Lecture 3 : Scalar and Matrix Concentration

Here, we will give an aside on probabilities, and in particular on various ways to establish what is know as concentration. Given some information about a distribution, e.g., its mean or its variance or information about higher moments, there are various ways to establish bounds on the tails of sums of random variables from that distribution. That is, there are various ways to establish that estimates are close to their expected value, with high probability. Today and next time, we will cover several of these methods, both in the case where the random variables are scalar or real-valued and when the random variables are matrix-valued. The former can be used to bound that latter, e.g., by bounding every element of the random matrix individually, but the latter often provide tighter bounds in those cases.