For symmetric Boolean functions, approximate signed-subcube weight is 2^Theta(D) and sparsity is 2^Theta(D) log n up to log factors, where D is the deepest transition depth.
Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques (APPROX/RANDOM 2012) , series =
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A Paturi Theorem for Signed Subcube Representations
For symmetric Boolean functions, approximate signed-subcube weight is 2^Theta(D) and sparsity is 2^Theta(D) log n up to log factors, where D is the deepest transition depth.