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Variance reduction is a crucial idea for Monte Carlo simulation and the stochastic Lanczos quadrature method is a dedicated method to approximate the trace of a matrix function. Inspired by their advantages, we combine these two techniques to approximate the log-determinant of large-scale symmetric positive definite matrices. Key questions to be answered for such a method are how to construct or choose an appropriate projection subspace and derive guaranteed theoretical analysis. This paper applies some probabilistic approaches including the projection-cost-preserving sketch and matrix concentration inequalities to construct a suboptimal subspace. Furthermore, we provide some insights on choosing design parameters in the underlying algorithm by deriving corresponding approximation error and probabilistic error estimations. Numerical experiments demonstrate our method's effectiveness and illustrate the quality of the derived error bounds.

Inverse reinforcement learning (IRL) algorithms often rely on (forward) reinforcement learning or planning over a given time horizon to compute an approximately optimal policy for a hypothesized reward function and then match this policy with expert demonstrations. The time horizon plays a critical role in determining both the accuracy of reward estimate and the computational efficiency of IRL algorithms. Interestingly, an effective time horizon shorter than the ground-truth value often produces better results faster. This work formally analyzes this phenomenon and provides an explanation: the time horizon controls the complexity of an induced policy class and mitigates overfitting with limited data. This analysis leads to a principled choice of the effective horizon for IRL. It also prompts us to reexamine the classic IRL formulation: it is more natural to learn jointly the reward and the effective horizon together rather than the reward alone with a given horizon. Our experimental results confirm the theoretical analysis.

We develop easily accessible quantities for bounding the almost sure exponential convergence rate of push-sum algorithms. We analyze the scenario of i.i.d. synchronous gossip, every agent communicating towards its single target at every step. Multiple bounding expressions are developed depending on the generality of the setup, all functions of the spectrum of the network. While the most general bound awaits further improvement, with more symmetries, close bounds can be established, as demonstrated by numerical simulations.

Straggler nodes are well-known bottlenecks of distributed matrix computations which induce reductions in computation/communication speeds. A common strategy for mitigating such stragglers is to incorporate Reed-Solomon based MDS (maximum distance separable) codes into the framework; this can achieve resilience against an optimal number of stragglers. However, these codes assign dense linear combinations of submatrices to the worker nodes. When the input matrices are sparse, these approaches increase the number of non-zero entries in the encoded matrices, which in turn adversely affects the worker computation time. In this work, we develop a distributed matrix computation approach where the assigned encoded submatrices are random linear combinations of a small number of submatrices. In addition to being well suited for sparse input matrices, our approach continues have the optimal straggler resilience in a certain range of problem parameters. Moreover, compared to recent sparse matrix computation approaches, the search for a ``good'' set of random coefficients to promote numerical stability in our method is much more computationally efficient. We show that our approach can efficiently utilize partial computations done by slower worker nodes in a heterogeneous system which can enhance the overall computation speed. Numerical experiments conducted through Amazon Web Services (AWS) demonstrate up to 30% reduction in per worker node computation time and 100x faster encoding compared to the available methods.

This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0,1). By providing an a priori estimate of the solution, we have established the existence and uniqueness of a numerical solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2 type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings.

Correlation matrices are an essential tool for investigating the dependency structures of random vectors or comparing them. We introduce an approach for testing a variety of null hypotheses that can be formulated based upon the correlation matrix. Examples cover MANOVA-type hypothesis of equal correlation matrices as well as testing for special correlation structures such as, e.g., sphericity. Apart from existing fourth moments, our approach requires no other assumptions, allowing applications in various settings. To improve the small sample performance, a bootstrap technique is proposed and theoretically justified. Based on this, we also present a procedure to simultaneously test the hypotheses of equal correlation and equal covariance matrices. The performance of all new test statistics is compared with existing procedures through extensive simulations.

In this paper, we provide a novel framework for the analysis of generalization error of first-order optimization algorithms for statistical learning when the gradient can only be accessed through partial observations given by an oracle. Our analysis relies on the regularity of the gradient w.r.t. the data samples, and allows to derive near matching upper and lower bounds for the generalization error of multiple learning problems, including supervised learning, transfer learning, robust learning, distributed learning and communication efficient learning using gradient quantization. These results hold for smooth and strongly-convex optimization problems, as well as smooth non-convex optimization problems verifying a Polyak-Lojasiewicz assumption. In particular, our upper and lower bounds depend on a novel quantity that extends the notion of conditional standard deviation, and is a measure of the extent to which the gradient can be approximated by having access to the oracle. As a consequence, our analysis provides a precise meaning to the intuition that optimization of the statistical learning objective is as hard as the estimation of its gradient. Finally, we show that, in the case of standard supervised learning, mini-batch gradient descent with increasing batch sizes and a warm start can reach a generalization error that is optimal up to a multiplicative factor, thus motivating the use of this optimization scheme in practical applications.

The strong convergence of numerical methods for stochastic differential equations (SDEs) for $t\in[0,\infty)$ is proved. The result is applicable to any one-step numerical methods with Markov property that have the finite time strong convergence and the uniformly bounded moment. In addition, the convergence of the numerical stationary distribution to the underlying one can be derived from this result. To demonstrate the application of this result, the strong convergence in the infinite horizon of the backward Euler-Maruyama method in the $L^p$ sense for some small $p\in (0,1)$ is proved for SDEs with super-linear coefficients, which is also a a standalone new result. Numerical simulations are provided to illustrate the theoretical results.

The Gaussian kernel and its traditional normalizations (e.g., row-stochastic) are popular approaches for assessing similarities between data points. Yet, they can be inaccurate under high-dimensional noise, especially if the noise magnitude varies considerably across the data, e.g., under heteroskedasticity or outliers. In this work, we investigate a more robust alternative -- the doubly stochastic normalization of the Gaussian kernel. We consider a setting where points are sampled from an unknown density on a low-dimensional manifold embedded in high-dimensional space and corrupted by possibly strong, non-identically distributed, sub-Gaussian noise. We establish that the doubly stochastic affinity matrix and its scaling factors concentrate around certain population forms, and provide corresponding finite-sample probabilistic error bounds. We then utilize these results to develop several tools for robust inference under general high-dimensional noise. First, we derive a robust density estimator that reliably infers the underlying sampling density and can substantially outperform the standard kernel density estimator under heteroskedasticity and outliers. Second, we obtain estimators for the pointwise noise magnitudes, the pointwise signal magnitudes, and the pairwise Euclidean distances between clean data points. Lastly, we derive robust graph Laplacian normalizations that accurately approximate various manifold Laplacians, including the Laplace Beltrami operator, improving over traditional normalizations in noisy settings. We exemplify our results in simulations and on real single-cell RNA-sequencing data. For the latter, we show that in contrast to traditional methods, our approach is robust to variability in technical noise levels across cell types.

We study the problem of estimating the convex hull of the image $f(X)\subset\mathbb{R}^n$ of a compact set $X\subset\mathbb{R}^m$ with smooth boundary through a smooth function $f:\mathbb{R}^m\to\mathbb{R}^n$. Assuming that $f$ is a submersion, we derive a new bound on the Hausdorff distance between the convex hull of $f(X)$ and the convex hull of the images $f(x_i)$ of $M$ sampled inputs $x_i$ on the boundary of $X$. When applied to the problem of geometric inference from a random sample, our results give tighter and more general error bounds than the state of the art. We present applications to the problems of robust optimization, of reachability analysis of dynamical systems, and of robust trajectory optimization under bounded uncertainty.