There are several ways to measure the compressibility of a random measure; they include general approaches, such as using the rate-distortion curve, as well as more specific notions, such as the Renyi information dimension (RID), and dimensional-rate bias (DRB). The RID parameter indicates the concentration of the measure around lower-dimensional subsets of the space while the DRB parameter specifies the compressibility of the distribution over these lower-dimensional subsets. While the evaluation of such compressibility parameters is well-studied for continuous and discrete measures (e.g., the DRB is closely related to the entropy and differential entropy in discrete and continuous cases, respectively), 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. This class of distributions naturally arises when considering linear transformation of component-wise independent discrete-continuous random variables. Here, we evaluate the RID and DRB for such probability measures. 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 AX, where A satisfies SPARK(A) = rank(A) + 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测度

有几种方法可以测量随机度量的可压缩性；它们包括一般方法，例如使用率失真曲线，以及更具体的概念，例如人一信息维度 (RID) 和维率偏差 (DRB)。RID 参数表示空间的低维子集周围的度量集中，而 DRB 参数指定分布在这些低维子集上的可压缩性。虽然对连续和离散度量对此类压缩性参数的评估进行了充分研究（例如，DRB 分别与离散和连续情况下的熵和微分熵密切相关），但离散-连续度量的情况非常微妙。在本文中，我们关注一类在仿射低维子集上具有奇点的多维随机度量。在考虑组件方式独立离散连续随机变量的线性变换时，自然会出现此类分布。在这里，我们针对此类概率度量评估 RID 和 DRB。我们进一步提供了多维随机度量的 RID 的上限，这些度量是通过按分量独立的离散连续随机变量 (X) 的 Lipschitz 函数获得的。当 Lipschitz 函数为 AX，其中 A 满足 SPARK(A) = rank(A) + 1（例如，Vandermonde 矩阵）时，显示上限是可以实现的。当考虑具有非高斯激励噪声的离散域移动平均过程时，上述结果允许我们评估块平均 RID 和 DRB，