2,689 research outputs found
On Lie Algebras Generated by Few Extremal Elements
We give an overview of some properties of Lie algebras generated by at most 5
extremal elements. In particular, for any finite graph {\Gamma} and any field K
of characteristic not 2, we consider an algebraic variety X over K whose
K-points parametrize Lie algebras generated by extremal elements. Here the
generators correspond to the vertices of the graph, and we prescribe
commutation relations corresponding to the nonedges of {\Gamma}. We show that,
for all connected undirected finite graphs on at most 5 vertices, X is a
finite-dimensional affine space. Furthermore, we show that for
maximal-dimensional Lie algebras generated by 5 extremal elements, X is a
point. The latter result implies that the bilinear map describing extremality
must be identically zero, so that all extremal elements are sandwich elements
and the only Lie algebra of this dimension that occurs is nilpotent. These
results were obtained by extensive computations with the Magma computational
algebra system. The algorithms developed can be applied to arbitrary {\Gamma}
(i.e., without restriction on the number of vertices), and may be of
independent interest.Comment: 19 page
Rational motivic path spaces and Kim's relative unipotent section conjecture
We initiate a study of path spaces in the nascent context of "motivic dga's",
under development in doctoral work by Gabriella Guzman. This enables us to
reconstruct the unipotent fundamental group of a pointed scheme from the
associated augmented motivic dga, and provides us with a factorization of Kim's
relative unipotent section conjecture into several smaller conjectures with a
homotopical flavor. Based on a conversation with Joseph Ayoub, we prove that
the path spaces of the punctured projective line over a number field are
concentrated in degree zero with respect to Levine's t-structure for mixed Tate
motives. This constitutes a step in the direction of Kim's conjecture.Comment: Minor corrections, details added, and major improvements to
exposition throughout. 52 page
Curriculum Learning by Transfer Learning: Theory and Experiments with Deep Networks
We provide theoretical investigation of curriculum learning in the context of
stochastic gradient descent when optimizing the convex linear regression loss.
We prove that the rate of convergence of an ideal curriculum learning method is
monotonically increasing with the difficulty of the examples. Moreover, among
all equally difficult points, convergence is faster when using points which
incur higher loss with respect to the current hypothesis. We then analyze
curriculum learning in the context of training a CNN. We describe a method
which infers the curriculum by way of transfer learning from another network,
pre-trained on a different task. While this approach can only approximate the
ideal curriculum, we observe empirically similar behavior to the one predicted
by the theory, namely, a significant boost in convergence speed at the
beginning of training. When the task is made more difficult, improvement in
generalization performance is also observed. Finally, curriculum learning
exhibits robustness against unfavorable conditions such as excessive
regularization.Comment: ICML 201
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