28,404 research outputs found
Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives
In this paper, we investigate a sheaf-theoretic interpretation of
stratification learning from geometric and topological perspectives. Our main
result is the construction of stratification learning algorithms framed in
terms of a sheaf on a partially ordered set with the Alexandroff topology. We
prove that the resulting decomposition is the unique minimal stratification for
which the strata are homogeneous and the given sheaf is constructible. In
particular, when we choose to work with the local homology sheaf, our algorithm
gives an alternative to the local homology transfer algorithm given in Bendich
et al. (2012), and the cohomology stratification algorithm given in Nanda
(2017). Additionally, we give examples of stratifications based on the
geometric techniques of Breiding et al. (2018), illustrating how the
sheaf-theoretic approach can be used to study stratifications from both
topological and geometric perspectives. This approach also points toward future
applications of sheaf theory in the study of topological data analysis by
illustrating the utility of the language of sheaf theory in generalizing
existing algorithms
A Novel A Priori Simulation Algorithm for Absorbing Receivers in Diffusion-Based Molecular Communication Systems
A novel a priori Monte Carlo (APMC) algorithm is proposed to accurately
simulate the molecules absorbed at spherical receiver(s) with low computational
complexity in diffusion-based molecular communication (MC) systems. It is
demonstrated that the APMC algorithm achieves high simulation efficiency since
by using this algorithm, the fraction of molecules absorbed for a relatively
large time step length precisely matches the analytical result. Therefore, the
APMC algorithm overcomes the shortcoming of the existing refined Monte Carlo
(RMC) algorithm which enables accurate simulation for a relatively small time
step length only. Moreover, for the RMC algorithm, an expression is proposed to
quickly predict the simulation accuracy as a function of the time step length
and system parameters, which facilitates the choice of simulation time step for
a given system. Furthermore, a rejection threshold is proposed for both the RMC
and APMC algorithms to significantly save computational complexity while
causing an extremely small loss in accuracy.Comment: 11 pages, 14 figures, submitted to IEEE Transactions on
NanoBioscience. arXiv admin note: text overlap with arXiv:1803.0463
Topological Paramagnetism in Frustrated Spin-One Mott Insulators
Time reversal protected three dimensional (3D) topological paramagnets are
magnetic analogs of the celebrated 3D topological insulators. Such paramagnets
have a bulk gap, no exotic bulk excitations, but non-trivial surface states
protected by symmetry. We propose that frustrated spin-1 quantum magnets are a
natural setting for realising such states in 3D. We describe a physical picture
of the ground state wavefunction for such a spin-1 topological paramagnet in
terms of loops of fluctuating Haldane chains with non-trivial linking phases.
We illustrate some aspects of such loop gases with simple exactly solvable
models. We also show how 3D topological paramagnets can be very naturally
accessed within a slave particle description of a spin-1 magnet. Specifically
we construct slave particle mean field states which are naturally driven into
the topological paramagnet upon including fluctuations. We propose bulk
projected wave functions for the topological paramagnet based on this slave
particle description. An alternate slave particle construction leads to a
stable U(1) quantum spin liquid from which a topological paramagnet may be
accessed by condensing the emergent magnetic monopole excitation of the spin
liquid.Comment: 16 pages, 5 figure
Parametric Local Metric Learning for Nearest Neighbor Classification
We study the problem of learning local metrics for nearest neighbor
classification. Most previous works on local metric learning learn a number of
local unrelated metrics. While this "independence" approach delivers an
increased flexibility its downside is the considerable risk of overfitting. We
present a new parametric local metric learning method in which we learn a
smooth metric matrix function over the data manifold. Using an approximation
error bound of the metric matrix function we learn local metrics as linear
combinations of basis metrics defined on anchor points over different regions
of the instance space. We constrain the metric matrix function by imposing on
the linear combinations manifold regularization which makes the learned metric
matrix function vary smoothly along the geodesics of the data manifold. Our
metric learning method has excellent performance both in terms of predictive
power and scalability. We experimented with several large-scale classification
problems, tens of thousands of instances, and compared it with several state of
the art metric learning methods, both global and local, as well as to SVM with
automatic kernel selection, all of which it outperforms in a significant
manner
Multigraded regularity: coarsenings and resolutions
Let S = k[x_1,...,x_n] be a Z^r-graded ring with deg (x_i) = a_i \in Z^r for
each i and suppose that M is a finitely generated Z^r-graded S-module. In this
paper we describe how to find finite subsets of Z^r containing the multidegrees
of the minimal multigraded syzygies of M. To find such a set, we first coarsen
the grading of M so that we can view M as a Z-graded S-module. We use a
generalized notion of Castelnuovo-Mumford regularity, which was introduced by
D. Maclagan and G. Smith, to associate to M a number which we call the
regularity number of M. The minimal degrees of the multigraded minimal syzygies
are bounded in terms of this invariant.Comment: 20 pages, 1 figure; small corrections made; final version; to appear
in J. of Algebr
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