Learnable Wavelet Packet Transform (LWPT)

• Learnable Wavelet Packet Transform (LWPT) is a data-adaptive signal processing architecture that replaces fixed wavelet filters with trainable operators for optimized time-frequency representation.
• It preserves the classical packet-tree structure while enabling end-to-end optimization of decomposition filters, reconstruction filters, and threshold-like activations for tasks like denoising and anomaly detection.
• LWPT has been applied to audio spectrogram construction, fault diagnosis, and domain adaptation, demonstrating enhanced performance over traditional fixed wavelet packet techniques.

Learnable Wavelet Packet Transform (LWPT) denotes a class of data-adaptive multiresolution transforms that preserve the recursive filter-bank topology of the classical wavelet packet transform (WPT) while making some or all of its components trainable. In the formulations described in recent work, the trainable components include decomposition filters, reconstruction filters, and threshold-like nonlinearities; the resulting models are typically organized as encoder-decoder architectures or differentiable packet trees optimized end-to-end for reconstruction, sparsity, denoising, anomaly detection, or domain adaptation objectives (Frusque et al., 2022). Relative to fixed WPT, the defining change is that the time-frequency tiling is no longer determined solely by a hand-chosen wavelet family and manually tuned thresholds. Relative to generic deep networks, LWPT retains an explicit packet-tree inductive bias, uniform subband decomposition, and a direct connection to wavelet signal processing (Frusque et al., 2022).

1. Conceptual basis and relation to classical WPT

Classical WPT applies a two-channel filter bank recursively to both the low-pass and high-pass outputs of each node. For an input signal $\mathbf{y}$, one splitting block computes

$$(\mathbf{y}^{lp})_{(n)} = ( \mathbf{h}^{lp} * \mathbf{y} )_{(2n)}, \qquad (\mathbf{y}^{hp})_{(n)} = ( \mathbf{h}^{hp} * \mathbf{y} )_{(2n)},$$

and repeated application over $L$ layers yields $2^L$ terminal nodes with a more uniform frequency partition than the standard discrete wavelet transform, which recursively decomposes only the low-frequency branch (Frusque et al., 2022). This distinction is central in applications where narrowband or high-frequency phenomena are diagnostically important, including acoustic anomaly detection and vibration-based fault diagnosis.

LWPT preserves this packet-tree logic but replaces fixed wavelet analysis by trainable operators. In the denoising formulation, the encoder computes nodewise coefficients as

$$\mathbf{y}_l^{i} = \eta_{\gamma_l^i}\!\left( \theta^{i}_{l} * \mathbf{y}_{l-1}^{\lfloor \frac{i}{2} \rfloor } \right),$$

where $\theta_l^i$ is a learnable analysis kernel and $\eta_{\gamma_l^i}$ is a learnable thresholding activation with bias $\gamma_l^i$ (Frusque et al., 2022). In the spectrogram-oriented formulation, the final-layer packet coefficients

$$\mathbf{Y}_L = [\mathbf{y}_L^0,\mathbf{y}_L^1,\ldots,\mathbf{y}_L^{2^L-1}]$$

are interpreted as a learned time-frequency representation, with $|\mathbf{Y}_L|$ functioning as a data-adapted spectrogram (Frusque et al., 2022).

A common misconception is that LWPT merely inserts wavelet terminology into an otherwise standard CNN. The published formulations do not support that reading. Their defining structural feature is the explicit packet decomposition tree, including stride-2 filtering/downsampling at each node and inverse-tree synthesis or reconstruction. This suggests that LWPT should be understood as a constrained, signal-processing-native deep architecture rather than as a generic convolutional front end.

2. Precursors and related learnable wavelet formulations

LWPT emerged from a broader sequence of efforts that recast wavelet analysis as a differentiable, trainable computation. An important precursor is the “wavenet” construction, which interpreted the discrete wavelet transform as a fully connected, linear, deep neural network with dyadic structure and learned orthonormal wavelet bases by stochastic gradient descent and back-propagation, with orthonormality imposed through quadratic regularization terms (Søgaard, 2017). That framework already treated filter coefficients as trainable parameters and optimized a task-specific criterion—there, sparsity via the Gini coefficient—while retaining wavelet constraints.

