Wavelet Methods

Wavelet-based financial forecasters typically use the transform only to denoise, or reduce it to a single spectral snapshot at the forecast origin, and the convolution that produces the coefficients is usually bilateral, so it can read past the forecast origin. DSTNet instead retains the recent evolution of filter-bank magnitudes as a causal Dynamic Spectral Trajectory, built from seven trailing technical indicators over a twenty-day lookback with a one-sided Morlet-derived filter bank and an explicit burn-in for the left-boundary transient. A factorized Scale-Temporal Spectral Transformer attends along the time and filter-bank axes separately, a learned gate fuses the spectral branch with a CNN-BiLSTM, and horizon-specific gates emit one, three, five, and ten day forecasts in a single pass. We evaluate seven equity indices and gold under a common expanding-window protocol and an untouched one-year hold-out, against nine learned baselines and a random-walk persistence benchmark. Under MAE and MAPE, persistence is the strongest of the ten fixed competitors in 29 of the 32 series-horizon cells and DSTNet is the only model below it in every cell, by 0.7 to 0.9 percent at one day and 3.4 to 4.5 percent at ten days. At one day, paired testing favours DSTNet against the weaker learned baselines but is inconclusive against persistence and the strongest learned forecasters. A downstream allocation diagnostic does not support an equity-timing advantage on any of the seven indices.

The acoustic front-end determines which forensic cues a speech deepfake detector can exploit. The wavelet scattering transform (WST) provides stable multiscale coefficients with explicit coordinates, yet direct flattening obscures the parent relation between paths. We introduce WST-Graph, reconstructing these paths as a sparse modulation-carrier grid for an AASIST graph backend. Modulation-level normalization and length-aware adaptive local attention pooling produce fixed relative-time representations while retaining the acoustic axes before learned adaptation. This yields a waveform-to-graph interface with a fixed, parameter-free WST. Our configurations remain competitive with AASIST while using approximately 60% fewer trainable parameters and show clear gains on selected out-of-domain benchmarks. These results underscore the value of preserving parent-child relations within the carrier-modulation topology when constructing a compact, physically grounded interface for graph-based speech deepfake detection. Code will be released at https://github.com/saki-ciallo/wst-graph.

Time series analysis is fundamental in domains such as finance, healthcare, and meteorology. Real-world time series often exhibit multiscale characteristics shaped by diverse latent factors, resulting in intricate temporal patterns and rich frequency structures. However, existing approaches typically focus on either frequency-domain decomposition or time-domain pattern extraction in isolation, neglecting their joint structure. This decoupled modeling limits representation expressiveness and undermines performance in tasks requiring simultaneous temporal and spectral reasoning. To address this gap, we propose m-WCN, a novel end-to-end deep learning framework that neuralizes multi-wavelet decomposition for joint extraction of temporal patterns and frequency components. By approximating the classical GHM multi-wavelet transform with trainable convolutional operators and enforcing orthogonality constraints, m-WCN produces interpretable multi-resolution representations. Built on this foundation, we introduce two task-specific architectures: TFBC for time series classification, which boosts discriminative features across frequency scales, and FTB for forecasting, which ensembles frequency-aware predictors. Extensive experiments on 64 UCR datasets and seven public forecasting benchmarks demonstrate the effectiveness of our approach. Built on the neuralized m-WCN, our TFBC and FTB outperform various baseline models across diverse datasets, achieving average improvements of 19.97% in classification and 19.92% in forecasting tasks.

