M-RWTL: Rational Wavelet Transform Learning

• M-RWTL is a data-adaptive approach that generalizes classical dyadic wavelets by using arbitrary rational subband decompositions with closed-form least-squares filter estimation.
• It integrates lifting frameworks with deep neural network techniques to achieve perfect reconstruction and significant PSNR gains (up to 1.8 dB for certain signals) in applications like ECG and image processing.
• The framework is modular and scalable, supporting nonlinear extensions and task-driven learning, which optimizes sparse representations for compressed sensing and biomedical signal analysis.

Rational Wavelet Transform Learning (M-RWTL) refers to the family of data-adaptive methodologies for learning rational wavelet transforms—i.e., transforms in which the subband decompositions occur at arbitrary rational (non-dyadic) rates—from observed signals, ensembles, or tasks. These approaches extend classical dyadic wavelet learning and lifting frameworks, enabling signal or class-matched basis design, closed-form adaptive filter estimation, and, in modern variants, fully differentiable integration with deep neural networks. M-RWTL frameworks guarantee perfect reconstruction (PR), support modular stacking and nonlinear extensions, and have demonstrated improved sparse representations and application performance relative to fixed or non-adaptive wavelets in compressed sensing, image and biomedical signal processing, and neural feature extraction.

1. Mathematical Formulation and Lifting Framework

Rational wavelet transforms generalize dyadic wavelets by partitioning the frequency axis into $M$ subbands, with each analysis branch downsampling by an arbitrary integer $q_i$ such that $\sum_{i} q_i = M$. The two-channel rational analysis filterbank employs analysis filters $G_l(z)$, $G_h(z)$ and corresponding synthesis filters $F_l(z)$, $F_h(z)$, with critical sampling enforced by $\frac{q_1}{M} + \frac{q_2}{M} = 1$. Filterbank realization can equivalently be phrased as merging branches of an $M$-band uniform filterbank to create lowpass and highpass branches of unequal width, with analysis filters given by

$$G_l(z) = \sum_{i=0}^{q_1-1} z^{-iM} G_i(z^{q_1}), \qquad G_h(z) = \sum_{i=0}^{q_2-1} z^{-iM} G_{i+q_1}(z^{q_2})$$

and similar structure for the synthesis bank.

In the lifting domain, the rational Lazy wavelet (“split” stage) generalizes the dyadic Lazy transform by distributing samples according to the $(q_1, q_2)$ split: $$G_l^{(0)}(z) = \sum_{i=0}^{q_1-1} z^{-i q_2}, \qquad G_h^{(0)}(z) = z^{q_1q_2} \sum_{i=0}^{q_2-1} z^{-i q_1}$$ Prediction (modifying $G_h$, $F_l$) and update (modifying $G_l$, $F_h$) stages employ rate converters to adaptively map between unequal branch rates, followed by short FIR filters $T(z)$ (prediction) and $S(z)$ (update). Each step's structure ensures invertibility and PR, with the overall polyphase matrices $E(z)$, $R(z)$ satisfying $R(z) E(z) = c z^{-n_0} I$.

For statistical learning or ensemble-matched rational wavelets, the above lifting framework can be extended by parameterizing the prediction and update filters and optimizing their coefficients with respect to a prescribed signal (or class) model, e.g., fractional Brownian motion for image rows/columns (Ansari et al., 2017).

2. Learning Algorithms and Signal/Statistical Matching

The key trait of M-RWTL is closed-form, direct derivation of the predictor and updater filter coefficients from data. The process proceeds as follows (Ansari et al., 2017):

• Prediction filter learning: From the observed signal, compute initial detail coefficients $d[n]$ via the rational Lazy wavelet. Construct the rate-converted branch and fit the coefficients $\mathbf{t}$ in $T(z)$ by least-squares minimization of the difference between the predicted and actual details:

$$\min_{\mathbf{t}} \sum_n \bigl(d[n] - a_4[n] \bigr)^2$$

The normal equations admit closed-form LS solutions.

• Update filter learning: Given the partial synthesis after the prediction stage, solve for $S(z)$ (parameterized by $\mathbf{s}$) to minimize the output reconstruction error:

$$\min_{\mathbf{s}} \sum_n \bigl( x_{rl}[n] - x[n] \bigr)^2$$

• Filter updates propagate through the system according to the established lifting and polyphase formulations, ensuring PR.

For statistical (ensemble) learning (Ansari et al., 2017), expected mean-square errors are minimized with respect to a process prior (e.g., fBm), yielding expectation-based normal equations. The result is a convex LS problem and closed-form predictors/updaters.

Recent deep learning extensions, such as variable-projection for rational Gaussian wavelets, treat all filter and scaling parameters (scales, translations, zeros, poles) as differentiable and optimize them jointly with neural network loss (Ámon et al., 3 Feb 2025, Siino et al., 22 Oct 2025).

