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We consider differentially private approximate singular vector computation. Known worst-case lower bounds show that the error of any differentially private algorithm must scale polynomially with the dimension of the singular vector. We are able to replace this dependence on the dimension by a natural parameter known as the coherence of the matrix that is often observed to be significantly smaller than the dimension both theoretically and empirically. We also prove a matching lower bound showingarXiv:1211.0975v1 fatcat:r6r2oia3tvavbdzjfay4cco2ka