Vast sea of numbers,
Can you be described by few,
As bones define flesh?
Many data objects, such as matrices, which are traditionally described by providing a high precision number for every (row,column) entry, can be represented to usefully high precision by many fewer numbers. This so-called “data sparsity” holds for the mathematical descriptions of many physical and statistical phenomena whose effects or correlations decay smoothly with distance.
An apparently complex interaction or relationship is, with a special perspective, much simpler. In an extreme limit, we can lump the moon’s gravitational effect on the Earth by assuming that all of its distributed mass is concentrated at a single point. The potential to represent dense matrices by products of a small number of vectors (the number needed is called the “rank”) is analogous to this and leads to huge savings in memory and operations when manipulating such objects. The effect of the whole can be represented by a carefully defined abstraction. One version of this technique is called “skeletonization,” which suggests the Sciku above. For an example of this philosophy, see Yokota et al, 2014.
Original research: https://doi.org/10.14529/jsfi140104
David Keyes directs the Extreme Computing Research Center at KAUST, where he was a founding dean in 2009. He inhabits the intersection of Mathematics, Computer Science, and applications, with a focus on colonizing emerging energy-efficient architectures for scientific computations. He is a Fellow of SIAM and AMS and has received the ACM Gordon Bell Prize and the IEEE Sidney Fernbach Award. As a lover of poetry, he is delighted to discover the Sciku community.
Enjoyed this sciku? Check out David’s other sciku: Algorithmic complexity.