# Fully Learnable Deep Wavelet Transform for Unsupervised Monitoring of High-Frequency Time Series

arXiv: https://arxiv.org/abs/2105.00899 · HTML: https://mlsanity.com/paper/2105.00899/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 44 references that are in the map.
- First posted on arXiv 3 May 2021.

## Builds on (most cited first, 15 of 15)

- [Deep learning for time series classification: a review](https://mlsanity.com/paper/1809.04356/) (2018)
- [1D Convolutional Neural Networks and Applications: A Survey](https://mlsanity.com/paper/1905.03554/) (2019)
- [Optimal Approximation with Sparsely Connected Deep Neural Networks](https://mlsanity.com/paper/1705.01714/) (2017)
- [Potential, Challenges and Future Directions for Deep Learning in Prognostics and Health Management Applications](https://mlsanity.com/paper/2005.02144/) (2020)
- [Multilevel Wavelet Decomposition Network for Interpretable Time Series Analysis](https://mlsanity.com/paper/1806.08946/) (2018)
- [Multi-level Wavelet Convolutional Neural Networks](https://mlsanity.com/paper/1907.03128/) (2019)
- [Kymatio: Scattering Transforms in Python](https://mlsanity.com/paper/1812.11214/) (2018)
- [MIMII Dataset: Sound Dataset for Malfunctioning Industrial Machine Investigation and Inspection](https://mlsanity.com/paper/1909.09347/) (2019)
- [Gabor Convolutional Networks](https://mlsanity.com/paper/1705.01450/) (2017)
- [Deep Scattering Spectrum](https://mlsanity.com/paper/1304.6763/) (2013)
- [Convolutional Neural Networks Analyzed via Convolutional Sparse Coding](https://mlsanity.com/paper/1607.08194/) (2016)
- [WaveletKernelNet: An Interpretable Deep Neural Network for Industrial Intelligent Diagnosis](https://mlsanity.com/paper/1911.07925/) (2019)
- [Fourier Neural Networks: A Comparative Study](https://mlsanity.com/paper/1902.03011/) (2019)
- [Missing-Class-Robust Domain Adaptation by Unilateral Alignment for Fault Diagnosis](https://mlsanity.com/paper/2001.02015/) (2020)
- [Feature Learning for Fault Detection in High-Dimensional Condition-Monitoring Signals](https://mlsanity.com/paper/1810.05550/) (2018)

