As a fundamental mathematical problem in the field of machine learning, the linear separability test still lacks a theoretically complete and computationally efficient method. This paper proposes and proves a sufficient and necessary condition for linear separability test based on a sphere model. The advantage of this test method is two-fold: (1) it provides not only a qualitative test of linear separability but also a quantitative analysis of the separability of linear separable instances; (2) it has low time cost and is more efficient than existing test methods. The proposed method is validated through a large number of experiments on benchmark datasets and artificial datasets, demonstrating both its correctness and efficiency.

Generative Adversarial Nets (GAN) has been a popular research topic in processing of images, speech, texts, and videos, and many other fields.However, GAN still has some drawbacks such as unstable training and mode collapse.To address these challenges, this paper proposes an auto-encoding GAN, which is composed of a set of generators, a discriminator, an encoder and a decoder.A set of generators is responsible for learning different modes, accelerating the convergence of the model and preventing model collapse.The discriminator is used to distinguish between real samples and generated ones.In order to improve feature representation of the encoder and prevent multiple generators from covering a certain mode, an approach consisting of three phases is proposed accordingly.First, a clustering algorithm is presented to perceive the distribution of real and generated samples.Then, cluster center matching is utilized to keep consistency of the distribution of real and generated samples.Finally, the encoder and decoder are jointly optimized by the generated and real samples.Therefore, the encoder can map the generated and real samples to the embedding space so as to encode distinguishable features, and the decoder can distinguish from which generator the generated samples come and from which mode the real samples come.Experiments are conducted on image datasets to verify effectiveness of the auto-encoding GAN for reducing mode collapse and enhancing feature representation.