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On the Evolution of the Capacity-Achieving Input Support for the Amplitude-Constrained AWGN Channel

Luca Barletta, Alex Dytso

arXiv:2609.10039v1cs.ITmath.IT

TL;DR

The paper asks how the support of the capacity-achieving input distribution changes with the amplitude constraint in an AWGN channel, a structure whose transitions remain unresolved. Using KKT-function zero multiplicities and continuity properties, it proves that nonzero support points persist locally and that local support-cardinality changes are confined to the origin. The direction of transitions as amplitude increases, and global monotonicity of support cardinality, remain open.

  • Problem

    For the peak-constrained AWGN channel, the exact CAID structure is generally unknown, and how its support evolves with amplitude remains unresolved.

    Source: §1 · p. 1

  • Method

    The paper combines KKT-function zero and multiplicity bounds with local geometry and continuity of the optimal input distribution and KKT function.

    Source: §5 · p. 21

  • Results

    Locally, nonzero support points persist, and support cardinality is either unchanged or larger by one, with any additional point originating at the origin.

    Source: §1 · p. 2, §5 · p. 21

  • Takeaways & Limitations

    All local changes in support cardinality occur at the origin, through its appearance or disappearance or its splitting or merging into a symmetric pair.

    Source: §1 · p. 2, §5 · p. 21

    The results do not determine transition direction or prove global monotonicity, and nonzero-point persistence is specific to this amplitude-constrained AWGN setting.

    Source: §4.2 · pp. 20–21

Abstract

from arXiv · show

We consider an additive white Gaussian noise channel subject to a peak-amplitude constraint and study how the support of the capacity-achieving input distribution (CAID) evolves as the amplitude constraint $A$ varies. Although the CAID is known to be unique, symmetric, discrete, and finitely supported, the structure of its support transitions has remained largely unresolved. We show that the origin is the only possible degenerate support point and the only possible inactive contact point of the KKT function. We then establish continuity properties of the optimal distribution and the KKT function and prove that every nonzero support point is locally stable: under small changes in $A$, it persists as a unique nearby atom whose location and probability mass vary continuously. Combining these results, we prove that, locally, the support cardinality at a nearby amplitude is either unchanged or larger by one and that any such increase can occur only at the origin. Consequently, all local changes in support cardinality are confined to the origin: locally, the only possible transition mechanisms are the appearance or disappearance of the point at the origin and the splitting or merging of the origin into a symmetric pair.

1 Introduction

For the peak-constrained AWGN channel, the paper studies how the capacity-achieving input distribution’s support changes with amplitude. It proves that nonzero support points persist locally and that local changes in support cardinality are confined to the origin.

1.1 Past Work

  • The exact capacity and CAID structure are generally unknown; prior work established a unique, symmetric, discrete CAID with finite support and bounds on support size.Sharma and Shamai conjectured that support cardinality is nondecreasing in A, a global property this work does not establish.Source: §1 · p. 1, §1.1 · pp. 2–3

1 Introduction

  • Figure 1 plots the CAID support against A, with marker size proportional to probability mass.The introduction uses the numerical figure to motivate its study of support evolution.Source: §1 · p. 1, Figure 1 · p. 2
  • Locally, support cardinality is either unchanged or larger by one, and every local change occurs at the origin while nonzero support points persist.The paper leaves the direction of transitions as A increases unresolved.Source: §1 · p. 2

1.2 Outline and Contributions

  • The analysis revisits KKT-function zero bounds with multiplicities, identifies the origin as the only possible degenerate interior support point or inactive contact, and establishes continuity and local stability.The paper’s outline also identifies the local increase-by-at-most-one result as a main contribution.Source: §1.2 · pp. 3–4

1.3 Notation

  • The notation uses supp(PX) for distribution support, ⇒ for weak convergence, and D(P∥Q) for relative entropy.Random variables use uppercase letters and their realizations use the corresponding lowercase letters.Source: §1.3 · p. 4

2 Preliminaries

The preliminaries use the KKT function to connect optimal support points with its zeros and local maxima, then develop zero classifications and analytic tools for studying their evolution. These tools include zero-counting results and stability under locally uniform convergence.

2.1 KKT Conditions

  • The KKT conditions make support points zeros and local maxima of G_A, so tracking its zeros helps study support evolution.Figure 2 depicts G_A as a function of x for different amplitude constraints A.Source: §2.1 · p. 4, Figure 2 · p. 5
  • G_A has a unique entire extension to the complex plane, and its zeros are isolated with finite multiplicity.The section also derives higher-order derivatives using conditional cumulants.Source: §2.1 · p. 6

2.2 Classification of Zeros of G_A

  • The paper classifies interior zeros and defines inactive contact points as interior, non-support points where the KKT inequality is an equality.A nondegenerate support point is also defined through a condition on G_A''.Source: §2.2 · p. 7

2.3 Zero-Counting Techniques and Stability of Zeros

  • The analysis combines Karlin’s oscillation theorem, Rolle’s theorem with multiplicities, exponential-polynomial zero bounds, and Hurwitz’s theorem for zero stability.Hurwitz’s theorem preserves the number of zeros, counted with multiplicity, near an isolated zero under locally uniform convergence.Source: §2.3 · pp. 7–8

3 Properties of Zeros of GA

The section analyzes the zeros of the KKT function, including their multiplicities, to establish structural constraints on support points. It shows that only the origin can be a degenerate support point or an inactive contact point.

