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.

翻译：投影维护是核心数据结构任务之一。高效的投影维护数据结构已推动了许多凸规划算法的最新突破。在本工作中，我们将此框架进一步扩展至Kronecker积结构。给定约束矩阵${\sf A}$和具有稀疏特征基的半正定矩阵$W\in \mathbb{R}^{n\times n}$，我们考虑维护形式为${\sf B}^\top({\sf B}{\sf B}^\top)^{-1}{\sf B}$的投影，其中${\sf B}={\sf A}(W\otimes I)$或${\sf B}={\sf A}(W^{1/2}\otimes W^{1/2})$。在每次迭代中，权重矩阵$W$接收一个低秩更新，同时我们接收到一个新向量$h$。目标是维护投影矩阵并以良好的近似保证应答查询${\sf B}^\top({\sf B}{\sf B}^\top)^{-1}{\sf B}h$。我们为此任务设计了一种快速的动态数据结构，且该结构能抵御自适应敌手。遵循[Beimel, Kaplan, Mansour, Nissim, Saranurak and Stemmer, STOC'22]那项优美而开创性的工作，我们利用差分隐私工具来减少数据结构所需的随机性，并进一步提升了运行时间。