Very often, in the course of uncertainty quantification tasks or data analysis, one has to deal with high-dimensional random variables (RVs). A high-dimensional RV can be described by its probability density (pdf) and/or by the corresponding probability characteristic functions (pcf), or by a polynomial chaos (PCE) or similar expansion. Here the interest is mainly to compute characterisations like the entropy, or relations between two distributions, like their Kullback-Leibler divergence. These are all computed from the pdf, which is often not available directly, and it is a computational challenge to even represent it in a numerically feasible fashion in case the dimension $d$ is even moderately large. In this regard, we propose to represent the density by a high order tensor product, and approximate this in a low-rank format. We show how to go from the pcf or functional representation to the pdf. This allows us to reduce the computational complexity and storage cost from an exponential to a linear. The characterisations such as entropy or the $f$-divergences need the possibility to compute point-wise functions of the pdf. This normally rather trivial task becomes more difficult when the pdf is approximated in a low-rank tensor format, as the point values are not directly accessible any more. The data is considered as an element of a high order tensor space. The considered algorithms are independent of the representation of the data as a tensor. All that we require is that the data can be considered as an element of an associative, commutative algebra with an inner product. Such an algebra is isomorphic to a commutative sub-algebra of the usual matrix algebra, allowing the use of matrix algorithms to accomplish the mentioned tasks.
翻译:在不确定性量化任务或数据分析过程中,通常需要处理高维随机变量(RVs)。高维RV可以用其概率密度(pdf)和(或)相应的概率特征函数(pcf)来描述,或者用一个多式混杂(PCE)或类似的扩展来描述。这里的利益主要在于计算像英特或功能表达式这样的特性,或者两个分布式之间的关系,比如其库尔回背-利耶的差异。所有这些都是从pdf中计算出来的,通常无法直接得到,如果维值甚至略大,则需要以数字上可行的方式来表示。在这方面,我们提议用高排序的格调产品来代表其密度,并且以低位格式来估计。这让我们能够将计算复杂性和存储成本从可读的指数降为线性。如果认为像英特或美的正值那样,则在数字值上以数字格式来代表它,则需要以数字值表示一个正常的直位值值值,当一个正常的正值值值值值值值值时,则需要将数据变成一个更难的正值值值值值值值值值值值值值值值值,作为一个亚值的正值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值,作为一个正常的值值值值值值值值值值的值值值值值值值值值值值作为一个正常的值值值值值值的值的值,作为一个直值值值值的值的值的值值值值,作为一个直值的值的值的值值值值值值的值的值的值的值值的值值值值值值值值值的值值值值值值值值值值值值值值的值值值的值值值值值,作为一个值的值的值的值的值的值的值,作为一个值值值,作为一个值值的值的值的值值值的值的值的值的值的值的值值值值值值值值值值值的值值作为一个值值的值值值值值的值值的值的值的值的值的值值值值值值值值值作为一个正常的值的值的值的值