We research the issue of personal vector imply estimation within the shuffle mannequin of privateness the place nn customers every have a unit vector in dd dimensions. We suggest a brand new multi-message protocol that achieves the optimum error utilizing O~(min(nε2,d))tilde{mathcal{O}}left(min(nvarepsilon^2,d)proper) messages per person. Furthermore, we present that any (unbiased) protocol that achieves optimum error requires every person to ship Ω(min(nε2,d)/log(n))Omega(min(nvarepsilon^2,d)/log(n)) messages, demonstrating the optimality of our message complexity as much as logarithmic elements.
Moreover, we research the single-message setting and design a protocol that achieves imply squared error O(dnd/(d+2)ε−4/(d+2))mathcal{O}(dn^{d/(d+2)}varepsilon^{-4/(d+2)}). Furthermore, we present that any single-message protocol should incur imply squared error Ω(dnd/(d+2))Omega(dn^{d/(d+2)}), exhibiting that our protocol is perfect within the commonplace setting the place ε=Θ(1)varepsilon = Theta(1). Lastly, we research robustness to malicious customers and present that malicious customers can incur giant additive error with a single shuffler.