Deep neural network (DNN) has been attracting a great attention in various applications. Network training algorithms play essential roles for the effectiveness of DNN. Although stochastic gradient descent (SGD) and other explicit gradient-based methods are the most popular algorithms, there are still many challenges such as gradient vanishing and explosion occurring in training a complex and deep neural networks. Motivated by the idea of error back propagation (BP) and proximal point methods (PPM), we propose a semi-implicit back propagation method for neural network training. Similar to the BP, the update on the neurons are propagated in a backward fashion and the parameters are optimized with proximal mapping. The implicit update for both hidden neurons and parameters allows to choose large step size in the training algorithm. Theoretically, we demonstrate the convergence of the proposed method under some standard assumptions. The experiments on illustrative examples, and two real data sets: MNIST and CIFAR-10, demonstrate that the proposed semi-implicit BP algorithm leads to better performance in terms of both loss decreasing and training/test accuracy, with a detail comparison to SGD/Adam and a similar algorithm proximal back propagation (ProxBP).

It is known that many statistical and machine learning approaches heavily rely on pairwise distance between data points.The choice of distance function on the underlying manifold has a fundamental impact on performance of these processes.This is closely related to questions of how to appropriately calculate distances, and hence, fundamental solutions (heat kernels) for heat operators can be obtained.In general, it is not so easy to obtain a closed form for heat kernels.We first survey results of heat kernels on radially symmetric Riemannian manifolds, e.g., Euclidean spaces and unit spheres in R n .For the cases n = 1, 2, 3, we may construct the heat kernel explicitly.But, the computation is much more complicated when n > 3.However, by results of Nagase, we may construct parametrices for the heat kernel by using elementary functions so that the error terms can be under controlled.In the second part of the paper, we discuss some results on subRiemannian manifolds, especially 3-dimensional sphere in C 2 as a CR-manifold.We study geodesics connecting two given points on S 3 respecting the Hopf fibration.This geodesic boundary value problem is completely solved in the case of S 3 and some partial results are obtained in the general case.The Carnot-Carathéodory distance is calculated.We also present some motivations related to quantum mechanics.Then we give a brief discussion of Greiner's methods on the heat kernel for the Cauchy-Riemann subLaplacian on S 2n+1 .We provide a brief discussion on applications of these heat kernels to graph kernels in the last part of the paper.

In this paper, a weighted set order relation given by the oriented distance function is introduced, which does not need any convexity and is applicable even the ordering cone has an empty interior.The Ekeland's variational principle, Caristi's fixed point theorem, and Takahashi's minimization theorem associated with the introduced weighted set order relation are constructed, and the equivalences among them are deduced.As an application, the existence of solutions to a set optimization problem is examined to verify the validity of the results obtained.

Producción Científica

In this article, we introduce and study a new q-fractional integral operator which essentially stems from a successive application of the Srivastava-Owa operator of fractional integration with the q-Ruscheweyh derivative operator. As an application of this new q-fractional integral operator, we investigate a coefficient problem involving the second Hankel determinant H 2 2 ( f ) for normalized analytic and univalent functions f (z) belonging to some normalized families of q-starlike and q-convex functions in the open unit disk U.

We investigate a class of Nash equilibrium problems on networks where both the utility function and the strategy space of each player are parametrized by means of continuous functions of a vector parameter. We provide a variational inequality reformulation of the Nash equilibrium problem which turns out to be much simpler than the original problem. Moreover, under additional assumptions, the continuity of the solution with respect to the parameter is proved.

The class of quasimonotone variational inequalities is more general and applicable than the class of pseudomonotone and monotone variational inequalities. However, few results can be found in the literature on quasimonotone variational inequalities and currently results are mostly on weak convergent methods in the framework of Hilbert spaces. In this paper, we study the class of non-Lipschitz quasimonotone variational inequalities and the class of non-Lipschitz variational inequalities without monotonicity in the framework of Banach spaces. We propose a new inertial Tseng's extragradient method and obtain some strong convergence results for the proposed algorithm under some mild conditions on the control parameters. While the cost operator is non-Lipschitz, our proposed method does not require any linesearch procedure but employs a more efficient and simple self-adaptive step sizes with known parameters. Finally, we present several numerical experiments to demonstrate the implementability of our proposed method.

In this paper, we investigate a split common fixed point problem with multiple output sets within a more general framework and we present a novel iterative method that boasts the advantage of the step size calculation that is independent of the norm of linear mappings. We prove the weak convergence of the method and the strong convergence of its variants under certain conditions. Furthermore, we apply our main results to the split feasibility problem with multiple output sets. Our numerical results indicate that our method is an effective approach to this problem.

To deal with nondifferentiable interval-valued functions (IVFs) (not necessarily convex), we present the notion of Fréchet subdifferentiability or gH-Fréchet subdifferentiability.We explore its relationship with gH-differentiability and develop various calculus results for gH-Fréchet subgradients of extended IVFs.By using the proposed notion of subdifferentiability, we derive new necessary optimality conditions for unconstrained interval optimization problems (IOPs) with nondifferentiable IVFs.We also provide a necessary condition for unconstrained weak sharp minima of an extended IVF in terms of the proposed notion of subdifferentiability. Examples are pesented to support the main results.