Distributed Shape Learning of Complex Objects Using Gaussian Kernel

TL;DR

The paper tackles distributed object-shape learning for multiple networked robots using LiDAR data, addressing the difficulty of sharing an infinite-dimensional Gaussian RKHS function. It replaces a finite-equality-sharing scheme based on polynomial kernels with a grid-based finite-basis approach in conjunction with a data-independent continuous-time ADMM that preserves performance guarantees. By fixing the function space to be spanned by a set of grid points $\{g_j\}$, each robot learns local coefficients $c_j^\ell$ and a common threshold $\gamma^\ell$ while enforcing $c^\ell\gamma^\ell = z$ for consensus, yielding a shared decision boundary $\{x : \sum_j c_j^* k(x, g_j) + \gamma^* = 0\}$. Simulations with three robots show overlapping learned boundaries that correctly separate the labeled regions, underscoring the practical viability and pointing to grid-point selection as a key design factor for future work.

Abstract

This paper addresses distributed learning of a complex object for multiple networked robots based on distributed optimization and kernel-based support vector machine. In order to overcome a fundamental limitation of polynomial kernels assumed in our antecessor, we employ Gaussian kernel as a kernel function for classification. The Gaussian kernel prohibits the robots to share the function through a finite number of equality constraints due to its infinite dimensionality of the function space. We thus reformulate the optimization problem assuming that the target function space is identified with the space spanned by the bases associated with not the data but a finite number of grid points. The above relaxation is shown to allow the robots to share the function by a finite number of equality constraints. We finally demonstrate the present approach through numerical simulations.

Figures (5)

• Figure 1: Object shape learning by multiple robots.
• Figure 2: Acquisition of data points by LiDAR sensor.
• Figure 3: The close curve with a dashed line is the target object.
• Figure 4: Trajectory of $[(x^{\ell})^\top \ \gamma^\ell]$ for robot $\ell=1$.
• Figure 5: The number and positions of grid points have been changed from Fig. \ref{['fig:learned_shape_5_3']}