Given a dense matrix $A \in \mathbb{R}^{n \times m}$, with $n < m$, I am interested in finding a good approximation by choosing $s$ rows and zeroing the rest. This leads me to the following optimization problem

$$\underset{A_I}{\min}\left\lVert A - A_I \right\rVert_F^2$$

where $I$ is the index set of $s$ selected rows and $A_I$ is the restriction to this set (with rows $I^c$ set to zero). Are there any known results in this direction?

Smola's paper seems to be close to what I want, but I can not follow its notation when performing column selection. Is $K_i$ a matrix or a column?

Given a dense matrix $A \in \mathbb{R}^{n \times m}$, with $n < m$, I am interested in finding a good approximation by choosing $s$ rows and zeroing the rest. This leads me to the following optimization problem

$$\underset{A_I}{\min}\left\lVert A - A_I \right\rVert_F^2$$

where $I$ is the index set of $s$ selected rows and $A_I$ is the restriction to this set (with rows $I^c$ set to zero). Are there any known results in this direction?

Smola's paper seems to be close to what I want, but I can not follow its notation when performing column selection. Is $K_i$ a matrix or a column in Equation (11)? At first I thought it was about columns, but then when discussing selection strategies in Equation (25) it seems that there is a Gram-Schmidt-like linear relationship between $K_i$'s even though the previous selected columns are already orthogonal.

Given a dense matrix $A \in \mathbb{R}^{n \times m}$ with $n < m$, I am interested in finding a good approximation by choosing $s$ rows and zeroing the rest. This leads me to the following optimization problem

$ \begin{equation} \underset{A_I}{min}\left\lVert A - A_I \right\rVert_F^2, \end{equation} $

where $I$ is the index set of $s$ selected rows and $A_I$ is the restriction to this set (with rows $I^c$ set to zero).

  Are there any known results in this direction?

Smola's paper seems to be close to what I want, but I can not follow its notation when performing column selection (is $K_i$ a matrix or a column?).

Given a dense matrix $A \in \mathbb{R}^{n \times m}$, with $n < m$, I am interested in finding a good approximation by choosing $s$ rows and zeroing the rest. This leads me to the following optimization problem

$$\underset{A_I}{\min}\left\lVert A - A_I \right\rVert_F^2$$

where $I$ is the index set of $s$ selected rows and $A_I$ is the restriction to this set (with rows $I^c$ set to zero). Are there any known results in this direction?

Smola's paper seems to be close to what I want, but I can not follow its notation when performing column selection. Is $K_i$ a matrix or a column?