Approximate herding data terms (negative result)

Kernel herding's data term (mean_l k(x_i, x_l), Chen, Welling & Smola 2010, Eq. 8) was tried with RandomSlices/RandomFeatures approximations and rejected: all candidate rows share the same random directions or features, so the estimator's noise is correlated across rows, and greedy argmax selection tracks that noise rather than averaging it out. In the table below the smallest budgets (k = 64 and 256, D = 512) select subsets worse than a random subset. Larger budgets beat random but stay 7-35× from exact herding, and only k = 8192 and D = 32768 come within about 3.5×. At that budget the estimator's own cost (O(kN log N) for slices, O(NDp) for Fourier features) matches the exact O(N²) data term for N around 10⁵. RandomSlices/RandomFeatures remain available for energydistance/mmd quality diagnostics; HerdingSplitter's data term is exact only.

N = 1500, p = 3, n = 300, 3 rng seeds per row.