In this manuscript we derive the optimal out-of-sample causal predictor for a linear system that has been observed in $k+1$ within-sample environments. In this model we consider $k$ shifted environments and one observational environment. Each environment corresponds to a linear structural equation model (SEM) with its own shift and noise vector, both in $L^2$. The strength of the shifts can be put in a certain order, and we may therefore speak of all shifts that are less or equally strong than a given shift. We consider the space of all shifts are $\gamma$ times less or equally strong than any weighted average of the observed shift vectors with weights on the unit sphere. For each $\beta\in\mathbb{R}^p$ we show that the supremum of the risk functions $R_{\tilde{A}}(\beta)$ over $\tilde{A}\in C^\gamma$ has a worst-risk decomposition into a (positive) linear combination of risk functions, depending on $\gamma$. We then define the causal regularizer, $\beta_\gamma$, as the argument $\beta$ that minimizes this risk. The main result of the paper is that this regularizer can be consistently estimated with a plug-in estimator outside a set of zero Lebesgue measure in the parameter space. A practical obstacle for such estimation is that it involves the solution of a general degree polynomial which cannot be done explicitly. Therefore we also prove that an approximate plug-in estimator using the bisection method is also consistent. An interesting by-product of the proof of the main result is that the plug-in estimation of the argmin of the maxima of a finite set of quadratic risk functions is consistent outside a set of zero Lebesgue measure in the parameter space.
相關內容
Ordinary state-based peridynamic (OSB-PD) models have an unparalleled capability to simulate crack propagation phenomena in solids with arbitrary Poisson's ratio. However, their non-locality also leads to prohibitively high computational cost. In this paper, a fast solution scheme for OSB-PD models based on matrix operation is introduced, with which, the graphics processing units (GPUs) are used to accelerate the computation. For the purpose of comparison and verification, a commonly used solution scheme based on loop operation is also presented. An in-house software is developed in MATLAB. Firstly, the vibration of a cantilever beam is solved for validating the loop- and matrix-based schemes by comparing the numerical solutions to those produced by a FEM software. Subsequently, two typical dynamic crack propagation problems are simulated to illustrate the effectiveness of the proposed schemes in solving dynamic fracture problems. Finally, the simulation of the Brokenshire torsion experiment is carried out by using the matrix-based scheme, and the similarity in the shapes of the experimental and numerical broken specimens further demonstrates the ability of the proposed approach to deal with 3D non-planar fracture problems. In addition, the speed-up of the matrix-based scheme with respect to the loop-based scheme and the performance of the GPU acceleration are investigated. The results emphasize the high computational efficiency of the matrix-based implementation scheme.
We study how to verify specific frequency distributions when we observe a stream of $N$ data items taken from a universe of $n$ distinct items. We introduce the \emph{relative Fr\'echet distance} to compare two frequency functions in a homogeneous manner. We consider two streaming models: insertions only and sliding windows. We present a Tester for a certain class of functions, which decides if $f $ is close to $g$ or if $f$ is far from $g$ with high probability, when $f$ is given and $g$ is defined by a stream. If $f$ is uniform we show a space $\Omega(n)$ lower bound. If $f$ decreases fast enough, we then only use space $O(\log^2 n\cdot \log\log n)$. The analysis relies on the Spacesaving algorithm \cite{MAE2005,Z22} and on sampling the stream.
Parametric mathematical models such as parameterizations of partial differential equations with random coefficients have received a lot of attention within the field of uncertainty quantification. The model uncertainties are often represented via a series expansion in terms of the parametric variables. In practice, this series expansion needs to be truncated to a finite number of terms, introducing a dimension truncation error to the numerical simulation of a parametric mathematical model. There have been several studies of the dimension truncation error corresponding to different models of the input random field in recent years, but many of these analyses have been carried out within the context of numerical integration. In this paper, we study the $L^2$ dimension truncation error of the parametric model problem. Estimates of this kind arise in the assessment of the dimension truncation error for function approximation in high dimensions. In addition, we show that the dimension truncation error rate is invariant with respect to certain transformations of the parametric variables. Numerical results are presented which showcase the sharpness of the theoretical results.
A family of stabilizer-free $P_k$ virtual elements are constructed on triangular meshes. When choosing an accurate and proper interpolation, the stabilizer of the virtual elements can be dropped while the quasi-optimality is kept. The interpolating space here is the space of continuous $P_k$ polynomials on the Hsieh-Clough-Tocher macro-triangle, where the macro-triangle is defined by connecting three vertices of a triangle with its barycenter. We show that such an interpolation preserves $P_k$ polynomials locally and enforces the coerciveness of the resulting bilinear form. Consequently the stabilizer-free virtual element solutions converge at the optimal order. Numerical tests are provided to confirm the theory and to be compared with existing virtual elements.
We show that the problem of counting the number of $n$-variable unate functions reduces to the problem of counting the number of $n$-variable monotone functions. Using recently obtained results on $n$-variable monotone functions, we obtain counts of $n$-variable unate functions up to $n=9$. We use an enumeration strategy to obtain the number of $n$-variable balanced monotone functions up to $n=7$. We show that the problem of counting the number of $n$-variable balanced unate functions reduces to the problem of counting the number of $n$-variable balanced monotone functions, and consequently, we obtain the number of $n$-variable balanced unate functions up to $n=7$. Using enumeration, we obtain the numbers of equivalence classes of $n$-variable balanced monotone functions, unate functions and balanced unate functions up to $n=6$. Further, for each of the considered sub-class of $n$-variable monotone and unate functions, we also obtain the corresponding numbers of $n$-variable non-degenerate functions.
