329,597 research outputs found
Local, Smooth, and Consistent Jacobi Set Simplification
The relation between two Morse functions defined on a common domain can be
studied in terms of their Jacobi set. The Jacobi set contains points in the
domain where the gradients of the functions are aligned. Both the Jacobi set
itself as well as the segmentation of the domain it induces have shown to be
useful in various applications. Unfortunately, in practice functions often
contain noise and discretization artifacts causing their Jacobi set to become
unmanageably large and complex. While there exist techniques to simplify Jacobi
sets, these are unsuitable for most applications as they lack fine-grained
control over the process and heavily restrict the type of simplifications
possible.
In this paper, we introduce a new framework that generalizes critical point
cancellations in scalar functions to Jacobi sets in two dimensions. We focus on
simplifications that can be realized by smooth approximations of the
corresponding functions and show how this implies simultaneously simplifying
contiguous subsets of the Jacobi set. These extended cancellations form the
atomic operations in our framework, and we introduce an algorithm to
successively cancel subsets of the Jacobi set with minimal modifications
according to some user-defined metric. We prove that the algorithm is correct
and terminates only once no more local, smooth and consistent simplifications
are possible. We disprove a previous claim on the minimal Jacobi set for
manifolds with arbitrary genus and show that for simply connected domains, our
algorithm reduces a given Jacobi set to its simplest configuration.Comment: 24 pages, 19 figure
Effect of Depth and Width on Local Minima in Deep Learning
In this paper, we analyze the effects of depth and width on the quality of
local minima, without strong over-parameterization and simplification
assumptions in the literature. Without any simplification assumption, for deep
nonlinear neural networks with the squared loss, we theoretically show that the
quality of local minima tends to improve towards the global minimum value as
depth and width increase. Furthermore, with a locally-induced structure on deep
nonlinear neural networks, the values of local minima of neural networks are
theoretically proven to be no worse than the globally optimal values of
corresponding classical machine learning models. We empirically support our
theoretical observation with a synthetic dataset as well as MNIST, CIFAR-10 and
SVHN datasets. When compared to previous studies with strong
over-parameterization assumptions, the results in this paper do not require
over-parameterization, and instead show the gradual effects of
over-parameterization as consequences of general results
Reductions of Galois representations for slopes in
We describe the semi-simplification of the mod reduction of certain
crystalline two dimensional local Galois representations of slopes in the
interval and all weights. The proof uses the mod Local Langlands
Correspondence for . We also give a complete description of the
submodules generated by the second highest monomial in the mod symmetric
power representations of .Comment: 41 page
The effect of tax simplification on state and local governments
Tax reform ; Municipal bonds
Residually reducible representations of algebras over local Artinian rings
In this paper we generalize a result of Urban on the structure of residually reducible representations on local Artinian rings from the case that the semi-simplification of the residual representation splits into 2 absolutely irreducible representations to the case where it splits into m ≥ 2 absolutely irreducible representations
Sliding elastic lattice: an explanation of the motion of superconducting vortices
We introduce a system where an elastic lattice of particles is moved slowly
at a constant velocity under the influence of a local external potential,
construct a rigid-body model through simplification processes, and show that
the two systems produce similar results. Then, we apply our model to a
superconducting vortex system and produce path patterns similar to the ones
reported in [Lee et al., Phys. Rev. B 84, 060515 (2011)] suggesting that the
reasoning of the simplification processes in this paper can be a possible
explanation of the experimentally observed phenomenon.Comment: 5 pages, 3 figures, Submitted to Physical Review Letters; Reference
[17] Lee et al., Phys. Rev. B Accepted changed to Lee et al., Phys. Rev. B
84, 060515 (2011
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