On the sum of the L1 influences of bounded functions

Let f: {-1, 1}n → [-1, 1] have degree d as a multilinear polynomial. It is well-known that the total influence of f is at most d. Aaronson and Ambainis asked whether the total L1 influence of f can also be bounded as a function of d. Bačkurs and Bavarian answered this question in the affirmative, providing a bound of O(d3) for general functions and O(d2) for homogeneous functions. We improve on their results by providing a bound of d2 for general functions and O(d log d) for homogeneous functions. In addition, we prove a bound of d/(2p) + o(d) for monotone functions, and provide a matching example.

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