Large datasets are often affected by cell-wise outliers in the form of missing or erroneous data. However, discarding any samples containing outliers may result in a dataset that is too small to accurately estimate the covariance matrix. Moreover, the robust procedures designed to address this problem require the invertibility of the covariance operator and thus are not effective on high-dimensional data. In this paper, we propose an unbiased estimator for the covariance in the presence of missing values that does not require any imputation step and still achieves near minimax statistical accuracy with the operator norm. We also advocate for its use in combination with cell-wise outlier detection methods to tackle cell-wise contamination in a high-dimensional and low-rank setting, where state-of-the-art methods may suffer from numerical instability and long computation times. To complement our theoretical findings, we conducted an experimental study which demonstrates the superiority of our approach over the state of the art both in low and high dimension settings.
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Sensory perception originates from the responses of sensory neurons, which react to a collection of sensory signals linked to various physical attributes of a singular perceptual object. Unraveling how the brain extracts perceptual information from these neuronal responses is a pivotal challenge in both computational neuroscience and machine learning. Here we introduce a statistical mechanical theory, where perceptual information is first encoded in the correlated variability of sensory neurons and then reformatted into the firing rates of downstream neurons. Applying this theory, we illustrate the encoding of motion direction using neural covariance and demonstrate high-fidelity direction recovery by spiking neural networks. Networks trained under this theory also show enhanced performance in classifying natural images, achieving higher accuracy and faster inference speed. Our results challenge the traditional view of neural covariance as a secondary factor in neural coding, highlighting its potential influence on brain function.
For large Reynolds number flows, it is typically necessary to perform simulations that are under-resolved with respect to the underlying flow physics. For nodal discontinuous spectral element approximations of these under-resolved flows, the collocation projection of the nonlinear flux can introduce aliasing errors which can result in numerical instabilities. In Dzanic and Witherden (J. Comput. Phys., 468, 2022), an entropy-based adaptive filtering approach was introduced as a robust, parameter-free shock-capturing method for discontinuous spectral element methods. This work explores the ability of entropy filtering for mitigating aliasing-driven instabilities in the simulation of under-resolved turbulent flows through high-order implicit large eddy simulations of a NACA0021 airfoil in deep stall at a Reynolds number of 270,000. It was observed that entropy filtering can adequately mitigate aliasing-driven instabilities without degrading the accuracy of the underlying high-order scheme on par with standard anti-aliasing methods such as over-integration, albeit with marginally worse performance at higher approximation orders.
Numerous statistical methods have been developed to explore genomic imprinting and maternal effects, which are causes of parent-of-origin patterns in complex human diseases. Most of the methods, however, either only model one of these two confounded epigenetic effects, or make strong yet unrealistic assumptions about the population to avoid over-parameterization. A recent partial likelihood method (LIMEDSP ) can identify both epigenetic effects based on discordant sibpair family data without those assumptions. Theoretical and empirical studies have shown its validity and robustness. As LIMEDSP method obtains parameter estimation by maximizing partial likelihood, it is interesting to compare its efficiency with full likelihood maximizer. To overcome the difficulty in over-parameterization when using full likelihood, this study proposes a discordant sib-pair design based Monte Carlo Expectation Maximization (MCEMDSP ) method to detect imprinting and maternal effects jointly. Those unknown mating type probabilities, the nuisance parameters, are considered as latent variables in EM algorithm. Monte Carlo samples are used to numerically approximate the expectation function that cannot be solved algebraically. Our simulation results show that though this MCEMDSP algorithm takes longer computation time, it can generally detect both epigenetic effects with higher power, which demonstrates that it can be a good complement of LIMEDSP method
To obtain strong convergence rates of numerical schemes, an overwhelming majority of existing works impose a global monotonicity condition on coefficients of SDEs. On the contrary, a majority of SDEs from applications do not have globally monotone coefficients. As a recent breakthrough, the authors of [Hutzenthaler, Jentzen, Ann. Probab., 2020] originally presented a perturbation theory for stochastic differential equations (SDEs), which is crucial to recovering strong convergence rates of numerical schemes in a non-globally monotone setting. However, only a convergence rate of order $1/2$ was obtained there for time-stepping schemes such as a stopped increment-tamed Euler-Maruyama (SITEM) method. As an open problem, a natural question was raised by the aforementioned work as to whether higher convergence rate than $1/2$ can be obtained when higher order schemes are used. The present work attempts to solve the tough problem. To this end, we develop some new perturbation estimates that are able to reveal the order-one strong convergence of numerical methods. As the first application of the newly developed estimates, we identify the expected order-one pathwise uniformly strong convergence of the SITEM method for additive noise driven SDEs and multiplicative noise driven second order SDEs with non-globally monotone coefficients. As the other application, we propose and analyze a positivity preserving explicit Milstein-type method for Lotka-Volterra competition model driven by multi-dimensional noise, with a pathwise uniformly strong convergence rate of order one recovered under mild assumptions. These obtained results are completely new and significantly improve the existing theory. Numerical experiments are also provided to confirm the theoretical findings.