Diffusion models have shown strong potential for kilometer-scale precipitation downscaling, but their performance in geographically unseen regions and event regimes remains insufficiently understood. Building on the wavelet diffusion model (WDM) framework, this study evaluates cross-region and cross-event generalization. Six 3 x 3 deg U.S. regions represent convective, winter, tropical, and atmospheric-river precipitation regimes. Low-resolution inputs are generated by block averaging NOAA Multi-Radar/Multi-Sensor (MRMS) composite reflectivity fields. A WDM trained only on Oklahoma (OK) samples and a WDM trained on all six regions are compared with nearest-neighbor and Bicubic interpolation. Model performance is evaluated using three metric families that measure image-domain reconstruction, spectral and distributional fidelity, and bin-wise precipitation detection. The OK-trained WDM remains competitive outside OK. Although the all-region WDM delivers the best and most consistent overall image-domain and detection performance, its gains are uneven across precipitation intensities. Bin-wise critical success index (CSI) over 5-dBZ reflectivity bins shows that WDM improvements concentrate in localized higher-reflectivity structures, which image-domain metrics partly obscure. In addition, the performance differences among samples are strongly associated with the spatial organization of the precipitation field, quantified by Moran's I as the spatial autocorrelation of each reflectivity bin. The sample-level Moran's I-CSI correlation stratified by sample intensity reaches 0.901 in all six regions, including regions unseen during training. Overall, these findings support future efforts to transfer downscaling models to regions with limited local training data and to generate globally consistent, high-resolution precipitation products.

These notes develop the mathematical foundations and construction of a Fourier-Bessel wavelet family inspired by the disk harmonics of Shaqfa et al.[9]. We begin with the relevant properties of Bessel and modified Bessel functions and introduce the wavelet properties required for the construction. We then derive the Fourier-Bessel disk harmonics as solutions to the Helmholtz equation on the unit disk subject to a Neumann boundary condition. Building on this basis, we construct a wavelet family by applying a Gaussian spatial envelope and introducing a zero-mean correction for the zeroth angular order. We derive the corresponding normalisation constants for $L^2$-based applications and discuss $L^1$-based normalisation for frequency-domain peak consistency. Finally, we derive a closed-form Fourier-domain representation of the resulting wavelets. The main motivation is the approximately linear spacing, which converges to $π$ between consecutive radial eigenvalues. Rather than replacing the conventional dyadic organisation of wavelet families, this construction lays out the foundation to explore whether a more uniform radial frequency allocation can be useful for applications in which broad and balanced frequency coverage is desirable.

This study proposes an integrated framework for the processing and classification of invasive bird species vocalizations within natural soundscapes, characterized by high levels of environmental noise. We address the challenge of signal degradation by employing a Bayesian wavelet shrinkage methodology based on the Epanechnikov kernel prior, which offers a closed form decision rule and high computational efficiency for processing large bioacoustic datasets. The methodology was applied to recordings of three species obtained from the iNaturalist platform: \textit{Euphonia violacea}, \textit{Leiothrix lutea}, and \textit{Passer domesticus}. After signal denoising, we extracted a comprehensive set of features, including Mel-Frequency Cepstral Coefficients (MFCCs) and spectral indices such as entropy and zero-crossing rate. Several supervised learning models: Random Forest, Multinomial Logistic Regression and Support Vector Machine (SVM) were evaluated across different feature dimensionalities. Our results demonstrate that the proposed wavelet based preprocessing significantly enhances classification performance, with the SVM model achieving the highest accuracy (up to 0.9398) under a 10-dimensional MFCC configuration. This research provides a robust statistical tool for automated ecological monitoring and the management of biological invasions.

This work introduces a family of tunable lifting schemes for biorthogonal wavelet filter banks. We propose three lifting strategies: low-pass tuning (LS-LayLatt-LP), high-pass tuning (LS-LayLatt-HP), and a sequential lifting scheme that jointly adapts low- and high-frequency branches (LS-LayLatt-Sequential). All proposed designs are formulated using a lattice-based lifting structure, which guarantees invertibility and stability for arbitrary parameter values within the lifting functions. We evaluated the proposed methods by integrating them into a ResNet-18 backbone for image classification on the Describable Textures Dataset (DTD), as well as for anomaly detection on hazelnut images from the MVTec-AD dataset and private KRC102S dataset. Experimental results demonstrate consistent performance improvements across all evaluated tasks.

The paper considers the construction of two new orthogonal multiwavelets with supercompact support by using the Fast Bauer's method for matrix spectral factorization on the matrix product filter of the orthogonal CL multiwavelet filter. The new multiwavelets possess orthogonality, symmetry/antisymmetry, and one of them provides better coding and smoothness than other supercompact multiwavelets. The performance of the new multiwavelet filters in subband-based edge detection, grayscale and color image compression and 1D and 2D signal denoising is compared with the GHM, SA4, CL, Integer Haar and Alpert multifilters. The comparative analysis shows that new multiwavelets can provides better human visual measures, SSIM and MS-SSIM in image compression and denoising applications.

