The notion of compressibility of a random measure is a rather general concept which find applications in many contexts from data compression, to signal quantization, and parameter estimation. While compressibility for discrete and continuous measures is generally well understood, the case of discrete-continuous measures is quite subtle. In this paper, we focus on a class of multi-dimensional random measures that have singularities on affine lower-dimensional subsets. We refer to this class of random variables as affinely singular . Affinely singular random vectors naturally arises when considering linear transformation of component-wise independent discrete-continuous random variables. To measure the compressibility of such distributions, we introduce the new notion of dimensional-rate bias (DRB) which is closely related to the entropy and differential entropy in discrete and continuous cases, respectively. Similar to entropy and differential entropy, DRB is useful in evaluating the mutual information between distributions of the aforementioned type. Besides the DRB, we also evaluate the the RID of these distributions. We further provide an upper-bound for the RID of multi-dimensional random measures that are obtained by Lipschitz functions of component-wise independent discrete-continuous random variables (X). The upper-bound is shown to be achievable when the Lipschitz function is $A \mathrm {X}$ , where $A$ satisfies ${\mathrm{ SPARK}}({A_{m\times n}}) = m+1$ (e.g., Vandermonde matrices). When considering discrete-domain moving-average processes with non-Gaussian excitation noise, the above results allow us to evaluate the block-average RID and DRB, as well as to determine a relationship between these parameters and other existing compressibility measures.

中文翻译：

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仿射奇异随机向量的可压缩性度量

随机测量的可压缩性的概念是一个相当普遍的概念，它在从数据压缩到信号量化和参数估计的许多情况下都有应用。虽然离散和连续测量的可压缩性通常很好理解，但离散连续测量的情况非常微妙。在本文中，我们关注一类在仿射低维子集上具有奇异性的多维随机度量。我们将这类随机变量称为仿射单数。当考虑按分量独立离散连续随机变量的线性变换时，自然会出现仿射奇异随机向量。为了测量这种分布的可压缩性，我们引入了维度率偏差 (DRB) 的新概念，它分别与离散和连续情况下的熵和微分熵密切相关。与熵和微分熵类似，DRB 可用于评估上述类型分布之间的互信息。除了 DRB，我们还评估了这些分布的 RID。我们进一步为通过分量独立离散连续随机变量 (X) 的 Lipschitz 函数获得的多维随机测量的 RID 提供了一个上限。 $A \mathrm {X}$ ， 在哪里 $澳元满足 ${\mathrm{ 火花}}({A_{m\times n}}) = m+1$（例如，范德蒙德矩阵）。当考虑具有非高斯激励噪声的离散域移动平均过程时，上述结果使我们能够评估块平均 RID 和 DRB，以及确定这些参数与其他现有可压缩性度量之间的关系。