1,592 research outputs found
Minimization via duality
We show how to use duality theory to construct minimized versions of a wide class of automata. We work out three cases in detail: (a variant of) ordinary automata, weighted automata and probabilistic automata. The basic idea is that instead of constructing a maximal quotient we go to the dual and look for a minimal subalgebra and then return to the original category. Duality ensures that the minimal subobject becomes the maximally quotiented object
Generalized Low Rank Models
Principal components analysis (PCA) is a well-known technique for
approximating a tabular data set by a low rank matrix. Here, we extend the idea
of PCA to handle arbitrary data sets consisting of numerical, Boolean,
categorical, ordinal, and other data types. This framework encompasses many
well known techniques in data analysis, such as nonnegative matrix
factorization, matrix completion, sparse and robust PCA, -means, -SVD,
and maximum margin matrix factorization. The method handles heterogeneous data
sets, and leads to coherent schemes for compressing, denoising, and imputing
missing entries across all data types simultaneously. It also admits a number
of interesting interpretations of the low rank factors, which allow clustering
of examples or of features. We propose several parallel algorithms for fitting
generalized low rank models, and describe implementations and numerical
results.Comment: 84 pages, 19 figure
P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches
While the P vs NP problem is mainly approached form the point of view of
discrete mathematics, this paper proposes reformulations into the field of
abstract algebra, geometry, fourier analysis and of continuous global
optimization - which advanced tools might bring new perspectives and approaches
for this question. The first one is equivalence of satisfaction of 3-SAT
problem with the question of reaching zero of a nonnegative degree 4
multivariate polynomial (sum of squares), what could be tested from the
perspective of algebra by using discriminant. It could be also approached as a
continuous global optimization problem inside , for example in
physical realizations like adiabatic quantum computers. However, the number of
local minima usually grows exponentially. Reducing to degree 2 polynomial plus
constraints of being in , we get geometric formulations as the
question if plane or sphere intersects with . There will be also
presented some non-standard perspectives for the Subset-Sum, like through
convergence of a series, or zeroing of fourier-type integral for some natural . The last discussed
approach is using anti-commuting Grassmann numbers , making nonzero only if has a Hamilton cycle. Hence,
the PNP assumption implies exponential growth of matrix representation of
Grassmann numbers. There will be also discussed a looking promising
algebraic/geometric approach to the graph isomorphism problem -- tested to
successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure
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