Consider the loss function: $$ \text{L} = \sum_{u=0}^{M-1}(X_{u}-(10-\sqrt{\frac{\sum_{i=0}^{N - 1} (Z_{u,i} - Y_{i})^2}{N}}))^2 $$
$$\text{where:} \\ Z = \mathbb{R}^{m{\times}n} \text{ matrix} $$ $$ X = \mathbb{R}^{m{\times}1} \text{ matrix} $$ $$ Y = \mathbb{R}^{1{\times}n} \text{ matrix} $$ $$0<=Z,X,Y<=10$$
This function is strictly convex in terms of X but not convex in Y. Is there a way to make an RMSE function convex in terms of X, Y or both without changing the RMSE aspect of the equation? I feel like it should be convex because: the rmse or l2 norm is convex -> 10 - concave l2 norm is concave -> affine variable minus concave function is convex - > convex squared is convex (above zero) -> sum of convex functions is convex. However the eigen vectors in matlab have negative solutions. Is this equation convex, strictly convex or not convex at all?
DCP seems to go wrong at squaring but matlab at sqrt.
Hessian values for x1...xm = [2, 0] [0, 2]
Cholesky Factorisation for x1...xm = [2^{1/2}, 0] [ 0, 2^{1/2}]
eigen values for x1...xm = 2 2
values from Y and both X and Y matrices from matlab is too complex to meaningfully include here