Lower Bounds for Linear Operators

2025-09-02

AI Summary

This work investigates lower bounds for computing linear operators in static data structures, focusing on the trade-off between preprocessing and query efficiency in the cell-probe model. Using a synthesis of linear algebra over finite fields, information theory, circuit complexity, and communication complexity, the paper establishes several fundamental results. First, it provides the first rigorous evidence for a long-standing folklore conjecture: nonlinear preprocessing cannot circumvent the classical logarithmic query-time lower bound for random linear operators. Second, it proves a new query-time lower bound $t geq Omega(min{log(m/s),, n/log s})$, breaking the “logarithmic barrier.” Third, it improves the wire complexity lower bound for depth-$d$ circuits to $Omega(n cdot log^{1/d} n)$. Finally, it resolves key special cases of the communication variant of the Multiphase Conjecture and the Jukna–Schnitger Conjecture.

Technology Category

Machine Learning: Other Foundations of Machine LearningKnowledge Representation and Reasoning: Computational Complexity of ReasoningSearch and Optimization: Non-convex Optimization

Application Category

Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphsSecurity and Privacy: Applications of cryptographyEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider a static data structure problem of computing a linear operator under cell-probe model. Given a linear operator $M in mathbb{F}_2^{m imes n}$, 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 $langle M_i , X angle$ in time $t$. We prove that for a random operator $M$, any such data structure requires: $$ t geq Ω( min { log (m/s) , n / log s } ).$$ 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 $Ω(n cdot log^{1/d}(n))$ for depth-$d$ circuits with arbitrary gates that compute a specific linear operator $M in mathbb{F}_2^{O(n) imes n}$, 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 $m$.

Problem

• Proving lower bounds for linear operator computation under cell-probe model
• Overcoming logarithmic barrier in static data structures with random operators
• Resolving conjectures on non-linear preprocessing and multiphase communication

Innovation

• Random linear operator overcomes logarithmic barrier
• Wire lower bound for depth-d circuits
• Progress on Multiphase and Jukna-Schnitger conjectures