Scale-covariant spiking wavelets

The gist The work establishes a theoretical connection between wavelet transforms and spiking neural networks through scale-space theory.

Wavelet Foundations

Wavelets decompose a signal f into components using the “mother” wavelet generating function ψ(x; a, b) = a−1/2ψ(x − b/a) (1) with a certain scale a and temporal shift b (T f)(x; a, b) = ⟨f, ψ(x; a, b)⟩ = Z ∞−∞ f(x)a−1/2ψ x − b/a dx (2). The resolution of identity ensures that the wavelet can represent any function transform f(t) = C−1 ψ Z ∞−∞ Z ∞−∞ a −2 ⟨f, ψ(t; a, b)⟩ ψ(t; a, b) da db (3).

Scale-Space Theory

Scale-space theory parameterizes a signal f: R → R over its scale in a scale-space representation L: R × R+ → R [20]. This conveniently corresponds to the convolution of smoothing kernels g(t; τ), where τ is the scale parameter describing the variance of the smoothing kernels and ∗ denotes convolution L(t; τ) = g(t; τ) ∗ f(x) = Z ∞ u=−∞ g(u; τ) f(t − u)du (4). For any kernel g to be variation diminishing, it must then adhere to the following Laplace transform Ψ(ˆ s) = C e−γs2+as/s m KY=∞ k=1 (1 + aks)e−aks (5). In this work, we space the scaling factors τk, in terms of the variance of the composed convolution kernel, logarithmically using powers of a distribution parameter c > 1 relative to a maximum scale level τmax [22, (18)].

Spiking Neural Networks and Wavelets

The relation between scale-space representations and spiking neural networks established in [3] is leveraged to construct a spiking neural network that can represent the wavelet system. The Spike Response Model (SRM) [23, pp. 154] is chosen to describe generic neuron models with a single integral of composed linear kernels, see Fig. 3. For the leaky integrate-and-fire model, the SRM yields a scale-space representation at scale µ for some input signal f [3, (23)]. The scaling factors cancel when considering temporal scaling operations t′ = st and t′f = st'f for the functions f′(t′) = f(t), where u' = su, du' = sdu, and µ' = sµ, demonstrating scale covariance for the LIF neuron model [3, (24)].

Spiking Mother Wavelet Construction

To address the sign problem inherent in outgoing spikes discarding negative values, a two-channel representation is constructed at some scale µ with two integrative terms µ u˙+ = −u+ + f(t) and µ u˙− = −u− − f(t) (21). The kernel κ at each scale k is implemented as a difference of exponentials κ(t; µk) = 1/Cκ (gexp(t; µk+1) − gexp(t; µk)) (22), where Cκ = (µk+1 − µk)/(µkµk+1) normalizes the kernel, and µk < µk+1 are adjacent time constants in the scale hierarchy. The full signal representation at scale µk combines both polarity channels: M(t; µk) = M+(t; µk) − M−(t; µk) (24). This difference kernel exhibits approximate scale covariance when both time constants scale proportionally, preserving the bandpass structure across scales, and since it relies on truncated exponentials, this is entirely representable with leaky integrators as physical primitives.

Experimental Validation and Conclusion

The experiment demonstrates the spiking neural network wavelets on a sinusoidal f(t) = sin(t) and an arbitrary composite sinusoidal f(t) = sin(2π·0.5 t) + 0.5 sin(2π 2 t) + 0.3 sin(2π 8 t) over time t, see Fig. 5. The truncated exponential wavelet achieves best reconstruction, followed by Morlet, and spiking wavelets [4, (3.3.26)]. The spiking wavelet performance plateaus beyond K = 3 channels, suggesting that the spike-count encoding scheme via (21) captures the frequency components without requiring additional channels; this plateau hints at the need for more sophisticated spike-based and spike-time encoding strategies that better exploit additional channels, including cross-channel dependencies, for richer signal representations. The approach demonstrates that neural integration can transform signals into wavelet representations, providing a foundation for investigating how biological constraints and neural dynamics influence wavelet analysis.

