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.