We will discuss a new kind of weak Szemeredi regularity lemma. It allows one to decompose a positive semidefinite matrix into a small number of "flat" matrices, up to a small error in the Frobenius norm. The proof utilizes randomized rounding based on Grothendieck’s identity. The regularity lemma can be interpreted as a probabilistic statement about "covariance loss" – the amount of covariance that is lost by taking conditional expectation of a random vector. This talk is based on a joint work with March Boedihardjo and Thomas Strohmer.

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