238 research outputs found
Non-convex Optimization for Machine Learning
A vast majority of machine learning algorithms train their models and perform
inference by solving optimization problems. In order to capture the learning
and prediction problems accurately, structural constraints such as sparsity or
low rank are frequently imposed or else the objective itself is designed to be
a non-convex function. This is especially true of algorithms that operate in
high-dimensional spaces or that train non-linear models such as tensor models
and deep networks.
The freedom to express the learning problem as a non-convex optimization
problem gives immense modeling power to the algorithm designer, but often such
problems are NP-hard to solve. A popular workaround to this has been to relax
non-convex problems to convex ones and use traditional methods to solve the
(convex) relaxed optimization problems. However this approach may be lossy and
nevertheless presents significant challenges for large scale optimization.
On the other hand, direct approaches to non-convex optimization have met with
resounding success in several domains and remain the methods of choice for the
practitioner, as they frequently outperform relaxation-based techniques -
popular heuristics include projected gradient descent and alternating
minimization. However, these are often poorly understood in terms of their
convergence and other properties.
This monograph presents a selection of recent advances that bridge a
long-standing gap in our understanding of these heuristics. The monograph will
lead the reader through several widely used non-convex optimization techniques,
as well as applications thereof. The goal of this monograph is to both,
introduce the rich literature in this area, as well as equip the reader with
the tools and techniques needed to analyze these simple procedures for
non-convex problems.Comment: The official publication is available from now publishers via
http://dx.doi.org/10.1561/220000005
Non-Local Robust Quaternion Matrix Completion for Large-Scale Color Images and Videos Inpainting
The image nonlocal self-similarity (NSS) prior refers to the fact that a
local patch often has many nonlocal similar patches to it across the image. In
this paper we apply such NSS prior to enhance the robust quaternion matrix
completion (QMC) method and significantly improve the inpainting performance. A
patch group based NSS prior learning scheme is proposed to learn explicit NSS
models from natural color images. The NSS-based QMC algorithm computes an
optimal low-rank approximation to the high-rank color image, resulting in high
PSNR and SSIM measures and particularly the better visual quality. A new joint
NSS-base QMC method is also presented to solve the color video inpainting
problem based quaternion tensor representation. The numerical experiments on
large-scale color images and videos indicate the advantages of NSS-based QMC
over the state-of-the-art methods.Comment: 22 pages, 10 figure
- …