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.

中文翻译：

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加权卷积差之和的最小化问题：和中给定数量的元素的情况

摘要

我们考虑求和两个数字序列中的元素的未学优化问题：\（Y \） 长度的\（N \） 和\（U \） 长度的\（Q \当量Ñ\） 。该问题的目的是使可变长度（不小于\（q \））的序列的加权卷积的差之和最小 。在每个差异中，第一个未加权卷积是序列\（U \） 由于其元素的多次重复而扩展为可变长度的自卷积，第二个未加权卷积是具有\\的子序列的扩展序列的加权卷积Y \）。我们用给定的输入差异数分析问题的一个变体。我们表明，该问题等效于 一些指数大小的序列集的 元素\（X \）对序列\（Y \）的近似。这样的集合由所有长度为\（N \）的序列组成，这些序列 包括给定数量 \（M \） 作为序列\（U \）的容许准周期性（波动）重复作为子序列 。每个准周期重复均来自以下对序列\（U \）的容许转换：（1）将\（U \）移位 一个变量，且不超过\（T _ {\ max} \ leq N \ 用于相邻的重复，（2）变量的扩展映射 \（U \） 到可变长度序列：的元素的可变多重重复\（U \） 。近似标准是元素方向差异平方和的最小化。我们证明了优化问题和相应的逼近问题在多项式时间内都可以解决。具体来说，我们表明存在一种精确的算法可以解决时间\（\ mathcal {O}（T _ {\ max} ^ {3} MN）\）的问题。如果\（T _ {\ max} \） 是问题的固定参数，则算法花费的时间为\（\ mathcal {O}（MN）\）。在数值建模示例中，我们展示了该算法在解决类似心电图（ECG）和类似光电体积描记图（PPG）信号的噪声鲁棒处理的模型应用问题中的适用性。