On multivariate discrete least squares

Yeon Ju Lee, Charles A. Micchelli, Jungho Yoon

Abstract: For a positive integer n∈N$n\in \mathbb{N}$ we introduce the index set Nn:={1,2,…,n}${\mathbb{N}}_n:=\{1,2,\dots ,n\}$. Let X:={xi:i∈Nn}$X:=\{x_i:i\in {\mathbb{N}}_n\}$ be a distinct set of vectors in Rd${\mathbb{R}}^d$, Y:={yi:i∈Nn}$Y:=\{y_i:i\in {\mathbb{N}}_n\}$ a prescribed data set of real numbers in R$\mathbb{R}$ and F:={fj:j∈Nm},m, a given set of real valued continuous functions defined on some neighborhood O$\mathcal{O}$ of Rd${\mathbb{R}}^d$ containing X$X$. The discrete least squares problem determines a (generally unique) function f=∑j∈Nmcj⋆fj∈spanF$f={\sum }_{j\in {\mathbb{N}}_m}{c}_j^{\star }{f}_j\in \text{span}\mathcal{F}$ which minimizes the square of the ℓ2−${\ell }^2-$norm ∑i∈Nn(∑j∈Nmcjfj(xi)−yi)2$\sum _{i\in {\mathbb{N}}_n}{(\sum _{j\in {\mathbb{N}}_m}{c}_j{f}_j\left(x_i\right)-y_i)}^2$ over all vectors (cj:j∈Nm)∈Rm$(c_j:j\in {\mathbb{N}}_m)\in {\mathbb{R}}^m$. The value of f$f$ at some s∈O$s\in \mathcal{O}$ may be viewed as the optimally predicted value (in the ℓ2−${\ell }^2-$sense) of all functions in spanF$\text{span}\mathcal{F}$ from the given data X={xi:i∈Nn}$X=\{x_i:i\in {\mathbb{N}}_n\}$ and Y={yi:i∈Nn}$Y=\{y_i:i\in {\mathbb{N}}_n\}$.We ask “What happens if the components of X$X$ and s$s$ are nearly the same”. For example, when all these vectors are near the origin in Rd${\mathbb{R}}^d$. From a practical point of view this problem comes up in image analysis when we wish to obtain a new pixel value from nearby available pixel values as was done in [2], for a specified set of functions F$\mathcal{F}$.This problem was satisfactorily solved in the univariate case in Section 6 of Lee and Micchelli (2013). Here, we treat the significantly more difficult multivariate case using an approach recently provided in Yeon Ju Lee, Charles A. Micchelli and Jungho Yoon (2015).