Refinement of Operator-valued Reproducing Kernels

This paper studies the construction of a refinement kernel for a given operator-valued reproducing kernel such that the vector-valued reproducing kernel Hilbert space of the refinement kernel contains that of the given one as a subspace. The study is motivated from the need of updating the current operator-valued reproducing kernel in multi-task learning when underfitting or overfitting occurs. Numerical simulations confirm that the established refinement kernel method is able to meet this need. Various characterizations are provided based on feature maps and vector-valued integral representations of operator-valued reproducing kernels. Concrete examples of refining translation invariant and finite Hilbert-Schmidt operator-valued reproducing kernels are provided. Other examples include refinement of Hessian of scalar-valued translation-invariant kernels and transformation kernels. Existence and properties of operator-valued reproducing kernels preserved during the refinement process are also investigated

A Primal-Dual Based Power Control Approach for Capacitated Edge Servers

The intensity of radio waves decays rapidly with increasing propagation distance, and an edge server's antenna needs more power to form a larger signal coverage area. Therefore, the power of the edge server should be controlled to reduce energy consumption. In addition, edge servers with capacitated resources provide services for only a limited number of users to ensure the quality of service (QoS). We set the signal transmission power for the antenna of each edge server and formed a signal disk, ensuring that all users were covered by the edge server signal and minimizing the total power of the system. This scenario is a typical geometric set covering problem, and even simple cases without capacity limits are NP-hard problems. In this paper, we propose a primal-dual-based algorithm and obtain an $m$-approximation result. We compare our algorithm with two other algorithms through simulation experiments. The results show that our algorithm obtains a result close to the optimal value in polynomial time

A Local-Ratio-Based Power Control Approach for Capacitated Access Points in Mobile Edge Computing

Terminal devices (TDs) connect to networks through access points (APs) integrated into the edge server. This provides a prerequisite for TDs to upload tasks to cloud data centers or offload them to edge servers for execution. In this process, signal coverage, data transmission, and task execution consume energy, and the energy consumption of signal coverage increases sharply as the radius increases. Lower power leads to less energy consumption in a given time segment. Thus, power control for APs is essential for reducing energy consumption. Our objective is to determine the power assignment for each AP with same capacity constraints such that all TDs are covered, and the total power is minimized. We define this problem as a \emph{minimum power capacitated cover } (MPCC) problem and present a \emph{minimum local ratio} (MLR) power control approach for this problem to obtain accurate results in polynomial time. Power assignments are chosen in a sequence of rounds. In each round, we choose the power assignment that minimizes the ratio of its power to the number of currently uncovered TDs it contains. In the event of a tie, we pick an arbitrary power assignment that achieves the minimum ratio. We continue choosing power assignments until all TDs are covered. Finally, various experiments verify that this method can outperform another greedy-based way

Mumford-Shah model and its application in image processing

The Mumford-Shah (MS) model has been studied in details in this thesis. It is found that the piecewise constant approximation MS model can not be used for images with large variation in the intensities. Therefore a linear approximation MS model is introduced. We have found that the linear approximation MS model provides better segmentation results than the piecewise constant MS model. The level set methods are used in the numerical computations. We have explicitly proved that the MS energy decreases with time (iterations) for all cases. The o and p dependence of the MS model is also studied. It is found that when o becomes large, the piecewise constant model is recovered. On the other hand, if o tends to zero, detailed structure of the input image can be obtained by the MS segmentation model. The MS and the Rudin-Osher-Fatemi (ROF) like models are generalized to include high order derivative terms. It is found that this kind of model can be used for edges with low contrast. The MS model is also generalized to a new model which can be used to detect roof edges which are difficult to detect by other models. Verification of the proposed models is done based on experimental result