V. Slavin

V. Slavin, O. Kryvchikov, D. Laptev

We present a graph-based deep learning framework for predicting the magnetic properties of quasi-one-dimensional Ising spin systems. The lattice geometry is encoded as a graph and processed by a graph neural network (GNN) followed by fully connected layers. The model is trained on Monte Carlo simulation data and accurately reproduces key features of the magnetization curve, including plateaus, critical transition points, and the effects of geometric frustration. It captures both local motifs and global symmetries, demonstrating that GNNs can infer magnetic behavior directly from structural connectivity. The proposed approach enables efficient prediction of magnetization without the need for additional Monte Carlo simulations.

L. Berlyand, O. Krupchytskyi, V. Slavin

This paper is concerned with a special class of deep neural networks (DNNs) where the input and the output vectors have the same dimension. Such DNNs are widely used in applications, e.g., autoencoders. The training of such networks can be characterized by their fixed points (FPs). We are concerned with the dependence of the FPs number and their stability on the distribution of randomly initialized DNNs' weight matrices. Specifically, we consider the i.i.d. random weights with heavy and light-tail distributions. Our objectives are twofold. First, the dependence of FPs number and stability of FPs on the type of the distribution tail. Second, the dependence of the number of FPs on the DNNs' architecture. We perform extensive simulations and show that for light tails (e.g., Gaussian), which are typically used for initialization, a single stable FP exists for broad types of architectures. In contrast, for heavy tail distributions (e.g., Cauchy), which typically appear in trained DNNs, a number of FPs emerge. We further observe that these FPs are stable attractors and their basins of attraction partition the domain of input vectors. Finally, we observe an intriguing non-monotone dependence of the number of fixed points $Q(L)$ on the DNNs' depth $L$. The above results were first obtained for untrained DNNs with two types of distributions at initialization and then verified by considering DNNs in which the heavy tail distributions arise in training.

L. Pastur, V. Slavin

We study the distribution of singular values of product of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the population covariance matrices assumed to be non-random or random but independent of the random data matrix in statistics and random matrix theory are now certain functions of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated in recent work [25, 13] by using the techniques of free probability theory. Since, however, free probability theory deals with population covariance matrices which are independent of the data matrices, its applicability has to be justified. The justification has been given in [22] for Gaussian data matrices with independent entries, a standard analytical model of free probability, by using a version of the techniques of random matrix theory. In this paper we use another, more streamlined, version of the techniques of random matrix theory to generalize the results of [22] to the case where the entries of the synaptic weight matrices are just independent identically distributed random variables with zero mean and finite fourth moment. This, in particular, extends the property of the so-called macroscopic universality on the considered random matrices.