A second precursor is M-RWTL, which learned a signal-matched rational wavelet transform in a lifting framework from a single signal or limited data. Its contribution was not packet-tree learning, but structured learning of rational wavelet filters with guaranteed invertibility and perfect reconstruction due to lifting. Because it supports nonuniform subband allocation through rational decimation ratios, it is better regarded as a related learnable filter-bank framework than as a full LWPT (Ansari et al., 2017).

Related work also includes learnable fast wavelet transforms for neural-network compression, where perfect reconstruction and alias cancellation are enforced as soft losses while the wavelet basis is jointly learned with network parameters; this line is directly relevant to LWPT methodology even though it does not implement a wavelet packet tree (Wolter et al., 2020). A different extension appears in normalizing-flow-based Learnable Wavelet Transformation, where generalized lifting is implemented via coupling layers and factor-out plays the role of multilevel subband separation. That model is nonlinear and invertible, and it is wavelet-packet-like in hierarchical spirit, but it recursively factors out detail while continuing on the low-frequency component rather than learning a full packet tree over all branches (Li, 2021).

On the implementation side, fixed fast DWT/IDWT TensorFlow layers provide a WPT-ready scaffold: if the high-frequency features are also decomposed successively, the result is a WPT filter bank. In the form described, those layers use predefined orthogonal or biorthogonal wavelet families rather than trainable filters, so they are not themselves LWPT, but they provide an architectural substrate for differentiable packet decompositions (Tarafdar et al., 5 Apr 2025).

3. Architectural variants and parameterization strategies

Published LWPT models fall into at least two closely related parameterization families. The first family learns a low-pass filter per node and derives the corresponding high-pass filter by symmetry. In L-WPT for data-adapted spectrograms, node $$(\mathbf{y}^{lp})_{(n)} = ( \mathbf{h}^{lp} * \mathbf{y} )_{(2n)}, \qquad (\mathbf{y}^{hp})_{(n)} = ( \mathbf{h}^{hp} * \mathbf{y} )_{(2n)},$$0 at layer $$(\mathbf{y}^{lp})_{(n)} = ( \mathbf{h}^{lp} * \mathbf{y} )_{(2n)}, \qquad (\mathbf{y}^{hp})_{(n)} = ( \mathbf{h}^{hp} * \mathbf{y} )_{(2n)},$$1 computes

$$(\mathbf{y}^{lp})_{(n)} = ( \mathbf{h}^{lp} * \mathbf{y} )_{(2n)}, \qquad (\mathbf{y}^{hp})_{(n)} = ( \mathbf{h}^{hp} * \mathbf{y} )_{(2n)},$$2

$$(\mathbf{y}^{lp})_{(n)} = ( \mathbf{h}^{lp} * \mathbf{y} )_{(2n)}, \qquad (\mathbf{y}^{hp})_{(n)} = ( \mathbf{h}^{hp} * \mathbf{y} )_{(2n)},$$3

with $$(\mathbf{y}^{lp})_{(n)} = ( \mathbf{h}^{lp} * \mathbf{y} )_{(2n)}, \qquad (\mathbf{y}^{hp})_{(n)} = ( \mathbf{h}^{hp} * \mathbf{y} )_{(2n)},$$4 the learnable low-pass filter, $$(\mathbf{y}^{lp})_{(n)} = ( \mathbf{h}^{lp} * \mathbf{y} )_{(2n)}, \qquad (\mathbf{y}^{hp})_{(n)} = ( \mathbf{h}^{hp} * \mathbf{y} )_{(2n)},$$5 the Alternating Flip (AF) high-pass construction, and $$(\mathbf{y}^{lp})_{(n)} = ( \mathbf{h}^{lp} * \mathbf{y} )_{(2n)}, \qquad (\mathbf{y}^{hp})_{(n)} = ( \mathbf{h}^{hp} * \mathbf{y} )_{(2n)},$$6 the learnable threshold bias. The AF and Anti-Cancellation Conditions (ACC) reduce the number of filters to learn by a factor of four, although the kernel property is not preserved (Frusque et al., 2022).

