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4.
Sparsity
: Discretization schemes are designed in such a way that most of the
coefficients of the discrete equations vanish.
5.
Patterns
: The nonzero coefficients are arranged in a way governed by a predict-
able pattern.
6.
Structure inheritance
: The discrete system inherits mathematical structure
from the continuous one. (For example, an elliptic differential equation may
be discretized in such a way that the discrete system is also elliptic, i.e., positive
definite.)
Sadly enough, except for sparsity, none of the above holds with respect to the
recommendation setting. First of all, there is no underlying continuous structure at
all, let alone a physical interpretation. The real-world phenomenon and the equation
of its model are genuinely discrete. Apart from this, most parameters of the model,
as, e.g., the transition probabilities, are not known in advance, but have to be
figured out empirically as one goes along. Furthermore, although the coefficient
matrices of the Bellman equations arising in recommendations are typically sparse,
the nonzero coefficients are distributed basically at random. Finally, there is no
structure to be inherited from a continuous model. For example, there is no reason
to assume positive definiteness, a crucial prerequisite of convergence results on
many multigrid-related methods, let alone that the considered Markov chains
be reversible.
All in all, this supports the impression we have gained through years of
research and practical experience in the field of mathematical data analysis: on
one hand, it seems that many approaches from the twentieth-century mathematics
are suitable to be carried over to problems arising in data analysis. On the other
hand, the underlying mathematical theory is designed for settings, the structure of
which differs in many essential respects from that encountered in data analysis-
related problems. Our colleague Mijail Guillemard recently put this as follows:
In hindsight, I recognize that in my years as a PhD-student, I wasted a lot of time immersing
myself in mathematical theories that are not quite suited for the problems the solution of
which I sought after.
Again, in hindsight, it does not come as a surprise that a major part of
state-of-the-art mathematical theory is hardly applicable to data analysis. After
all, until lately, the development of mathematics was predominantly driven by
problems encountered in science and engineering, to which data analysis was
added only recently. As a consequence, data analysis requires outright novel
extensions and generalizations of classical mathematical theories, the development
of which will certainly keep researches occupied for decades to come.
Let us conclude this outlook by a brief philosophical remark: historically,
computers and computational mathematics were primarily designed for solving
numerical problems related to differential equations. The development which took
place in the course of the following decades, however, is an instance of what the
American biologist Stephen Jay Gould refers to as an
exaptation:
computers were
gradually transformed into general-purpose information processing, storage, and
communication systems and began to figure increasingly in the organization of
