Matrix Chernoff Bound
Rudolf Ahlswede and Andreas Winter introduced (Ahlswede & Winter 2003) a Chernoff bound for matrix-valued random variables.
If is distributed according to some distribution over matrices with zero mean, and if are independent copies of then for any ,
where holds almost surely and is an absolute constant.
Notice that the number of samples in the inequality depends logarithmically on . In general, unfortunately, such a dependency is inevitable: take for example a diagonal random sign matrix of dimension . The operator norm of the sum of independent samples is precisely the maximum deviation among independent random walks of length . In order to achieve a fixed bound on the maximum deviation with constant probability, it is easy to see that should grow logarithmically with in this scenario.
The following theorem can be obtained by assuming has low rank, in order to avoid the dependency on the dimensions.
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