Maximum likelihood estimation (MLE) is a well-known estimation method used in many robotic and computer vision applications. Under Gaussian assumption, the MLE converts to a nonlinear least squares (NLS) problem. Efficient solutions to NLS exist and they are based on iteratively solving sparse linear systems until convergence. In general, the existing solutions provide only an estimation of the mean state vector, the resulting covariance being computationally too expensive to recover. Nevertheless, in many simultaneous localisation and mapping (SLAM) applications, knowing only the mean vector is not enough. Data association, obtaining reduced state representations, active decisions and next best view are only a few of the applications that require fast state covariance recovery. Furthermore, computer vision and robotic applications are in general performed online. In this case, the state is updated and recomputed every step and its size is continuously growing, therefore, the estimation process may become highly computationally demanding. This paper introduces a general framework for incremental MLE called SLAM++, which fully benefits from the incremental nature of the online applications, and provides efficient estimation of both the mean and the covariance of the estimate. Based on that, we propose a strategy for maintaining a sparse and scalable state representation for large scale mapping, which uses information theory measures to integrate only informative and non-redundant contributions to the state representation. SLAM++ differs from existing implementations by performing all the matrix operations by blocks. This led to extremely fast matrix manipulation and arithmetic operations. Even though this paper tests SLAM++ efficiency on SLAM problems, its applicability remains general.
Several approximations for marginal covariance recovery have been proposed in the literature. @cite_5 suggested using conditional covariances, which are inversions of sub-blocks of @math called the Markov blankets. The result is an overconfident approximation of the marginal covariances. Online, conservative approximations were proposed in @cite_33 , where at every step, the covariances corresponding to the new variables are computed by solving the augmented system with a set of basis vectors. The recovered covariance column is passed to a Kalman filter bank, which updates the rest of the covariance matrix. The filtering is reported to run in constant time, and the recovery speed is bounded by the linear solving. In the context of MLE, belief propagation over a spanning tree or loopy intersection propagation can be used to obtain conservative approximations suitable for data association, @cite_18 .
@cite_5 @cite_18 @cite_33
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In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings. Many estimation problems in robotics rely on efficiently solving nonlinear least squares (NLS). For example, it is well known that the simultaneous localisation and mapping (SLAM) problem can be formulated as a maximum likelihood estimation (MLE) and solved using NLS, yielding a mean state vector. However, for many applications recovering only the mean vector is not enough. Data association, active decisions, next best view, are only few of the applications that require fast state covariance recovery. The problem is not simple since, in general, the covariance is obtained by inverting the system matrix and the result is dense. The main contribution of this paper is a novel algorithm for fast incremental covariance update, complemented by a highly efficient implementation of the covariance recovery. This combination yields to two orders of magnitude reduction in computation time, compared to the other state of the art solutions. The proposed algorithm is applicable to any NLS solver implementation, and does not depend on incremental strategies described in our previous papers, which are not a subject of this paper. Most of the existing sparse-recovery methods assume a static system: the signal is a finite-length vector for which a fixed set of measurements and sparse representation are available and an l1 problem is solved for the reconstruction. However, the same representation and reconstruction framework is not readily applicable in a streaming system: the signal changes over time, and it is measured and reconstructed sequentially over small intervals. This is particularly desired when dividing signals into disjoint blocks and processing each block separately is infeasible or inefficient. In this paper, we discuss two streaming systems and a new homotopy algorithm for quickly solving the associated l1 problems: 1) recovery of smooth, time-varying signals for which, instead of using block transforms, we use lapped orthogonal transforms for sparse representation and 2) recovery of sparse, time-varying signals that follows a linear dynamic model. For both systems, we iteratively process measurements over a sliding interval and solve a weighted l1-norm minimization problem for estimating sparse coefficients. Since we estimate overlapping portions of the signal while adding and removing measurements, instead of solving a new l1 program as the system changes, we use available signal estimates as starting point in a homotopy formulation and update the solution in a few simple steps. We demonstrate with numerical experiments that our proposed streaming recovery framework provides better reconstruction compared to the methods that represent and reconstruct signals as independent, disjoint blocks, and that our proposed homotopy algorithm updates the solution faster than the current state-of-the-art solvers. The goal of sparse recovery is to recover a k-sparse signal x e Rn from (possibly noisy) linear measurements of the form y = Ax, where A e Rmxn describes the measurement process. Standard results in compressive sensing show that it is possible to recover the signal x from m = O(k log(n k)) measurements, and that this bound is tight. The framework of model-based compressive sensing [BCDH10] overcomes the lower bound and reduces the number of measurements further to O(k) by limiting the supports of x to a subset M of the (nk) possible supports. This has led to many measurement-efficient algorithms for a wide variety of signal models, including block-sparsity and tree-sparsity. Unfortunately, extending the framework to other, more general models has been stymied by the following obstacle: for the framework to apply, one needs an algorithm that, given a signal x, solves the following optimization problem exactly: [EQUATION] (here x[n] denotes the projection of x on coordinates not in Ω). However, an approximation algorithm for this optimization task is not sufficient. Since many problems of this form are not known to have exact polynomial-time algorithms, this requirement poses an obstacle for extending the framework to a richer class of models. In this paper, we remove this obstacle and show how to extend the model-based compressive sensing framework so that it requires only approximate solutions to the aforementioned optimization problems. Interestingly, our extension requires the existence of approximation algorithms for both the maximization and the minimization variants of the optimization problem. Further, we apply our framework to the Constrained Earth Mover's Distance (CEMD) model introduced in [SHI13], obtaining a sparse recovery scheme that uses significantly less than O(k log(n k)) measurements. This is the first non-trivial theoretical bound for this model, since the validation of the approach presented in [SHI13] was purely empirical. The result is obtained by designing a novel approximation algorithm for the maximization version of the problem and proving approximation guarantees for the minimization algorithm described in [SHI13].