IEEE Signal Processing - July 2018 - 72
Exploiting Structure in Big Data Analytics:
Sparse and Low-Rank Structures
Vardan Papyan, Yaniv Romano,
Jeremias Sulam, and Michael Elad
Theoretical Foundations of Deep Learning
via Sparse Representations
A multilayer sparse model and its connection to convolutional neural networks
M
odeling data is the way we-scientists-believe that information should be explained and handled. Indeed, models
play a central role in practically every task in signal and
image processing and machine learning. Sparse representation
theory (we shall refer to it as Sparseland) puts forward an emerging, highly effective, and universal model. Its core idea is the
description of data as a linear combination of few atoms taken
from a dictionary of such fundamental elements.
Our prime objective in this article is to review a recently
introduced [1] model-based explanation of deep learning, which
relies on sparse modeling of data. We start by presenting the
general story of Sparseland, describing its key achievements.
We then turn to describe the convolutional-sparse-coding (CSC)
model and present a multilayered (ML) extension of it, both
being special cases of the general sparsity-inspired model. We
show how ML-CSC leads to a solid and systematic theoretical
justification of the architectures used in deep learning, along
with a novel ability of analyzing the performance of the resulting machines. As such, this work offers a unique and first of
its kind theoretical view for a field that has been, until recently,
considered as purely heuristic.
Introduction
©istockphoto.com/shuoshu
Digital Object Identifier 10.1109/MSP.2018.2820224
Date of publication: 27 June 2018
72
The field of sparse and redundant representations has made a
major leap in the past two decades. Starting with a series of
infant ideas and few mathematical observations, it grew to
become a mature and highly influential discipline. Sparseland
puts forward a universal mathematical model for describing the
inherent low dimensionality that may exist in natural data sources. This model suggests a description of signals as linear combinations of few columns, called atoms, from a given redundant
matrix, termed dictionary. In other words, these signals admit
sparse representations with respect to their corresponding dictionary of prototype signals.
The Sparseland model gradually became central in signal and
image processing and machine-learning applications, leading to
state-of-the-art results in a wide variety of tasks and across many
different domains. A partial explanation for the great appeal that
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July 2018
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