# Wavelet Design in a Learning Framework

arXiv: https://arxiv.org/abs/2107.11225 · HTML: https://mlsanity.com/paper/2107.11225/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 63 references that are in the map.
- First posted on arXiv 23 Jul 2021.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Lossy Image Compression with Compressive Autoencoders](https://mlsanity.com/paper/1703.00395/) (2017)
- [Multi-level Wavelet-CNN for Image Restoration](https://mlsanity.com/paper/1805.07071/) (2018)
- [A deep convolutional neural network using directional wavelets for low-dose X-ray CT reconstruction](https://mlsanity.com/paper/1610.09736/) (2016)
- [Scattering Networks for Hybrid Representation Learning](https://mlsanity.com/paper/1809.06367/) (2018)
- [Multiscale Adaptive Representation of Signals: I. The Basic Framework](https://mlsanity.com/paper/1507.04835/) (2015)
- [Learning Filter Bank Sparsifying Transforms](https://mlsanity.com/paper/1803.01980/) (2018)

## Abstract (arXiv)

Wavelets have proven to be highly successful in several signal and image processing applications. Wavelet design has been an active field of research for over two decades, with the problem often being approached from an analytical perspective. In this paper, we introduce a learning based approach to wavelet design. We draw a parallel between convolutional autoencoders and wavelet multiresolution approximation, and show how the learning angle provides a coherent computational framework for addressing the design problem. We aim at designing data-independent wavelets by training filterbank autoencoders, which precludes the need for customized datasets. In fact, we use high-dimensional Gaussian vectors for training filterbank autoencoders, and show that a near-zero training loss implies that the learnt filters satisfy the perfect reconstruction property with very high probability. Properties of a wavelet such as orthogonality, compact support, smoothness, symmetry, and vanishing moments can be incorporated by designing the autoencoder architecture appropriately and with a suitable regularization term added to the mean-squared error cost used in the learning process. Our approach not only recovers the well known Daubechies family of orthogonal wavelets and the Cohen-Daubechies-Feauveau family of symmetric biorthogonal wavelets, but also learns wavelets outside these families.