219 research outputs found
A prismatic classifying space
A qualgebra is a set having two binary operations that satisfy
compatibility conditions which are modeled upon a group under conjugation and
multiplication. We develop a homology theory for qualgebras and describe a
classifying space for it. This space is constructed from -colored prisms
(products of simplices) and simultaneously generalizes (and includes)
simplicial classifying spaces for groups and cubical classifying spaces for
quandles. Degenerate cells of several types are added to the regular prismatic
cells; by duality, these correspond to "non-rigid" Reidemeister moves and their
higher dimensional analogues. Coupled with -coloring techniques, our
homology theory yields invariants of knotted trivalent graphs in
and knotted foams in . We re-interpret these invariants as
homotopy classes of maps from or to the classifying space of .Comment: 28 pages, 24 figure
A counterexample to Thiagarajan's conjecture on regular event structures
We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002)
that regular event structures correspond exactly to event structures obtained
as unfoldings of finite 1-safe Petri nets. The same counterexample is used to
disprove a closely related conjecture by Badouel, Darondeau, and Raoult (1999)
that domains of regular event structures with bounded -cliques are
recognizable by finite trace automata. Event structures, trace automata, and
Petri nets are fundamental models in concurrency theory. There exist nice
interpretations of these structures as combinatorial and geometric objects.
Namely, from a graph theoretical point of view, the domains of prime event
structures correspond exactly to median graphs; from a geometric point of view,
these domains are in bijection with CAT(0) cube complexes.
A necessary condition for both conjectures to be true is that domains of
regular event structures (with bounded -cliques) admit a regular nice
labeling. To disprove these conjectures, we describe a regular event domain
(with bounded -cliques) that does not admit a regular nice labeling.
Our counterexample is derived from an example by Wise (1996 and 2007) of a
nonpositively curved square complex whose universal cover is a CAT(0) square
complex containing a particular plane with an aperiodic tiling. We prove that
other counterexamples to Thiagarajan's conjecture arise from aperiodic 4-way
deterministic tile sets of Kari and Papasoglu (1999) and Lukkarila (2009).
On the positive side, using breakthrough results by Agol (2013) and Haglund
and Wise (2008, 2012) from geometric group theory, we prove that Thiagarajan's
conjecture is true for regular event structures whose domains occur as
principal filters of hyperbolic CAT(0) cube complexes which are universal
covers of finite nonpositively curved cube complexes
Recovering Structured Probability Matrices
We consider the problem of accurately recovering a matrix B of size M by M ,
which represents a probability distribution over M2 outcomes, given access to
an observed matrix of "counts" generated by taking independent samples from the
distribution B. How can structural properties of the underlying matrix B be
leveraged to yield computationally efficient and information theoretically
optimal reconstruction algorithms? When can accurate reconstruction be
accomplished in the sparse data regime? This basic problem lies at the core of
a number of questions that are currently being considered by different
communities, including building recommendation systems and collaborative
filtering in the sparse data regime, community detection in sparse random
graphs, learning structured models such as topic models or hidden Markov
models, and the efforts from the natural language processing community to
compute "word embeddings".
Our results apply to the setting where B has a low rank structure. For this
setting, we propose an efficient algorithm that accurately recovers the
underlying M by M matrix using Theta(M) samples. This result easily translates
to Theta(M) sample algorithms for learning topic models and learning hidden
Markov Models. These linear sample complexities are optimal, up to constant
factors, in an extremely strong sense: even testing basic properties of the
underlying matrix (such as whether it has rank 1 or 2) requires Omega(M)
samples. We provide an even stronger lower bound where distinguishing whether a
sequence of observations were drawn from the uniform distribution over M
observations versus being generated by an HMM with two hidden states requires
Omega(M) observations. This precludes sublinear-sample hypothesis tests for
basic properties, such as identity or uniformity, as well as sublinear sample
estimators for quantities such as the entropy rate of HMMs
Recognizing Partial Cubes in Quadratic Time
We show how to test whether a graph with n vertices and m edges is a partial
cube, and if so how to find a distance-preserving embedding of the graph into a
hypercube, in the near-optimal time bound O(n^2), improving previous O(nm)-time
solutions.Comment: 25 pages, five figures. This version significantly expands previous
versions, including a new report on an implementation of the algorithm and
experiments with i
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