566 research outputs found
The structure and stability of persistence modules
We give a self-contained treatment of the theory of persistence modules
indexed over the real line. We give new proofs of the standard results.
Persistence diagrams are constructed using measure theory. Linear algebra
lemmas are simplified using a new notation for calculations on quiver
representations. We show that the stringent finiteness conditions required by
traditional methods are not necessary to prove the existence and stability of
the persistence diagram. We introduce weaker hypotheses for taming persistence
modules, which are met in practice and are strong enough for the theory still
to work. The constructions and proofs enabled by our framework are, we claim,
cleaner and simpler.Comment: New version. We discuss in greater depth the interpolation lemma for
persistence module
Entity-Linking via Graph-Distance Minimization
Entity-linking is a natural-language-processing task that consists in
identifying the entities mentioned in a piece of text, linking each to an
appropriate item in some knowledge base; when the knowledge base is Wikipedia,
the problem comes to be known as wikification (in this case, items are
wikipedia articles). One instance of entity-linking can be formalized as an
optimization problem on the underlying concept graph, where the quantity to be
optimized is the average distance between chosen items. Inspired by this
application, we define a new graph problem which is a natural variant of the
Maximum Capacity Representative Set. We prove that our problem is NP-hard for
general graphs; nonetheless, under some restrictive assumptions, it turns out
to be solvable in linear time. For the general case, we propose two heuristics:
one tries to enforce the above assumptions and another one is based on the
notion of hitting distance; we show experimentally how these approaches perform
with respect to some baselines on a real-world dataset.Comment: In Proceedings GRAPHITE 2014, arXiv:1407.7671. The second and third
authors were supported by the EU-FET grant NADINE (GA 288956
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