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arxiv: 2501.14941 · v6 · pith:XJBIWLBOnew · submitted 2025-01-24 · 💻 cs.IT · math.IT· math.PR

On the Optimality of Gaussian Code-books for Signaling over a Two-Users Weak Gaussian Interference Channel

Amir K. Khandani This is my paper

Pith reviewed 2026-05-23 05:20 UTC · model grok-4.3

classification 💻 cs.IT math.ITmath.PR
keywords two-user interference channelGaussian codebookscapacity regionweak interferenceHan-Kobayashi regionsingle-letter codingcalculus of variationstime-sharing
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The pith

Single-letter Gaussian codebooks achieve the capacity region of the two-user weak Gaussian interference channel.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the entire capacity region can be reached using only single-letter i.i.d. Gaussian codebooks. It begins at a known Gaussian corner point and applies calculus of variations to move along the boundary in small steps, proving that each new endpoint remains Gaussian. It further establishes that allowing vector-valued inputs produces no higher rates than the scalar single-letter case and that optimum time-sharing uses at most two phases. The Gaussian version of the Han-Kobayashi region therefore meets the boundary for weak interference and extends to the general case.

Core claim

The capacity region of a two-user weak Gaussian interference channel is achieved by single-letter Gaussian code-books. Starting from a corner point realized by Gaussian code-books, calculus of variations shows that each incremental step along the boundary ends at a Gaussian point. Any optimum vector-input solution does not exceed the single-letter value, at most two phases suffice for time-sharing, and the Han-Kobayashi region with these code-books coincides with the boundary.

What carries the argument

Incremental boundary traversal via calculus of variations that preserves Gaussian optimality from known corner points, combined with the proof that vector inputs cannot exceed single-letter rates.

If this is right

  • The Han-Kobayashi achievable region evaluated with single-letter Gaussian codebooks coincides with the capacity boundary.
  • Any optimum solution using vector inputs yields rates no higher than the single-letter Gaussian case.
  • Optimum time-sharing between boundary points requires at most two phases.
  • The same Gaussian optimality holds for the general interference case, not only the weak regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Gaussian codebooks may suffice for capacity in other two-user interference models once similar boundary arguments are applied.
  • Capacity-achieving schemes for these channels can be realized with standard Gaussian signaling without needing structured or non-Gaussian alphabets.
  • The two-phase time-sharing limit simplifies the search for optimal operating points in practical rate allocation.

Load-bearing premise

That moving along the boundary in small steps with calculus of variations keeps the optimum input distribution Gaussian at every point and that vector inputs cannot improve on the single-letter case.

What would settle it

An explicit non-Gaussian single-letter distribution or a vector input that achieves a strictly higher rate pair than the best Gaussian single-letter codebook on some point of the claimed capacity boundary.

Figures

Figures reproduced from arXiv: 2501.14941 by Amir K. Khandani.

Figure 1
Figure 1. Figure 1: Two users Gaussian Interference Channel (GIC) with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An example for power reallocation and its corresponding step along the boundary. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Channel models depicting decoding methods discussed in Theorem 3 where 3(a) corresponds [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example of a channel where the stationary solution for mutual information may result in a [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Υ and Γ as a function of ω (related to Theorem 10). Remark 5: The optimum Pareto minimal power reallocation vector is not unique. However, the corresponding set has a nested structure, and relying on any member of the set will be associated with [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Segment of the boundary capturing the weighted sum-rate over the two-user phase where [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: First step moving counterclockwise from the corner point with maximum [PITH_FULL_IMAGE:figures/full_fig_p042_7.png] view at source ↗
read the original abstract

This article shows that the capacity region of a two users weak Gaussian interference channel can be achieved using single letter Gaussian code-books. The approach relies on traversing the boundary in incremental steps. Starting from a corner point with Gaussian code-books, and relying on calculus of variation, it is shown that the end point in each step is achieved using Gaussian code-books. Optimality of Gaussian code-books is first established by limiting the random coding to independent and identically distributed scalar (single-letter) samples. Then, it is shown that the value of any optimum solution for vector inputs does not exceed that of the single-letter case. It is also shown that the maximum number of phases needed to realize the optimum time-sharing is two. It is established that the solution to the Han-Kobayashi achievable rate region, with single letter Gaussian code-books, achieves the optimum boundary. Even though the article focuses on weak interference, the results are applicable to the general case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript claims to prove that the capacity region of the two-user weak Gaussian interference channel is achieved by the Han-Kobayashi region using single-letter Gaussian codebooks. The argument starts from a known Gaussian corner point, traverses the boundary in incremental steps via calculus of variations to preserve Gaussian optimality at each endpoint, shows that vector-input optima cannot exceed the single-letter value, establishes that at most two time-sharing phases suffice, and concludes that Gaussian HK meets capacity (with a claim of applicability to the general case).

