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.

In this paper, we will show that the width of simplices defined by systems of linear inequalities can be computed in polynomial time if some minors of their constraint matrices are bounded. Additionally, we present some quasi-polynomial-time and polynomial-time algorithms to solve the integer linear optimization problem defined on simplices minus all their integer vertices assuming that some minors of the constraint matrices of the simplices are bounded.

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In this paper, we present fixed-parameter tractable algorithms for special cases of the shortest lattice vector, integer linear programming, and simplex width computation problems, when matrices included in the problems' formulations are near square. The parameter is the maximum absolute value of the rank minors in the corresponding matrices. Additionally, we present fixed-parameter tractable algorithms with respect to the same parameter for the problems, when the matrices have no singular rank submatrices.

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