Understand
Given an unknown $n \times n$ matrix $A$ having non-negative entries, the \emph{inner product} (IP) oracle takes as inputs a specified row (or a column) of $A$ and a vector $v \in \mathbb{R}^{n}$, and returns their inner product.
- A derivative of IP is the induced degree query in an unknown graph $G=(V(G), E(G))$ that takes a vertex $u \in V(G)$ and a subset $S \subseteq V(G)$ as input and reports the number of neighbors of $u$ that are present in $S$.
- The goal of this paper is to understand the strength of the inner product oracle.
- Our results in that direction are as follows: (I) IP oracle can solve bilinear form estimation, i.e., estimate the value of ${\bf x}^{T}A\bf{y}$ given two vectors ${\bf x},\, {\bf y} \in \mathbb{R}^{n}$ with non-negative entries and can sample almost uniformly entries of a matrix with non-negative entries; (ii) We tackle for the first time weighted edge estimation and weighted sampling of edges that follow as an application to the bilinear form estimation and almost uniform sampling problems, respectively; (iii) induced degree query, a derivative of IP can solve edge estimation and an almost uniform edge sampling in induced subgraphs.