In highly diffusion regimes when the mean free path $\varepsilon$ tends to zero, the radiative transfer equation has an asymptotic behavior which is governed by a diffusion equation and the corresponding boundary condition. Generally, a numerical scheme for solving this problem has the truncation error containing an $\varepsilon^{-1}$ contribution, that leads to a nonuniform convergence for small $\varepsilon$. Such phenomenons require high resolutions of discretizations, which degrades the performance of the numerical scheme in the diffusion limit. In this paper, we first provide a--priori estimates for the scaled spherical harmonic ($P_N$) radiative transfer equation. Then we present an error analysis for the spherical harmonic discontinuous Galerkin (DG) method of the scaled radiative transfer equation showing that, under some mild assumptions, its solutions converge uniformly in $\varepsilon$ to the solution of the scaled radiative transfer equation. We further present an optimal convergence result for the DG method with the upwind flux on Cartesian grids. Error estimates of $\left(1+\mathcal{O}(\varepsilon)\right)h^{k+1}$ (where $h$ is the maximum element length) are obtained when tensor product polynomials of degree at most $k$ are used.
We consider the problem of sequential change detection, where the goal is to design a scheme for detecting any changes in a parameter or functional $\theta$ of the data stream distribution that has small detection delay, but guarantees control on the frequency of false alarms in the absence of changes. In this paper, we describe a simple reduction from sequential change detection to sequential estimation using confidence sequences: we begin a new $(1-\alpha)$-confidence sequence at each time step, and proclaim a change when the intersection of all active confidence sequences becomes empty. We prove that the average run length is at least $1/\alpha$, resulting in a change detection scheme with minimal structural assumptions~(thus allowing for possibly dependent observations, and nonparametric distribution classes), but strong guarantees. Our approach bears an interesting parallel with the reduction from change detection to sequential testing of Lorden (1971) and the e-detector of Shin et al. (2022).
A cross-bifix-free code of length $n$ over $\mathbb{Z}_q$ is defined as a non-empty subset of $\mathbb{Z}_q^n$ satisfying that the prefix set of each codeword is disjoint from the suffix set of every codeword. Cross-bifix-free codes have found important applications in digital communication systems. One of the main research problems on cross-bifix-free codes is to construct cross-bifix-free codes as large as possible in size. Recently, Wang and Wang introduced a family of cross-bifix-free codes $S_{I,J}^{(k)}(n)$, which is a generalization of the classical cross-bifix-free codes studied early by Lvenshtein, Gilbert and Chee {\it et al.}. It is known that $S_{I,J}^{(k)}(n)$ is nearly optimal in size and $S_{I,J}^{(k)}(n)$ is non-expandable if $k=n-1$ or $1\leq k<n/2$. In this paper, we first show that $S_{I,J}^{(k)}(n)$ is non-expandable if and only if $k=n-1$ or $1\leq k<n/2$, thereby improving the results in [Chee {\it et al.}, IEEE-TIT, 2013] and [Wang and Wang, IEEE-TIT, 2022]. We then construct a new family of cross-bifix-free codes $U^{(t)}_{I,J}(n)$ to expand $S_{I,J}^{(k)}(n)$ such that the resulting larger code $S_{I,J}^{(k)}(n)\bigcup U^{(t)}_{I,J}(n)$ is a non-expandable cross-bifix-free code whenever $S_{I,J}^{(k)}(n)$ is expandable. Finally, we present an explicit formula for the size of $S_{I,J}^{(k)}(n)\bigcup U^{(t)}_{I,J}(n)$.
Given an observational study with $n$ independent but heterogeneous units, our goal is to learn the counterfactual distribution for each unit using only one $p$-dimensional sample per unit containing covariates, interventions, and outcomes. Specifically, we allow for unobserved confounding that introduces statistical biases between interventions and outcomes as well as exacerbates the heterogeneity across units. Modeling the conditional distribution of the outcomes as an exponential family, we reduce learning the unit-level counterfactual distributions to learning $n$ exponential family distributions with heterogeneous parameters and only one sample per distribution. We introduce a convex objective that pools all $n$ samples to jointly learn all $n$ parameter vectors, and provide a unit-wise mean squared error bound that scales linearly with the metric entropy of the parameter space. For example, when the parameters are $s$-sparse linear combination of $k$ known vectors, the error is $O(s\log k/p)$. En route, we derive sufficient conditions for compactly supported distributions to satisfy the logarithmic Sobolev inequality. As an application of the framework, our results enable consistent imputation of sparsely missing covariates.
When modelling discontinuities (interfaces) using the finite element method, the standard approach is to use a conforming finite-element mesh in which the mesh matches the interfaces. However, this approach can prove cumbersome if the geometry is complex, in particular in 3D. In this work, we develop an efficient technique for a non-conforming finite-element treatment of weak discontinuities by using laminated microstructures. The approach is inspired by the so-called composite voxel technique that has been developed for FFT-based spectral solvers in computational homogenization. The idea behind the method is rather simple. Each finite element that is cut by an interface is treated as a simple laminate with the volume fraction of the phases and the lamination orientation determined in terms of the actual geometrical arrangement of the interface within the element. The approach is illustrated by several computational examples relevant to the micromechanics of heterogeneous materials. Elastic and elastic-plastic materials at small and finite strain are considered in the examples. The performance of the proposed method is compared to two alternative, simple methods showing that the new approach is in most cases superior to them while maintaining the simplicity.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191