Hilbert maps: Scalable continuous occupancy mapping with stochastic gradient descent

The vast amount of data robots can capture today motivates the development of fast and scalable statistical tools to model the space the robot operates in. We devise a new technique for environment representation through continuous occupancy mapping that improves on the popular occupancy grip maps in two fundamental aspects: (1) it does not assume an a priori discrimination of the world into grid cells and therefore can provide maps at an arbitrary resolution; (2) it captures spatial relationships between measurements naturally, thus being more robust to outliers and possessing better generalization performance. The technique, named Hilbert maps, is based on the computation of fast kernel approximations that project the data in a Hilbert space where a logistic regression classifier is learnt. We show that this approach allows for efficient stochastic gradient optimization where each measurement is only processed once during learning in an online manner. We present results with three types of approximations: random Fourier; Nyström; and a novel sparse projection. We also extend the approach to accept probability distributions as inputs, for example, due to uncertainty over the position of laser scans due to sensor or localization errors. In this extended version, experiments were conducted in two dimensions and three dimensions, using popular benchmark datasets. Furthermore, an analysis of the adaptive capabilities of the technique to handle large changes in the data, such as trajectory update before and after loop closure during simultaneous localization and mapping, is also included.

[1]  Benjamin Recht,et al.  Weighted Sums of Random Kitchen Sinks: Replacing minimization with randomization in learning , 2008, NIPS.

[2]  Yoram Singer,et al.  Pegasos: primal estimated sub-gradient solver for SVM , 2007, ICML '07.

[3]  Le Song,et al.  A Hilbert Space Embedding for Distributions , 2007, Discovery Science.

[4]  David R. Cox,et al.  The Theory of Stochastic Processes , 1967, The Mathematical Gazette.

[5]  Jasper Snoek,et al.  Practical Bayesian Optimization of Machine Learning Algorithms , 2012, NIPS.

[6]  Danica Kragic,et al.  Strategies for multi-modal scene exploration , 2010, 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[7]  Sergei Vassilvitskii,et al.  k-means++: the advantages of careful seeding , 2007, SODA '07.

[8]  Andrew Zisserman,et al.  Efficient additive kernels via explicit feature maps , 2010, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[9]  Yoram Singer,et al.  Pegasos: primal estimated sub-gradient solver for SVM , 2011, Math. Program..

[10]  Alexander J. Smola,et al.  Fastfood: Approximate Kernel Expansions in Loglinear Time , 2014, ArXiv.

[11]  Léon Bottou,et al.  The Tradeoffs of Large Scale Learning , 2007, NIPS.

[12]  Wolfram Burgard,et al.  OctoMap: an efficient probabilistic 3D mapping framework based on octrees , 2013, Autonomous Robots.

[13]  Benjamin Recht,et al.  Random Features for Large-Scale Kernel Machines , 2007, NIPS.

[14]  AI Koan,et al.  Weighted Sums of Random Kitchen Sinks: Replacing minimization with randomization in learning , 2008, NIPS.

[15]  Léon Bottou,et al.  Large-Scale Machine Learning with Stochastic Gradient Descent , 2010, COMPSTAT.

[16]  Iain Murray.  Introduction To Gaussian Processes , 2008 .

[17]  Fabio Tozeto Ramos,et al.  Particle Filter Localization on Continuous Occupancy Maps , 2016, ISER.

[18]  John Platt,et al.  Probabilistic Outputs for Support vector Machines and Comparisons to Regularized Likelihood Methods , 1999 .

[19]  Le Song,et al.  A unified kernel framework for nonparametric inference in graphical models ] Kernel Embeddings of Conditional Distributions , 2013 .

[20]  Léon Bottou,et al.  Stochastic Gradient Descent Tricks , 2012, Neural Networks: Tricks of the Trade.

[21]  Wei Xu,et al.  Towards Optimal One Pass Large Scale Learning with Averaged Stochastic Gradient Descent , 2011, ArXiv.

[22]  Fabio Tozeto Ramos,et al.  Gaussian process occupancy maps* , 2012, Int. J. Robotics Res..

[23]  H. Zou,et al.  Regularization and variable selection via the elastic net , 2005 .

[24]  Tong Zhang,et al.  Solving large scale linear prediction problems using stochastic gradient descent algorithms , 2004, ICML.

[25]  Klaus-Robert Müller,et al.  Covariate Shift Adaptation by Importance Weighted Cross Validation , 2007, J. Mach. Learn. Res..

[26]  Matthias W. Seeger,et al.  Using the Nyström Method to Speed Up Kernel Machines , 2000, NIPS.

[27]  Rong Jin,et al.  Nyström Method vs Random Fourier Features: A Theoretical and Empirical Comparison , 2012, NIPS.

[28]  Viii Supervisor.  Sonar-Based Real-World Mapping and Navigation , 2001 .

[29]  Hugh F. Durrant-Whyte,et al.  Contextual occupancy maps using Gaussian processes , 2009, 2009 IEEE International Conference on Robotics and Automation.

[30]  James Hensman,et al.  Scalable Variational Gaussian Process Classification , 2014, AISTATS.

[31]  Salah Sukkarieh,et al.  A Gaussian process-based RRT planner for the exploration of an unknown and cluttered environment with a UAV , 2013, Adv. Robotics.

[32]  Fabio Tozeto Ramos,et al.  A Sparse Covariance Function for Exact Gaussian Process Inference in Large Datasets , 2009, IJCAI.

[33]  Anthony Widjaja,et al.  Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond , 2003, IEEE Transactions on Neural Networks.

[34]  Alberto Elfes,et al.  Occupancy grids: a probabilistic framework for robot perception and navigation , 1989 .