37.1: Least Squares Fit
- Page ID
- 70388
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Review Chapters Chapter 13 pg 225-239 of the Boyd textbook.
In this first part of this course, we try to solve the system of linear equations \(Ax=b\) with an \(m \times n\) matrix \(A\) and a column vector \(b\).
There are three possible outcomes: an unique solution, no solution, and infinite many solutions. (Review the material on this part if you are no familiar with when the three types of outcomes happen.)
When \(m<n\), we call the matrix \(A\) underdeterminated, because we can not have an unique solution for it. When \(m>n\), we call the matrix \(A\) overdeterminated, becasue we may not have a solution with high probability.
However, if we still need to find a best \(x\), even when there is no solution or infinite many solutions we use a technique called least squares fit (LSF). Least squares fit find \(x\) such that \(\|Ax-b\|\) is the smallest (i.e. we try to minimize the estimation error).
- When there is no solution, we want to find \(x\) such that \(Ax−b\) is small (here, we want \(\|Ax-b\|\) to be small).
- If the null space of \(A\) is just \(\{0\}\), we can find an unique \(x\) to obtain the smallest \(\|Ax-b\|\).
- If there is a unique solution \(x^*\) for \(Ax=b\), then \(x^*\) is the optimal \(x\) to obtain the smallest \(\|Ax-b\|\), which is 0.
- Because the null space of \(A\) is just \(\{0\}\), you can not have infinite many solutions for \(Ax=b\).
- If the null space of \(A\) is not just \(\{0\}\), we know that we can always add a nonzero point \(x_0\) in the null space of \(A\) to a best \(x^*\), and \(\|A(x^*+x_0)-b\|=\|Ax^*-b\|\). Therefore, when we have multiple best solutions, we choose to find the \(x\) in the rowspace of \(A\), and this is unique.
Let \(A=\begin{bmatrix}1\\2\end{bmatrix},\quad b=\begin{bmatrix}1.5 \\ 2\end{bmatrix}\). Find the best \(x\) such that \(\|Ax-b\|\) has the smallest value.
Compute \((A^\top A)^{-1}A^\top b\).