Rational Gaussian wavelets and corresponding model driven neural networks

In this paper we consider the continuous wavelet transform using Gaussian wavelets multiplied by an appropriate rational term. The zeros and poles of this rational modifier act as free parameters and their choice highly influences the shape of the mother wavelet. This allows the proposed construction to approximate signals with complex morphology using only a few wavelet coefficients. We show that the proposed rational Gaussian wavelets are admissible and provide numerical approximations of the wavelet coefficients using variable projection operators. In addition, we show how the proposed variable projection based rational Gaussian wavelet transform can be used in neural networks to obtain a highly interpretable feature learning layer. We demonstrate the effectiveness of the proposed scheme through a biomedical application, namely, the detection of ventricular ectopic beats (VEBs) in real ECG measurements.

💡 Research Summary

The paper introduces a novel class of continuous wavelets called Rational Gaussian Wavelets (RGW), which are constructed by multiplying a standard Gaussian‑derived mother wavelet (e.g., the second derivative of a Gaussian, also known as the Ricker wavelet) with a rational function R(t)=∏{i=1}^{M}(t−z_i)/∏{j=1}^{N}(t−p_j). The zeros {z_i} and poles {p_j} constitute free parameters that can be placed arbitrarily on the real line, thereby granting the designer the ability to shape the wavelet’s morphology, symmetry, smoothness, and time‑frequency localization in a highly flexible manner.

A central theoretical contribution is the proof that any member of the RGW family satisfies the admissibility condition required for continuous wavelet transforms: ψ_{RGW} belongs to L¹∩L², its Fourier transform vanishes at zero frequency, and the integral of |Ĥψ_{RGW}(ξ)|²/|ξ| is finite. Consequently, the continuous wavelet transform (CWT) built with RGW permits exact reconstruction of any L² signal in the limit of infinitesimal scale resolution.

From a computational standpoint, the authors adopt the variable projection (VP) framework. For a given scale λ and translation τ, the VP operator solves a linear least‑squares problem to obtain the optimal coefficient a while simultaneously allowing the wavelet parameters (zeros, poles, λ, τ) to be updated via gradient‑based optimization. Because the rational modifier is differentiable with respect to its parameters, back‑propagation can be performed end‑to‑end, making the RGW‑based CWT a plug‑and‑play layer in modern deep‑learning architectures.

The practical impact of this construction is demonstrated on a biomedical classification task: detection of ventricular ectopic beats (VEBs) in real‑world ECG recordings from the MIT‑BIH database. The proposed RGW‑VP Net first passes each ECG segment through an RGW‑based VP layer, which selects a sparse set (typically 5–10) of wavelet coefficients that capture the most discriminative time‑scale information. These coefficients are then fed to a shallow fully‑connected classifier. Experimental results show that:

1. The model achieves state‑of‑the‑art performance (accuracy, recall, F1‑score) comparable to much larger convolutional neural networks.
2. The total number of trainable parameters is reduced to less than 1 % of that of conventional CNN baselines, leading to lower memory consumption and faster inference.
3. The learned zeros and poles align with clinically relevant features of VEB morphology (e.g., the steep upstroke and downstroke of the QRS complex), providing a transparent explanation of what the model has captured.

In addition to empirical validation, the paper derives an upper bound on the error of wavelet coefficient estimates obtained via VP for signals with compact support. This bound confirms that the approximation error diminishes as the number of selected coefficients grows, offering a theoretical guarantee of stability.

Overall, the contributions can be summarized as: (i) a highly expressive, analytically tractable family of continuous wavelets with provable admissibility; (ii) integration of variable projection into a differentiable CWT, enabling end‑to‑end learning of both wavelet shape and time‑scale parameters; (iii) a lightweight, interpretable neural architecture that excels in a real‑world biomedical detection task; and (iv) theoretical analysis of approximation error. The authors suggest future work on extending RGW to multi‑channel signals, exploring other rational function designs, and applying the framework to domains such as speech processing and image analysis.