedure.
We consider the problem of approximating a function from $L^2$ by an element of a given $m$-dimensional space $V_m$, associated with some feature map $\varphi$, using evaluations of the function at random points $x_1,\dots,x_n$. After recalling some results on optimal weighted least-squares using independent and identically distributed points, we consider weighted least-squares using projection determinantal point processes (DPP) or volume sampling. These distributions introduce dependence between the points that promotes diversity in the selected features $\varphi(x_i)$. We first provide a generalized version of volume-rescaled sampling yielding quasi-optimality results in expectation with a number of samples $n = O(m\log(m))$, that means that the expected $L^2$ error is bounded by a constant times the best approximation error in $L^2$. Also, further assuming that the function is in some normed vector space $H$ continuously embedded in $L^2$, we further prove that the approximation is almost surely bounded by the best approximation error measured in the $H$-norm. This includes the cases of functions from $L^\infty$ or reproducing kernel Hilbert spaces. Finally, we present an alternative strategy consisting in using independent repetitions of projection DPP (or volume sampling), yielding similar error bounds as with i.i.d. or volume sampling, but in practice with a much lower number of samples. Numerical experiments illustrate the performance of the different strategies.
Regression with random data objects is becoming increasingly common in modern data analysis. Unfortunately, like the traditional regression setting with Euclidean data, random response regression is not immune to the trouble caused by unusual observations. A metric Cook's distance extending the classical Cook's distances of Cook (1977) to general metric-valued response objects is proposed. The performance of the metric Cook's distance in both Euclidean and non-Euclidean response regression with Euclidean predictors is demonstrated in an extensive experimental study. A real data analysis of county-level COVID-19 transmission in the United States also illustrates the usefulness of this method in practice.
We demonstrate that large language models can produce reasonable numerical ratings of the logical consistency of claims. We also outline a mathematical approach based on sheaf theory for lifting such ratings to hypertexts such as laws, jurisprudence, and social media and evaluating their consistency globally. This approach is a promising avenue to increasing consistency in and of government, as well as to combating mis- and disinformation and related ills.
We consider a general multivariate model where univariate marginal distributions are known up to a parameter vector and we are interested in estimating that parameter vector without specifying the joint distribution, except for the marginals. If we assume independence between the marginals and maximize the resulting quasi-likelihood, we obtain a consistent but inefficient QMLE estimator. If we assume a parametric copula (other than independence) we obtain a full MLE, which is efficient but only under a correct copula specification and may be biased if the copula is misspecified. Instead we propose a sieve MLE estimator (SMLE) which improves over QMLE but does not have the drawbacks of full MLE. We model the unknown part of the joint distribution using the Bernstein-Kantorovich polynomial copula and assess the resulting improvement over QMLE and over misspecified FMLE in terms of relative efficiency and robustness. We derive the asymptotic distribution of the new estimator and show that it reaches the relevant semiparametric efficiency bound. Simulations suggest that the sieve MLE can be almost as efficient as FMLE relative to QMLE provided there is enough dependence between the marginals. We demonstrate practical value of the new estimator with several applications. First, we apply SMLE in an insurance context where we build a flexible semi-parametric claim loss model for a scenario where one of the variables is censored. As in simulations, the use of SMLE leads to tighter parameter estimates. Next, we consider financial risk management examples and show how the use of SMLE leads to superior Value-at-Risk predictions. The paper comes with an online archive which contains all codes and datasets.
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