## Built on by (most cited first, 29 of 29)

- [A Survey on Deep Learning based Time Series Analysis with Frequency Transformation](https://mlsanity.com/paper/2302.02173/) (2023)
- [Robust Time Series Denoising with Learnable Wavelet Packet Transform](https://mlsanity.com/paper/2206.06126/) (2022)
- [TCCT: Tightly-Coupled Convolutional Transformer on Time Series Forecasting](https://mlsanity.com/paper/2108.12784/) (2021)
- [Time Series Embedding Methods for Classification Tasks: A Review](https://mlsanity.com/paper/2501.13392/) (2025)
- [Learnable Wavelet Packet Transform for Data-Adapted Spectrograms](https://mlsanity.com/paper/2201.11069/) (2022)
- [FTMixer: Frequency and Time Domain Representations Fusion for Time Series Modeling](https://mlsanity.com/paper/2405.15256/) (2024)
- [Learning Informative Health Indicators Through Unsupervised Contrastive Learning](https://mlsanity.com/paper/2208.13288/) (2022)
- [TFN: An Interpretable Neural Network with Time-Frequency Transform Embedded for Intelligent Fault Diagnosis](https://mlsanity.com/paper/2209.01992/) (2022)
- [WavSpA: Wavelet Space Attention for Boosting Transformers' Long Sequence Learning Ability](https://mlsanity.com/paper/2210.01989/) (2022)
- [WaveRoRA: Wavelet Rotary Route Attention for Multivariate Time Series Forecasting](https://mlsanity.com/paper/2410.22649/) (2024)
- [Domain knowledge-informed Synthetic fault sample generation with Health Data Map for cross-domain Planetary Gearbox Fault Diagnosis](https://mlsanity.com/paper/2305.19569/) (2023)
- [Interpreting What Typical Fault Signals Look Like via Prototype-matching](https://mlsanity.com/paper/2403.07033/) (2024)
- [A Comparison of Deep Learning Architectures for Spacecraft Anomaly Detection](https://mlsanity.com/paper/2403.12864/) (2024)
- [Beyond Fixed Patches: Enhancing GPTs for Financial Prediction with Adaptive Segmentation and Learnable Wavelets](https://mlsanity.com/paper/2505.02880/) (2025)
- [How (Not) to Hybridize Neural and Mechanistic Models for Epidemiological Forecasting](https://mlsanity.com/paper/2602.06323/) (2026)
- [A Bidirectional Long Short Term Memory Approach for Infrastructure Health Monitoring Using On-board Vibration Response](https://mlsanity.com/paper/2412.02643/) (2024)
- [SHapley Estimated Explanation (SHEP): A Fast Post-Hoc Attribution Method for Interpreting Intelligent Fault Diagnosis](https://mlsanity.com/paper/2504.03773/) (2025)
- [RINS-T: Robust Implicit Neural Solvers for Time Series Linear Inverse Problems](https://mlsanity.com/paper/2510.17396/) (2025)
- [WavPool: A New Block for Deep Neural Networks](https://mlsanity.com/paper/2306.08734/) (2023)
- [Smart filter aided domain adversarial neural network for fault diagnosis in noisy industrial scenarios](https://mlsanity.com/paper/2307.01429/) (2023)
- [WCCNet: Wavelet-context Cooperative Network for Efficient Multispectral Pedestrian Detection](https://mlsanity.com/paper/2308.01042/) (2023)
- [A Generic Machine Learning Framework for Fully-Unsupervised Anomaly Detection with Contaminated Data](https://mlsanity.com/paper/2308.13352/) (2023)
- [WaveletInception Networks for on-board Vibration-Based Infrastructure Health Monitoring](https://mlsanity.com/paper/2507.12969/) (2025)
- [Physics-Constrained Denoising Autoencoders for Data-Scarce Wildfire UAV Sensing](https://mlsanity.com/paper/2601.11794/) (2026)
- [HPMixer: Hierarchical Patching for Multivariate Time Series Forecasting](https://mlsanity.com/paper/2602.16468/) (2026)
- [Towards Real-Time ECG and EMG Modeling on μNPUs](https://mlsanity.com/paper/2604.18067/) (2026)
- [NPMixer: Hierarchical Neighboring Patch Mixing for Time Series Forecasting](https://mlsanity.com/paper/2605.07476/) (2026)
- [Task-Directed Residual AddUNet:Perfect-Reconstruction Routing for Full-Rate Representations](https://mlsanity.com/paper/2609.15857/) (2026)
- [Wavelet Flow Matching for Time Series](https://mlsanity.com/paper/2609.39374/) (2026)

## Abstract (arXiv)

High-Frequency (HF) signals are ubiquitous in the industrial world and are of great use for monitoring of industrial assets. Most deep learning tools are designed for inputs of fixed and/or very limited size and many successful applications of deep learning to the industrial context use as inputs extracted features, which is a manually and often arduously obtained compact representation of the original signal. In this paper, we propose a fully unsupervised deep learning framework that is able to extract a meaningful and sparse representation of raw HF signals. We embed in our architecture important properties of the fast discrete wavelet transformation (FDWT) such as (1) the cascade algorithm, (2) the conjugate quadrature filter property that links together the wavelet, the scaling and transposed filter functions, and (3) the coefficient denoising. Using deep learning, we make this architecture fully learnable: both the wavelet bases and the wavelet coefficient denoising are learnable. To achieve this objective, we propose a new activation function that performs a learnable hard-thresholding of the wavelet coefficients. With our framework, the denoising FDWT becomes a fully learnable unsupervised tool that does neither require any type of pre- nor post-processing, nor any prior knowledge on wavelet transform. We demonstrate the benefits of embedding all these properties on three machine-learning tasks performed on open source sound datasets. We perform an ablation study of the impact of each property on the performance of the architecture, achieve results well above baseline and outperform other state-of-the-art methods.