具有记忆的高斯信源的色散及对抽象信源的推广
Dispersion of Gaussian Sources with Memory and an Extension to Abstract Sources
AI总结：
针对独立非同分布分量信源（包括高斯记忆信源）的有限块长有损压缩，推导了可达速率的二阶渐近表达式，并推广了i.i.d.抽象信源的色散结果。
摘要：
我们考虑分量独立但非同分布的信息源的有限块长有损压缩。关键在于，具有记忆的高斯信源可以转化为这种形式。我们证明，在超过失真$d$的概率至多为$\epsilon$的操作约束下，块长$n$时的最小可达速率满足$R(n, d, \epsilon)=\mathbb{R}_n(d)+\sqrt{\frac{\mathbb{V}_n(d)}{n}}Q^{-1}(\epsilon)+O \left(\frac{\log n}{n}\right)$，其中$Q^{-1}(\cdot)$是逆$Q$函数，而$\mathbb{R}_n(d)$和$\mathbb{V}_n(d)$是利用信源的$n$字母联合分布和失真度量计算的信源基本特征，分别称为$n$阶信息率失真函数和信源色散。该结果推广了现有针对i.i.d.分量抽象信源的色散结果。我们分析中的关键新型技术工具是点质量乘积代理测度，它使得典型集的构造成为可能。该代理通过保持坐标间的可加性并促进独立非同质项之和的典型性分析，将经验分布推广到i.i.d.场景之外。此外，对于高斯自回归信源，我们通过近似$n$字母协方差矩阵的特征值，量化了$\mathbb{R}_n(d)$和$\mathbb{V}_n(d)$随块长$n$趋于无穷时趋近其极限值的速度。利用这些收敛结果，我们将唯一已知的具有记忆的信源色散结果（即标量高斯-马尔可夫信源）加以精炼并推广到更一般的有限记忆高斯自回归信源。
英文摘要：
We consider finite blocklength lossy compression of information sources whose components are independent but non-identically distributed. Crucially, Gaussian sources with memory can be cast in this form. We show that under the operational constraint of exceeding distortion $d$ with probability at most $ε$, the minimum achievable rate at blocklength $n$ satisfies $R(n, d, ε)=\mathbb{R}_n(d)+\sqrt{\frac{\mathbb{V}_n(d)}{n}}Q^{-1}(ε)+O \left(\frac{\log n}{n}\right)$, where $Q^{-1}(\cdot)$ is the inverse $ Q$-function, while $\mathbb{R}_n(d)$ and $\mathbb{V}_n(d)$ are fundamental characteristics of the source computed using its $n$-letter joint distribution and the distortion measure, called the $n$th-order informational rate-distortion function and the source dispersion, respectively. This result generalizes the existing dispersion result for abstract sources with i.i.d. components. The key novel technical tool in our analysis is the point-mass product proxy measure, which enables the construction of typical sets. This proxy generalizes the empirical distribution beyond the i.i.d. setting by preserving additivity across coordinates and facilitating a typicality analysis for sums of independent, non-identical terms. Furthermore, for Gaussian autoregressive sources, we quantify how fast $\mathbb{R}_n(d)$ and $\mathbb{V}_n(d)$ approach their limiting values as the blocklength $n$ tends to infinity, by approximating the eigenvalues of the $n$th-letter covariance matrix. Using these convergence results, we sharpen and extend the only known dispersion result for a source with memory, namely the scalar Gauss-Markov source, to more general Gaussian autoregressive sources with finite memory.
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