We consider the problem of characterizing entrywise functions that preserve the cone of positive definite matrices when applied to every off-diagonal element. Our results extend theorems of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J. 26], Christensen and Ressel [Trans. Amer. Math. Soc., 243], and others, where similar problems were studied when the function is applied to all elements, including the diagonal ones. It is shown that functions that are guaranteed to preserve positive definiteness cannot at the same time induce sparsity, i.e., set elements to zero. These results have important implications for the regularization of positive definite matrices, where functions are often applied to only the off-diagonal elements to obtain sparse matrices with better properties (e.g., Markov random field/graphical model structure, better condition number). As a particular case, it is shown that emph{soft-thresholding}, a commonly used operation in modern high-dimensional probability and statistics, is not guaranteed to maintain positive definiteness, even if the original matrix is sparse. This result has a deep connection to graphs, and in particular, to the class of trees. We then proceed to fully characterize functions which do preserve positive definiteness. This characterization is in terms of absolutely monotonic functions and turns out to be quite different from the case when the function is also applied to diagonal elements. We conclude by giving bounds on the condition number of a matrix which guarantee that the regularized matrix is positive definite.
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