474 research outputs found
Extensions by Antiderivatives, Exponentials of Integrals and by Iterated Logarithms
Let F be a characteristic zero differential field with an algebraically
closed field of constants, E be a no-new-constant extension of F by
antiderivatives of F and let y1, ..., yn be antiderivatives of E. The
antiderivatives y1, ..., yn of E are called J-I-E antiderivatives if the
derivatives of yi in E satisfies certain conditions. We will discuss a new
proof for the Kolchin-Ostrowski theorem and generalize this theorem for a tower
of extensions by J-I-E antiderivatives and use this generalized version of the
theorem to classify the finitely differentially generated subfields of this
tower. In the process, we will show that the J-I-E antiderivatives are
algebraically independent over the ground differential field. An example of a
J-I-E tower is extensions by iterated logarithms. We will discuss the normality
of extensions by iterated logarithms and produce an algorithm to compute its
finitely differentially generated subfields.Comment: 66 pages, 1 figur
Iterated Antiderivative Extensions
Let be a characteristic zero differential field with an algebraically
closed field of constants and let be a no new constants extension of .
We say that is an \textsl{iterated antiderivative extension} of if
is a liouvillian extension of obtained by adjoining antiderivatives alone.
In this article, we will show that if is an iterated antiderivative
extension of and is a differential subfield of that contains
then is an iterated antiderivative extension of .Comment: 15 pages, 0 figure
Light Field Blind Motion Deblurring
We study the problem of deblurring light fields of general 3D scenes captured
under 3D camera motion and present both theoretical and practical
contributions. By analyzing the motion-blurred light field in the primal and
Fourier domains, we develop intuition into the effects of camera motion on the
light field, show the advantages of capturing a 4D light field instead of a
conventional 2D image for motion deblurring, and derive simple methods of
motion deblurring in certain cases. We then present an algorithm to blindly
deblur light fields of general scenes without any estimation of scene geometry,
and demonstrate that we can recover both the sharp light field and the 3D
camera motion path of real and synthetically-blurred light fields.Comment: To be presented at CVPR 201
On the Analysis of Trajectories of Gradient Descent in the Optimization of Deep Neural Networks
Theoretical analysis of the error landscape of deep neural networks has
garnered significant interest in recent years. In this work, we theoretically
study the importance of noise in the trajectories of gradient descent towards
optimal solutions in multi-layer neural networks. We show that adding noise (in
different ways) to a neural network while training increases the rank of the
product of weight matrices of a multi-layer linear neural network. We thus
study how adding noise can assist reaching a global optimum when the product
matrix is full-rank (under certain conditions). We establish theoretical
foundations between the noise induced into the neural network - either to the
gradient, to the architecture, or to the input/output to a neural network - and
the rank of product of weight matrices. We corroborate our theoretical findings
with empirical results.Comment: 4 pages + 1 figure (main, excluding references), 5 pages + 4 figures
(appendix
ADINE: An Adaptive Momentum Method for Stochastic Gradient Descent
Two major momentum-based techniques that have achieved tremendous success in
optimization are Polyak's heavy ball method and Nesterov's accelerated
gradient. A crucial step in all momentum-based methods is the choice of the
momentum parameter which is always suggested to be set to less than .
Although the choice of is justified only under very strong theoretical
assumptions, it works well in practice even when the assumptions do not
necessarily hold. In this paper, we propose a new momentum based method
, which relaxes the constraint of and allows the
learning algorithm to use adaptive higher momentum. We motivate our hypothesis
on by experimentally verifying that a higher momentum () can help
escape saddles much faster. Using this motivation, we propose our method
that helps weigh the previous updates more (by setting the
momentum parameter ), evaluate our proposed algorithm on deep neural
networks and show that helps the learning algorithm to
converge much faster without compromising on the generalization error.Comment: 8 + 1 pages, 12 figures, accepted at CoDS-COMAD 201
Primary osteogenic sarcoma of the breast: A case report
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