515 research outputs found
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem
Labeling the Features Not the Samples: Efficient Video Classification with Minimal Supervision
Feature selection is essential for effective visual recognition. We propose
an efficient joint classifier learning and feature selection method that
discovers sparse, compact representations of input features from a vast sea of
candidates, with an almost unsupervised formulation. Our method requires only
the following knowledge, which we call the \emph{feature sign}---whether or not
a particular feature has on average stronger values over positive samples than
over negatives. We show how this can be estimated using as few as a single
labeled training sample per class. Then, using these feature signs, we extend
an initial supervised learning problem into an (almost) unsupervised clustering
formulation that can incorporate new data without requiring ground truth
labels. Our method works both as a feature selection mechanism and as a fully
competitive classifier. It has important properties, low computational cost and
excellent accuracy, especially in difficult cases of very limited training
data. We experiment on large-scale recognition in video and show superior speed
and performance to established feature selection approaches such as AdaBoost,
Lasso, greedy forward-backward selection, and powerful classifiers such as SVM.Comment: arXiv admin note: text overlap with arXiv:1411.771
Mathematical Optimization Algorithms for Model Compression and Adversarial Learning in Deep Neural Networks
Large-scale deep neural networks (DNNs) have made breakthroughs in a variety of tasks, such as image recognition, speech recognition and self-driving cars. However, their large model size and computational requirements add a significant burden to state-of-the-art computing systems. Weight pruning is an effective approach to reduce the model size and computational requirements of DNNs. However, prior works in this area are mainly heuristic methods. As a result, the performance of a DNN cannot maintain for a high weight pruning ratio. To mitigate this limitation, we propose a systematic weight pruning framework for DNNs based on mathematical optimization. We first formulate the weight pruning for DNNs as a non-convex optimization problem, and then systematically solve it using alternating direction method of multipliers (ADMM). Our work achieves a higher weight pruning ratio on DNNs without accuracy loss and a higher acceleration on the inference of DNNs on CPU and GPU platforms compared with prior works.
Besides the issue of model size, DNNs are also sensitive to adversarial attacks, a small invisible noise on the input data can fully mislead a DNN. Research on the robustness of DNNs follows two directions in general. The first is to enhance the robustness of DNNs, which increases the degree of difficulty for adversarial attacks to fool DNNs. The second is to design adversarial attack methods to test the robustness of DNNs. These two aspects reciprocally benefit each other towards hardening DNNs. In our work, we propose to generate adversarial attacks with low distortion via convex optimization, which achieves 100% attack success rate with lower distortion compared with prior works. We also propose a unified min-max optimization framework for the adversarial attack and defense on DNNs over multiple domains. Our proposed method performs better compared with the prior works, which use average-based strategies to solve the problems over multiple domains
Sharp recovery bounds for convex demixing, with applications
Demixing refers to the challenge of identifying two structured signals given
only the sum of the two signals and prior information about their structures.
Examples include the problem of separating a signal that is sparse with respect
to one basis from a signal that is sparse with respect to a second basis, and
the problem of decomposing an observed matrix into a low-rank matrix plus a
sparse matrix. This paper describes and analyzes a framework, based on convex
optimization, for solving these demixing problems, and many others. This work
introduces a randomized signal model which ensures that the two structures are
incoherent, i.e., generically oriented. For an observation from this model,
this approach identifies a summary statistic that reflects the complexity of a
particular signal. The difficulty of separating two structured, incoherent
signals depends only on the total complexity of the two structures. Some
applications include (i) demixing two signals that are sparse in mutually
incoherent bases; (ii) decoding spread-spectrum transmissions in the presence
of impulsive errors; and (iii) removing sparse corruptions from a low-rank
matrix. In each case, the theoretical analysis of the convex demixing method
closely matches its empirical behavior.Comment: 51 pages, 13 figures, 2 tables. This version accepted to J. Found.
Comput. Mat
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