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.