Problem 78: Linear Sketching Over $F_2$

Suggested by	Grigory Yaroslavtsev
Source	Banff 2017
Short link	https://sublinear.info/78

For a function $f:\{0,1\}^n\rightarrow\{0,1\}$, we define its deterministic linear sketch complexity $D^\text{lin}(f)$ as a the smallest number $k$ such that there exist $k$ sets $S_1,\ldots,S_k \subseteq [n]$ and for any $x\in \{0,1\}^n$ we can determine $f(x)$ using $\sum_{i\in S_1} x_i, \ldots, \sum_{i\in S_k} x_i$, where the sum is mod $2$. Randomized linear sketch complexity is denoted by $R^\text{lin}(f)$.

Given $f$, we also define $f^+:\{0,1\}^n\times \{0,1\}^n \rightarrow\{0,1\}$ as follows. For any $x,y\in\{0,1\}^n$, define $f^+ (x,y) = f(x\oplus y)$, where $\oplus$ denotes bitwise XOR. It is known that $D^\text{lin}(f) = D^\rightarrow(f^+)$ [Montanaro-O-09], where $D^\rightarrow$ denotes one-way communication complexity (Alice sends one message to Bob).

Prove (or disprove) the following conjecture: $R^\text{lin}(f) = \tilde{\Theta}(R^\rightarrow(f^+))$.