There are three underlying quantities $x$, $y$, and $a$, where $x$ and $y$ are vectors, and $a$ is a scalar. They are related by $x = ay$.
We get noisy observations, $x_0,y_0$. We want to find $a$, and better estimates of $x,y$. Any suggestions on how to do it?
I could think of one possible idea:
minimize: $||x-ay|| + ||x - x_0|| + ||y-y_0||$.
Can anyone direct me to a paper/ book which discusses such ideas?
I don't think there's any way to get an estimate of all three unknowns $(a, x_0, y_0)$ given only one data point. Trivially, $a = x_0 / y_0$ is a zero-error solution.
However, if you have multiple data points, you can use the method of "Total Least Squares" to find $a$.