# Structure Spaces

Brijnesh J. Jain, Klaus Obermayer.

Year: 2009, Volume: 10, Issue: 93, Pages: 2667−2714

#### Abstract

Finite structures such as point patterns, strings, trees, and graphs occur
as "natural" representations of structured data in different application
areas of machine learning. We develop the theory of *structure spaces*
and derive geometrical and analytical concepts such as the angle between
structures and the derivative of functions on structures. In particular, we
show that the gradient of a differentiable structural function is a
well-defined structure pointing in the direction of steepest
ascent. Exploiting the properties of structure spaces, it will turn out that
a number of problems in structural pattern recognition such as central
clustering or learning in structured output spaces can be formulated as
optimization problems with cost functions that are locally Lipschitz. Hence,
methods from nonsmooth analysis are applicable to optimize those cost
functions.