This paper studies and applies methods for diagnosing cardiovascular diseases on the basis of electrocardiogram data using neural networks. We will use recurrent and convolutional neural networks. The key goal of this work is to build a neural network model for solving problems of classifying a set of various cardiovascular diseases. The initial data with which we will work and on the basis of which we will draw conclusions are one-dimensional electrocardiogram signals. These are waves with a duration of 10 s and a sampling frequency of 100 and 500 Hz. Also, on the basis of our signal, segmentation for this signal will be calculated. It displays the belonging of the signal element to one of the main peaks (segments) of the signal. As an extension of the sample and augmentation, we will not supply the entire signal, but its parts lasting 6 and 9 s, and will also periodically add normal noise. To evaluate the results of the model, we will use the following quality metrics: precision, recall, specificity, and F1-measure. The tasks we are considering are binary and multiclass classifications. We will see that many diseases described in the task are determined with fairly good metric values. We will also compare the results of each model. The work is designed to speed up and improve the work of doctors and help them in solving the problems of diagnosing heart pathologies.

The problem of explainability of artificial intelligence is extremely relevant at the present time. The most acute need for this tool is felt in the areas of practical application of intelligent systems. One of these areas is medicine. Extracting knowledge and understanding the reason for the model's prediction will allow us to provide in a human-readable form the main factors used by the model to predict. The article will consider a method for highlighting areas of the electrocardiogram associated with signs of cardiovascular diseases using the transformer machine learning architecture. Visualization of the attention of the transformer to the ECG signal allows you to highlight the segments of the electrocardiogram, which may show signs of various cardiovascular diseases. For the diagnosis of inferior myocardial infarction, the algorithm for labeling the data extracted from the trained model identified Q and R waves, for anterior septal myocardial infarction, Q and T waves, as well as the ST segment, and for left ventricular hypertrophy, the ST segment. Areas of attraction of "attention" of the neural network are shown graphically and displayed in the article.

In this paper, we consider the counting function $E_P(y) = |P_{y} \cap Z^{n_x}|$ for a parametric polyhedron $P_{y} = \{x \in R^{n_x} \colon A x \leq b + B y\}$, where $y \in R^{n_y}$. We give a new representation of $E_P(y)$, called a \emph{piece-wise step-polynomial with periodic coefficients}, which is a generalization of piece-wise step-polynomials and integer/rational Ehrhart's quasi-polynomials. It gives the fastest way to calculate $E_P(y)$ in certain scenarios. The most important cases are the following: 1) We show that, for the parametric polyhedron $P_y$ defined by a standard-form system $A x = y,\, x \geq 0$ with a fixed number of equalities, the function $E_P(y)$ can be represented by a polynomial-time computable function. In turn, such a representation of $E_P(y)$ can be constructed by an $poly\bigl(n, \|A\|_{\infty}\bigr)$-time algorithm; 2) Assuming again that the number of equalities is fixed, we show that integer/rational Ehrhart's quasi-polynomials of a polytope can be computed by FPT-algorithms, parameterized by sub-determinants of $A$ or its elements; 3) Our representation of $E_P$ is more efficient than other known approaches, if $A$ has bounded elements, especially if it is sparse in addition. Additionally, we provide a discussion about possible applications in the area of compiler optimization. In some "natural" assumptions on a program code, our approach has the fastest complexity bounds.

