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Scale-covariant spiking wavelets

Jens Egholm Pedersen, Tony Lindeberg, Peter Gerstoft

TL;DR

Addresses energy and scalability challenges in deep learning by linking wavelet signal processing with spiking neural networks through scale-space theory. Proposes a method in which time-causal limit kernels and scale covariance enable spiking neurons to realize mother wavelets and compute wavelet-like representations. Demonstrates reconstruction of simple signals using truncated exponentials and spiking wavelets, indicating tradeoffs with scale and encoding schemes. This framework provides a principled, energy-efficient pathway for neuromorphic signal processing with potential applications in edge sensing, audio, and compression.

Abstract

We establish a theoretical connection between wavelet transforms and spiking neural networks through scale-space theory. We rely on the scale-covariant guarantees in the leaky integrate-and-fire neurons to implement discrete mother wavelets that approximate continuous wavelets. A reconstruction experiment demonstrates the feasibility of the approach and warrants further analysis to mitigate current approximation errors. Our work suggests a novel spiking signal representation that could enable more energy-efficient signal processing algorithms.

Scale-covariant spiking wavelets

TL;DR

Addresses energy and scalability challenges in deep learning by linking wavelet signal processing with spiking neural networks through scale-space theory. Proposes a method in which time-causal limit kernels and scale covariance enable spiking neurons to realize mother wavelets and compute wavelet-like representations. Demonstrates reconstruction of simple signals using truncated exponentials and spiking wavelets, indicating tradeoffs with scale and encoding schemes. This framework provides a principled, energy-efficient pathway for neuromorphic signal processing with potential applications in edge sensing, audio, and compression.

Abstract

We establish a theoretical connection between wavelet transforms and spiking neural networks through scale-space theory. We rely on the scale-covariant guarantees in the leaky integrate-and-fire neurons to implement discrete mother wavelets that approximate continuous wavelets. A reconstruction experiment demonstrates the feasibility of the approach and warrants further analysis to mitigate current approximation errors. Our work suggests a novel spiking signal representation that could enable more energy-efficient signal processing algorithms.
Paper Structure (11 sections, 25 equations, 5 figures)

This paper contains 11 sections, 25 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Method
  3. Wavelets
  4. Scale-space theory
  5. Scale space representation
  6. Scale covariance
  7. Relation between scale-spaces and wavelets
  8. Spiking neural networks
  9. Spiking mother wavelet
  10. Experiment
  11. Discussion

Figures (5)

  • Figure 1: Transformational invariance (left) and covariance (right). When $L$ is invariant to a scaling in time by a factor $s$, the scaled representations $L$ and $L'$ do not distinguish between the original signal $f(t)$ and the transformed signal $f(t')$. The covariant representation retains the structure of the transformation, so that the right diagram commutes.
  • Figure 2: A simulated approximation of the time-causal limit kernel \ref{['eq:limitkernel']} approximated with the truncated exponential kernel $g_{\rm exp}$\ref{['eq:truncated_exponential']} and its first and second derivatives for $\tau_{\rm max} = 1$ for $K=7$. The kernels integrate to $(1 - 10^{-5})$, $2 \times 10^{-5}$, and $-2 \times 10^{-5}$, respectively.
  • Figure 3: The spike response model (SRM) of a leaky integrate-and-fire system governed by $u$. SRM models neurons as compositions of linear systems, in this case an integrator $\kappa$, that generates spikes $z$ when $u \geqslant \theta_{\text{thr}}$, and a reset kernel $\eta$.
  • Figure 4: Two-channel spiking kernel representation for a signal across temporal scales ($\mu_0$ and $\mu_1$). Signal $f(t) = \sin(t)$ (orange) is integrated \ref{['eq:srm']} by positive neurons (blue) and negative neurons (green), with $\theta_{\rm thr} = 0.5$.
  • Figure 5: Reconstruction of a sinusoidal (top) and a composite sinusoidal (bottom) with different wavelets. The left panels show the raw signals. From left to right, the wavelets are: Morlet, truncated exponential \ref{['eq:truncated_exponential']} with $K = 5$, and spiking wavelets from \ref{['eq:positive_negative_decomposition']} with $K = \{3, 6, 12\}$. The original signal is superimposed on the reconstructions as a dashed line for comparison.