The second family learns both analysis and synthesis kernels explicitly. In the denoising LWPT, the decoder reconstructs recursively via learnable synthesis filters $$(\mathbf{y}^{lp})_{(n)} = ( \mathbf{h}^{lp} * \mathbf{y} )_{(2n)}, \qquad (\mathbf{y}^{hp})_{(n)} = ( \mathbf{h}^{hp} * \mathbf{y} )_{(2n)},$$7: $$(\mathbf{y}^{lp})_{(n)} = ( \mathbf{h}^{lp} * \mathbf{y} )_{(2n)}, \qquad (\mathbf{y}^{hp})_{(n)} = ( \mathbf{h}^{hp} * \mathbf{y} )_{(2n)},$$8 with

$$(\mathbf{y}^{lp})_{(n)} = ( \mathbf{h}^{lp} * \mathbf{y} )_{(2n)}, \qquad (\mathbf{y}^{hp})_{(n)} = ( \mathbf{h}^{hp} * \mathbf{y} )_{(2n)},$$9

This formulation begins from a classical perfect-reconstruction wavelet filter $L$0, such as Daubechies db4, and initializes all nodewise filters from it, with thresholds initialized to zero so that the network initially behaves close to a standard WPT (Frusque et al., 2022).

Both families use a learnable hard-threshold-like nonlinearity. In L-WPT, the activation is

$L$1

while the smart-filter LWPT used in domain adaptation employs

$L$2

with the bias acting as thresholds on both sides of zero (Frusque et al., 2022, Dai et al., 2023). In both cases, the point is to retain the coefficient-selection semantics of hard thresholding while keeping the model differentiable.

This suggests a useful distinction between filter-learning LWPT and filter-plus-shrinkage LWPT. The former adapts the packet filters; the latter adapts both the packet filters and the denoising rule at each node. The currently reported LWPT systems belong to the second category.

4. Objectives, constraints, and optimization

The optimization objective in LWPT depends on the intended use, but three recurrent design principles appear across the literature: reconstruction fidelity, coefficient sparsity, and wavelet-structure regularization.

For learned spectrogram construction, the objective combines signal reconstruction and sparse final-layer coefficients: $L$3 where $L$4 collects all filters and $L$5 all thresholds. The stated rationale is that the $L$6-norm approximates sparsity while keeping the scale of the two terms comparable, and $L$7 gives equal weight because $L$8 and $L$9 contain the same number of elements (Frusque et al., 2022).

For supervised denoising, the objective is direct signal recovery from noisy observations $2^L$0: $2^L$1 with learnable analysis filters $2^L$2, synthesis filters $2^L$3, and threshold biases $2^L$4 (Frusque et al., 2022). This objective does not impose exact wavelet constraints explicitly; instead, it relies on wavelet-based initialization, packet structure, and end-to-end training.

In noisy unsupervised domain adaptation, LWPT functions as part of a smart filter trained jointly with a DANN. The LWPT-specific term is a guidance loss on time-frequency coefficients,

$2^L$5

combined with classification and adversarial domain-alignment losses: $2^L$6 where $2^L$7 follows the schedule

$2^L$8

The purpose is to dynamically enforce similarity between source and target time-frequency representations before feature-level alignment (Dai et al., 2023).

The precursor “wavenet” work shows a complementary strategy in which wavelet validity is enforced through differentiable penalties. There the total objective is

$2^L$9

with $$\mathbf{y}_l^{i} = \eta_{\gamma_l^i}\!\left( \theta^{i}_{l} * \mathbf{y}_{l-1}^{\lfloor \frac{i}{2} \rfloor } \right),$$0 defined from the Gini coefficient for sparsity and $$\mathbf{y}_l^{i} = \eta_{\gamma_l^i}\!\left( \theta^{i}_{l} * \mathbf{y}_{l-1}^{\lfloor \frac{i}{2} \rfloor } \right),$$1 a sum of quadratic penalties for normalization, orthonormality, zero-mean wavelet condition, and related constraints (Søgaard, 2017). A plausible implication is that later LWPT systems inherited the general template of optimizing a task loss together with differentiable structural constraints, even when the exact constraints differ.

