The paper investigates a mixture of expert models. The mixture of experts is a combination of experts, local approximation model, and a gate function, which weighs these experts and forms their ensemble. In this work, each expert is a linear model. The gate function is a neural network with softmax on the last layer. The paper analyzes various prior distributions for each expert. The authors propose a method that takes into account the relationship between prior distributions of different experts. The EM algorithm optimises both parameters of the local models and parameters of the gate function. As an application problem, the paper solves a problem of shape recognition on images. Each expert fits one circle in an image and recovers its parameters: the coordinates of the center and the radius. The computational experiment uses synthetic and real data to test the proposed method. The real data is a human eye image from the iris detection problem.

The article deals with methods for reducing the complexity of approximating models.Probabilistic substantiation of distillation and privileged teaching methods is proposed. General conclusions are given for an arbitrary parametric function with a predetermined structure. A theoretical basis is demonstrated for the special cases of linear and logistic regression. The analysis of the considered models is carried out in a computational experiment on synthetic samples and real data. The FashionMNIST and Twitter Sentiment Analysis samples are considered as real data.

The loss landscape of neural networks is a critical aspect of their training, and understanding its properties is essential for improving their performance. In this paper, we investigate how the loss surface changes when the sample size increases, a previously unexplored issue. We theoretically analyze the convergence of the loss landscape in a fully connected neural network and derive upper bounds for the difference in loss function values when adding a new object to the sample. Our empirical study confirms these results on various datasets, demonstrating the convergence of the loss function surface for image classification tasks. Our findings provide insights into the local geometry of neural loss landscapes and have implications for the development of sample size determination techniques.

Controllability and observability Gramians, along with their inverses, are widely used to solve various problems in control theory. This paper proposes spectral decompositions of the controllability Gramian and its inverse based on system eigenvalues for a continuous LTI dynamical system in the controllability canonical (companion) form. The Gramian and its inverse are represented as sums of Hermitian matrices, each corresponding to individual system eigenvalues or their pairwise combinations. These decompositions are obtained for the solutions of both algebraic and differential Lyapunov and Riccati equations with arbitrary initial conditions, allowing for the estimation of system spectral properties over an arbitrary time interval and their prediction at future moments. The derived decompositions are also generalized to the case of multiple eigenvalues in the dynamics matrix spectrum, enabling a closed-form estimation of the effects of resonant interactions with the system's eigenmodes. The spectral components are interpreted as measurable quantities in the minimum energy control problem. Therefore, they are unambiguously defined and can quantitatively characterize the influence of individual eigenmodes and associated system devices on controllability, observability, and the asymptotic dynamics of perturbation energy. The additional information obtained from these decompositions can improve the accuracy of algorithms in solving various practical problems, such as stability analysis, minimum energy control, structural design, tuning regulators, optimal placement of actuators and sensors, network analysis, and model order reduction.

Grammatical error correction is one of the core natural language processing tasks. Presently, the open-source state-of-the-art sequence tagging for English is the GECToR model. For Russian, this problem does not have equally effective solutions due to the lack of annotated datasets, which motivated the current research. In this paper, we describe the process of creating a synthetic dataset and training the model on it. The GECToR architecture is adapted for the Russian language, and it is called RuGECToR. This architecture is chosen because, unlike the sequence-to-sequence approach, it is easy to interpret and does not require a lot of training data. The aim is to train the model in such a way that it generalizes the morphological properties of the language rather than adapts to a specific training sample. The presented model achieves the quality of 82.5 in the metric on synthetic data and 22.2 on the RULEC dataset, which was not used at the training stage.