  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.
