Approximate update methods for the singular value decomposition

During Part C^U I worked with Prof. Yuji Nakatsukasa for my dissertation on approximate update methods for the singular value decomposition. I looked at the following problem: if you have a matrix $A \in \mathbb R^{n \times n}$ whose singular value decomposition $A = U \Sigma V^\top$ is already known, how can you leverage this information to approximate the singular values of a perturbation $A + E$, where $E$ is small?

You can download my dissertation here.