Wavelet convolution (WTConv) has emerged as an increasingly popular drop-in replacement for standard convolutions, expanding a network's receptive field exponentially with the number of decomposition levels while keeping the parameter count linear. However, its reference implementation is severely memory-bound due to excessive data movement through high-bandwidth memory (HBM). We develop an I/O model of WTConv to characterize this bottleneck and use it to guide three algebraic reformulations: (1) recomputing the inexpensive Haar analysis butterfly on chip, (2) collapsing the multi-level synthesis cascade into a single closed-form pass indexed by output-coordinate bits, and (3) folding learned per-channel scales into the convolution weights. Together, these reformulations enable an I/O-aware fused implementation that substantially reduces HBM traffic. We evaluate the WTConvNeXt configuration across decomposition levels and a broad range of tensor shapes. Despite performing comparable arithmetic, the reference WTConv is substantially slower than the depthwise convolution it replaces. Our reformulation reduces modeled HBM traffic by approximately $2.55\times$, yielding up to a $4.35\times$ training speedup over the reference while roughly halving peak memory usage. Thus, our reformulation preserves the benefits of WTConv while substantially reducing its execution time and memory footprint, removing the systems overhead that previously limited its practical efficiency.

Rip currents are recurrent coastal natural hazards that threaten beachgoers and create operational challenges for lifeguards and coastal managers. Reliable monitoring from standard RGB (red-green-blue) imagery acquired by unmanned aerial vehicles (UAVs) remains difficult because hazardous channels often appear as subtle gaps in breaking waves, foam texture, or sediment patterns, and these signatures are affected by illumination, sea state, and environmental noise. This study presents a physically informed coastal environmental monitoring workflow for detecting visually expressed rip-current indicators that integrates wavelet-derived spatial-frequency texture features with deep learning. We evaluate multiple strategies for incorporating Discrete Wavelet Transform features into convolutional architectures, from computationally efficient channel replacement to dual-stream fusion with attention mechanisms. Performance is assessed against a standard RGB baseline using a task specific convolutional neural network for image-level presence classification and a YOLOv8 model for object-level localization. Under the evaluated dataset conditions, integrating wavelet derived texture features improves performance over RGB-only models. The dual-stream architecture achieves the strongest classification performance, exceeding 95% accuracy with high recall, while channel replacement is most effective for YOLOv8 object detection, reaching 94% mAP@50 for localization. Explainable artificial intelligence analyses provide qualitative evidence that the models attend to visually plausible wave-gap regions associated with rip currents. These results suggest that under the conditions of the evaluated dataset, physically informed wavelet integration may support UAV-based decision-support tools for interpretable beach-safety risk mitigation.

Scientific machine learning methods such as neural operators and physics-informed neural networks have advanced engineering applications and inverse problems, but their training typically requires large volumes of simulated data. This makes data preparation and model training expensive. We propose Graph Wavelet Compressed Sensing (GWCS), a learning-based framework for offline compression of graph signals by representing them as sparse, interpretable wavelet-domain representations using the spectral graph wavelet transform. The framework combines a nonparametric multilevel importance sampler, which retains high-energy wavelet coefficients within each scale for a given compression ratio, with a scale-aware graph neural network that reconstructs the signal from the sparse coefficients. We evaluate the proposed framework on synthetic approximately band-limited graph signals over random graphs and four PDE simulation datasets over meshes, which include Turbulent Radiative Layer, Viscoelastic Instability, Kolmogorov Flow, and Dynamic Stall. We compare against graph signal sampling methods and graph autoencoder baselines. Results demonstrate that the framework achieves high reconstruction fidelity and substantial data compression compared to existing benchmarks.

