9,387 research outputs found
Restricted Space Algorithms for Isomorphism on Bounded Treewidth Graphs
The Graph Isomorphism problem restricted to graphs of bounded treewidth or
bounded tree distance width are known to be solvable in polynomial time
[Bod90],[YBFT99]. We give restricted space algorithms for these problems
proving the following results: - Isomorphism for bounded tree distance width
graphs is in L and thus complete for the class. We also show that for this kind
of graphs a canon can be computed within logspace. - For bounded treewidth
graphs, when both input graphs are given together with a tree decomposition,
the problem of whether there is an isomorphism which respects the
decompositions (i.e. considering only isomorphisms mapping bags in one
decomposition blockwise onto bags in the other decomposition) is in L. - For
bounded treewidth graphs, when one of the input graphs is given with a tree
decomposition the isomorphism problem is in LogCFL. - As a corollary the
isomorphism problem for bounded treewidth graphs is in LogCFL. This improves
the known TC1 upper bound for the problem given by Grohe and Verbitsky
[GroVer06].Comment: STACS conference 2010, 12 page
Privately Releasing Conjunctions and the Statistical Query Barrier
Suppose we would like to know all answers to a set of statistical queries C
on a data set up to small error, but we can only access the data itself using
statistical queries. A trivial solution is to exhaustively ask all queries in
C. Can we do any better?
+ We show that the number of statistical queries necessary and sufficient for
this task is---up to polynomial factors---equal to the agnostic learning
complexity of C in Kearns' statistical query (SQ) model. This gives a complete
answer to the question when running time is not a concern.
+ We then show that the problem can be solved efficiently (allowing arbitrary
error on a small fraction of queries) whenever the answers to C can be
described by a submodular function. This includes many natural concept classes,
such as graph cuts and Boolean disjunctions and conjunctions.
While interesting from a learning theoretic point of view, our main
applications are in privacy-preserving data analysis:
Here, our second result leads to the first algorithm that efficiently
releases differentially private answers to of all Boolean conjunctions with 1%
average error. This presents significant progress on a key open problem in
privacy-preserving data analysis.
Our first result on the other hand gives unconditional lower bounds on any
differentially private algorithm that admits a (potentially
non-privacy-preserving) implementation using only statistical queries. Not only
our algorithms, but also most known private algorithms can be implemented using
only statistical queries, and hence are constrained by these lower bounds. Our
result therefore isolates the complexity of agnostic learning in the SQ-model
as a new barrier in the design of differentially private algorithms
Full Flow: Optical Flow Estimation By Global Optimization over Regular Grids
We present a global optimization approach to optical flow estimation. The
approach optimizes a classical optical flow objective over the full space of
mappings between discrete grids. No descriptor matching is used. The highly
regular structure of the space of mappings enables optimizations that reduce
the computational complexity of the algorithm's inner loop from quadratic to
linear and support efficient matching of tens of thousands of nodes to tens of
thousands of displacements. We show that one-shot global optimization of a
classical Horn-Schunck-type objective over regular grids at a single resolution
is sufficient to initialize continuous interpolation and achieve
state-of-the-art performance on challenging modern benchmarks.Comment: To be presented at CVPR 201
Landauer's principle as a special case of Galois connection
It is demonstrated how to construct a Galois connection between two related
systems with entropy. The construction, called the Landauer's connection,
describes coupling between two systems with entropy. It is straightforward and
transfers changes in one system to the other one preserving ordering structure
induced by entropy. The Landauer's connection simplifies the description of the
classical Landauer's principle for computational systems. Categorification and
generalization of the Landauer's principle opens area of modelling of various
systems in presence of entropy in abstract terms.Comment: 24 pages, 3 figure
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