3.1 Bounds on the Number of Zeros of GA

  • Section 3.1 re-derives a bound on KKT-function zeros while explicitly accounting for their multiplicities.The argument uses a Gaussian-transform representation and zero-counting results.Source: §3.1 · pp. 9–10

3.2 Order of Zeros of GA

  • The origin is the only possible degenerate support point and the only possible inactive contact point of the KKT function.Interior zeros have even multiplicity; an inactive contact at zero has multiplicity two, while a degenerate support point there has multiplicity four.Source: §3.2 · pp. 10–11
  • Every interior zero of the KKT function has even multiplicity, and a degenerate interior support point has multiplicity at least four.The paper’s multiplicity count implies that degeneracy is possible only at the origin.Source: §3.2 · pp. 10–11
  • Several zero properties hold for any channel with a real-analytic, nonzero KKT function; Claim 2 is suggested to extend to a broader family of additive channels.The paper gives noise distributions with log-concave densities as an example of that family.Source: §3.2 · p. 11
  • Across A−, A0, and A+, one example turns an inactive multiplicity-two origin contact into an active nondegenerate support point; another replaces the degenerate origin contact with two symmetric nondegenerate support points, increasing cardinality exactly one.Source: §3.2 · pp. 12–13

4 Transitions of the Support Points

The section studies how optimal support points respond to changes in the amplitude constraint. It establishes local persistence for nondegenerate points and presents numerical illustrations of origin transitions.

4.1 Local Stability of Nondegenerate Support Points

  • Each nondegenerate support point persists locally as a unique nearby support point whose location and probability mass vary continuously with the amplitude constraint.The result concerns sufficiently small changes in A.Source: §4.1 · p. 13
  • The numerical illustrations show an origin support point appearing near A0 ≈ 1.6659 and splitting into a symmetric pair near A0 ≈ 2.9075.These are illustrations of two possible origin transitions.Source: Figure 3 · p. 14, Figure 4 · p. 15
  • The subsection reports continuity results for the optimal input and output distributions and capacity as A varies.The stated continuity result includes locally uniform convergence.Source: §4.1 · p. 16

3. Convergence of G′′ A:

The paper establishes local continuity and persistence properties for the optimal support, then confines possible local support changes to the origin. The resulting transition geometry is local; it does not establish that support cardinality is nondecreasing as A increases.

3. Convergence of G′′ A:

  • Locally uniform convergence of G′′_A and holomorphic continuity of G_A support the analysis of nearby support points.The section states that G′′_A converges locally uniformly and treats G_A as an entire function continuous on compact sets.Source: §3 · p. 16
  • Each nondegenerate nonzero support point has exactly one nearby continuation under sufficiently small changes in A, and its location converges to the original point.The local uniqueness and convergence follow from the neighborhood argument around x0.Source: §3 · pp. 16–17
  • At the boundary, the nearby support points remain ±A, and their probability masses converge to those at the reference amplitude.The boundary neighborhoods contain the unique nearby points ±A.Source: §3 · p. 17, §4.2 · p. 19

4.2 Support Size Transitions

  • Near the origin, an inactive contact can contribute at most one support point, while a nondegenerate origin atom can persist as one point or be replaced by two.If the KKT function is strictly negative at zero, the nearby origin neighborhood contributes no support points; an origin atom cannot disappear in the stated local case.Source: §4.2 · pp. 19–20
  • Every possible local change in support configuration is confined to a neighborhood of the origin, since nonzero support points have unique nearby continuations.The theorem’s local cardinality bound does not establish monotonicity; a decrease as A increases is not ruled out.Source: §4.2 · pp. 18, 20
  • The local cardinality result leaves the direction of transitions unresolved, so it does not prove the conjectured nondecreasing behavior of K(A).A decrease in support cardinality as A increases is not ruled out.Source: §4.2 · p. 20
  • The only local origin transitions are appearance or disappearance of zero and splitting or merging between zero and a symmetric pair.The local persistence result is specific to this amplitude-constrained AWGN setting; the paper notes that other channel models or constraints may permit positive branches to merge.Source: §4.2 · p. 21

5 Conclusion and Outlook

The paper uses multiplicity bounds for the KKT function together with local continuity to characterize support changes near a fixed amplitude. It confines local changes to the origin, while leaving monotonicity of support cardinality open.

