I have a high-dimensional space, say $\mathbb{R}^{1000}$, and I have samples $y_1, \ldots , y_n \in \mathbb{R}^{1000}$ and $\hat{y}_1, \ldots , \hat{y}_n \in \mathbb{R}^{1000}$. Would $$ R^2 = 1 - \frac{\sum_i || y_i - \hat{y}_i||^2}{\sum_i || y_i - \mu ||^2},$$ where $\mu = \frac{1}{n} \sum_i y_i$, be a reasonable expression, for the some kind of "variance explained"? Is there a mathematical reasoning regarding what it captures? Suppose that the $\hat{y}_i$ were learned by some norm-square minimizing procedure ("least squares"), so the particular choice of Euclidian norm is at least partially relevant.
Or, what measure of fit would you use in such an higher-dimensional setting?