Traffic-utilisation measurements for network monitoring are corrupted by additive noise and statistical drift: time-dependent change in the signal's mean, variance, distributional shape, or tail behaviour. Static wavelet denoising, calibrated under stationary independent and identically distributed (i.i.d.) Gaussian assumptions, becomes mismatched under drift and, at moderate-to-high signal-to-noise ratio (SNR), over-suppresses useful structure and degrades monitoring decisions. We propose a drift-aware framework treating adaptive wavelet denoising as a preprocessing layer optimised for two tasks: anomaly detection, recovering the multi-scale transient load bursts that noise and drift obscure, and capacity estimation, recovering the operational required capacity $C_{95}$ (95th percentile of utilisation). Because localised bursts are multi-scale structure a wavelet preserves but a low-pass filter removes, detection discriminates denoiser families. A four-detector gate (Page-Hinkley, variance-ratio, Jensen-Shannon, Anderson-Darling) determines when to invoke a learned policy, and a Proximal Policy Optimization agent selects a per-window wavelet configuration over a mixed discrete-continuous action space. Unlike prior work, the reward is downstream task utility, not reconstruction fidelity. The denoiser is benchmarked, per drift type and input SNR, against a low-pass moving-average filter, VisuShrink, SureShrink, BayesShrink, and a Wiener filter. Defining the anomaly target on the clean signal and the drift gate on the corruption keeps both stages non-circular.

Recently, neural operators have shown promising outcomes for learning solution operators of differential equations directly from data. This framework learns a functional mapping from the parameter field to the solution field, enabling the prediction of an entire class of solutions rather than a specific instance. However, existing operators often struggle to capture both global dynamics and fine-scale structure simultaneously. To design an effective operator capable of representing multiscale features, a hierarchical multiscale decomposition framework is required. In this study, we develop the Lifting Neural Operator (LiNO), a multiresolution operator built on the second-generation wavelet lifting scheme. LiNO learns a multiresolution decomposition directly from data by parameterizing the lifting transform. This lifting transformation is adaptive to the underlying solution function and exactly invertible by construction, enabling information-preserving multiscale operator learning. In the lifted multiresolution space, the operator evolves coarse and directional detail coefficients separately, resulting in scale-aware modeling of the underlying physics. We evaluate LiNO on several benchmarks, including Darcy flow, the Poisson equation, the Allen-Cahn equation, the compressible Navier-Stokes equation, and the Gray-Scott reaction-diffusion system. Together, these benchmarks cover a wide range of physical behaviors, including multiscale phenomena, transport-dominated dynamics, and chaotic systems. LiNO demonstrates strong performance on these challenging benchmarks compared with state-of-the-art neural operators. These results suggest that adaptive multiresolution operators provide a promising direction for scientific machine learning.

Large language models (LLMs) are attractive for context-aware time series forecasting because they can integrate heterogeneous textual signals, yet their discrete, language-oriented tokenization and embedding interfaces are misaligned with continuous numerical values, often harming numerical ordering and forecasting reliability. We propose TempoWave, a plug-and-play temporal wavelet digit interface that maps each scalar observation into digit-wise embeddings constructed from multi-wavelet, multi-scale coefficients. By directly overriding standard token representations, TempoWave seamlessly exposes both fine-grained local fluctuations and macro global structures in a transformer-compatible form, ensuring that precise numerical formatting, distinct digit identity, and robustness to common normalization operations are maintained throughout the LLM pipeline. Experiments across five context-enriched forecasting benchmarks demonstrate that TempoWave consistently improves LLM-based forecasters over standard numeric tokenization and alternative embedding interfaces, achieving a new state-of-the-art. These results highlight the numeric interface as a key bottleneck and suggest that principled multi-resolution embeddings can better couple LLMs' contextual reasoning with precise forecasting. Our code is available at https://github.com/DC-research/TempoWAVE and our model can be accessed at https://huggingface.co/Melady/TempoWAVE.