In our paper, we consider the following general problems: check feasibility, count the number of feasible solutions, find an optimal solution, and count the number of optimal solutions in 𝒫 ∩ ℤ ^n , assuming that 𝒫 is a polyhedron, defined by systems A x ≤ b or Ax = b, x ≥ 0 with a sparse matrix A. We develop algorithms for these problems that outperform state-of-the-art ILP and counting algorithms on sparse instances with bounded elements in terms of the computational complexity. Assuming that the matrix A has bounded elements, our complexity bounds have the form s^O(n) , where s is the minimum between numbers of non-zeroes in columns and rows of A, respectively. For s = o (log n ) , this bound outperforms the state-of-the-art ILP feasibility complexity bound (log n)^O(n) , due to Reis Rothvoss (in: 2023 IEEE 64th Annual symposium on foundations of computer science (FOCS), IEEE, pp. 974–988). For s = ϕ ^o(log n) , where ϕ denotes the input bit-encoding length, it outperforms the state-of-the-art ILP counting complexity bound ϕ ^O(n log n) , due to Barvinok et al. (in: Proceedings of 1993 IEEE 34th annual foundations of computer science, pp. 566–572, https://doi.org/10.1109/SFCS.1993.366830 , 1993), Dyer, Kannan (Math Oper Res 22(3):545–549, https://doi.org/10.1287/moor.22.3.545 , 1997), Barvinok, Pommersheim (Algebr Combin 38:91–147, 1999), Barvinok (in: European Mathematical Society, ETH-Zentrum, Zurich, 2008). We use known and new methods to develop new exponential algorithms for Edge/Vertex Multi-Packing/Multi-Cover Problems on graphs and hypergraphs. This framework consists of many different problems, such as the Stable Multi-set, Vertex Multi-cover, Dominating Multi-set, Set Multi-cover, Multi-set Multi-cover, and Hypergraph Multi-matching problems, which are natural generalizations of the standard Stable Set, Vertex Cover, Dominating Set, Set Cover, and Maximum Matching problems.

This study presents the development of a tool for identifying named entities in Uzbek legal texts. It should be noted, that besides of detecting named entities, the authors developed an algorithm, which is able to standardize word forms by replacing the detected dialect words (Karluk, Kypchak and Oghuz) with their formal forms. This will help to fix popular grammatical mistakes among native speakers from different regions of the Uzbekistan. The proposed hybrid approach combines the traditional approach, which is used in the preprocessing (standardization of word forms), where a dictionary with more than 10 thousand marked words is actively used. At the same time, a custom language model is used to work with detecting named entities, which was trained on 2000 legal sentences. The testing results showed quite high indicators, in particular, the language model detected named entities with an accuracy of 90%, and the recall reached 94%. Moreover, the algorithm used to standardize dialect word forms showed even higher rates, ranging from 90% to 100 % depending on the dialect.

Diagnosis by electrocardiogram (ECG) is an extremely urgent and important task, the quality, timeliness and speed of which people’s lives and health depend on. To date, a large number of researchers treat neural networks as a panacea, hoping that any task can be solved quickly and without problems. Often, this approach does not lead to the best results. The article explores the use of deep learning as a universal tool for solving the problem in determining pathological ECG signals with markers of myocardial infarction, hypertrophy, conduction disturbances, and changes in ST segment morphology. During the experiments, the positive impact of using thresholding and replacements to increase the predictive ability of the network, the use of various ensembles on trained deep learning models was established. The addition of artificial models also improves the classifying ability of ensembles. Returning a random number in the absence of a single mode also makes it possible to increase the accuracy of the ensemble.

We propose a cut-based algorithm for finding all vertices and all facets of the convex hull of all integer points of a polyhedron defined by a system of linear inequalities. Our algorithm, DDMCuts , is based on the Gomory cuts and the dynamic version of the double description method. We describe the computer implementation of the algorithm and present the results of computational experiments comparing our algorithm with a naive one and an algorithm implemented in Normaliz.

The Rosenblatt’s first theorem about the omnipotence of shallow networks states that elementary perceptrons can solve any classification problem if there are no discrepancies in the training set. Minsky and Papert considered elementary perceptrons with restrictions on the neural inputs: a bounded number of connections or a relatively small diameter of the receptive field for each neuron at the hidden layer. They proved that under these constraints, an elementary perceptron cannot solve some problems, such as the connectivity of input images or the parity of pixels in them. In this note, we demonstrated Rosenblatt’s first theorem at work, showed how an elementary perceptron can solve a version of the travel maze problem, and analysed the complexity of that solution. We also constructed a deep network algorithm for the same problem. It is much more efficient. The shallow network uses an exponentially large number of neurons on the hidden layer (Rosenblatt’s A-elements), whereas for the deep network, the second-order polynomial complexity is sufficient. We demonstrated that for the same complex problem, the deep network can be much smaller and reveal a heuristic behind this effect.

In this work we provide an algorithm for the detection of cardiac diseases that operates directly on ECG data without any preprocessing and interpret the results.

