Information Complexity versus Corruption and Applications to Orthogonality and Gap-Hamming
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 $\Omega(n)$ 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.
💡 Research Summary
The paper establishes a fundamental link between two major strands of randomized communication‑complexity lower‑bound techniques: information complexity (IC) and the smooth corruption (or smooth rectangle) bound, a powerful recent “rectangular” method. The authors first prove that, under rectangular input distributions (i.e., product distributions that factor across the two parties), the IC of any protocol dominates the smooth corruption bound. In technical terms, for any protocol π and any rectangle‑based distribution μ that is a product of marginal distributions, the information revealed by π, I(π; μ), is at least as large as the smooth corruption value C_smooth(μ). This result shows that the smooth corruption bound is subsumed by IC in this restricted setting, providing a unified perspective that had been missing: while most classical lower‑bound tools (discrepancy, corruption, partition bound, etc.) rely on large monochromatic or low‑error rectangles, IC is an information‑theoretic quantity that does not explicitly reference rectangles. By proving the inclusion, the paper bridges the gap between these viewpoints.
The authors then conjecture that the same inclusion holds for arbitrary (not necessarily product) input distributions. If true, this would resolve the long‑standing direct‑sum question for randomized communication: the total communication needed to solve k independent copies of a function would be Θ(k) times the communication needed for one copy, because the smooth corruption bound is known to tensorize under arbitrary distributions. Proving the conjecture would therefore give a clean, distribution‑independent proof of direct‑sum for randomized protocols.
Armed with the inclusion result, the paper turns to an application: the orthogonality problem (ORT). In ORT, Alice and Bob each receive an n‑bit string x and y, and must decide whether the inner product ⟨x, y⟩ (over the reals) equals zero. ORT is closely related to Gap‑Hamming‑Distance (GHD), a benchmark problem for communication‑complexity lower bounds. Prior work had established an Ω(n) lower bound on the randomized communication complexity of GHD using smooth corruption and related techniques, but no comparable lower bound was known for the information complexity of ORT under the uniform distribution.
The authors adapt the recent GHD lower‑bound machinery to the IC setting. The key technical challenge is to control the amount of information that a protocol can leak while still maintaining a low error on a large rectangle. They first bound the probability mass of any “smooth” rectangle under the uniform distribution, showing it must be exponentially small (≈2^{‑Ω(n)}). Then they perform a delicate entropy analysis: any protocol that succeeds with constant error on such a rectangle must convey at least Ω(n) bits of mutual information between the inputs and the transcript. This involves a refined use of chain‑rule decompositions, conditional mutual information bounds, and a careful handling of the “smoothness” parameter that allows a small fraction of the rectangle to be corrupted.
The result is an optimal Ω(n) lower bound on the uniform‑distribution information complexity of ORT. Because ORT reduces to GHD (by padding and thresholding the inner product), this also yields an Ω(n) IC lower bound for GHD under uniform inputs, matching the known communication‑complexity bound. The proof, while conceptually similar to the GHD communication lower bound, requires substantially more technical detail to translate rectangle‑size arguments into information‑theoretic ones.
Beyond ORT, the paper outlines several future directions. Extending the inclusion of smooth corruption into IC to non‑product distributions would immediately give a distribution‑independent direct‑sum theorem. Moreover, the techniques could be adapted to other functions where smooth corruption is known to be tight (e.g., set‑disjointness, Gap‑Inner‑Product). Finally, the authors suggest that the bridge between IC and rectangular methods may open pathways to lower bounds in multi‑party, quantum, or streaming models, where information‑theoretic arguments are already powerful but have lacked a clean connection to rectangle‑based combinatorial techniques.
In summary, the paper makes three major contributions: (1) it proves that information complexity subsumes the smooth corruption bound under product distributions; (2) it conjectures a full‑distribution version that would resolve the direct‑sum problem; and (3) it leverages this connection to obtain an optimal Ω(n) information‑complexity lower bound for the orthogonality problem (and consequently for Gap‑Hamming‑Distance) under the uniform distribution, thereby deepening our understanding of the interplay between information‑theoretic and combinatorial lower‑bound methods in communication complexity.