I'm trying to solve the following equation for $\beta$, the only unknown in the equation. I have two questions.
- I'm not quite sure how a matrix could fit into the Huber loss function and subsequent argmin operation. I am thinking I'm supposed to treat $\dfrac{1}{2}x$ and $\zeta|x|$ like a scalar multiplication on a matrix, but I'm not sure what I am minimizing in this case because the norm of the matrix is taken outside of the argmin function.
- Is my goal to find a single $\beta$ that minimizes over all k or $N^{'}-1$ number of $\beta_k$s for each k? I am thinking it might be the later because argmin operation is within $P_k^M$ instead of $P^M$.
$ k = 0, 1, 2, \dots{}, N^{'}-1 $
$ N^{'} = 2N $
$ P_k^M = \dfrac{N^{'}}{4} |argmin_\beta \Gamma({\phi_k \beta - \mathrm{x}})|^2 $
$\Gamma (x) = \begin{cases}\dfrac{1}{2}x & |x| \leq \zeta \\ \zeta|x|-\dfrac{1}{2}\zeta^2 & |x| > \zeta \end{cases}$ for some hyperparameter $\zeta$
$ \phi_k= \begin{bmatrix} cos(0) & sin(0) \\ cos(2\pi k / N^{'}) & sin(2\pi k / N^{'}) \\ cos(4\pi k / N^{'}) & sin(4\pi k / N^{'}) \\ \vdots{} & \vdots{} \\ cos(2(N^{'}-1)\pi k / N^{'}) & sin(2(N^{'}-1)\pi k / N^{'}) \end{bmatrix} $
$ \beta = \begin{bmatrix} \beta_{0, t = 0} & \beta_{0, 1} & \dots{} & \beta_{0, N^{'}-1} \\ \beta_{1, t = 0} & \beta_{1, 1} & \dots{} & \beta_{1, N^{'}-1} \end{bmatrix} $
$ \mathrm{x} = \begin{bmatrix} w_{j=0, t=0} & \dots{} & w_{0, N-1} & 0 & \dots{} & 0 \\ & \vdots{}& \ddots{} & & \vdots{} & \\ w_{N^{'}-1, 0} & \dots{} & w_{N^{'}-1, N-1} & 0 & \dots{} & 0 \end{bmatrix} $
The algorithm I am trying to implement is RobustPeriod: Time-Frequency Mining for Robust Multiple Periodicity Detection