In this paper,an accelerated proximal gradient algorithm is proposed for Hankel tensor completion problems.In our method,the iterative completion tensors generated by the new algorithm keep Hankel structure based on p...In this paper,an accelerated proximal gradient algorithm is proposed for Hankel tensor completion problems.In our method,the iterative completion tensors generated by the new algorithm keep Hankel structure based on projection on the Hankel tensor set.Moreover,due to the special properties of Hankel structure,using the fast singular value thresholding operator of the mode-s unfolding of a Hankel tensor can decrease the computational cost.Meanwhile,the convergence of the new algorithm is discussed under some reasonable conditions.Finally,the numerical experiments show the effectiveness of the proposed algorithm.展开更多
Many machine learning problems can be formulated as minimizing the sum of a function and a non-smooth regularization term.Proximal stochastic gradient methods are popular for solving such composite optimization proble...Many machine learning problems can be formulated as minimizing the sum of a function and a non-smooth regularization term.Proximal stochastic gradient methods are popular for solving such composite optimization problems.We propose a minibatch proximal stochastic recursive gradient algorithm SRG-DBB,which incorporates the diagonal Barzilai–Borwein(DBB)stepsize strategy to capture the local geometry of the problem.The linear convergence and complexity of SRG-DBB are analyzed for strongly convex functions.We further establish the linear convergence of SRGDBB under the non-strong convexity condition.Moreover,it is proved that SRG-DBB converges sublinearly in the convex case.Numerical experiments on standard data sets indicate that the performance of SRG-DBB is better than or comparable to the proximal stochastic recursive gradient algorithm with best-tuned scalar stepsizes or BB stepsizes.Furthermore,SRG-DBB is superior to some advanced mini-batch proximal stochastic gradient methods.展开更多
Support vector machine(SVM)is a widely used method for classification.Proximal support vector machine(PSVM)is an extension of SVM and a promisingmethod to lead to a fast and simple algorithm for generating a classifie...Support vector machine(SVM)is a widely used method for classification.Proximal support vector machine(PSVM)is an extension of SVM and a promisingmethod to lead to a fast and simple algorithm for generating a classifier.Motivated by the fast computational efforts of PSVM and the properties of sparse solution yielded by l1-norm,in this paper,we first propose a PSVM with a cardinality constraint which is eventually relaxed byl1-norm and leads to a trade-offl1−l2 regularized sparse PSVM.Next we convert thisl1−l2 regularized sparse PSVM into an equivalent form of1 regularized least squares(LS)and solve it by a specialized interior-point method proposed by Kim et al.(J SelTop Signal Process 12:1932–4553,2007).Finally,l1−l2 regularized sparse PSVM is illustrated by means of a real-world dataset taken from the University of California,Irvine Machine Learning Repository(UCI Repository).Moreover,we compare the numerical results with the existing models such as generalized eigenvalue proximal SVM(GEPSVM),PSVM,and SVM-Light.The numerical results showthat thel1−l2 regularized sparsePSVMachieves not only better accuracy rate of classification than those of GEPSVM,PSVM,and SVM-Light,but also a sparser classifier compared with the1-PSVM.展开更多
文摘In this paper,an accelerated proximal gradient algorithm is proposed for Hankel tensor completion problems.In our method,the iterative completion tensors generated by the new algorithm keep Hankel structure based on projection on the Hankel tensor set.Moreover,due to the special properties of Hankel structure,using the fast singular value thresholding operator of the mode-s unfolding of a Hankel tensor can decrease the computational cost.Meanwhile,the convergence of the new algorithm is discussed under some reasonable conditions.Finally,the numerical experiments show the effectiveness of the proposed algorithm.
基金the National Natural Science Foundation of China(Nos.11671116,11701137,12071108,11991020,11991021 and 12021001)the Major Research Plan of the NSFC(No.91630202)+1 种基金the Strategic Priority Research Program of Chinese Academy of Sciences(No.XDA27000000)the Natural Science Foundation of Hebei Province(No.A2021202010)。
文摘Many machine learning problems can be formulated as minimizing the sum of a function and a non-smooth regularization term.Proximal stochastic gradient methods are popular for solving such composite optimization problems.We propose a minibatch proximal stochastic recursive gradient algorithm SRG-DBB,which incorporates the diagonal Barzilai–Borwein(DBB)stepsize strategy to capture the local geometry of the problem.The linear convergence and complexity of SRG-DBB are analyzed for strongly convex functions.We further establish the linear convergence of SRGDBB under the non-strong convexity condition.Moreover,it is proved that SRG-DBB converges sublinearly in the convex case.Numerical experiments on standard data sets indicate that the performance of SRG-DBB is better than or comparable to the proximal stochastic recursive gradient algorithm with best-tuned scalar stepsizes or BB stepsizes.Furthermore,SRG-DBB is superior to some advanced mini-batch proximal stochastic gradient methods.
基金This research was supported by the National Natural Science Foundation of China(No.11371242).
文摘Support vector machine(SVM)is a widely used method for classification.Proximal support vector machine(PSVM)is an extension of SVM and a promisingmethod to lead to a fast and simple algorithm for generating a classifier.Motivated by the fast computational efforts of PSVM and the properties of sparse solution yielded by l1-norm,in this paper,we first propose a PSVM with a cardinality constraint which is eventually relaxed byl1-norm and leads to a trade-offl1−l2 regularized sparse PSVM.Next we convert thisl1−l2 regularized sparse PSVM into an equivalent form of1 regularized least squares(LS)and solve it by a specialized interior-point method proposed by Kim et al.(J SelTop Signal Process 12:1932–4553,2007).Finally,l1−l2 regularized sparse PSVM is illustrated by means of a real-world dataset taken from the University of California,Irvine Machine Learning Repository(UCI Repository).Moreover,we compare the numerical results with the existing models such as generalized eigenvalue proximal SVM(GEPSVM),PSVM,and SVM-Light.The numerical results showthat thel1−l2 regularized sparsePSVMachieves not only better accuracy rate of classification than those of GEPSVM,PSVM,and SVM-Light,but also a sparser classifier compared with the1-PSVM.