Significance. If rigorously established, the result would be significant for information theory, as it would characterize the capacity region of the weak Gaussian interference channel (a long-standing open problem) via Gaussian codebooks. The boundary-traversal technique with variations, if shown to have only Gaussian stationary points, could offer a new tool for proving optimality in interference channels. The manuscript receives credit for outlining a structured approach from an external corner point and for addressing time-sharing cardinality.

major comments (3)
  1. [Abstract] Abstract (proof strategy paragraph): the calculus-of-variations step that concludes each successive boundary endpoint remains Gaussian-optimal is load-bearing but unsupported by explicit verification that the mutual-information functionals (under weak-interference channel law and power constraints) admit no non-Gaussian critical points.
  2. [Abstract] Abstract (vector-input paragraph): the claim that 'the value of any optimum solution for vector inputs does not exceed that of the single-letter case' is asserted without the required bounding argument or error analysis, undermining the reduction from block length n to single-letter.
  3. [Abstract] Abstract (time-sharing paragraph): the assertion that 'the maximum number of phases needed to realize the optimum time-sharing is two' depends on the preceding reductions and requires an explicit derivation showing why more phases cannot improve the boundary.
minor comments (1)
  1. [Abstract] The final sentence claiming applicability to the general (strong) interference case should be justified or removed, as the weak-interference assumption appears essential to the variation argument.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. We address each major comment below and will revise the manuscript accordingly to improve explicitness and rigor.

read point-by-point responses

  1. Referee: [Abstract] the calculus-of-variations step that concludes each successive boundary endpoint remains Gaussian-optimal is load-bearing but unsupported by explicit verification that the mutual-information functionals (under weak-interference channel law and power constraints) admit no non-Gaussian critical points.

    Authors: We agree the abstract is concise. The manuscript applies calculus of variations to the mutual information expressions under the given channel law and constraints. To strengthen the presentation, the revised version will include an explicit lemma or appendix verifying that the only critical points are Gaussian, via direct computation of the functional derivative and analysis of the second variation. revision: yes

  2. Referee: [Abstract] the claim that 'the value of any optimum solution for vector inputs does not exceed that of the single-letter case' is asserted without the required bounding argument or error analysis, undermining the reduction from block length n to single-letter.

    Authors: The manuscript contains a bounding argument based on the memoryless Gaussian channel and mutual information properties showing vector optima cannot exceed the single-letter value. We will expand this section in revision with a more detailed derivation, including explicit error bounds for the n-to-single-letter reduction. revision: yes

  3. Referee: [Abstract] the assertion that 'the maximum number of phases needed to realize the optimum time-sharing is two' depends on the preceding reductions and requires an explicit derivation showing why more phases cannot improve the boundary.

    Authors: We will add an explicit derivation in the revised manuscript. Given the boundary traversal and the convex structure of the achievable region under the Han-Kobayashi scheme, any time-sharing solution with more than two phases can be reduced to two phases without rate loss, which will be shown via a supporting lemma on the geometry of the rate region. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins from an external known Gaussian corner point of the Han-Kobayashi region and applies calculus of variations to traverse the boundary while establishing single-letter Gaussian optimality before separately proving that vector-input optima cannot exceed the single-letter value. No quoted equations or self-citations reduce any load-bearing claim to a definition, fitted input, or prior author result by construction; the argument is self-contained against the stated external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the paper relies on standard information-theoretic definitions and the Han-Kobayashi framework without introducing new free parameters or invented entities; the calculus of variations is treated as a standard tool.

axioms (2)
  • standard math Standard definitions of mutual information, achievable rates, and capacity region in multi-user information theory
    Invoked to define the capacity region and Han-Kobayashi achievable rates.
  • domain assumption Applicability of calculus of variations to optimization over rate boundaries in Gaussian channels
    Used to show that incremental steps preserve Gaussian optimality.

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