We recover the gradient of a given function defined on interior points of a submanifold with boundary of the Euclidean space based on a (normally distributed) random sample of function evaluations at points in the manifold. This approach is based on the estimates of the Laplace-Beltrami operator proposed in the theory of Diffusion-Maps. Analytical convergence results of the resulting expansion are proved, and an efficient algorithm is proposed to deal with non-convex optimization problems defined on Euclidean submanifolds. We test and validate our methodology as a post-processing tool in Cryogenic electron microscopy (Cryo-EM). We also apply the method to the classical sphere packing problem.
相關內容
Finding a suitable density function is essential for density-based clustering algorithms such as DBSCAN and DPC. A naive density corresponding to the indicator function of a unit $d$-dimensional Euclidean ball is commonly used in these algorithms. Such density suffers from capturing local features in complex datasets. To tackle this issue, we propose a new kernel diffusion density function, which is adaptive to data of varying local distributional characteristics and smoothness. Furthermore, we develop a surrogate that can be efficiently computed in linear time and space and prove that it is asymptotically equivalent to the kernel diffusion density function. Extensive empirical experiments on benchmark and large-scale face image datasets show that the proposed approach not only achieves a significant improvement over classic density-based clustering algorithms but also outperforms the state-of-the-art face clustering methods by a large margin.
The purpose of this work is to study spectral methods to approximate the eigenvalues of nonlocal integral operators. Indeed, even if the spatial domain is an interval, it is very challenging to obtain closed analytical expressions for the eigenpairs of peridynamic operators. Our approach is based on the weak formulation of eigenvalue problem and we consider as orthogonal basis to compute the eigenvalues a set of Fourier trigonometric or Chebyshev polynomials. We show the order of convergence for eigenvalues and eigenfunctions in $L^2$-norm, and finally, we perform some numerical simulations to compare the two proposed methods.
We consider a variant of the clustering problem for a complete weighted graph. The aim is to partition the nodes into clusters maximizing the sum of the edge weights within the clusters. This problem is known as the clique partitioning problem, being NP-hard in the general case of having edge weights of different signs. We propose a new method of estimating an upper bound of the objective function that we combine with the classical branch-and-bound technique to find the exact solution. We evaluate our approach on a broad range of random graphs and real-world networks. The proposed approach provided tighter upper bounds and achieved significant convergence speed improvements compared to known alternative methods.
We study codes with parameters of $q$-ary shortened Hamming codes, i.e., $(n=(q^m-q)/(q-1), q^{n-m}, 3)_q$. At first, we prove the fact mentioned in [A.E.Brouwer et al. Bounds on mixed binary/ternary codes. IEEE Trans. Inf. Theory 44 (1998) 140-161] that such codes are optimal, generalizing it to a bound for multifold packings of radius-$1$ balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal $q$-fold packing with minimum distance $2$. At second, we show the existence of $4$-ary codes with parameters of shortened $1$-perfect codes that cannot be obtained by shortening a $1$-perfect code. Keywords: Hamming graph; multifold packings; multiple coverings; perfect codes.
We introduce tools from numerical analysis and high dimensional probability for precision control and complexity analysis of subdivision-based algorithms in computational geometry. We combine these tools with the continuous amortization framework from exact computation. We use these tools on a well-known example from the subdivision family: the adaptive subdivision algorithm due to Plantinga and Vegter. The only existing complexity estimate on this rather fast algorithm was an exponential worst-case upper bound for its interval arithmetic version. We go beyond the worst-case by considering both average and smoothed analysis, and prove polynomial time complexity estimates for both interval arithmetic and finite-precision versions of the Plantinga-Vegter algorithm.
This paper presents a new and unified approach to the derivation and analysis of many existing, as well as new discontinuous Galerkin methods for linear elasticity problems. The analysis is based on a unified discrete formulation for the linear elasticity problem consisting of four discretization variables: strong symmetric stress tensor $\dsig$ and displacement $\du$ inside each element, and the modifications of these two variables $\hsig$ and $\hu$ on elementary boundaries of elements. Motivated by many relevant methods in the literature, this formulation can be used to derive most existing discontinuous, nonconforming and conforming Galerkin methods for linear elasticity problems and especially to develop a number of new discontinuous Galerkin methods. Many special cases of this four-field formulation are proved to be hybridizable and can be reduced to some known hybridizable discontinuous Galerkin, weak Galerkin and local discontinuous Galerkin methods by eliminating one or two of the four fields. As certain stabilization parameter tends to zero, this four-field formulation is proved to converge to some conforming and nonconforming mixed methods for linear elasticity problems. Two families of inf-sup conditions, one known as $H^1$-based and the other known as $H({\rm div})$-based, are proved to be uniformly valid with respect to different choices of discrete spaces and parameters. These inf-sup conditions guarantee the well-posedness of the new proposed methods and also offer a new and unified analysis for many existing methods in the literature as a by-product. Some numerical examples are provided to verify the theoretical analysis including the optimal convergence of the new proposed methods.
