154 research outputs found
Duality and interval analysis over idempotent semirings
In this paper semirings with an idempotent addition are considered. These
algebraic structures are endowed with a partial order. This allows to consider
residuated maps to solve systems of inequalities . The
purpose of this paper is to consider a dual product, denoted , and the
dual residuation of matrices, in order to solve the following inequality . Sufficient conditions ensuring the
existence of a non-linear projector in the solution set are proposed. The
results are extended to semirings of intervals
Graph Kernels
We present a unified framework to study graph kernels, special cases of which include the random
walk (Gärtner et al., 2003; Borgwardt et al., 2005) and marginalized (Kashima et al., 2003, 2004;
Mahé et al., 2004) graph kernels. Through reduction to a Sylvester equation we improve the time
complexity of kernel computation between unlabeled graphs with n vertices from O(n^6) to O(n^3).
We find a spectral decomposition approach even more efficient when computing entire kernel matrices.
For labeled graphs we develop conjugate gradient and fixed-point methods that take O(dn^3)
time per iteration, where d is the size of the label set. By extending the necessary linear algebra to
Reproducing Kernel Hilbert Spaces (RKHS) we obtain the same result for d-dimensional edge kernels,
and O(n^4) in the infinite-dimensional case; on sparse graphs these algorithms only take O(n^2)
time per iteration in all cases. Experiments on graphs from bioinformatics and other application
domains show that these techniques can speed up computation of the kernel by an order of magnitude
or more. We also show that certain rational kernels (Cortes et al., 2002, 2003, 2004) when
specialized to graphs reduce to our random walk graph kernel. Finally, we relate our framework to
R-convolution kernels (Haussler, 1999) and provide a kernel that is close to the optimal assignment
kernel of Fröhlich et al. (2006) yet provably positive semi-definite
Presenting Distributive Laws
Distributive laws of a monad T over a functor F are categorical tools for
specifying algebra-coalgebra interaction. They proved to be important for
solving systems of corecursive equations, for the specification of well-behaved
structural operational semantics and, more recently, also for enhancements of
the bisimulation proof method. If T is a free monad, then such distributive
laws correspond to simple natural transformations. However, when T is not free
it can be rather difficult to prove the defining axioms of a distributive law.
In this paper we describe how to obtain a distributive law for a monad with an
equational presentation from a distributive law for the underlying free monad.
We apply this result to show the equivalence between two different
representations of context-free languages
Machine Learning Techniques as Applied to Discrete and Combinatorial Structures
Machine Learning Techniques have been used on a wide array of input types: images, sound waves, text, and so forth. In articulating these input types to the almighty machine, there have been all sorts of amazing problems that have been solved for many practical purposes.
Nevertheless, there are some input types which don’t lend themselves nicely to the standard set of machine learning tools we have. Moreover, there are some provably difficult problems which are abysmally hard to solve within a reasonable time frame.
This thesis addresses several of these difficult problems. It frames these problems such that we can then attempt to marry the allegedly powerful utility of existing machine learning techniques to the practical solvability of said problems
Distributed-Memory Breadth-First Search on Massive Graphs
This chapter studies the problem of traversing large graphs using the
breadth-first search order on distributed-memory supercomputers. We consider
both the traditional level-synchronous top-down algorithm as well as the
recently discovered direction optimizing algorithm. We analyze the performance
and scalability trade-offs in using different local data structures such as CSR
and DCSC, enabling in-node multithreading, and graph decompositions such as 1D
and 2D decomposition.Comment: arXiv admin note: text overlap with arXiv:1104.451
Finally, a Polymorphic Linear Algebra Language (Pearl)
Many different data analytics tasks boil down to linear algebra primitives. In practice, for each different type of workload, data scientists use a particular specialised library. In this paper, we present Pilatus, a polymorphic iterative linear algebra language, applicable to various types of data analytics workloads. The design of this domain-specific language (DSL) is inspired by both mathematics and programming languages: its basic constructs are borrowed from abstract algebra, whereas the key technology behind its polymorphic design uses the tagless final approach (a.k.a. polymorphic embedding/object algebras). This design enables us to change the behaviour of arithmetic operations to express matrix algebra, graph algorithms, logical probabilistic programs, and differentiable programs. Crucially, the polymorphic design of Pilatus allows us to use multi-stage programming and rewrite-based optimisation to recover the performance of specialised code, supporting fixed sized matrices, algebraic optimisations, and fusion
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