Physics-informed surrogate models can accelerate computational fluid dynamics simulations. However, many existing methods reproduce global flow patterns more reliably than localized multiscale structures. This study presents a physics-informed Fourier-wavelet transformer for next-step velocity-field reconstruction in real-world flow benchmarks. The proposed formulation combines hybrid Fourier-wavelet spectral encoding with physics-biased self-attention based on partial differential equation residual diagnostics. It also uses self-supervised pretraining through Masked Physics Prediction and Equation Consistency Prediction. The experiments are conducted on two real benchmark cases: cylinder-wake flow and fluid-structure interaction. All approaches are evaluated under a shared local protocol and compared with spectral, transformer-based, operator-learning, and physics-informed neural-network baselines. On the cylinder-wake benchmark, the proposed model achieves the best aggregate accuracy, with an all-channel normalized mean-squared error of 0.05875 and an all-channel Pearson correlation coefficient of 0.97019. On the fluid-structure-interaction benchmark, it gives the lowest all-channel normalized mean-squared error of $2.70 \times 10^{-4}$, compared with $4.02 \times 10^{-4}$ for the strongest baseline. Component-wise field comparisons and scale-separated diagnostics further show stronger recovery of localized wake structures, including near-body, wake-core, and far-wake features. The results demonstrate improved real-world flow reconstruction while maintaining a practical accuracy-cost tradeoff.

We introduce PHAST-Net, an attention-guided, physics-informed network for unified estimation of Ideal Time-Frequency Representations (ITFRs), spanning spectral, tempo-based, metrical, and harmonic representations such as Spectrograms, Tempograms, and Metrograms. PHAST-Net learns an application-general mapping from a constellation of wavelet transforms, the proposed Continuous Log-frequency Adaptive Wavelet Transform (CLAWT), to high-resolution, cross-term-suppressed time-frequency (T-F) representations. The proposed constellation of CLAWTs is selected through Cohen's class kernel analysis to maximise curvature coverage in a logarithmic-frequency T-F plane tailored to harmonic signal structure. PHAST-Net further incorporates a proposed physics-informed auxiliary reprojection loss designed to reconstruct the idealised observed CLAWT constellation from the predicted ITFR and the corresponding Cohen's class kernels during training. This auxiliary objective promotes transform consistency and energy conservation, mitigates pathological target sparsity, and enhances optimisation stability. Attention layers further promote effective cross-term suppression across the input constellation. The log-frequency formulation also enables Harmonic PHAST-Net, which estimates a Harmonic ITFR that isolates fundamental structure, supporting robust fundamental-only representations for speech and music, such as derived fundamental Tempograms and Metrograms. We further introduce Spline-PHAST-Net, which parameterises detected and associated T-F ridges as continuous spline trajectories, enabling arbitrary-grid re-rendering and signal reconstruction. Trained on an effectively unbounded procedurally generated dataset, PHAST-Net demonstrates improved accuracy over established approaches, providing a unified framework for high-resolution, cross-term-robust analysis of speech, music, and broader nonstationary signals.

Spiking neural networks (SNNs) have garnered significant interest in computer vision due to their potential for efficiency and biological inspiration. While spiking CNN-based methods have shown promise for image restoration (IR) tasks, their performance is constrained by the inherent receptive field limitations of CNN operations. In the paper, we explore the benefits of discrete wavelet transformation and propose a spiking pyramid wavelet-based model (SPWM) for high-efficient and low-energy target. Specifically, we develop a spiking dual pyramid wavelet (SDPW) block to model long-range dependency and exploit the properties of the degradation in the wavelet domain. Experimental results on several benchmarks demonstrate that SPWM significantly lowers computational costs and energy consumption while maintaining image quality. Our method showcases the potential of SNNs in the field of IR, offering new insights for future applications of resource-limited devices.

Interferometric Synthetic Aperture Radar (InSAR) enables effective monitoring of volcanic deformation; however, the observed signals are often corrupted by atmospheric phase delays, seasonal surface changes, and decorrelation effects. Existing atmospheric correction methods, such as numerical weather model-based methods, can reduce these effects but do not consistently remove atmospheric artefacts and may introduce residual biases. To address these limitations, we propose a novel learning-based method for denoising unwrapped InSAR interferograms, using a hybrid training strategy that combines physically motivated synthetic deformation with real atmospheric noise. Specifically, we introduce WaveDINO, a wavelet-based multi-scale denoising framework conditioned on frozen DINOv3 foundation-model features and terrain information. Training uses synthetic magma-source deformation superimposed on short-term interferograms to expose the network to realistic atmospheric statistics while retaining known ground truth. Performance is evaluated on both controlled synthetic data and long-term real interferograms from Laguna del Maule (Chile) and Campi Flegrei (Italy), with independent GNSS measurements used for validation. WaveDINO consistently outperforms competing models, improving agreement with GNSS measurements, and reducing mean GNSS misfit by approximately 3% and 19% at two sites, respectively, while surpassing weather-model-based corrections.