It is known that the most dangerous predictors of sudden cardiac death is the appearance in a person of signs of atrial and ventricular fibrillation. At the same time, this type of rhythm can rarely be detected during a quick screening examination, which greatly complicates the diagnosis and, as a result, the choice of effective treatment. Within the framework of this article, the experience of using the most popular architectures of convolutional neural networks adapted for the analysis of the electrocardiogram signal is presented to localize areas of sinus rhythm and fibrillation. As the networks under study, such architectures as ResNet, DenseNet and XceptionNet were considered. Each of their architectures showed good results, but the DenseNet network showed the best results. Due to a number of advantages described in the article, the DenseNet network was chosen. It should be noted that the chosen architecture of the neural network, after a number of necessary modifications and improvements, will be integrated into the existing diagnostic complex for cardiovascular diseases, developed on the basis of Lobachevsky University.

In this study, we perform a comparative experimental analysis of several modifications of the dynamic double description method for generating the extreme rays of a polyhedral cone. The modifications under consideration include using graph adjacency test, bit pattern trees and maintaining the set of all adjacent extreme rays. Results of computational experiments on several classes of problems are presented.

A set S of Boolean points is a specifying set for a threshold function f if the only threshold function consistent with f on S is f itself. The minimal cardinality of a specifying set for f is the specification number of f and it is never smaller than n+1 for a function with n relevant variables. In the present paper, we develop an inductive approach to describing the set of Boolean threshold functions with minimum specification number by means of operations that allow us to extend functions of n variables in this set to functions of n+1 variables.

We propose a method for generating an electrocardiogram (ECG) signal for one cardiac cycle using a variational autoencoder. Our goal was to encode the original ECG signal using as few features as possible. Using this method we extracted a vector of new 25 features, which in many cases can be interpreted. The generated ECG has quite natural appearance. The low value of the Maximum Mean Discrepancy metric, 3.83 × 10−3, indicates good quality of ECG generation too. The extracted new features will help to improve the quality of automatic diagnostics of cardiovascular diseases. Generating new synthetic ECGs will allow us to solve the issue of the lack of labeled ECG for using them in supervised learning.

The paper concerns the elimination of a set of variables from a system of linear inequalities. We employ the widely used Fourier–Motzkin elimination method extended with the Chernikov rules. A straightforward implementation of the algorithm results in extensive enumeration during the most computationally demanding stage. We propose a new way of checking Chernikov rules using bit pattern trees as an accelerating data structure to avoid extensive enumeration. The bit pattern tree is a data structure based on k-d tree used to accelerate the double description method. First we describe an adaptation of that approach to check the second Chernikov rule in Fourier–Motzkin elimination. We also propose a new algorithm that employs bit pattern trees to accelerate both Chernikov rules. Presented results of computational evaluation prove competitiveness of the proposed algorithms.

Let a polyhedron P be defined by one of the following ways: and let all rank order minors of A be bounded by $$\varDelta $$ in absolute values. We show that the short rational generating function for the power series $$\begin{aligned} \sum \limits _{m \in P \cap {{\,\mathrm{{\mathbb {Z}}}\,}}^n} {{\,\mathrm{{\mathbf {x}}}\,}}^m \end{aligned}$$ can be computed with the arithmetical complexity $$ O\left( T_{{\mathrm{SNF}}}(d) \cdot d^{k} \cdot d^{\log _2 \varDelta }\right) , $$ where k and $$\varDelta $$ are fixed, $$d = \dim P$$ , and $$T_{{\mathrm{SNF}}}(m)$$ is the complexity of computing the Smith Normal Form for $$m \times m$$ integer matrices. In particular, $$d = n$$ , for the case (i), and $$d = n-k$$ , for the case (ii). The simplest examples of polyhedra that meet the conditions (i) or (ii) are the simplices, the subset sum polytope and the knapsack or multidimensional knapsack polytopes. Previously, the existence of a polynomial time algorithm in varying dimension for the considered class of problems was unknown already for simplicies ( $$k = 1$$ ). We apply these results to parametric polytopes and show that the step polynomial representation of the function $$c_P({{\,\mathrm{{\mathbf {y}}}\,}}) = |P_{{{\,\mathrm{{\mathbf {y}}}\,}}} \cap {{\,\mathrm{{\mathbb {Z}}}\,}}^n|$$ , where $$P_{{{\,\mathrm{{\mathbf {y}}}\,}}}$$ is a parametric polytope, whose structure is close to the cases (i) or (ii), can be computed in polynomial time even if the dimension of $$P_{{{\,\mathrm{{\mathbf {y}}}\,}}}$$ is not fixed. As another consequence, we show that the coefficients $$e_i(P,m)$$ of the Ehrhart quasi-polynomial $$\begin{aligned} \left| mP \cap {{\,\mathrm{{\mathbb {Z}}}\,}}^n\right| = \sum \limits _{j = 0}^n e_j(P,m)m^j \end{aligned}$$ can be computed with a polynomial-time algorithm, for fixed k and $$\varDelta $$ .

