Toroidal Hitomezashi Patterns

We define Hitomezashi patterns and loops on a torus and provide several structural results for such loops. For a given pattern, our main theorems give optimal residual information regarding the Hitomezashi loop length, loop count, as well as possible homology classes of such loops. Special attention is paid to toroidal Hitomezashi patterns that are symmetric with respect to the diagonal x = y.Comment: 15 pages, 9 figure

A Classification Supervised Auto-Encoder Based on Predefined Evenly-Distributed Class Centroids

Classic Autoencoders and variational autoencoders are used to learn complex data distributions, that are built on standard function approximators, such as neural networks, which can be trained by stochastic gradient descent methods. Especially, VAE has shown promise on a lot of complex task. In this paper, a new autoencoder model - classification supervised autoencoder (CSAE) based on predefined evenly-distributed class centroids (PEDCC) is proposed. To carry out the supervised learning for autoencoder, we use PEDCC of latent variables to train the network to ensure the maximization of inter-class distance and the minimization of inner-class distance. Instead of learning mean/variance of latent variables distribution and taking reparameterization of VAE, latent variables of CSAE are directly used to classify and as input of decoder. In addition, a new loss function is proposed to combine the loss function of classification, the loss function of image codec error and the loss function for enhancing subjective quality of decoded image. Based on the basic structure of the universal autoencoder, we realized the comprehensive optimal results of encoding, decoding and classification, and good model generalization performance at the same time. Theoretical advantages are reflected in experimental results.Comment: 17 pages,9 figures, 5 table

A Short Proof to Defant and Kravitz's theorem on the Length of Hitomezashi Loops

We provide a shorter proof to Defant and Kravitz's theorem (arXiv:2201.03461, Theorem 1.2) on the length of Hitomezashi loops modulo 8.Comment: 4 pages, 2 figure