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We develop two new methods for selecting the penalty parameter for the $\ell^1$-penalized high-dimensional M-estimator, which we refer to as the analytic and bootstrap-after-cross-validation methods. For both methods, we derive nonasymptotic error bounds for the corresponding $\ell^1$-penalized M-estimator and show that the bounds converge to zero under mild conditions, thus providing a theoretical justification for these methods. We demonstrate via simulations that the finite-sample performance of our methods is much better than that of previously available and theoretically justified methods.

In this work we present the first initialization methods equipped with explicit performance guarantees adapted to the pose-graph simultaneous localization and mapping (SLAM) and rotation averaging (RA) problems. SLAM and rotation averaging are typically formalized as large-scale nonconvex point estimation problems, with many bad local minima that can entrap the smooth optimization methods typically applied to solve them; the performance of standard SLAM and RA algorithms thus crucially depends upon the quality of the estimates used to initialize this local search. While many initialization methods for SLAM and RA have appeared in the literature, these are typically obtained as purely heuristic approximations, making it difficult to determine whether (or under what circumstances) these techniques can be reliably deployed. In contrast, in this work we study the problem of initialization through the lens of spectral relaxation. Specifically, we derive a simple spectral relaxation of SLAM and RA, the form of which enables us to exploit classical linear-algebraic techniques (eigenvector perturbation bounds) to control the distance from our spectral estimate to both the (unknown) ground-truth and the global minimizer of the estimation problem as a function of measurement noise. Our results reveal the critical role that spectral graph-theoretic properties of the measurement network play in controlling estimation accuracy; moreover, as a by-product of our analysis we obtain new bounds on the estimation error for the maximum likelihood estimators in SLAM and RA, which are likely to be of independent interest. Finally, we show experimentally that our spectral estimator is very effective in practice, producing initializations of comparable or superior quality at lower computational cost compared to existing state-of-the-art techniques.

In this paper we analyze a posteriori error estimates for a mixed formulation of the linear elasticity eigenvalue problem. A posteriori estimators for the nearly and perfectly compressible elasticity spectral problems are proposed. With a post-process argument, we are able to prove reliability and efficiency for the proposed estimators. The numerical method is based in Raviart-Thomas elements to approximate the pseudostress and piecewise polynomials for the displacement. We illustrate our results with numerical tests.

Two-grid methods with exact solution of the Galerkin coarse-grid system have been well studied by the multigrid community: an elegant identity has been established to characterize the convergence factor of exact two-grid methods. In practice, however, it is often too costly to solve the Galerkin coarse-grid system exactly, especially when its size is large. Instead, without essential loss of convergence speed, one may solve the coarse-grid system approximately. In this paper, we develop a new framework for analyzing the convergence of inexact two-grid methods: two-sided bounds for the energy norm of the error propagation matrix of inexact two-grid methods are presented. In the framework, a restricted smoother involved in the identity for exact two-grid convergence is used to measure how far the actual coarse-grid matrix deviates from the Galerkin one. As an application, we establish a unified convergence theory for multigrid methods.

We propose a uniform block diagonal preconditioner for condensed H(div)-conforming HDG schemes for parameter-dependent saddle point problems including the generalized Stokes problem and the linear elasticity. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning (ASP) technique [49]. A robust preconditioner spectrally equivalent to the Schur complement on the element-wise constant pressure space is also constructed. Finally, numerical results of the generalized Stokes and the steady linear elasticity equations verify the robustness of our proposed preconditioner with respect to model parameters and mesh size.

Domain decomposition methods are among the most efficient for solving sparse linear systems of equations. Their effectiveness relies on a judiciously chosen coarse space. Originally introduced and theoretically proved to be efficient for self-adjoint operators, spectral coarse spaces have been proposed in the past few years for indefinite and non-self-adjoint operators. This paper presents a new spectral coarse space that can be constructed in a fully-algebraic way unlike most existing spectral coarse spaces. We present theoretical convergence result for Hermitian positive definite diagonally dominant matrices. Numerical experiments and comparisons against state-of-the-art preconditioners in the multigrid community show that the resulting two-level Schwarz preconditioner is efficient especially for non-self-adjoint operators. Furthermore, in this case, our proposed preconditioner outperforms state-of-the-art preconditioners.

This paper is concerned with the efficient spectral solutions for weakly singular nonlocal diffusion equations with Dirichlet-type volume constraints. This type of equation contains an integral operator which typically has a singularity at the midpoint of the integral domain, and the approximation of such the integral operator is one of the essential difficulties in solving the nonlocal equations. To overcome this problem, two-sided Jacobi spectral quadrature rules are proposed to develop a Jacobi spectral collocation method for the nonlocal diffusion equations. Rigorous convergence analysis of the proposed method is presented in $L^\infty$ norms, and we further prove that the Jacobi collocation solution converges to its corresponding local limit as nonlocal interactions vanish. Numerical examples are given to verify the theoretical results.

Maximum likelihood estimation of generalized linear mixed models(GLMMs) is difficult due to marginalization of the random effects. Computing derivatives of a fitted GLMM's likelihood (with respect to model parameters) is also difficult, especially because the derivatives are not by-products of popular estimation algorithms. In this paper, we describe GLMM derivatives along with a quadrature method to efficiently compute them, focusing on lme4 models with a single clustering variable. We describe how psychometric results related to IRT are helpful for obtaining these derivatives, as well as for verifying the derivatives' accuracies. After describing the derivative computation methods, we illustrate the many possible uses of these derivatives, including robust standard errors, score tests of fixed effect parameters, and likelihood ratio tests of non-nested models. The derivative computation methods and applications described in the paper are all available in easily-obtained R packages.

Comparing structured objects such as graphs is a fundamental operation involved in many learning tasks. To this end, the Gromov-Wasserstein (GW) distance, based on Optimal Transport (OT), has proven to be successful in handling the specific nature of the associated objects. More specifically, through the nodes connectivity relations, GW operates on graphs, seen as probability measures over specific spaces. At the core of OT is the idea of conservation of mass, which imposes a coupling between all the nodes from the two considered graphs. We argue in this paper that this property can be detrimental for tasks such as graph dictionary or partition learning, and we relax it by proposing a new semi-relaxed Gromov-Wasserstein divergence. Aside from immediate computational benefits, we discuss its properties, and show that it can lead to an efficient graph dictionary learning algorithm. We empirically demonstrate its relevance for complex tasks on graphs such as partitioning, clustering and completion.

The main contribution of this paper is a new submap joining based approach for solving large-scale Simultaneous Localization and Mapping (SLAM) problems. Each local submap is independently built using the local information through solving a small-scale SLAM; the joining of submaps mainly involves solving linear least squares and performing nonlinear coordinate transformations. Through approximating the local submap information as the state estimate and its corresponding information matrix, judiciously selecting the submap coordinate frames, and approximating the joining of a large number of submaps by joining only two maps at a time, either sequentially or in a more efficient Divide and Conquer manner, the nonlinear optimization process involved in most of the existing submap joining approaches is avoided. Thus the proposed submap joining algorithm does not require initial guess or iterations since linear least squares problems have closed-form solutions. The proposed Linear SLAM technique is applicable to feature-based SLAM, pose graph SLAM and D-SLAM, in both two and three dimensions, and does not require any assumption on the character of the covariance matrices. Simulations and experiments are performed to evaluate the proposed Linear SLAM algorithm. Results using publicly available datasets in 2D and 3D show that Linear SLAM produces results that are very close to the best solutions that can be obtained using full nonlinear optimization algorithm started from an accurate initial guess. The C/C++ and MATLAB source codes of Linear SLAM are available on OpenSLAM.