The following snapshot comes from the paper Latent Variable Graphical Model Selection Via Convex Optimization:
I know little about algebraic geometry so I have several basic questions:
- How is the dimension of $\mathcal{S}(k)$ and $\mathcal{L}(r)$ calculated? I've found an explanation for the latter one, but it's quite complicated. Can you give hints for me?
- Are some points nonsmooth because they have a smaller dimension?
- How is the tangent space calculated?
- Finally, how is the curvature calculated?
Thank you very much for answering. Reference materials are also welcomed.