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 ($\mathbf{X}$). The upper-bound is shown to be achievable when the Lipschitz function is $A \mathbf{X}$, where $A$ satisfies {\changed$\spark({A_{m\times n}}) = m+1$} (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. {\changed Finally, we provide an application of the notion of DRB for calculating the mutual information.}

翻译：有几种方法可以测量随机度量的压缩性; 它们包括一般方法, 如使用率扭曲曲线, 以及更具体的概念, 如 Renyi 信息维度(RID) 和 维度偏差(DRB) 。 RID 参数表示测量在空间的低维子集周围的集中度, 而 DRB 参数则指定这些低维子集的分布性可压缩性。 虽然对这种压缩参数的评估是针对连续和离散测量的( 例如, DRB 与离散和连续案例的螺旋和差变异性通度密切相关), 离散- 信息维度度度( RID) 和 离散和 异性( 异性) (DRB) 相近, 我们进一步提供了一个不连续的不连续度数据维度AADRID 的上位值 。 在本文中, 直位值的 RIDD 函数是可实现的。