The increasing need for intelligent sensors in a wide range of everyday objects requires the existence of low power information processing systems which can operate autonomously in their environment. In particular, merging and processing the outputs
of different sensors efficiently is a necessary requirement for mobile agents with cognitive abilities. In this work, we present a multi-layer spiking neural network for inference of relations between stimuli patterns in dedicated neuromorphic systems. The system is trained with a new version of the backpropagation algorithm adapted to on-chip learning in neuromorphic hardware: Error gradients are encoded as spike signals which are propagated through symmetric synapses, using the same integrate-and-fire hardware infrastructure as used during forward propagation. We demonstrate the strength of the approach on an arithmetic relation inference task and on visual XOR on the MNIST dataset. Compared to previous, biologically-inspired implementations of networks for learning and inference of relations, our approach is able to achieve better performance with less neurons. Our architecture is the first spiking neural network architecture with on-chip learning capabilities, which is able to perform relational inference on complex visual stimuli. These features make our system interesting for sensor fusion applications and embedded learning in autonomous neuromorphic agents.
We investigate the unconditional basis property of martingale differences in weighted $L^2$ spaces in the non-homogeneous situation (i.e. when the reference measure is not doubling). Specifically, we prove that finiteness of the quantity $[w]_{A_2}
=sup_I , < w>_I < w^{-1}>_I$, defined through averages $ <cdot >_I$ relative to the reference measure $ u$, implies that each martingale transform relative to $ u$ is bounded in $L^2(w, d u)$. Moreover, we prove the linear in $[w]_{A_2}$ estimate of the unconditional basis constant of the Haar system. Even in the classical case of the standard dyadic lattice in $mathbb{R}^n$, where the results about unconditional basis and linear in $[w]_{A_2}$ estimates are known, our result gives something new, because all the estimates are independent of the dimension $n$. Our approach combines the technique of outer measure spaces with the Bellman function argument.
