Rank-sensitive bounds for higher-order Hoeffding components

Establish whether the dimension-dependent factors appearing in the higher-order Hoeffding bounds for complete U-statistics of trace-polynomial estimators can be replaced by rank-sensitive constants at every Hoeffding order.

Background

The paper derives bounds for every Hoeffding order of complete U-statistics estimating trace polynomials from global classical shadows. At order j, the generic bound contains a factor of the form (d+1)^{2(j-1)}, where d is the ambient Hilbert-space dimension. This dependence limits the result in growing-dimensional regimes and prevents the established bound from covering logarithmically increasing polynomial degrees.

The degree-two analysis achieves a sharper dimension-free degenerate-term bound involving the projected-block rank s. The unresolved problem is to determine whether analogous rank-sensitive replacements are possible for all higher Hoeffding orders, which could extend useful risk guarantees beyond the fixed-degree setting and potentially toward logarithmically increasing degrees.