49,980 research outputs found
New algorithms for the dual of the convex cost network flow problem with application to computer vision
Motivated by various applications to computer vision, we consider an integer convex optimization problem which is the dual of the convex cost network flow problem. In this
paper, we first propose a new primal algorithm for computing an optimal solution of the problem. Our primal algorithm iteratively updates primal variables by solving associated
minimum cut problems. The main contribution in this paper is to provide a tight bound for the number of the iterations. We show that the time complexity of the primal algorithm is K ¢ T(n;m) where K is the range of primal variables and T(n;m) is the time needed to compute a minimum cut in a graph with n nodes and m edges.
We then propose a primal-dual algorithm for the dual of the convex cost network flow problem. The primal-dual algorithm can be seen as a refined version of the primal algorithm by maintaining dual variables (flow) in addition to primal variables. Although its time complexity is the same as that for the primal algorithm, we can expect a better performance
practically.
We finally consider an application to a computer vision problem called the panoramic stitching problem. We apply several implementations of our primal-dual algorithm to some
instances of the panoramic stitching problem and test their practical performance.
We also show that our primal algorithm as well as the proofs can be applied to the L\-convex function minimization problem which is a more general problem than the dual of the convex cost network flow problem
Stochastic Primal-Dual Coordinate Method for Regularized Empirical Risk Minimization
We consider a generic convex optimization problem associated with regularized
empirical risk minimization of linear predictors. The problem structure allows
us to reformulate it as a convex-concave saddle point problem. We propose a
stochastic primal-dual coordinate (SPDC) method, which alternates between
maximizing over a randomly chosen dual variable and minimizing over the primal
variable. An extrapolation step on the primal variable is performed to obtain
accelerated convergence rate. We also develop a mini-batch version of the SPDC
method which facilitates parallel computing, and an extension with weighted
sampling probabilities on the dual variables, which has a better complexity
than uniform sampling on unnormalized data. Both theoretically and empirically,
we show that the SPDC method has comparable or better performance than several
state-of-the-art optimization methods
A Primal Dual Smoothing Framework for Max-Structured Nonconvex Optimization
We propose a primal dual first-order smoothing framework for solving a class
of nonsmooth nonconvex optimization problems with max-structure. We analyze the
primal and dual oracle complexities of the framework via two approaches, i.e.,
the dual-then-primal and primal-the-dual smoothing approaches. Our framework
improves the best-known oracle complexities of the existing methods, even in
the restricted problem setting. As the cornerstone of our framework, we propose
a conceptually simple primal dual method for solving a class of convex-concave
saddle-point problems with primal strong convexity, which is based on a newly
developed non-Hilbertian inexact accelerated proximal gradient algorithm. This
primal dual method has a dual oracle complexity that is significantly better
than the previous ones, and a primal oracle complexity that matches the
best-known, up to logarithmic factor. Finally, we extend our framework to the
stochastic case, and demonstrate that the oracle complexities of this extension
indeed match the state-of-the-art.Comment: 37 page
Regularization and Kernelization of the Maximin Correlation Approach
Robust classification becomes challenging when each class consists of
multiple subclasses. Examples include multi-font optical character recognition
and automated protein function prediction. In correlation-based
nearest-neighbor classification, the maximin correlation approach (MCA)
provides the worst-case optimal solution by minimizing the maximum
misclassification risk through an iterative procedure. Despite the optimality,
the original MCA has drawbacks that have limited its wide applicability in
practice. That is, the MCA tends to be sensitive to outliers, cannot
effectively handle nonlinearities in datasets, and suffers from having high
computational complexity. To address these limitations, we propose an improved
solution, named regularized maximin correlation approach (R-MCA). We first
reformulate MCA as a quadratically constrained linear programming (QCLP)
problem, incorporate regularization by introducing slack variables in the
primal problem of the QCLP, and derive the corresponding Lagrangian dual. The
dual formulation enables us to apply the kernel trick to R-MCA so that it can
better handle nonlinearities. Our experimental results demonstrate that the
regularization and kernelization make the proposed R-MCA more robust and
accurate for various classification tasks than the original MCA. Furthermore,
when the data size or dimensionality grows, R-MCA runs substantially faster by
solving either the primal or dual (whichever has a smaller variable dimension)
of the QCLP.Comment: Submitted to IEEE Acces
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