Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower bound on the smallest eigenvalue of a finite sum of random positive semidefinite (PSD) matrices.
For PSD matrices, the eigenvalues are the same as the singular values. My question is:
Is there any tool, or any other special cases (apart from the PSD case) for which there is a tool, to obtain lower bounds on the singular values of a finite sum of random matrices, or random symmetrical matrices?