In the article a new sparse low-rank matrix decomposition model is proposed based on the smoothly clipped absolute deviation (SCAD) penalty. In order to overcome the computational hurdle we generalize the alternating direction method of multipliers (ADMM) algorithm to develop an alternative algorithm to solve the model. The algorithm we designed alternatively renew the sparse matrix and low-rank matrix in terms of the closed form of SCAD penalty. Thus, the algorithm reduces the computational complexity while at the same time to keep the computational accuracy. A series of simulations have been designed to demonstrate the performances of the algorithm with comparing with the Augmented Lagrange Multiplier (ALM) algorithm. Ultimately, we apply the model to an on- board video background modeling problem. According to model the on-board video background, we can separate the video background and passenger's actions. Thus, the model can help us to identify the abnormal action of train passengers. The experiments show the background matrix we estimated is not only sparser, but the computational efficiency is also improved.
In this paper we introduce a new type of orthogonality for real normed planes which coincides with usual orthogonality in the Euclidean situation. With the help of this type of orthogonality we derive several characterizations of the Euclidean plane among all normed planes, all of them yielding also characteristic properties of inner product spaces among real normed linear spaces of dimensions $d\geq 3$.
Pseudo $\star $-autonomous lattices are non-commutative generalizations of $\star $-autonomous lattices. It is proved that the class of pseudo $\star $-autonomous lattices is a variety of algebras which is term equivalent to the class of dualizing residuated lattices. It is shown that the kernels of congruences of pseudo $\star $-autonomous lattices can be described as their normal ideals.
We prove a non-archimedean Dugundji extension theorem for the spaces $C^{\ast }(X,\mathbb {K})$ of continuous bounded functions on an ultranormal space $X$ with values in a non-archimedean non-trivially valued complete field $\mathbb {K}$. Assuming that $\mathbb {K}$ is discretely valued and $Y$ is a closed subspace of $X$ we show that there exists an isometric linear extender $T\colon C^{\ast }(Y,\mathbb {K})\rightarrow C^{\ast }(X,\mathbb {K})$ if $X$ is collectionwise normal or $Y$ is Lindelöf or $\mathbb {K}$ is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace $Y$ of an ultraregular space $X$ is a retract of $X$.
It is shown that there exist a continuous function f and a regulated function g defined on the interval [0,1] such that g vanishes everywhere except for a countable set, and the K *-integral of f with respect to g does not exist. The problem was motivated by extensions of evolution variational inequalities to the space of regulated functions.
In this paper a nonmonotone limited memory BFGS (NLBFGS) method is applied for approximately solving optimal control problems (OCPs) governed by one-dimensional parabolic partial differential equations. A discretized optimal control problem is obtained by using piecewise linear finite element and well-known backward Euler methods. Afterwards, regarding the implicit function theorem, the optimal control problem is transformed into an unconstrained nonlinear optimization problem (UNOP). Finally the obtained UNOP is solved by utilizing the NLBFGS method. In comparison to other existing methods, the NLBFGS method shows a significant improvement especially for nonlinear and ill-posed control problems.
In this paper we consider Neumann noncoercive hemivariational inequalities, focusing on nontrivial solutions. We use the critical point theory for locally Lipschitz functionals.
We investigate a relation about subadditivity of functions. Based on subadditivity of functions, we consider some conditions for continuous t-norms to act as the weakest t-norm TW-based addition. This work extends some results of Marková-Stupňanová [15], Mesiar [18].
In this paper, we give the mapping theorems on $\aleph $-spaces and $g$-metrizable spaces by means of some sequence-covering mappings, mssc-mappings and $\pi $-mappings.