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SCALE-COVARIANT SPIKING WAVELETS Jens Egholm Pedersen1, Tony Lindeberg2, Peter Gerstoft1 1: DTU Electro, Technical University of Denmark, Denmark 2: Department of Computational Science and Technology, KTH Royal Institute of Technology, Sweden 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. Index Terms— Wavelets, spiking neural networks, scale space, signal processing 1. INTRODUCTION The deep learning paradigm is facing challenging scaling issues, currently due to energy consumption and transistor scaling [1]. Biologically-inspired spiking neural networks have been hypothesized to address some limitations in deep learning, including time, space, and energy [1, 2]. However, current computational models have not convincingly realized the promising gains, and we lack theories to guide better implementations [3, 2]. A key challenge for spiking neural systems is the choice of coding. Neural systems exhibit singular, digital activations in massively parallel networks that are highly sparse in both space and time [1]. This leaves a vast search space for possible encodings, and discretization is challenging for training spiking networks at scale [3]. Signals are efficiently represented by the wavelet transform [4], based on orthogonal or semi-overlapping bases for signals at varying resolutions and scales. However, conventional wavelet implementations on digital hardware have fundamental energy constraints in always-on, edge, or real-time systems. Neuromorphic hardware offers orders-of-magnitude energy improvements through event-driven computation [5], but lacks theoretical foundations that connect the neural primitives to signal processing. Establishing that spiking neural networks can implement wavelets enables rigorous, energy-efficient signal processing on neuromorphic substrates. This work bridges signal processing and neuromorphic computing by leveraging the connection between scale-space representations and wavelets [6]. We demonstrate that leaky integrate-and-fire neurons can form mother wavelets, enabling applications in edge sensing (continuous processing with microJoule budgets), audio processing (native spike-domain computation for event-based microphones), and signal compression (exploiting spatio-temporal sparsity). Through standardized interfaces such as the Neuromorphic Intermediate Representation [7], these theoretical foundations Support from the Novo Nordisk Foundation (NNF24OC0089302) and the Swedish Research Council (2022-02969) is gratefully acknowledged. The admissibility criteria for a mother wavelet were extended to scale-space representations, a covariant re-parameterization of a signal across varying scales [6]. Covariance for signal transformations is an interesting property leveraged in many domains [8], including computer vision [9], PDE solving [10], graph convolutional neural networks [11], and recently for spiking neural networks [3]. In this paper, we leverage the connection between scale-space and wavelets to study spiking wavelets as the basis for mathematically sound and compact representations. The relation between wavelets and scale-spaces was formally introduced in [6] and extended to Gabor kernels in [12]. Wavelets have been used in neural networks to model complex processes [13]. Spiking networks implement wavelets in experiments described in [14], and [15] leveraged Hilbert transforms for beamforming using spiking neurons. Spiking neural networks are brain-inspired (neuromorphic) neural networks that co-locate memory and computation, avoiding the von Neumann bottleneck, and have high performance and low energy consumption. Due to their compact hardware implementations, neuromorphic computing can run machine learning on the edge rather than on a central computer. Neuromorphic computing is based on impulses and is more energy-efficient [5], but challenging to represent on conventional digital systems as they are inherently continuous [7]. They have been demonstrated for speech detection [16, 17, 18] and require gradient approximations due to the discontinuities of the spikes in backpropagation [19].

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L(t; τ) = L′ (t; τ′) f(t) f'(t′) t' = st g(t; τ)∗g(t')∗g(t'; τ') t' = sτ'=s 2τ Fig. 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').

1. Scale-covariant signal representation enables more energy-efficient signal processing algorithms by leveraging the connection between scale-space representations and wavelets. This allows for rigorous, energy-efficient signal processing on neuromorphic substrates with microJoule budgets.

2. Implement edge sensing with continuous processing by having leaky integrate-and-fire neurons form mother wavelets, which enables applications in audio processing (native spike-domain computation for event-based microphones).

3. Develop signal compression techniques by exploiting spatio-temporal sparsity using the derived spiking wavelet representations, leading to more energy efficient signal representation.

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.