5 Conclusion and Outlook

  • The KKT function’s multiplicity budget, combined with its inequality and continuity properties, yields local stability of nonzero support points and confines support changes to the origin.The paper identifies the origin as the only possible inactive contact point or degenerate interior support point.Source: §5 · p. 21
  • The zero-counting argument exploits the Gaussian kernel, while the multiplicity analysis may extend to channels with suitable zero-counting and regularity properties.The paper presents this extension as a possibility, not an established result.Source: §5 · p. 21
  • The local transition theorem does not rule out a decrease in K(A) as A increases, leaving monotonicity unresolved.Proving nondecreasing support cardinality remains necessary to establish the full conjecture of Sharma and Shamai.Source: §5 · pp. 21–22
  • Numerical evidence suggests that the optimal input’s probability masses become progressively less concentrated as the amplitude constraint increases.Figure 5 plots partial sums in (165) against A as numerical evidence for the proposed majorization property.Source: §5 · p. 22, Figure 5 · p. 22

A Proof of Lemma 4

The proof relates zero multiplicities of a function to the number of zeros of its derivative, using multiplicity reductions at existing zeros and Rolle’s theorem between them.

A Proof of Lemma 4

  • Together, multiplicity reductions and intervening zeros give a lower bound on the number of zeros of f′ counted with multiplicity.The argument combines contributions at the original zeros with at least r − 1 additional zeros.Source: A Proof of Lemma 4 · pp. 22–23
  • A zero of f with multiplicity m_i contributes a zero of f′ with multiplicity m_i − 1.At x_i, the derivatives through order m_i − 1 vanish, while the derivative of order m_i does not.Source: A Proof of Lemma 4 · pp. 22–23
  • Rolle’s theorem supplies at least one additional zero of f′ between each consecutive pair of distinct zeros of f.The intervals between consecutive zeros are disjoint, so these are distinct additional zeros.Source: A Proof of Lemma 4 · p. 23

B.1 Proof of Continuity of the Input and Output Distributions and Capacity

The appendix proves continuity of the capacity-achieving input distribution as the amplitude varies, then uses continuity of the mutual-information functional to obtain continuity of capacity and the induced output distribution.

B.1 Proof of Continuity of the Input and Output Distributions and Capacity

  • Any weakly convergent subsequential limit of feasible optimal inputs is feasible at the limiting amplitude.The proof uses the Portmanteau theorem to establish feasibility.Source: §B.1 · p. 23
  • Continuity of mutual information on a common compact support shows that the weak limit is capacity-achieving, and uniqueness identifies it with the limiting CAID.The proof then concludes convergence of the full sequence of optimal input distributions.Source: §B.1 · p. 24
  • The input-distribution convergence and continuity of mutual information also yield continuity of capacity with respect to the amplitude.The passage states this capacity-continuity result for use in the subsequent argument.Source: §B.1 · pp. 24–25
  • The induced output distributions converge weakly by testing bounded continuous functions through the Gaussian channel.The proof shows that the transformed test function is bounded and continuous, then applies dominated convergence.Source: §B.1 · p. 25

B.2 Proof of Convergence of the Output Densities

Weak convergence yields convergence of the output densities, which is uniform on compact subsets of the real line.

B.2 Proof of Convergence of the Output Densities

  • Weak convergence gives pointwise convergence of the output densities through the bounded continuous Gaussian kernel.Source: §B.2 · p. 25
  • Uniform Lipschitz bounds make the output densities equicontinuous on R.Source: §B.2 · p. 25
  • The Arzelà–Ascoli theorem upgrades pointwise convergence to uniform convergence on every compact subset of R.The argument uses equicontinuity and compact-subset convergence.Source: §B.2 · p. 25

B.3 Convergence of G′′

The proof uses compact support and uniformity arguments to establish convergence of G′′ on compact subsets of R.

B.3 Convergence of G′′

  • The input distributions are supported on a common compact interval [−B, B].Source: §B.3 · pp. 25–26
  • A bounded, uniformly continuous function on [−B, B]×L and a finite-net argument support uniform convergence on L.Source: §B.3 · p. 26
  • The resulting convergence is uniform on every compact subset of R, with dominated convergence used in the argument.Source: §B.3 · p. 26

B.4 Proof of Holomorphic Continuity of GA

The KKT function is extended to the complex plane, and the proof establishes convergence uniformly on compact subsets.

B.4 Proof of Holomorphic Continuity of GA

  • For each compact K in C, the KKT function admits an entire extension.Source: §B.4 · p. 26
  • Locally uniform convergence of the output densities and dominated convergence support the compact-set convergence argument.Source: §B.4 · p. 27
  • The convergence is uniform on every compact subset of C.Source: §B.4 · p. 27
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