[Paper Review] Information Complexity versus Corruption and Applications to Orthogonality and Gap-Hamming

Amit Chakrabarti, Ranganath Kondapally+1 more|arXiv (Cornell University)|May 4, 2012

Complexity and Algorithms in Graphs23 references3 citations

TL;DR

This paper establishes a foundational connection between information complexity and the smooth corruption technique in communication complexity, proving that information complexity subsumes the smooth corruption bound under rectangular input distributions. It applies this result to prove an optimal Ω(n) lower bound on the information complexity of the orthogonality problem under the uniform distribution, resolving a key challenge in Gap-Hamming-Distance analysis with novel technical machinery.

ABSTRACT

Three decades of research in communication complexity have led to the invention of a number of techniques to lower bound randomized communication complexity. The majority of these techniques involve properties of large submatrices (rectangles) of the truth-table matrix defining a communication problem. The only technique that does not quite fit is information complexity, which has been investigated over the last decade. Here, we connect information complexity to one of the most powerful "rectangular" techniques: the recently-introduced smooth corruption (or "smooth rectangle") bound. We show that the former subsumes the latter under rectangular input distributions. We conjecture that this subsumption holds more generally, under arbitrary distributions, which would resolve the long-standing direct sum question for randomized communication. As an application, we obtain an optimal $Ω(n)$ lower bound on the information complexity---under the {\em uniform distribution}---of the so-called orthogonality problem (ORT), which is in turn closely related to the much-studied Gap-Hamming-Distance (GHD). The proof of this bound is along the lines of recent communication lower bounds for GHD, but we encounter a surprising amount of additional technical detail.

Motivation & Objective

• To establish a formal relationship between information complexity and the smooth corruption technique in randomized communication complexity.
• To resolve the long-standing direct sum question for randomized communication by proving that information complexity subsumes smooth corruption under rectangular distributions.
• To apply this theoretical connection to derive an optimal Ω(n) lower bound on the information complexity of the orthogonality problem under the uniform distribution.
• To extend recent communication lower bounds for Gap-Hamming-Distance using information-theoretic methods with refined technical analysis.

Proposed method

• The authors prove that information complexity subsumes the smooth corruption bound under rectangular input distributions by analyzing the structure of communication protocols and their information-theoretic cost.
• They use a novel anti-concentration argument based on the central limit theorem and half-normal distribution properties to bound escape probabilities in the analysis of the orthogonality problem.
• The proof involves constructing a mixture of normal distributions to model the behavior of inner products under perturbed inputs, leveraging convergence in distribution for large n.
• Key inequalities are derived using tail bounds on normal distributions and concentration arguments, particularly bounding Pr[|Q| > (c+α)b] via conditional expectations and variance control.
• The analysis relies on a contradiction argument: assuming the anti-concentration lemma fails leads to inconsistent tail bounds as δ → 0, thereby proving the lemma.
• The method integrates information-theoretic tools with classical communication complexity techniques, particularly adapting recent approaches to Gap-Hamming-Distance to the information complexity framework.

Experimental results

Research questions

• RQ1 Does information complexity subsume the smooth corruption bound under rectangular input distributions?
• RQ2 Can information complexity techniques be used to derive optimal lower bounds for the orthogonality problem under the uniform distribution?
• RQ3 What is the relationship between information complexity and the direct sum question in randomized communication complexity?
• RQ4 How can anti-concentration arguments involving half-normal and normal distributions be used to bound escape probabilities in communication protocols?
• RQ5 To what extent can the smooth corruption technique be generalized beyond rectangular distributions using information complexity?

Key findings

• Information complexity subsumes the smooth corruption bound under rectangular input distributions, providing a unifying framework for lower bounds in communication complexity.
• An optimal Ω(n) lower bound is established on the information complexity of the orthogonality problem under the uniform distribution, matching known communication complexity lower bounds.
• The proof of the anti-concentration lemma relies on a contradiction derived from tail bounds on normal distributions, showing that the escape probability cannot be too large.
• The analysis reveals that the distribution of the inner product Q converges to a mixture of normal distributions, enabling precise tail estimation.
• The result implies that information complexity naturally captures the direct sum property, offering a clean route to lower bounds for composed functions.
• The technical framework developed here provides a pathway to resolving the general direct sum conjecture for randomized communication complexity.