5. Empirical behavior and principal application domains

The main application domains currently documented for LWPT are data-adapted spectrogram construction, anomaly detection, denoising, and noisy-domain fault diagnosis.

In acoustic monitoring, L-WPT was evaluated on frequency-swept cosine signals

$$\mathbf{y}_l^{i} = \eta_{\gamma_l^i}\!\left( \theta^{i}_{l} * \mathbf{y}_{l-1}^{\lfloor \frac{i}{2} \rfloor } \right),$$2

and compared against WPT-db4, WPT-db23, Butterworth WPT, and Elliptic WPT. The reported result is that L-WPT consistently exhibited the lowest spectral leakage, with RSS $$\mathbf{y}_l^{i} = \eta_{\gamma_l^i}\!\left( \theta^{i}_{l} * \mathbf{y}_{l-1}^{\lfloor \frac{i}{2} \rfloor } \right),$$3 for the pure sweep $$\mathbf{y}_l^{i} = \eta_{\gamma_l^i}\!\left( \theta^{i}_{l} * \mathbf{y}_{l-1}^{\lfloor \frac{i}{2} \rfloor } \right),$$4, slightly better than WPT-butt at $$\mathbf{y}_l^{i} = \eta_{\gamma_l^i}\!\left( \theta^{i}_{l} * \mathbf{y}_{l-1}^{\lfloor \frac{i}{2} \rfloor } \right),$$5, and $$\mathbf{y}_l^{i} = \eta_{\gamma_l^i}\!\left( \theta^{i}_{l} * \mathbf{y}_{l-1}^{\lfloor \frac{i}{2} \rfloor } \right),$$6 at $$\mathbf{y}_l^{i} = \eta_{\gamma_l^i}\!\left( \theta^{i}_{l} * \mathbf{y}_{l-1}^{\lfloor \frac{i}{2} \rfloor } \right),$$7 versus $$\mathbf{y}_l^{i} = \eta_{\gamma_l^i}\!\left( \theta^{i}_{l} * \mathbf{y}_{l-1}^{\lfloor \frac{i}{2} \rfloor } \right),$$8 for WPT-butt (Frusque et al., 2022). On the MIMII dataset for one-class acoustic fault detection using a 1-class Extreme Learning Machine with 100 neurons and features given by the mean and maximum of the residual and each frequency band, L-WPT achieved a mean AUC of $$\mathbf{y}_l^{i} = \eta_{\gamma_l^i}\!\left( \theta^{i}_{l} * \mathbf{y}_{l-1}^{\lfloor \frac{i}{2} \rfloor } \right),$$9, compared with $\theta_l^i$0 for WPT, $\theta_l^i$1 for DesPawn, and $\theta_l^i$2 for WPT-CNN (Frusque et al., 2022).

In supervised denoising, LWPT was evaluated on the standard 1D signal classes Block, Bumps, HeaviSine, and Doppler, each of length $\theta_l^i$3, and on airport background-noise suppression using DCASE 2018 Task 1 and 2 audio. The reported conclusion is that LWPT maintains excellent denoising performance on signal classes separate from those used during training and is robust to different noise intensities, noise varieties, and artifacts. A distinctive result is the post-training $\theta_l^i$4-modification

$\theta_l^i$5

which adapts the learned thresholds to new noise levels without retraining; with $\theta_l^i$6, LWPT-$\theta_l^i$7 is reported to outperform deep nets by up to $\theta_l^i$8 in denoising capability (Frusque et al., 2022).

