372 research outputs found
Choose your path wisely: gradient descent in a Bregman distance framework
We propose an extension of a special form of gradient descent --- in the
literature known as linearised Bregman iteration -- to a larger class of
non-convex functions. We replace the classical (squared) two norm metric in the
gradient descent setting with a generalised Bregman distance, based on a
proper, convex and lower semi-continuous function. The algorithm's global
convergence is proven for functions that satisfy the Kurdyka-\L ojasiewicz
property. Examples illustrate that features of different scale are being
introduced throughout the iteration, transitioning from coarse to fine. This
coarse-to-fine approach with respect to scale allows to recover solutions of
non-convex optimisation problems that are superior to those obtained with
conventional gradient descent, or even projected and proximal gradient descent.
The effectiveness of the linearised Bregman iteration in combination with early
stopping is illustrated for the applications of parallel magnetic resonance
imaging, blind deconvolution as well as image classification with neural
networks
Enhanced CNN for image denoising
Owing to flexible architectures of deep convolutional neural networks (CNNs),
CNNs are successfully used for image denoising. However, they suffer from the
following drawbacks: (i) deep network architecture is very difficult to train.
(ii) Deeper networks face the challenge of performance saturation. In this
study, the authors propose a novel method called enhanced convolutional neural
denoising network (ECNDNet). Specifically, they use residual learning and batch
normalisation techniques to address the problem of training difficulties and
accelerate the convergence of the network. In addition, dilated convolutions
are used in the proposed network to enlarge the context information and reduce
the computational cost. Extensive experiments demonstrate that the ECNDNet
outperforms the state-of-the-art methods for image denoising.Comment: CAAI Transactions on Intelligence Technology[J], 201
On the convergence analysis of DCA
In this paper, we propose a clean and general proof framework to establish
the convergence analysis of the Difference-of-Convex (DC) programming algorithm
(DCA) for both standard DC program and convex constrained DC program. We first
discuss suitable assumptions for the well-definiteness of DCA. Then, we focus
on the convergence analysis of DCA, in particular, the global convergence of
the sequence generated by DCA under the Lojasiewicz subgradient
inequality and the Kurdyka-Lojasiewicz property respectively. Moreover, the
convergence rate for the sequences and are also
investigated. We hope that the proof framework presented in this article will
be a useful tool to conveniently establish the convergence analysis for many
variants of DCA and new DCA-type algorithms
Riemannian Optimization via Frank-Wolfe Methods
We study projection-free methods for constrained Riemannian optimization. In
particular, we propose the Riemannian Frank-Wolfe (RFW) method. We analyze
non-asymptotic convergence rates of RFW to an optimum for (geodesically) convex
problems, and to a critical point for nonconvex objectives. We also present a
practical setting under which RFW can attain a linear convergence rate. As a
concrete example, we specialize Rfw to the manifold of positive definite
matrices and apply it to two tasks: (i) computing the matrix geometric mean
(Riemannian centroid); and (ii) computing the Bures-Wasserstein barycenter.
Both tasks involve geodesically convex interval constraints, for which we show
that the Riemannian "linear oracle" required by RFW admits a closed-form
solution; this result may be of independent interest. We further specialize RFW
to the special orthogonal group and show that here too, the Riemannian "linear
oracle" can be solved in closed form. Here, we describe an application to the
synchronization of data matrices (Procrustes problem). We complement our
theoretical results with an empirical comparison of Rfw against
state-of-the-art Riemannian optimization methods and observe that RFW performs
competitively on the task of computing Riemannian centroids.Comment: Under Review. Largely revised version, including an extended
experimental section and an application to the special orthogonal group and
the Procrustes proble
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