  Projection maintenance is one of the core data structure tasks. Efficient
data structures for projection maintenance have led to recent breakthroughs in
many convex programming algorithms. In this work, we further extend this
framework to the Kronecker product structure. Given a constraint matrix ${\sf
A}$ and a positive semi-definite matrix $W\in \mathbb{R}^{n\times n}$ with a
sparse eigenbasis, we consider the task of maintaining the projection in the
form of ${\sf B}^\top({\sf B}{\sf B}^\top)^{-1}{\sf B}$, where ${\sf B}={\sf
A}(W\otimes I)$ or ${\sf B}={\sf A}(W^{1/2}\otimes W^{1/2})$. At each
iteration, the weight matrix $W$ receives a low rank change and we receive a
new vector $h$. The goal is to maintain the projection matrix and answer the
query ${\sf B}^\top({\sf B}{\sf B}^\top)^{-1}{\sf B}h$ with good approximation
guarantees. We design a fast dynamic data structure for this task and it is
robust against an adaptive adversary. Following the beautiful and pioneering
work of [Beimel, Kaplan, Mansour, Nissim, Saranurak and Stemmer, STOC'22], we
use tools from differential privacy to reduce the randomness required by the
data structure and further improve the running time.
