We study the amplitude-constrained additive white Gaussian noise channel. It is well known that the capacity-achieving input distribution for this channel is discrete and supported on finitely many points. The best known bounds show that the support size of the capacity-achieving distribution is lower-bounded by a term of order $A$ and upper-bounded by a term of order $A^2$, where $A$ denotes the amplitude constraint. It was conjectured in [1] that the linear scaling is optimal. In this work, we establish a new lower bound of order $A\sqrt{\log A}$, improving the known bound and ruling out the conjectured linear scaling. To obtain this result, we quantify the fact that the capacity-achieving output distribution is close to the uniform distribution in the interior of the amplitude constraint. Next, we introduce a wrapping operation that maps the problem to a compact domain and develop a theory of best approximation of the uniform distribution by finite Gaussian mixtures. These approximation bounds are then combined with stability properties of capacity-achieving distributions to yield the final support-size lower bound.

翻译：本文研究幅度受限加性高斯白噪声信道。已知该信道的容量可达输入分布是离散的，且支撑在有限个点上。现有最佳界表明容量可达分布的支撑集规模下界为$A$阶，上界为$A^2$阶，其中$A$表示幅度约束。文献[1]曾猜想线性缩放是最优的。本工作建立了$A\sqrt{\log A}$阶的新下界，改进了已知界并否定了线性缩放的猜想。为获得此结果，我们量化了容量可达输出分布在幅度约束内部接近均匀分布的特性。接着引入将问题映射到紧致域的环绕操作，并发展了用有限高斯混合逼近均匀分布的最佳逼近理论。这些逼近界再与容量可达分布的稳定性相结合，最终导出支撑集规模的下界。