We study the statistical capacity of the classical binary perceptrons with general thresholds $\kappa$. After recognizing the connection between the capacity and the bilinearly indexed (bli) random processes, we utilize a recent progress in studying such processes to characterize the capacity. In particular, we rely on \emph{fully lifted} random duality theory (fl RDT) established in \cite{Stojnicflrdt23} to create a general framework for studying the perceptrons' capacities. Successful underlying numerical evaluations are required for the framework (and ultimately the entire fl RDT machinery) to become fully practically operational. We present results obtained in that directions and uncover that the capacity characterizations are achieved on the second (first non-trivial) level of \emph{stationarized} full lifting. The obtained results \emph{exactly} match the replica symmetry breaking predictions obtained through statistical physics replica methods in \cite{KraMez89}. Most notably, for the famous zero-threshold scenario, $\kappa=0$, we uncover the well known $\alpha\approx0.8330786$ scaled capacity.
相關內容
We propose a new stabilised finite element method for the classical Kolmogorov equation. The latter serves as a basic model problem for large classes of kinetic-type equations and, crucially, is characterised by degenerate diffusion. The stabilisation is constructed so that the resulting method admits a \emph{numerical hypocoercivity} property, analogous to the corresponding property of the PDE problem. More specifically, the stabilisation is constructed so that spectral gap is possible in the resulting ``stronger-than-energy'' stabilisation norm, despite the degenerate nature of the diffusion in Kolmogorov, thereby the method has a provably robust behaviour as the ``time'' variable goes to infinity. We consider both a spatially discrete version of the stabilised finite element method and a fully discrete version, with the time discretisation realised by discontinuous Galerkin timestepping. Both stability and a priori error bounds are proven in all cases. Numerical experiments verify the theoretical findings.
Extensions of earlier algorithms and enhanced visualization techniques for approximating a correlation matrix are presented. The visualization problems that result from using column or colum--and--row adjusted correlation matrices, which give numerically a better fit, are addressed. For visualization of a correlation matrix a weighted alternating least squares algorithm is used, with either a single scalar adjustment, or a column-only adjustment with symmetric factorization; these choices form a compromise between the numerical accuracy of the approximation and the comprehensibility of the obtained correlation biplots. Some illustrative examples are discussed.
This work proposes a novel variational approximation of partial differential equations on moving geometries determined by explicit boundary representations. The benefits of the proposed formulation are the ability to handle large displacements of explicitly represented domain boundaries without generating body-fitted meshes and remeshing techniques. For the space discretization, we use a background mesh and an unfitted method that relies on integration on cut cells only. We perform this intersection by using clipping algorithms. To deal with the mesh movement, we pullback the equations to a reference configuration (the spatial mesh at the initial time slab times the time interval) that is constant in time. This way, the geometrical intersection algorithm is only required in 3D, another key property of the proposed scheme. At the end of the time slab, we compute the deformed mesh, intersect the deformed boundary with the background mesh, and consider an exact transfer operator between meshes to compute jump terms in the time discontinuous Galerkin integration. The transfer is also computed using geometrical intersection algorithms. We demonstrate the applicability of the method to fluid problems around rotating (2D and 3D) geometries described by oriented boundary meshes. We also provide a set of numerical experiments that show the optimal convergence of the method.
Recently, addressing spatial confounding has become a major topic in spatial statistics. However, the literature has provided conflicting definitions, and many proposed definitions do not address the issue of confounding as it is understood in causal inference. We define spatial confounding as the existence of an unmeasured causal confounder with a spatial structure. We present a causal inference framework for nonparametric identification of the causal effect of a continuous exposure on an outcome in the presence of spatial confounding. We propose double machine learning (DML), a procedure in which flexible models are used to regress both the exposure and outcome variables on confounders to arrive at a causal estimator with favorable robustness properties and convergence rates, and we prove that this approach is consistent and asymptotically normal under spatial dependence. As far as we are aware, this is the first approach to spatial confounding that does not rely on restrictive parametric assumptions (such as linearity, effect homogeneity, or Gaussianity) for both identification and estimation. We demonstrate the advantages of the DML approach analytically and in simulations. We apply our methods and reasoning to a study of the effect of fine particulate matter exposure during pregnancy on birthweight in California.
The sample correlation coefficient $R$ plays an important role in many statistical analyses. We study the moments of $R$ under the bivariate Gaussian model assumption, provide a novel approximation for its finite sample mean and connect it with known results for the variance. We exploit these approximations to present non-asymptotic concentration inequalities for $R$. Finally, we illustrate our results in a simulation experiment that further validates the approximations presented in this work.
We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of non-local in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and error analysis of the semi-discrete system. The error analysis includes the consideration of the singularity at $t = 0$ which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.
We study the complexity (that is, the weight of the multiplication table) of the elliptic normal bases introduced by Couveignes and Lercier. We give an upper bound on the complexity of these elliptic normal bases, and we analyze the weight of some special vectors related to the multiplication table of those bases. This analysis leads us to some perspectives on the search for low complexity normal bases from elliptic periods.
The mean residual life function is a key functional for a survival distribution. It has practically useful interpretation as the expected remaining lifetime given survival up to a particular time point, and it also characterizes the survival distribution. However, it has received limited attention in terms of inference methods under a probabilistic modeling framework. In this paper, we seek to provide general inference methodology for mean residual life regression. Survival data often include a set of predictor variables for the survival response distribution, and in many cases it is natural to include the covariates as random variables into the modeling. We thus propose a Dirichlet process mixture modeling approach for the joint stochastic mechanism of the covariates and survival responses. This approach implies a flexible model structure for the mean residual life of the conditional response distribution, allowing general shapes for mean residual life as a function of covariates given a specific time point, as well as a function of time given particular values of the covariate vector. To expand the scope of the modeling framework, we extend the mixture model to incorporate dependence across experimental groups, such as treatment and control groups. This extension is built from a dependent Dirichlet process prior for the group-specific mixing distributions, with common locations and weights that vary across groups through latent bivariate beta distributed random variables. We develop properties of the proposed regression models, and discuss methods for prior specification and posterior inference. The different components of the methodology are illustrated with simulated data sets. Moreover, the modeling approach is applied to a data set comprising right censored survival times of patients with small cell lung cancer.
We give a lower bound of $\Omega(\sqrt n)$ on the unambiguous randomised parity-query complexity of the approximate majority problem -- that is, on the lowest randomised parity-query complexity of any function over $\{0,1\}^n$ whose value is "0" if the Hamming weight of the input is at most n/3, is "1" if the weight is at least 2n/3, and may be arbitrary otherwise.
We introduce a single-set axiomatisation of cubical $\omega$-categories, including connections and inverses. We justify these axioms by establishing a series of equivalences between the category of single-set cubical $\omega$-categories, and their variants with connections and inverses, and the corresponding cubical $\omega$-categories. We also report on the formalisation of cubical $\omega$-categories with the Isabelle/HOL proof assistant, which has been instrumental in finding the single-set axioms.
小貼士
登錄享
相關主題
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191