The matrices in question are represented using $\sigma I + VV^T$ where $\sigma$ is a scalar, $I$ is the identity matrix and $V$ is of dimensions $n,s$ where $n$ denotes the length of a matrix and $s$ the approximation fidelity. I know that a low rank approximation means something different, but I do not know what the term for matrices in the above form is.
Also despite the question, I am interested in how they work in general.
This is not intended as a direct answer to your first question, but could provide some insight applicable to your more general interest. Let $$\underline {\overline {\bf{Q}} } = \sigma {\underline {\overline {\bf{I}} } _{\left[ n \right]}} + \underline {\overline {\bf{V}} } \,{\underline {\overline {\bf{V}} } ^T} = \sigma \left( {{{\underline {\overline {\bf{I}} } }_{\left[ n \right]}} + \underline {\overline {\bf{X}} } \,{{\underline {\overline {\bf{X}} } }^T}} \right)$$ where $$\underline {\overline {\bf{X}} } = \frac{1}{{\sqrt \sigma }}\underline {\overline {\bf{V}} }$$ It may (or may not) be well known that two consecutive uses of the generalized Woodbury identity yield $$\begin{array}{c} {\left( {{{\underline {\overline {\bf{I}} } }_{\left[ n \right]}} + \underline {\overline {\bf{X}} } \,{{\underline {\overline {\bf{X}} } }^T}} \right)^{ - 1}} = {\underline {\overline {\bf{I}} } _{\left[ n \right]}} - \underline {\overline {\bf{X}} } {\left( {{{\underline {\overline {\bf{I}} } }_{\left[ s \right]}} + {{\underline {\overline {\bf{X}} } }^T}\underline {\overline {\bf{X}} } } \right)^{ - 1}}{\underline {\overline {\bf{X}} } ^T}\\ = {\underline {\overline {\bf{I}} } _{\left[ n \right]}} - \underline {\overline {\bf{X}} } \,{\underline {\overline {\bf{X}} } ^T} + \underline {\overline {\bf{X}} } {\underline {\overline {\bf{X}} } ^T}{\left( {{{\underline {\overline {\bf{I}} } }_{\left[ n \right]}} + \underline {\overline {\bf{X}} } \,{{\underline {\overline {\bf{X}} } }^T}} \right)^{ - 1}}\underline {\overline {\bf{X}} } \,{\underline {\overline {\bf{X}} } ^T} \end{array}$$ This may then be recursively expanded as many times as desired to obtain $${\left( {{{\underline {\overline {\bf{I}} } }_{\left[ n \right]}} + \underline {\overline {\bf{X}} } \,{{\underline {\overline {\bf{X}} } }^T}} \right)^{ - 1}} = \sum\limits_{i = 0}^{2m - 1} {{{\left( { - 1} \right)}^i}{{\left( {\underline {\overline {\bf{X}} } \,{{\underline {\overline {\bf{X}} } }^T}} \right)}^i}} + {\left( {\underline {\overline {\bf{X}} } {{\underline {\overline {\bf{X}} } }^T}} \right)^m}{\left( {{{\underline {\overline {\bf{I}} } }_{\left[ n \right]}} + \underline {\overline {\bf{X}} } \,{{\underline {\overline {\bf{X}} } }^T}} \right)^{ - 1}}{\left( {\underline {\overline {\bf{X}} } \,{{\underline {\overline {\bf{X}} } }^T}} \right)^m}$$ Thus, in the limit $${\left( {{{\underline {\overline {\bf{I}} } }_{\left[ n \right]}} + \underline {\overline {\bf{X}} } \,{{\underline {\overline {\bf{X}} } }^T}} \right)^{ - 1}} = \sum\limits_{i = 0}^\infty {{{\left( { - 1} \right)}^i}{{\left( {\underline {\overline {\bf{X}} } \,{{\underline {\overline {\bf{X}} } }^T}} \right)}^i}}$$ as long as the series actually converges, which requires the largest singular value of $\underline {\overline {\bf{X}} }$ to be less than 1, or, equivalently, the largest singular value of $\underline {\overline {\bf{V}} }$ to be less than $\sqrt \sigma$.