固定节点多项式样条的线性最小二乘问题及其奇异性研究
Linear least squares problems involving fixed knots polynomial splines and their singularity study
摘要：
本文研究了数据逼近和信号处理中出现的一类逼近问题。近似值被构造为多项式样条（分段多项式）和所谓的原型函数的组合，多项式样条的参数受到优化，原型函数的选择基于应用而不是优化。相应的优化问题可以用公式表示为线性最小二乘问题（LLSP）。如果系统矩阵是非奇异的，那么相应的问题可以使用正规方程方法来解决，而对于奇异情况，必须使用较慢（但更鲁棒）的方法。本文给出了非奇异性的充分条件。这些条件的验证比系统矩阵的直接奇异性验证快得多。因此，可以通过选择更合适的方法来求解相应的LLSP来提高算法的效率。
英文摘要：
In this paper, we study a class of approximation problems, appearing in data approximation and signal processing. The approximations are constructed as combinations of polynomial splines (piecewise polynomials), whose parameters are subject to optimisation, and so called prototype functions, whose choice is based on the application, rather than optimisation. The corresponding optimisation problems can be formulated as Linear Least Squares Problems (LLSPs). If the system matrix is non-singular, then the corresponding problem can be solved using the normal equations method, while for singular cases slower (but more robust) methods have to be used. In this paper we develop sufficient conditions for non-singularity. These conditions can be verified much faster than the direct singularity verification of the system matrices. Therefore, the algorithm efficiency can be improved by choosing a better suited method for solving the corresponding LLSPs.
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