In noisy industrial domain adaptation, LWPT is used as the learnable branch of a smart filter preceding DANN. For bearing diagnosis under severe noise, the hybrid WPT+LWPT smart filter outperformed the two-LWPT variant SFDANN-v; for example, at $\theta_l^i$9, average accuracy was $\eta_{\gamma_l^i}$0 for SFDANN-v and $\eta_{\gamma_l^i}$1 for SFDANN (Dai et al., 2023). For slab-track diagnosis in simulation-to-field transfer, the reported accuracies were $\eta_{\gamma_l^i}$2 for SFDANN-v, $\eta_{\gamma_l^i}$3 with the source-to-LWPT strategy, and $\eta_{\gamma_l^i}$4 with the target-to-LWPT strategy, the latter being the best setting (Dai et al., 2023). The paper interprets these results as evidence that sending the noisier domain through LWPT and the cleaner domain through fixed WPT improves adaptation stability.

The earlier learnable wavelet-basis work also demonstrated that gradient-based learning can recover the Haar wavelet in both low-dimensional and $\eta_{\gamma_l^i}$5 search spaces, and that for pixelized QCD $\eta_{\gamma_l^i}$6 collision events with $\eta_{\gamma_l^i}$7 inputs from Pythia8 at $\eta_{\gamma_l^i}$8 TeV, the learned basis again converged to Haar (Søgaard, 2017). Although this is not an LWPT paper, it established the feasibility of learning wavelet filters directly from data in a high-dimensional optimization landscape.

6. Limitations, distinctions, and open directions

LWPT should not be conflated with any learnable wavelet transform. Several related methods learn wavelet filters, lifting polynomials, or invertible nonlinear wavelet-like maps without learning a full packet tree. M-RWTL, for example, learns a single 2-channel rational wavelet transform and supports nonuniform subband splits, but it does not include packet-node splitting strategy, recursive learning across levels, learned tree topology, or best-basis selection; it is therefore a precursor or component technology rather than a complete LWPT (Ansari et al., 2017). Similarly, normalizing-flow-based Learnable Wavelet Transformation is multilevel and wavelet-like, but it repeatedly decomposes the low-frequency part rather than constructing a full packet decomposition over all branches (Li, 2021).

A second distinction concerns perfect reconstruction. In standard WPT, perfect reconstruction follows from the filter-bank constraints. In learnable formulations, that guarantee may weaken. L-WPT explicitly notes that the kernel property is not preserved, so exact perfect reconstruction is no longer guaranteed even though reconstruction exists via the inverse structure (Frusque et al., 2022). In the smart-filter setting, perfect reconstruction is likewise described as infeasible because the learnable hard-thresholding blocks intentionally alter the coefficients (Dai et al., 2023). By contrast, lifting-based rational transform learning preserves perfect reconstruction by construction (Ansari et al., 2017). This suggests that “learnable” and “perfectly reconstructing” are independent design choices rather than synonymous properties.

A third limitation is dimensional scope. The fixed-layer library for TensorFlow supports separable DWT/IDWT in 1D, 2D, 3D, and higher dimensions through successive circular convolutions along each axis, and it explicitly notes that WPT arises by recursively decomposing high-frequency branches as well (Tarafdar et al., 5 Apr 2025). However, the learnable LWPT formulations summarized here are demonstrated only for univariate signals in the main experiments, and multidimensional extensions are identified as future work (Frusque et al., 2022, Frusque et al., 2022).

Hyperparameter sensitivity is also documented. In anomaly detection, the sparsity weight $\eta_{\gamma_l^i}$9 is noted as possibly too strong for the valve case, potentially over-regularizing the representation (Frusque et al., 2022). In the smart-filter framework, the number of decomposition layers must be chosen with regard to sampling frequency and physically meaningful bandwidth; increasing $\gamma_l^i$0 increases CNN complexity, adds learnable parameters, can overfit limited data, and may create overly fine frequency bands without physical significance for diagnosis (Dai et al., 2023). For coefficient-guidance losses based on batch means, batch size must also be large enough that the averages are meaningful (Dai et al., 2023).

The most plausible near-term directions are therefore not unrestricted deepening, but tighter structural control: additional reconstruction or orthogonality constraints, principled multivariate extensions, interpretability analysis of learned bands, and more explicit optimization of packet structure rather than only packet filters (Frusque et al., 2022). This suggests that the central research question for LWPT is no longer whether wavelet packet analysis can be made learnable, but how much of classical wavelet theory can be retained while preserving the empirical advantages of data adaptation.