We formalize and generalize the notation of Section 4.1 and prove the results. For the remainder of this section, we fix a dimension D (e.g. The D-dimensional SSM will be a map from a function u: D! to y: D!. Definition 2 (Indexing notation). Given a subset I [D], let I denote its complement [D]\I. Given a 2 N1 and b 2 N2, let a b 2 N1 N2 be defined as (a b)n1,n2 = an1bn2.

We describe a new software framework for fast training of generalized linear models. Theframework,named Snap Machine Learning (Snap ML), combines recent advances inmachine learning systems andalgorithms inanested manner to reflect the hierarchical architecture of modern computing systems.