3. Theoretical Properties and Generalizations

M-RWTL frameworks guarantee:

• Perfect reconstruction: Each lifting stage (predict or update) and the composite system remain invertible by construction, with PR following from polyphase matrix relations.
• Modularity: Multiple lifting stages may be stacked; predictors/updaters (T, S) may be linear or nonlinear (e.g., local adaptive, neural), with invertibility maintained.
• Specialization: As $M = 2, q_1 = q_2 = 1$, the standard dyadic (e.g., 5/3, 9/7) lifting factorizations are recovered as a particular case.
• Arbitrary rational splits and finite impulse response (FIR) filters: Flexible allocation of spectral content and consistent FIR hardware compatibility.

Extensions include higher-tap or longer-support predictors/updaters, full M-band (multichannel) rational lifting, and incorporation of nonlinear adaptive filters (Ansari et al., 2017).

4. Implementation and Computational Aspects

The classical M-RWTL lifting algorithm is computationally efficient:

• Each LS system (for prediction and update) is solved in $O(N \cdot \text{tap-length})$.
• The entire process involves FIR convolutions, resampling, and small-matrix algebra.
• For 2-tap filters, all learning is completed in two steps; increased tap-length or higher-order moments require correspondingly higher computation.

Pseudocode involves:

• Initial rational Lazy wavelet analysis.
• Blockwise construction of predictor matrix and least-squares solve for $T(z)$.
• Filter updates and repetition for the update stage $S(z)$.
• Output of four learned rational wavelet analysis/synthesis filters.

Modern model-driven or deep learning implementations introduce parameters for rational dilations, soft-threshold biases, and amplitude gains per level, which are optimized end-to-end via autodiff in a neural context (Siino et al., 22 Oct 2025).

5. Application Results and Empirical Performance

M-RWTL excels in sparse signal representation and inverse problems, notably compressed sensing (CS):

• 1D signals: On music, ECG, and speech, M-RWTL achieves up to ~1.8 dB PSNR improvements over standard (e.g., Daubechies, biorthogonal) dyadic wavelets on music, with >4–10 dB gain for ECG and speech at low (<50%) sampling ratios.
• Natural images: Statistically learned separable rational wavelets provide 0.6–0.8 dB PSNR gain at 50% sampling, and still ≈0.5 dB at 30% over 5/3 and 9/7 wavelets; these gains are achieved both on training-set and out-of-sample test images (Ansari et al., 2017).
• Neurophysiological signals: Learnable Rationally-Dilated Wavelet Transform (RDWT) front ends have improved worst-case subject accuracy for EEG motor-imagery by +0.17 to +2.54 points across protocols and datasets, with minimal computational overhead—demonstrating enhanced robustness without sacrificing efficiency (Siino et al., 22 Oct 2025).
• Biomedical signals: Rational Gaussian continuous wavelets, learned via variable projection, enable highly efficient, interpretable representations for ECG classification within neural architectures (Ámon et al., 3 Feb 2025).

6. Limitations, Open Questions, and Ongoing Directions

Despite substantial theoretical and empirical advances, several challenges remain:

• Optimal rational split selection: Currently, (q₁, q₂) are chosen based on prior spectral intuition. Automated, data-driven selection remains an open research direction.
• Partial data learning: For compressed sensing, training M-RWTL from only compressive (not fully sampled) measurements is not yet realized.
• Nonlinear and higher-order lifting: Systematic design and optimization of nonlinear, adaptive, or deep predictors/updaters within the rational lifting framework require further exploration.
• Extensions to higher-band and longer-support: Scaling to full M-band rational decomposition and longer predictor/updater filters involves complexity-performance tradeoffs; the impact on sparsity and reconstruction quality is not fully charted.
• Task-driven learning: Integration of task loss (e.g., classification, detection) with rational wavelet parameter learning, as in model-driven neural architectures, is an active area (Ámon et al., 3 Feb 2025, Siino et al., 22 Oct 2025).

A plausible implication is that further advances in adaptive rational wavelet learning will lead to more effective, interpretable, and application-specific transform-based representations at the interface of classical signal processing and modern machine learning.

7. Comparison with Related Rational Wavelet and Transform Learning Approaches

Prior to M-RWTL, rational wavelet design techniques (e.g., spectral-matching, frequency-domain optimization) typically relied on fixed, signal-agnostic filters and offered limited adaptivity (Ansari et al., 2017). Deterministic rational lifting from a single signal was previously proposed, but lacked closed-form ensemble solutions or PR guarantees. The statistical learning of rational wavelets via convex optimization over stochastic models (Ansari et al., 2017) and the signal-matched direct lifting approach (Ansari et al., 2017) represent the first frameworks to unify closed-form, data-driven, and modular rational wavelet design.

Recent work with rational Gaussian wavelets and trainable rational dilations within neural models further expands the scope of M-RWTL, offering fully interpretable, mathematically grounded, and empirically validated tools for both classical and deep learning paradigms (Ámon et al., 3 Feb 2025, Siino et al., 22 Oct 2025).