Assume you need to design an image retrieval system where you want to find similar images to a query image $X \in \Bbb{R}^{N \times M}$ among the database images $Y_{k} \in \Bbb{R}^{N \times M}$
You have two options, using the Frobenius norm: $\operatorname{dis}(X,Y_{k}) = \left \| X-Y_{k} \right \|_{F}$ or the normalized inner product: $$\operatorname{sim}(X,Y_{k}) = \frac{\operatorname{tr}(X^{T}Y_{k})}{\left \|X \right \|_{F}\left \|Y_{k} \right \|_{F}}$$
My Question is:
Under which condition will a system using the Frobenius norm yield the same result than a system that uses the normalized inner product?
Any help is greatly appreciated!
Both systems will return the same results if the “image matrices” would already be normalized to unit norm. In this case, one can easily show that the $|| X – Y_k||_F^2 = 2 – 2*tr({X^T * Y_k})$, i.e., they will result in the same ranking.