Title: Dynamic Weighted Projection Maintenance with $\ell^p$-Lewis Weight
Primary area: alignment, fairness, safety, privacy, and societal considerations
Keywords: data structure, lewis weight, differential privacy, projection maintenance

Abstract: We introduce a new data–structure problem—\emph{Dynamic $\ell_{p}$-Lewis Weight Projection Maintenance}—that asks us to maintain the projection
\begin{align*}
    P(W) = W^{1/2-1/p}\,A(A^{\top}W^{1-2/p}A)^{-1}A^{\top}W^{1/2-1/p}
\end{align*}
under a stream of diagonal weight updates and to support fast matrix–vector
products with $P(W)$. This setting strictly generalizes the $\sqrt{W}A$ projection, which is at the heart of state-of-the-art linear programming and interior point methods, and it captures a wide range of algorithms that rely on leverage scores or Lewis weights for sampling and preconditioning. We provide a deterministic projection-maintenance data structure with sublinear amortized updates. Moreover, we extend it to the differential privacy setting.