In this paper, we introduce adaptive network enhancement (ANE) method for the best least-squares approximation using two-layer ReLU neural networks (NNs). For a given function f(x), ANE generates NN and numerical integration mesh such that accuracy is within prescribed tolerance. The provides natural process obtaining good initialization which crucial training nonlinear optimization problems. N...

One of the most commonly used methods for the analysis of experimental data in the biochemical literature is nonlinear least squares (regression). This group of methods are also commonly misused. The purpose of this article is to review the assumptions inherent in the use of least-squares techniques and how these assumptions govern the ways that least-squares techniques can and should be used. ...

The mathematical problem of approximating one matrix by another of lower rank is closely related to the fundamental postulate of factor-theory. When formulated as a least-squares problem, the normal equations cannot be immediately written do~vn, since the elements of the approximate matrix are not independent of one another. The solution of the problem is simplified by first expressing the matr...

Using 55 years of daily average temperatures from a local weather station, I made a least-absolute-deviations regression model that accounts for three effects: seasonal variations, the 11-year solar cycle, and a linear trend. The solution correctly identifies the period and phase of the solar cycle. It also indicates that temperatures have gone up by about 2 ◦F over the 55 years covered by the ...

The standard approaches to solving overdetermined linear systems Ax ≈ b construct minimal corrections to the vector b and/or the matrix A such that the corrected system is compatible. In ordinary least squares (LS) the correction is restricted to b, while in data least squares (DLS) it is restricted to A. In scaled total least squares (Scaled TLS) [15], corrections to both b and A are allowed, ...

We define texture mapping as an optimization problem for which the goal is to preserve the maximum amount of information in the mapped texture. We derive a solution that is optimal in the least-squares sense and that corresponds to the pseudo-inverse of the texturemapping transformation. In practice, a first-order approximation of the least-squares solution is used as an initial estimate for th...