Support Size of $\varepsilon$-Capacity-Achieving Inputs for the Amplitude-Constrained AWGN Channel
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| Format: | Preprint |
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2026
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| _version_ | 1866918450620792832 |
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| author | Barletta, Luca Dytso, Alex |
| author_facet | Barletta, Luca Dytso, Alex |
| contents | We study the amplitude-constrained additive white Gaussian noise (AWGN) channel from the perspective of near-optimal input distributions. While it is known that the capacity-achieving input is discrete with finitely many mass points, the precise scaling of its support size as a function of the amplitude constraint remains an open problem. In this work, we instead consider the minimal support size required to achieve capacity up to an $\varepsilon$-gap. We introduce the quantity $K_\varepsilon(A)$, defined as the smallest support size among discrete inputs supported on $[-A,A]$ that achieves mutual information within $\varepsilon$ of capacity. We show that this relaxed formulation is significantly more tractable and admits sharp characterizations across different regimes of $\varepsilon$. In particular, when $\varepsilon$ decays polynomially with $A$, i.e., $\varepsilon = A^{-β}$ for $β\geq 1$, we establish that $K_\varepsilon(A) = Θ(A\sqrt{\log A})$. For exponentially small gaps, we obtain bounds of order between $A\sqrt{\log A}$ and $A^{3/2}$. Our approach combines approximation-theoretic bounds for Gaussian mixtures with information-theoretic control of entropy via $χ^2$-divergence, together with a wrapping argument that relates the problem to approximating the uniform distribution on the circle. Beyond the technical results, our framework provides a conceptual explanation for the variety of scaling laws observed in prior numerical studies, showing that these correspond to different regimes of $\varepsilon$-optimality rather than intrinsic properties of the exact optimizer. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_14915 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Support Size of $\varepsilon$-Capacity-Achieving Inputs for the Amplitude-Constrained AWGN Channel Barletta, Luca Dytso, Alex Information Theory We study the amplitude-constrained additive white Gaussian noise (AWGN) channel from the perspective of near-optimal input distributions. While it is known that the capacity-achieving input is discrete with finitely many mass points, the precise scaling of its support size as a function of the amplitude constraint remains an open problem. In this work, we instead consider the minimal support size required to achieve capacity up to an $\varepsilon$-gap. We introduce the quantity $K_\varepsilon(A)$, defined as the smallest support size among discrete inputs supported on $[-A,A]$ that achieves mutual information within $\varepsilon$ of capacity. We show that this relaxed formulation is significantly more tractable and admits sharp characterizations across different regimes of $\varepsilon$. In particular, when $\varepsilon$ decays polynomially with $A$, i.e., $\varepsilon = A^{-β}$ for $β\geq 1$, we establish that $K_\varepsilon(A) = Θ(A\sqrt{\log A})$. For exponentially small gaps, we obtain bounds of order between $A\sqrt{\log A}$ and $A^{3/2}$. Our approach combines approximation-theoretic bounds for Gaussian mixtures with information-theoretic control of entropy via $χ^2$-divergence, together with a wrapping argument that relates the problem to approximating the uniform distribution on the circle. Beyond the technical results, our framework provides a conceptual explanation for the variety of scaling laws observed in prior numerical studies, showing that these correspond to different regimes of $\varepsilon$-optimality rather than intrinsic properties of the exact optimizer. |
| title | Support Size of $\varepsilon$-Capacity-Achieving Inputs for the Amplitude-Constrained AWGN Channel |
| topic | Information Theory |
| url | https://arxiv.org/abs/2604.14915 |