# Minimization Problem for Sum of Weighted Convolution Differences: The Case of a Given Number of Elements in the Sum

### ABSTRACT

We consider an unstudied optimization problem of summing elements of two numerical sequences: $$Y$$ of length $$N$$ and $$U$$ of length $$q\leq N$$. The objective of the problem is minimization of the sum of differences of weighted convolutions of sequences of variable length (not less than $$q$$). In each difference, the first unweighted convolution is the autoconvolution of the sequence $$U$$ expanded to a variable length due to multiple repetitions of its elements, and the second one is the weighted convolution of the expanded sequence with a subsequence from $$Y$$. We analyze a variant of the problem with a given input number of differences. We show that the problem is equivalent to that of approximation of the sequence $$Y$$ by an element $$X$$ of some exponentially-sized set of sequences. Such a set consists of all the sequences of length $$N$$ that include as subsequences a given number $$M$$ of admissible quasi-periodic (fluctuating) repetitions of the sequence $$U$$. Each quasi-periodic repetition results from the following admissible transformations of the sequence $$U$$: (1) shift of $$U$$ by a variable, which do not exceed $$T_{\max}\leq N$$ for neighboring repetitions, (2) variable expanding mapping of $$U$$ to a variable-length sequence: variable-multiplicity repetitions of elements of $$U$$. The approximation criterion is minimization of the sum of the squares of element-wise differences. We demonstrate that the optimization problem and the respective approximation problem are solvable in a polynomial time. Specifically, we show that there exists an exact algorithm that solves the problem in the time $$\mathcal{O} (T_{\max}^{3}MN)$$. If $$T_{\max}$$ is a fixed parameter of the problem, then the time taken by the algorithm is $$\mathcal{O} (MN)$$. In examples of numerical modeling, we show the applicability of the algorithm to solving model applied problems of noise-robust processing of electrocardiogram (ECG)-like and photoplethysmogram (PPG)-like signals.

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## Funding

This work was supported by the Russian Foundation for Basic Research (project nos. 19-07-00397 and 19-01-00308), by RAS Basic Research Program (project no. 0314-2019-0015), and by “Top-5-100” Program of the Ministry of Education and Science of the Russian Federation.

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Correspondence to L. V. Mikhailova or P. S. Ruzankin or S. A. Khamidullin.

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