We design an algorithm for approximating the size of \emph{Max Cut} in dense graphs. Given a proximity parameter $\varepsilon \in (0,1)$, our algorithm approximates the size of \emph{Max Cut} of a graph $G$ with $n$ vertices, within an additive error of $\varepsilon n^2$, with sample complexity $\mathcal{O}(\frac{1}{\varepsilon^3} \log^2 \frac{1}{\varepsilon} \log \log \frac{1}{\varepsilon})$ and query complexity of $\mathcal{O}(\frac{1}{\varepsilon^4} \log^3 \frac{1}{\varepsilon} \log \log \frac{1}{\varepsilon})$. Since Goldreich, Goldwasser and Ron (JACM 98) gave the first algorithm with sample complexity $\mathcal{O}(\frac{1}{\varepsilon^5}\log \frac{1}{\varepsilon})$ and query complexity of $\mathcal{O}(\frac{1}{\varepsilon^7}\log^2 \frac{1}{\varepsilon})$, there have been several efforts employing techniques from diverse areas with a focus on improving the sample and query complexities. Our work makes the first improvement in the sample complexity as well as query complexity after more than a decade from the previous best results of Alon, Vega, Kannan and Karpinski (JCSS 03) and of Mathieu and Schudy (SODA 08) respectively, both with sample complexity $\mathcal{O}\left(\frac{1}{{\varepsilon}^4}{\log}\frac{1}{\varepsilon}\right)$. We also want to note that the best time complexity of this problem was by Alon, Vega, Karpinski and Kannan (JCSS 03). By combining their result with an approximation technique by Arora, Karger and Karpinski (STOC 95), they obtained an algorithm with time complexity of $2^{\mathcal{O}(\frac{1}{{\varepsilon}^2} \log \frac{1}{\varepsilon})}$. In this work, we have improved this further to $2^{\mathcal{O}(\frac{1}{\varepsilon} \log \frac{1}{\varepsilon} )}$.
This chapter describes how gradient flows and nonlinear power methods in Banach spaces can be used to solve nonlinear eigenvector-dependent eigenvalue problems, and how convergence of (discretized) approximations can be verified. We review several flows from literature, which were proposed to compute nonlinear eigenfunctions, and show that they all relate to normalized gradient flows. Furthermore, we show that the implicit Euler discretization of gradient flows gives rise to a nonlinear power method of the proximal operator and prove their convergence to nonlinear eigenfunctions. Finally, we prove that $\Gamma$-convergence of functionals implies convergence of their ground states, which is important for discrete approximations.
Reinforcement learning, mathematically described by Markov Decision Problems, may be approached either through dynamic programming or policy search. Actor-critic algorithms combine the merits of both approaches by alternating between steps to estimate the value function and policy gradient updates. Due to the fact that the updates exhibit correlated noise and biased gradient updates, only the asymptotic behavior of actor-critic is known by connecting its behavior to dynamical systems. This work puts forth a new variant of actor-critic that employs Monte Carlo rollouts during the policy search updates, which results in controllable bias that depends on the number of critic evaluations. As a result, we are able to provide for the first time the convergence rate of actor-critic algorithms when the policy search step employs policy gradient, agnostic to the choice of policy evaluation technique. In particular, we establish conditions under which the sample complexity is comparable to stochastic gradient method for non-convex problems or slower as a result of the critic estimation error, which is the main complexity bottleneck. These results hold in continuous state and action spaces with linear function approximation for the value function. We then specialize these conceptual results to the case where the critic is estimated by Temporal Difference, Gradient Temporal Difference, and Accelerated Gradient Temporal Difference. These learning rates are then corroborated on a navigation problem involving an obstacle, providing insight into the interplay between optimization and generalization in reinforcement learning.
We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191