Let $\mathbf{A} \in \mathbb{R}^{m\times n}$ where $m<n$. Also $rank(A)=m$ and $svd(\mathbf{A})=\mathbf{USV^{T}}$, where $\mathbf{U}\in\mathbb{R}^{m\times n}$, $\mathbf{S}\in\mathbb{R}^{n\times n}$, and $\mathbf{V}\in\mathbb{R}^{n\times n}$.
I know that $rank$-$k$ approximation of $\mathbf{A}$ can be written as
\begin{equation} \mathbf{A}_k = \sum_{i=1}^{k} \mathbf{u_i}s_{ii}\mathbf{v_i^T} \end{equation}
Notice that $\mathbf{A}_k \in \mathbb{R}^{m\times n}$ i-e dimensions remains unchanged.
However for my problem I need a new approximation $\mathbf{A}_{k}^{'}$ that is both
- low-rank
- low-dimension
By low-dimension I mean $\mathbf{A}_{k}^{'}\in\mathbb{R}^{m^{'}\times n^{'}}$ such that $m^{'}<m$ and $n^{'}<n$.
Since this is not a standard thing, I am looking for right
- Mathematical expression
- Paraphrase it in English language.
I would appreciate any help here from the experts.
Example:
