Title: Volume-Based Lower Bounds to the Capacity of the Gaussian Channel Under Pointwise Additive Input Constraints Show Chinese title

Title: 基于体积的高斯信道在逐点加性输入约束下的容量下界

Abstract: We present a family of relatively simple and unified lower bounds on the capacity of the Gaussian channel under a set of pointwise additive input constraints. Specifically, the admissible channel input vectors $\bx = (x_1, \ldots, x_n)$ must satisfy $k$ additive cost constraints of the form $\sum_{i=1}^n \phi_j(x_i) \le n \Gamma_j$, $j = 1,2,\ldots,k$, which are enforced pointwise for every $\bx$, rather than merely in expectation. More generally, we also consider cost functions that depend on a sliding window of fixed length $m$, namely, $\sum_{i=m}^n \phi_j(x_i, x_{i-1}, \ldots, x_{i-m+1}) \le n \Gamma_j$, $j = 1,2,\ldots,k$, a formulation that naturally accommodates correlation constraints as well as a broad range of other constraints of practical relevance. We propose two classes of lower bounds, derived by two methodologies that both rely on the exact evaluation of the volume exponent associated with the set of input vectors satisfying the given constraints. This evaluation exploits extensions of the method of types to continuous alphabets, the saddle-point method of integration, and basic tools from large deviations theory. The first class of bounds is obtained via the entropy power inequality (EPI), and therefore applies exclusively to continuous-valued inputs. The second class, by contrast, is more general, and it applies to discrete input alphabets as well. It is based on a direct manipulation of mutual information, and it yields stronger and tighter bounds, though at the cost of greater technical complexity. Numerical examples illustrating both types of bounds are provided, and several extensions and refinements are also discussed. Show Chinese abstract

Abstract: 我们提出了一类相对简单且统一的下界，用于高斯信道在一组点对点加法输入约束下的容量。 具体来说，允许的信道输入向量$\bx = (x_1, \ldots, x_n)$必须满足$k$个形式为$\sum_{i=1}^n \phi_j(x_i) \le n \Gamma_j$，$j = 1,2,\ldots,k$的加法代价约束，这些约束对于每个$\bx$都是逐点强制执行的，而不仅仅是在期望意义上。 更一般地，我们还考虑依赖于固定长度滑动窗口$m$的成本函数，即$\sum_{i=m}^n \phi_j(x_i, x_{i-1}, \ldots, x_{i-m+1}) \le n \Gamma_j$，$j = 1,2,\ldots,k$，这种表述自然地适应了相关性约束以及各种其他实际相关的约束。 我们提出了两类下界，这两类下界通过两种方法推导得出，这两种方法都依赖于对满足给定约束的输入向量集的相关体积指数的精确评估。 这种评估利用了类型方法在连续字母表上的扩展、积分的鞍点方法以及大偏差理论的基本工具。 第一类界限是通过熵功率不等式 (EPI) 得到的，因此仅适用于连续值输入。 相比之下，第二类界限更为通用，也适用于离散输入字母表。 它基于对互信息的直接操作，并且给出了更强和更紧的界限，尽管代价是更大的技术复杂性。 提供了说明这两种界限的数值例子，并讨论了几种扩展和改进。