In this work we propose a method for learning wavelet filters directly from data. We accomplish this by framing the discrete wavelet transform as a modified convolutional neural network. We introduce an autoencoder wavelet transform network that is trained using gradient descent. We show that the model is capable of learning structured wavelet filters from synthetic and real data. The learned w...

The Sparse Evaluation Graph has emerged over the past several years as an intermediate representation that captures the data ow information in a program compactly and helps perform data ow analysis e ciently. The contributions of this paper1 are three-fold: We present a linear time algorithm for constructing a variant of the Sparse Evaluation Graph for any data ow analysis problem. Our algorith...

Abstract In image compression and feature extraction, linear expansions are standardly used. It was pointed out by Lee and Seung that the positivity or non-negativity of a linear expansion is a very powerful constraint, that seems to lead to sparse representations for the images. Their technique, called Non-negative Matrix Factorization (NMF), was shown to be useful in approximating high dimens...

An elementary proof of a basic uncertainty principle concerning pairs of representations of WN vectors in different orthonormal bases is provided. The result, slightly stronger than stated before, has a direct impact on the uniqueness property of the sparse representation of such vectors using pairs of orthonormal bases as overcomplete dictionaries. The main contribution in this paper is the im...

In this chapter we review a variety of 3D sparse representations developed in recent years and adapted to different kinds of 3D signals. In particular, we describe 3D wavelets, ridgelets, beamlets and curvelets. We also present very recent 3D sparse representations on the 3D ball adapted to 3D signal naturally observed in spherical coordinates. Illustrative examples are provided for the differe...

This paper is an attempt to both expound and expand upon, from an approximation theorist’s point of view, some of the theoretical results that have been obtained in the sparse representation (compressed sensing) literature. In particular, we consider in detail l1 -approximation, which is fundamental in the theory of sparse representations, and the connection between the theory of sparse represe...