Computational modeling of charged species transport has enabled the analysis, design, and optimization of a diverse array of electrochemical and electrokinetic devices. These systems are represented by the Poisson-Nernst-Planck (PNP) equations coupled with the Navier-Stokes (NS) equation. Direct numerical simulation (DNS) to accurately capture the spatio-temporal variation of ion concentration and current flux remains challenging due to the (a) small critical dimension of the diffuse charge layer (DCL), (b) stiff coupling due to fast charge relaxation times, large advective effects, and steep gradients close to boundaries, and (c) complex geometries exhibited by electrochemical devices. In the current study, we address these challenges by presenting a direct numerical simulation framework that incorporates (a) a variational multiscale (VMS) treatment, (b) a block-iterative strategy in conjunction with semi-implicit (for NS) and implicit (for PNP) time integrators, and (c) octree based adaptive mesh refinement. The VMS formulation provides numerical stabilization critical for capturing the electro-convective flows often observed in engineered devices. The block-iterative strategy decouples the difficulty of non-linear coupling between the NS and PNP equations and allows the use of tailored numerical schemes separately for NS and PNP equations. The carefully designed second-order, hybrid implicit methods circumvent the harsh timestep requirements of explicit time steppers, thus enabling simulations over longer time horizons. Finally, the octree-based meshing allows efficient and targeted spatial resolution of the DCL. These features are incorporated into a massively parallel computational framework, enabling the simulation of realistic engineering electrochemical devices. The numerical framework is illustrated using several challenging canonical examples.
A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the famous Grothendieck constant. Here we show that such a result does not generalize to quartic polynomials and 2-query algorithms, even when we allow for additive approximations. We also show that the additive approximation implied by their result is tight for bounded bilinear forms, which gives a new characterization of the Grothendieck constant in terms of 1-query quantum algorithms. Along the way we provide reformulations of the completely bounded norm of a form, and its dual norm.
We propose and analyze a unified structure-preserving parametric finite element method (SP-PFEM) for the anisotropic surface diffusion of curves in two dimensions $(d=2)$ and surfaces in three dimensions $(d=3)$ with an arbitrary anisotropic surface energy density $\gamma(\boldsymbol{n})$, where $\boldsymbol{n}\in \mathbb{S}^{d-1}$ represents the outward unit vector. By introducing a novel unified surface energy matrix $\boldsymbol{G}_k(\boldsymbol{n})$ depending on $\gamma(\boldsymbol{n})$, the Cahn--Hoffman $\boldsymbol{\xi}$-vector and a stabilizing function $k(\boldsymbol{n}):\ \mathbb{S}^{d-1}\to {\mathbb R}$, we obtain a unified and conservative variational formulation for the anisotropic surface diffusion via different surface differential operators including the surface gradient operator, the surface divergence operator and the surface Laplace--Beltrami operator. A SP-PFEM discretization is presented for the variational problem. In order to establish the unconditional energy stability of the proposed SP-PFEM under a very mild condition on $\gamma(\boldsymbol{n})$, we propose a new framework via {\sl local energy estimate} for proving energy stability/structure-preserving properties of the parametric finite element method for the anisotropic surface diffusion. This framework sheds light on how to prove unconditional energy stability of other numerical methods for geometric partial differential equations. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as structure-preserving properties of the proposed SP-PFEM for the anisotropic surface diffusion with arbitrary anisotropic surface energy density $\gamma(\boldsymbol{n})$ arising from different applications.
In fitting a continuous bounded data, the generalized beta (and several variants of this distribution) and the two-parameter Kumaraswamy (KW) distributions are the two most prominent univariate continuous distributions that come to our mind. There are some common features between these two rival probability models and to select one of them in a practical situation can be of great interest. Consequently, in this paper, we discuss various methods of selection between the generalized beta proposed by Libby and Novick (1982) (LNGB) and the KW distributions, such as the criteria based on probability of correct selection which is an improvement over the likelihood ratio statistic approach, and also based on pseudo-distance measures. We obtain an approximation for the probability of correct selection under the hypotheses HLNGB and HKW , and select the model that maximizes it. However, our proposal is more appealing in the sense that we provide the comparison study for the LNGB distribution that subsumes both types of classical beta and exponentiated generators (see, for details, Cordeiro et al. 2014; Libby and Novick 1982) which can be a natural competitor of a two-parameter KW distribution in an appropriate scenario.
We propose a novel test procedure for comparing mean functions across two groups within the reproducing kernel Hilbert space (RKHS) framework. Our proposed method is adept at handling sparsely and irregularly sampled functional data when observation times are random for each subject. Conventional approaches, which are built upon functional principal components analysis, usually assume a homogeneous covariance structure across groups. Nonetheless, justifying this assumption in real-world scenarios can be challenging. To eliminate the need for a homogeneous covariance structure, we first develop the functional Bahadur representation for the mean estimator under the RKHS framework; this representation naturally leads to the desirable pointwise limiting distributions. Moreover, we establish weak convergence for the mean estimator, allowing us to construct a test statistic for the mean difference. Our method is easily implementable and outperforms some conventional tests in controlling type I errors across various settings. We demonstrate the finite sample performance of our approach through extensive simulations and two real-world applications.
Detecting differences in gene expression is an important part of single-cell RNA sequencing experiments, and many statistical methods have been developed for this aim. Most differential expression analyses focus on comparing expression between two groups (e.g., treatment vs. control). But there is increasing interest in multi-condition differential expression analyses in which expression is measured in many conditions, and the aim is to accurately detect and estimate expression differences in all conditions. We show that directly modeling single-cell RNA-seq counts in all conditions simultaneously, while also inferring how expression differences are shared across conditions, leads to greatly improved performance for detecting and estimating expression differences compared to existing methods. We illustrate the potential of this new approach by analyzing data from a single-cell experiment studying the effects of cytokine stimulation on gene expression. We call our new method "Poisson multivariate adaptive shrinkage", and it is implemented in an R package available online at //github.com/stephenslab/poisson.mash.alpha.
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