204,314 research outputs found
Algorithmic statistics: forty years later
Algorithmic statistics has two different (and almost orthogonal) motivations.
From the philosophical point of view, it tries to formalize how the statistics
works and why some statistical models are better than others. After this notion
of a "good model" is introduced, a natural question arises: it is possible that
for some piece of data there is no good model? If yes, how often these bad
("non-stochastic") data appear "in real life"?
Another, more technical motivation comes from algorithmic information theory.
In this theory a notion of complexity of a finite object (=amount of
information in this object) is introduced; it assigns to every object some
number, called its algorithmic complexity (or Kolmogorov complexity).
Algorithmic statistic provides a more fine-grained classification: for each
finite object some curve is defined that characterizes its behavior. It turns
out that several different definitions give (approximately) the same curve.
In this survey we try to provide an exposition of the main results in the
field (including full proofs for the most important ones), as well as some
historical comments. We assume that the reader is familiar with the main
notions of algorithmic information (Kolmogorov complexity) theory.Comment: Missing proofs adde
Limits of permutation sequences
A permutation sequence is said to be convergent if the density of occurrences
of every fixed permutation in the elements of the sequence converges. We prove
that such a convergent sequence has a natural limit object, namely a Lebesgue
measurable function with the additional properties that,
for every fixed , the restriction is a cumulative
distribution function and, for every , the restriction
satisfies a "mass" condition. This limit process is well-behaved:
every function in the class of limit objects is a limit of some permutation
sequence, and two of these functions are limits of the same sequence if and
only if they are equal almost everywhere. An ingredient in the proofs is a new
model of random permutations, which generalizes previous models and might be
interesting for its own sake.Comment: accepted for publication in the Journal of Combinatorial Theory,
Series B. arXiv admin note: text overlap with arXiv:1106.166
-rectifiability in low codimension in Heisenberg groups
A natural notion of higher order rectifiability is introduced for subsets of
Heisenberg groups in terms of covering a set almost everywhere
by a countable union of -regular
surfaces, for some . We prove that a sufficient condition
for -rectifiability of low-codimensional subsets in Heisenberg
groups is the almost everywhere existence of suitable approximate tangent
paraboloids.Comment: Corrected typos. Added more information in Section 2.1
(preliminaries) and detailed proofs in Section 2.2. Added Lemma 3.3 and
modified the main proof (Section 3.1) accordingl
Orbit closures in the enhanced nilpotent cone
We study the orbits of in the enhanced nilpotent cone
, where is the variety of nilpotent
endomorphisms of . These orbits are parametrized by bipartitions of , and we prove that the closure ordering corresponds to a natural partial
order on bipartitions. Moreover, we prove that the local intersection
cohomology of the orbit closures is given by certain bipartition analogues of
Kostka polynomials, defined by Shoji. Finally, we make a connection with Kato's
exotic nilpotent cone in type C, proving that the closure ordering is the same,
and conjecturing that the intersection cohomology is the same but with degrees
doubled.Comment: 32 pages. Update (August 2010): There is an error in the proof of
Theorem 4.7, in this version and the almost-identical published version. See
the corrigendum arXiv:1008.1117 for independent proofs of later results that
depend on that statemen
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