We present methods for offline generation of sparse roadmap spanners that result in graphs 79 smaller than existing approaches while returning solutions of equivalent path quality. Our method uses a hybrid approach to sampling that combines traditional graph discretization with random sampling. We present techniques that optimize the graph for the L1-norm metric function commonly used in joint-based robotic planning, purposefully choosing a @math -stretch factor based on the geometry of the space, and removing redundant edges that do not contribute to the graph quality. A high-quality pre-processed sparse roadmap is then available for re-use across many different planning scenarios using standard repair and re-plan methods. Pre-computing the roadmap offline results in more deterministic solutions, reduces the memory requirements by affording complex rejection criteria, and increases the speed of planning in high-dimensional spaces allowing more complex problems to be solved such as multi-modal task planning. Our method is validated through simulated benchmarks against the SPARS2 algorithm. The source code is freely available online as an open source extension to OMPL.
Our hybrid approach of combining discretized lattices with random sampling is similar to the extensive work on the subject in @cite_3 . They surprisingly found that deterministic sampling methods are superior the original PRM, noting that by definition a collection of pseudo-random samples should have too many points in some places, and not enough in others.'' Our approach is most similar to their proposed subsampled grid search (SGS), where discretized vertices along a grid are coarsely spaced and a local planner is used to collision check the edges between the grid. The unique aspect of our approach is that we size the grid optimally for the requirements of the spanning graph, and additionally perform random sampling as a second step. To the best of our knowledge this hybrid approach is unique in the literature.
@cite_3
2158207287 2787223504 2950132609 2616038372
We present, implement, and analyze a spectrum of closely-related planners, designed to gain insight into the relationship between classical grid search and probabilistic roadmaps (PRMs). Building on quasi-Monte Carlo sampling literature, we have developed deterministic variants of the PRM that use low-discrepancy and low-dispersion samples, including lattices. Classical grid search is extended using subsampling for collision detection and also the optimal-dispersion Sukharev grid, which can be considered as a kind of lattice-based roadmap to complete the spectrum. Our experimental results show that the deterministic variants of the PRM offer performance advantages in comparison to the original PRM and the recent Lazy PRM. This even includes searching using a grid with subsampled collision checking. Our theoretical analysis shows that all of our deterministic PRM variants are resolution complete and achieve the best possible asymptotic convergence rate, which is shown superior to that obtained by random sampling. Thus, in surprising contrast to recent trends, there is both experimental and theoretical evidence that some forms of grid search are superior to the original PRM. We propose in this paper a new approach to train the Generative Adversarial Nets (GANs) with a mixture of generators to overcome the mode collapsing problem. The main intuition is to employ multiple generators, instead of using a single one as in the original GAN. The idea is simple, yet proven to be extremely effective at covering diverse data modes, easily overcoming the mode collapsing problem and delivering state-of-the-art results. A minimax formulation was able to establish among a classifier, a discriminator, and a set of generators in a similar spirit with GAN. Generators create samples that are intended to come from the same distribution as the training data, whilst the discriminator determines whether samples are true data or generated by generators, and the classifier specifies which generator a sample comes from. The distinguishing feature is that internal samples are created from multiple generators, and then one of them will be randomly selected as final output similar to the mechanism of a probabilistic mixture model. We term our method Mixture Generative Adversarial Nets (MGAN). We develop theoretical analysis to prove that, at the equilibrium, the Jensen-Shannon divergence (JSD) between the mixture of generators’ distributions and the empirical data distribution is minimal, whilst the JSD among generators’ distributions is maximal, hence effectively avoiding the mode collapsing problem. By utilizing parameter sharing, our proposed model adds minimal computational cost to the standard GAN, and thus can also efficiently scale to large-scale datasets. We conduct extensive experiments on synthetic 2D data and natural image databases (CIFAR-10, STL-10 and ImageNet) to demonstrate the superior performance of our MGAN in achieving state-of-the-art Inception scores over latest baselines, generating diverse and appealing recognizable objects at different resolutions, and specializing in capturing different types of objects by the generators. We obtain an improved finite-sample guarantee on the linear convergence of stochastic gradient descent for smooth and strongly convex objectives, improving from a quadratic dependence on the conditioning @math (where @math is a bound on the smoothness and @math on the strong convexity) to a linear dependence on @math . Furthermore, we show how reweighting the sampling distribution (i.e. importance sampling) is necessary in order to further improve convergence, and obtain a linear dependence in the average smoothness, dominating previous results. We also discuss importance sampling for SGD more broadly and show how it can improve convergence also in other scenarios. Our results are based on a connection we make between SGD and the randomized Kaczmarz algorithm, which allows us to transfer ideas between the separate bodies of literature studying each of the two methods. In particular, we recast the randomized Kaczmarz algorithm as an instance of SGD, and apply our results to prove its exponential convergence, but to the solution of a weighted least squares problem rather than the original least squares problem. We then present a modified Kaczmarz algorithm with partially biased sampling which does converge to the original least squares solution with the same exponential convergence rate. We reconsider randomized algorithms for the low-rank approximation of symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results consist of an empirical evaluation of the performance quality and running time of sampling and projection methods on a diverse suite of SPSD matrices. Our results highlight complementary aspects of sampling versus projection methods; they characterize the effects of common data preprocessing steps on the performance of these algorithms; and they point to important differences between uniform sampling and nonuniform sampling methods based on leverage scores. In addition, our empirical results illustrate that existing theory is so weak that it does not provide even a qualitative guide to practice. Thus, we complement our empirical results with a suite of worst-case theoretical bounds for both random sampling and random projection methods. These bounds are qualitatively superior to existing bounds--e.g., improved additive-error bounds for spectral and Frobenius norm error and relative-error bounds for trace norm error--and they point to future directions to make these algorithms useful in even larger-scale machine learning applications.