CORE Seminar: Joel Tropp

Ran­dom matri­ces now play a role in many areas of the­o­ret­i­cal, applied, and com­pu­ta­tional math­e­mat­ics. There­fore, it is desir­able to have tools for study­ing ran­dom matri­ces that are flex­i­ble, easy to use, and pow­er­ful. Over the last fif­teen years, researchers have devel­oped a remark­able fam­ily of results, called matrix con­cen­tra­tion inequal­i­ties, that bal­ance these criteria.

This talk offers an invi­ta­tion to the field of matrix con­cen­tra­tion inequal­i­ties. The pre­sen­ta­tion begins with some his­tory of ran­dom matrix the­ory, and it intro­duces an impor­tant prob­a­bil­ity inequal­ity for scalar ran­dom vari­ables. It describes a flex­i­ble model for ran­dom matri­ces that is suit­able for many prob­lems, and it dis­cusses one of the most impor­tant matrix con­cen­tra­tion results, the matrix Bern­stein inequal­ity. The talk con­cludes with some appli­ca­tions drawn from algo­rithms, com­bi­na­torics, sta­tis­tics, sig­nal pro­cess­ing, sci­en­tific com­put­ing, and beyond.