We report Lobachevsky University Database (LUDB) of ECG signals, an open tool for validating ECG delineation algorithms, that is superior to the existing publicly available data bases in several aspects. LUDB contains 200 recordings of 10-second 12-lead electrocardiograms (ECG) from different subjects, representative of a variety of signal morfologies. The boundaries and peaks of QRS complexes and P and T waves are manually annotated by cardiologists for all recordings and independently for each lead, and all records received an expert classification by abnormalities. We present a case study for the recently proposed wavelet-based algorithm and the broadly used ecg-kit tool, and demonstrate the advantage of multi-lead ECG data analysis. LUDB contributes to the diversity of public databases employed in developing and validating novel ECG analysis algorithms, including the most advanced based on deep learning neural networks.

In Chirkov et al., (2019), classes of conic and discrete conic functions were introduced. In this paper we use the term convic instead conic. The class of convic functions properly includes the classes of convex functions, strictly quasiconvex functions and the class of quasiconvex polynomials. On the other hand, the class of convic functions is properly included in the class of quasiconvex functions. The discrete convic function is a discrete analogue of the convic function. In Chirkov et al., (2019), the lower bound 3(n-1) log(2 rho-1) for the number of calls to the comparison oracle needed to find the minimum of the discrete convic function defined on integer points of some n-dimensional ball with radius rho was obtained. But the problem of the existence of a polynomial (in log. for fixed n) algorithm for minimizing such functions has remained open. In this paper, we answer positively the question of the existence of such an algorithm. Namely, we propose an algorithm for minimizing discrete convic functions that uses 2(O)(n(2) log n) log rho calls to the comparison oracle and has 2(O()n(2) (log n)) poly(log rho) bit complexity. (C) 2019 Published by Elsevier B.V.

We propose a method for generating an electrocardiogram (ECG) signal for one cardiac cycle using a variational autoencoder. Using this method we extracted a vector of new 25 features, which in many cases can be interpreted. The generated ECG has quite natural appearance. The low value of the Maximum Mean Discrepancy metric, 0.00383, indicates good quality of ECG generation too. The extracted new features will help to improve the quality of automatic diagnostics of cardiovascular diseases. Also, generating new synthetic ECGs will allow us to solve the issue of the lack of labeled ECG for use them in supervised learning.

Stochastic separation theorems play important roles in high-dimensional data analysis and machine learning. It turns out that in high dimensional space, any point of a random set of points can be separated from other points by a hyperplane with high probability, even if the number of points is exponential in terms of dimensions. This and similar facts can be used for constructing correctors for artificial intelligent systems, for determining the intrinsic dimensionality of data and for explaining various natural intelligence phenomena. In this paper, we refine the estimations for the number of points and for the probability in stochastic separation theorems, thereby strengthening some results obtained earlier. We propose the boundaries for linear and Fisher separability, when the points are drawn randomly, independently and uniformly from a d-dimensional spherical layer and from the cube. These results allow us to better outline the applicability limits of the stochastic separation theorems in applications.

We study the proximity of the optimal value of the m-dimensional knapsack problem to the optimal value of that problem with the additional restriction that only one type of items is allowed to include in the solution. We derive exact and asymptotic formulas for the precision of such approximation, i.e. for the infinum of the ratio of the optimal value for the objective functions of the problem with the cardinality constraint and without it. In particular, we prove that the precision tends to 0.59136.../m if n tends to infinity and m is fixed. Also, we give the class of the worst multi-dimensional knapsack problems for which the bound is attained. Previously, similar results were known only for the case m=1.