Fully sampled MRI requires dense k-space acquisition, leading to long scan times, reduced clinical throughput, and increased sensitivity to patient motion. Accelerated MRI addresses this by acquiring undersampled k-space data and reconstructing the missing information computationally. However, reconstruction from undersampled measurements is highly ill-posed and can introduce aliasing artifacts, noise amplification, and loss of anatomical detail. Although conventional parallel imaging and compressed sensing methods mitigate these issues, and deep learning methods have further improved reconstruction quality, preserving high-frequency structures under aggressive undersampling remains challenging. In this work, we propose a Variational Network with a Wavelet-based U-Net (W-UNet) for accelerated MRI reconstruction. The framework combines physics-guided iterative reconstruction with learnable multi-scale frequency representations. Standard pooling operations are replaced with Discrete Wavelet Transform and Inverse Wavelet Transform modules, enabling lossless downsampling while preserving low-frequency structure and high-frequency edge details. Integrated into the refinement and sensitivity map estimation stages, the proposed design improves artifact suppression, feature preservation, and reconstruction fidelity in both single-coil and multi-coil settings. Experiments on fastMRI knee and M4Raw brain datasets show state-of-the-art performance. Ablation studies further confirm the effectiveness of wavelet-based feature decomposition for accelerated MRI reconstruction.

Deep learning has become prevalent in computational pathology pipelines that support tasks such as cancer screening and digital pathology analysis. However, the susceptibility of neural networks to adversarial perturbations raises safety concerns for reliable deployment in clinical practice. In histopathological images, this challenge is exacerbated by the difficulty of distinguishing high-frequency adversarial noise from subtle and diagnostically relevant tissue structures. To address this issue, we propose Stain-Aware Wavelet Regularization (SAWR), an adversarial purification framework that leverages multi-level wavelet-domain regularization based on Haar transform to hierarchically disentangle adversarial perturbations from diagnostic structural information. This spectral constraint is further extended to individual histological channels, enabling stain-specific frequency regulation consistent with the biological properties of Hematoxylin and Eosin. Extensive experiments demonstrate that SAWR improves adversarial robustness by up to 10.69% over the baseline approach, while maintaining texture and spectral fidelity under adversarial perturbations.

This paper studies whether audio, images, and video can share a common wavelet token schema rather than relying on separate modality-specific latent grids. It introduces a preliminary continuous-token model built around a one-level Haar DWT/IDWT frontend, a shared coefficient-token layout, optional structural metadata, lightweight modality value adapters, and a shared token-wise encoder-decoder trunk. On Speech Commands, EuroSAT RGB, and DAVIS 2017 data, a dense shared model reaches 39.92 dB audio, 29.37 dB image, and 23.93 dB video PSNR. A matched-rate sweep under continuous latent scalar budgets indicates that the visual gains are not explained solely by latent capacity, while also showing that additive metadata embeddings are not a universal source of improvement. Finally, fixed-rate energy selection provides a strong non-parametric baseline: energy_global improves average PSNR over uniform selection by 16.73 dB for audio, 16.90 dB for images, and 15.86 dB for video under compressed keep ratios. Masked sparse training reaches 34.45 dB video PSNR with 50% of dense tokens. The results support a unified wavelet token schema and sparse token interface, while stopping short of establishing a universal discrete vocabulary.

