Renyi维数与高斯滤波II
Renyi Dimension and Gaussian Filtering II
摘要：
我们考虑卷积高斯的一个不同的规模对波莱尔措施μ欧几里德δ维空间。结果的Lq范数在π中是可微的。我们计算这个衍生物，并显示如何其增长的高阶关系到其较低的Renyi维。我们假设q严格介于1和infty之间，并且mu是有限的，具有紧支撑。考虑为高斯函数选择一个标度序列n。在相邻尺度上，通常的Lq范数之间的差异可以通过幂律规则来设置n_n，使其增长得比n的任何正幂都慢。幂规则中的正确指数由较低的Renyi维数确定。我们计算并找到相关积分的高斯核版本的衍生物的界限。我们表明，高斯核版本的Renyi熵和连续。
英文摘要：
We consider convolving a Gaussian of a varying scale epsilon against a Borel measure mu on Euclidean delta-dimensional space. The Lq norm of the result is differentiable in epsilon. We calculate this derivative and show how the upper order of its growth relates to its lower Renyi dimension. We assume q is strictly between 1 and infty and that mu is finite with compact support. Consider choosing a sequence epsilon_n of scales for the Gaussians. The differences between the usual Lq norms at adjacent scales can be made to grow more slowly than any positive power of n by setting the epsilon_n by a power rule. The correct exponent in the power rule is determined by the lower Renyi dimension. We calculate and find bounds on the derivative of the Gaussian kernel versions of the correlation integral. We show that a Gaussian Kernel version of the Renyi entropy sum in continuous.
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