Optimal Transport (OT) has become a powerful tool to compare probability distributions. However, it suffers from a severe computational burden for high dimensional and continuous distributions. To this end, we develop two novel methods for the Kantorovich and Monge formulations, which are the fundamental problems in OT. First, we learn the optimal joint distribution in the Kantorovich formulation and propose an algorithm, namely Cop-OT, which transforms the primal objective of the Kantorovich problem into a tractable objective with respect to the copula parameter. Second, based on the copula formulation of the joint distribution, we learn the optimal map in the Monge problem and propose an algorithm, namely Map-OT, which describes the optimal map using a parameterized function estimated by approximating the barycentric projection of the optimal joint distribution and then obtains a tractable objective with respect to parameters of interest. Both of them can be solved by stochastic optimization with a stable optimizing process. Empirical results demonstrate that Cop-OT and Map-OT can gain more accurate approximations of the Kantorovich and Monge problems compared with the baseline methods.

Grey wolf optimization (GWO) is a widely used meta-heuristic method. It has limited searching potential when solving the majority of function optimization problems. This paper proposes a new variant of GWO, named SCGWO, which combines GWO with an improved spread strategy and a chaotic local search (CLS) mechanism to overcome these performance limitations. In detail, a spread strategy is introduced into the basic GWO to change the search agent's ability to avoid the local optima, the global exploration capability, and the individual movement's randomness. Then, a CLS mechanism is adopted to accelerate the convergence rate of the evolving agents. This method's effectiveness is illustrated by comparing the proposed SCGWO method with various algorithms, including seven GWO variants and eight well-known state-of-the-art algorithms on a comprehensive set of benchmark functions with the type of unimodal, multimodal, and composition functions. The experimental results confirmed that the established SCGWO algorithm has apparent advantages in processing unimodal, multimodal, and composition functions. Additionally, the proposed algorithm was utilized for finding the approximate optimal feature subset when applied to the feature selection problems on a set of 32 real-world data sets from the UCI machine learning repository. The results show that the binary variant also reveals a very competitive performance in dealing with feature selection. Our findings and analysis suggest that the proposed method can be a very suitable tool for realizing the optimal solutions to global optimization and wrapper-based feature selection tasks.

The traditional covert communication that relies on a central node is vulnerable to detection and attack. Applying blockchain to covert communication can improve the channel's anti-interference and antitampering. Whisper is the communication protocol of Ethereum, which mainly relies on payload to store information and padding to expand. These two fields can store a large amount of information, creating conditions for the realization of covert communication. In this paper, we propose a covert communication method based on the whisper protocol to covertly transfer information in the blockchain. To implement this method, we use payload to store the carrier information, matching it with the secret message. The generated index is recorded in the padding field. To improve the concealment of communication, we simulate the default filling rules of the protocol to maintain the message size. A new topic–key pair interaction method is also proposed to improve the security of the model. Moreover, the anti-interference, antitampering and antidetection of the newly proposed model are verified through theoretical analysis and experiment. The experimental findings show that the amount of information in the proposed method is 4.7 times that of the traditional time-based covert communication. The time consumption of information transmission is reduced to 52.25% under the same settings and even less in actual use. The cost of the new topic–key pair interaction is reduced by nearly 50% compared with the original method.