Consider the problem of shape averaging. In particular, suppose $X_i\ (i = 1,\dots, M$) are the input matrices, $X_i\in \mathbb R^{N\times2}$, with each sampled from corresponding 2D positions of shapes (or handwritten letters). We seek a shape average $V \in \mathbb R^{N\times 2}$; $V^T V = I$, of the $M$ input $X_i$ which minimizes the following optimization problem. $$ \min_{V,A_j} \sum_{j=1}^M \|X_j-VA_j\|_F^2,\ \text{s.t. } V^TV=I $$ where $A_j \in \mathbb R^{2\times2}$ are arbitrary nonsingular matrices. For simplicity, we assume all data $X_i$ are aligned so we do not have a translation term in the problem. An alternative approach is to instead minimize: $$ \min_{V,A_j} \sum_{j=1}^M \|X_jA_j-V\|_F^2,\ \text{s.t. } V^TV=I $$ where $A_j$ should also be arbitrary nonsingular matrices. Derive the solution for two optimizations. How do the solutions differ? You can assume all data matrices $X_j$ has rank $2$.
Hint: Refer to the PCA problem. Calculate its dual form and you’ll get the answer.
I have some knowledge of PCA, but don't know its duality, so it doesn't help. Hope someone could solve this problem...
Thanks in advance.
