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 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。 用于投影维护的有效数据结构导致许多 convex 编程算法最近出现突破。 在这项工作中, 我们进一步将这个框架扩展至 Kronecker 产品结构。 基于一个制约矩阵$( fsf A) $ 美元和正的半确定矩阵 $( w\\ in mathbb{ R ⁇ n\ time n} 美元, 并带有稀薄的偏差, 我们考虑维持投影的任务 : $( fsf Bunsf Bättop) ⁇ -1 unsf B} 。 其中, $( w\\\ otime I) 或 $( fs B ⁇ fs f 美元) 产品结构。 在每次试调时, 权重矩阵得到低级变化, 我们得到一个新的矢量 。 目标是维持投影矩阵, 并回答 $( wef B\fs lex ) 的调 B++ 数据设计, 需要一个动态、 快速的S&real 数据。