The other day I thought about the problem
$$\min_{\bf a}\{\|{\bf Ma}-\lambda {\bf a}\|_2^2+\epsilon\|{\bf a-d}\|_2^2\}$$
For a known triplet ${\bf M}, \lambda, {\bf d \neq 0}$
Minimum would be $0$ if $\bf d=a$ is an eigenvector with eigenvalue $\lambda$.
Could this be a start for (eigenvalue,eigenvector) approximation or am I just extra christmas-time wishful?
Best Answer
We can indeed use the question as outline for an algorithm:
$${\bf a_i} = \min_{\bf a}\{\|{\bf Ma}-\lambda_i {\bf a}\|_2^2+\epsilon\|{\bf a-d}_i\|_2^2\}$$
Let us start with ${\bf d}_1, \lambda_1$ either random if we have no clue or as some first guess.
We can devise the following update:
$${\bf d_{i+1} }= {\bf d_{i}- a_{i}}\\\lambda_{i+1} = \delta \cdot \lambda_{i} + (1-\delta)\cdot \text{mean}\left(\frac{\bf Ma_i}{\bf a_i}\right)\\ s.t. \delta \in [0,1]$$ We use $\epsilon = 10^{-1}, \delta = 1/4$ in following experiment. Below is example log2 plot of error:
Merry Christmas!