Lower Bounds for Linear Operators

We consider a static data structure problem of computing a linear operator under cell-probe model. Given a linear operator $M \in \mathbb{F}_2^{m \times n}$, the goal is to pre-process a vector $X \in \mathbb{F}_2^n$ into a data structure of size $s$ to answer any query $\langle M_i , X \rangle$ in time $t$. We prove that for a random operator $M$, any such data structure requires: $$t \geq \Omega ( \min \{ \log (m/s) , n / \log s \} ).$$ This result overcomes the well-known logarithmic barrier in static data structures [MNSW98, Sie04, PD06, PTW08, Pat11, DGW19] by using a random linear operator. Furthermore, it provides the first significant progress toward confirming a decades-old folklore conjecture: that non-linear pre-processing does not substantially help in computing most linear operators. A straightforward modification of our proof also yields a wire lower bound of $\Omega(n \cdot \log^{1/d}(n))$ for depth-$d$ circuits with arbitrary gates that compute a specific linear operator $M \in \mathbb{F}_2^{O(n) \times n}$, even against some small constant advantage over random guessing. This bound holds even for circuits with only a small constant advantage over random guessing, improving upon longstanding results [RS03, Che08a, Che08b, GHK+13] for a random operator. Finally, our work partially resolves the communication form of the Multiphase Conjecture [Pat10] and makes progress on Jukna-Schnitger's Conjecture [JS11, Juk12]. We address the former by considering the Inner Product (mod 2) problem (instead of Set Disjointness) when the number of queries $m$ is super-polynomial (e.g., $2^{n^{1/3}}$), and the total update time is $m^{0.99}$. Our result for the latter also applies to cases with super-polynomial $m$.