State-of-the-art learned image compression (LIC) schemes are increasingly based on hybrid CNN-transformer architectures. To further improve rate-distortion performance, we introduce channel-wise wavelet transforms into both the transformer and entropy-coding components. First, we propose a channel-wise wavelet-domain transformer attention (ChWDTA) mechanism. ChWDTA keeps the efficient windowed spatial self-attention used in modern LIC backbones, but computes the Q/K/V projections on channel-wise wavelet-transformed features before mapping the attention output back with the inverse transform. The resulting Channel-wise Wavelet-Domain Transformer Block (ChWDTB) therefore preserves the spatial tokenization pattern of windowed attention while sparsifying the channel covariance seen by the attention projections. Second, in the entropy-coding stage, we introduce a channel-wise wavelet packet (ChWP) decomposition that produces four equal-sized subbands, which better fit channel-wise slice-based autoregressive entropy modeling. When each channel-wise subband is divided into two slices, we use eight slices for entropy coding. With this configuration, the proposed scheme obtains BD-rate reductions of -17.82%, -19.15%, and -22.56% on the Kodak, CLIC Professional Validation, and Tecnick test sets, respectively. Even when each channel-wise subband is coded as a single slice, the scheme still retains most of the coding gains with lower complexity. The results confirm the advantage of introducing wavelet transform in CNN-transformer-based LIC schemes.

The detection of unmanned aerial vehicles (UAVs) is important for the protection of civilian and military infrastructure. In this paper we propose a cost effective UAV detection system using sound signals obtained from microphones. The recorded signals are passed through a signal processing pipeline which employs interpretable adaptive feature extractors using so-called rational Gaussian wavelets. These adaptive wavelet transformations are embedded into and trained together with an underlying small neural network which detects and classifies UAVs based on the obtained features. This leads to a physically interpretable machine learning algorithm that in addition to classifying UAVs is also capable of detecting UAV swarms. We demonstrate our results using data collected in indoor studio and noisy outdoor environments. We conclude that the proposed method outperforms traditional machine learning approaches for detecting and classifying single UAVs as well as drone swarms, while retaining a high degree of interpretability. Our implementation of the proposed methods is made publicly available for reproducibility.

This work introduces the Wavelet-Laplace Neural Operator (WLNO), a novel neural operator that fuses Haar wavelet multi-scale spatial decomposition with the Laplace-domain pole-residue formulation of the Laplace Neural Operator (LNO). While LNO captures transient and steady-state dynamics through learnable system poles and residues, it lacks an explicit mechanism for extracting spatially localized multi-scale features inherent in complex PDE solutions. WLNO addresses this by augmenting the LNO core with a parallel single-level Haar discrete wavelet transform (DWT) branch that decomposes the lifted feature map into four frequency subbands: approximation (LL), horizontal detail (LH), vertical detail (HL), and diagonal detail (HH) and applies independent learned $1\times1$ convolutions to each subband before reconstruction via the inverse DWT. The two branches are fused through a learnable sigmoid-gated weight $α_\mathrm{wav}$, initialized to give a small initial contribution to the wavelet branch, allowing the model to adaptively balance Laplace-domain dynamics against spatial multi-scale features throughout training. WLNO is evaluated against LNO on five benchmark PDE problems using identical hyperparameters, training data, and evaluation protocols: the diffusion equation, the Burgers equation, the reaction-diffusion system, Darcy flow, and the two-dimensional Navier-Stokes equation. WLNO consistently outperforms LNO on all five problems, with the most pronounced improvement on problems with strong spatial multi-scale structure, such as the Burgers equation with sharp shock fronts and the Navier-Stokes equation with coherent vortical structures, while remaining consistent across smoother and elliptic problems. These results demonstrate that wavelet-based multi-scale spatial decomposition is a principled and effective complement to Laplace-domain operator learning.

Scattering transforms achieve Lipschitz stability and translation invariance, but dense prediction tasks require preserving spatial structure lost in global averaging. We propose Phase-Aware Scattering Encoder-Decoder, which restores this information by explicitly preserving phase in skip connections. On image denoising (BSD68), breaking translation invariance improves PSNR by $+2.17$~dB; phase preservation adds $+1.03$~dB. A novel spatial shuffling ablation ($-1.26$~dB penalty) demonstrates phase encodes location-dependent structure. We conduct a preliminary extensibility study on a second dense prediction task (ISIC skin lesion segmentation), with full cross-validation as ongoing work. This work advances principled wavelet-deep learning integration, showing how phase information complements scattering's stability-expressiveness trade-off in pixel-level prediction.

