Manifold optimization is ubiquitous in computational and appliedmathematics,statistics,engineering,machine learning,physics,chemistry,etc.One of the main challenges usually is the non-convexity of the manifold constraints.By utilizing the geometry of manifold,a large class of constrained optimization problems can be viewed as unconstrained optimization problems on manifold.From this perspective,intrinsic structures,optimality conditions and numerical algorithms for manifold optimization are investigated.Some recent progress on the theoretical results of manifold optimization is also presented.
In this paper,we consider the problem of computing the smallest enclosing ball(SEB)of a set of m balls in Rn,where the product mn is large.We first approximate the non-differentiable SEB problem by its log-exponential aggregation function and then propose a computationally efficient inexact Newton-CG algorithm for the smoothing approximation problem by exploiting its special(approximate)sparsity structure.The key difference between the proposed inexact Newton-CG algorithm and the classical Newton-CG algorithm is that the gradient and the Hessian-vector product are inexactly computed in the proposed algorithm,which makes it capable of solving the large-scale SEB problem.We give an adaptive criterion of inexactly computing the gradient/Hessian and establish global convergence of the proposed algorithm.We illustrate the efficiency of the proposed algorithm by using the classical Newton-CG algorithm as well as the algorithm from Zhou et al.(Comput Optim Appl 30:147–160,2005)as benchmarks.
We present a partial first-order affine-scaling method for solving smooth optimization with linear inequality constraints. At each iteration, the algorithm considers a subset of the constraints to reduce the complexity. We prove the global convergence of the algorithm for general smooth objective functions, and show it converges at sublinear rate when the objective function is quadratic. Numerical experiments indicate that our algorithm is efficient.
With the rapid growth in fingerprint databases, it has become necessary to develop excellent fingerprint indexing to achieve efficiency and accuracy. Fingerprint indexing has been widely studied with real-valued features,but few studies focus on binary feature representation, which is more suitable to identify fingerprints efficiently in large-scale fingerprint databases. In this study, we propose a deep compact binary minutia cylinder code（DCBMCC）as an effective and discriminative feature representation for fingerprint indexing. Specifically, the minutia cylinder code（MCC）, as the state-of-the-art fingerprint representation, is analyzed and its shortcomings are revealed.Accordingly, we propose a novel fingerprint indexing method based on deep neural networks to learn DCBMCC.Our novel network restricts the penultimate layer to directly output binary codes. Moreover, we incorporate independence, balance, quantization-loss-minimum, and similarity-preservation properties in this learning process.Eventually, a multi-index hashing（MIH） based fingerprint indexing scheme further speeds up the exact search in the Hamming space by building multiple hash tables on binary code substrings. Furthermore, numerous experiments on public databases show that the proposed approach is an outstanding fingerprint indexing method since it has an extremely small error rate with a very low penetration rate.
Chao-chao BAIWei-qiang WANGTong ZHAORu-xin WANGMing-qiang LI
In this paper,we investigate a parallel subspace correction framework for composite convex optimization.The variables are first divided into a few blocks based on certain rules.At each iteration,the algorithms solve a suitable subproblem on each block simultaneously,construct a search direction by combining their solutions on all blocks,then identify a new point along this direction using a step size satisfying the Armijo line search condition.They are called PSCLN and PSCLO,respectively,depending on whether there are overlapping regions between two imme-diately adjacent blocks of variables.Their convergence is established under mild assumptions.We compare PSCLN and PSCLO with the parallel version of the fast iterative thresholding algorithm and the fixed-point continuation method using the Barzilai-Borwein step size and the greedy coordinate block descent method for solving the l1-regularized minimization problems.Our numerical results showthatPSCLN andPSCLOcan run fast and return solutions notworse than those from the state-of-theart algorithms on most test problems.It is also observed that the overlapping domain decomposition scheme is helpful when the data of the problem has certain special structures.