Accurate emulation of multi-scale physical systems governed by PDEs demands models that remain stable over long autoregressive rollouts while preserving fine-scale structures. Deterministic emulators produce overly-smoothed predictions, while generative approaches better capture details but are costly. Latent-space generative models have emerged as a compromise but with the additional cost of separately pre-trained autoencoders. We propose Wavelet Flow Matching (WFM), a novel generative emulator that overcomes current trade-offs between cost and skill by performing optimal-transport directly in the multi-scale wavelet space. Rather than learning a latent compression, WFM leverages the hierarchical structure of a U-Net to jointly predict transport velocities of a prescribed wavelet representation. On three challenging systems of chaotic fluid dynamics, WFM achieves superior long-horizon stability, accuracy and spectral coherence compared to state-of-the-art models. Our results clearly position the wavelet space as an effective training-free representation for generative emulation of complex physical dynamics.

We present an in-depth analysis of the Koopman semigroup via wavelet transform. Towards this goal, we start by introducing the wavelet-based observables and show that they are eigenfunctions of the Koopman semigroup when this semigroup is considered over the Banach space of continuous functions on a compact forward-invariant set endowed with the supremum norm. We then construct closed-form expressions of the action of the Koopman semigroup and its resolvent in terms of these observables. To approximate the action of Koopman semigroup numerically, we combine Extended Dynamic Mode Decomposition (EDMD) with the proposed wavelet-based observables leading to the Wavelet Dynamic Mode Decomposition via Continuous Wavelet Transform (cWDMD) algorithm. We validate our theoretical results on two numerical examples.

Determinantal point processes (DPPs) have emerged as a kernelized alternative to vanilla independent sampling for generating efficient minibatches, coresets and other parsimonious representations of large-scale datasets. While theoretical foundations and promising empirical performance have been demonstrated, there are two challenges for current proposals for DPP-based coresets or minibatches. The first is the need for families of DPPs with certain key variance reduction properties, usually constructed in a continuous setting, of which there are few known examples. The second is the need for an ad-hoc construction of a discrete DPP defined on a given dataset, that inherits such variance reduction. In this work, we contribute to the programme of establishing DPPs as a subsampling toolbox for ML by advancing on these two fronts. First, we propose new DPPs on the Euclidean space based on wavelets, with provably better accuracy guarantees than the best known rates. Second, we introduce a general method to convert such continuous DPPs, which are more amenable to proving analytical statements, into discrete kernels, which are pertinent for subsampling tasks such as minibatch and coreset constructions. This conversion mechanism simultaneously preserves the desired variance decay and reveals a low-rank decomposition of the discrete kernel, which makes sampling the corresponding DPP computationally inexpensive. En route, we enlarge the class of ML tasks amenable to improvements via DPP-based minibatches and coresets to include objective functions with arbitrarily low regularity, and rate guarantees that explicitly adapt to this regularity.

Spike-based encodings are sparse and energy-efficient, but have largely been formulated probabilistically, disconnected from most signal processing literature. We recast spike encoders as time-causal wavelet frames with quantitative bandwidths and reconstruction error bounds. The proposed wavelets preserve the sparsity and locality of spiking representations, with reconstruction up to spike quantization and time discretization. We demonstrate reconstruction on ECG and audio datasets, achieving a normalized RMSE comparable to continuous wavelet transforms. The spiking wavelets map directly to neuromorphic hardware.

Multivariate time series forecasting remains a challenge due to the complexity of local temporal dynamics and global dependencies across multiple variables. In this paper, we propose \textbf{N}eighboring \textbf{P}atching \textbf{Mixer} (\textbf{NPMixer}), a hierarchical architecture featuring a Learnable Stationary Wavelet Transform that adaptively learns filter coefficients to decompose signals into trend and detail components in a data-dependent manner. Our framework introduces a Neighboring Mixer Block that captures local temporal dynamics through a series of hierarchical MLP layers operating on non-overlapping patches. Specifically, the mixer block utilizes MLPs to learn temporal patterns within and across these patches, expanding the receptive field to capture multi-scale dependencies. A Channel-Mixing Encoder is applied to high-frequency components to learn channel correlations while preserving the stability of the underlying global trend. Extensive experiments on seven benchmark datasets demonstrate that NPMixer consistently outperforms state-of-the-art models, achieving better performance in 20 out of 28 ($71.4\%$) evaluated experimental setups for MSE.