6,691 research outputs found
Surprises in High-Dimensional Ridgeless Least Squares Interpolation
Interpolators -- estimators that achieve zero training error -- have
attracted growing attention in machine learning, mainly because state-of-the
art neural networks appear to be models of this type. In this paper, we study
minimum norm (``ridgeless'') interpolation in high-dimensional least
squares regression. We consider two different models for the feature
distribution: a linear model, where the feature vectors
are obtained by applying a linear transform to a vector of i.i.d.\ entries,
(with ); and a nonlinear model,
where the feature vectors are obtained by passing the input through a random
one-layer neural network, (with ,
a matrix of i.i.d.\ entries, and an
activation function acting componentwise on ). We recover -- in a
precise quantitative way -- several phenomena that have been observed in
large-scale neural networks and kernel machines, including the "double descent"
behavior of the prediction risk, and the potential benefits of
overparametrization.Comment: 68 pages; 16 figures. This revision contains non-asymptotic version
of earlier results, and results for general coefficient
Developmental Programming: Granulosa cell mRNA expression of differentiation, growth and apoptosis genes in abnormally large progestagenic follicles from androgenized ewes
High-m Kink/Tearing Modes in Cylindrical Geometry
The global ideal kink equation, for cylindrical geometry and zero beta, is
simplified in the high poloidal mode number limit and used to determine the
tearing stability parameter, . In the presence of a steep
monotonic current gradient, becomes a function of a parameter,
, characterising the ratio of the maximum current gradient to
magnetic shear, and , characterising the separation of the resonant
surface from the maximum of the current gradient. In equilibria containing a
current "spike", so that there is a non-monotonic current profile,
also depends on two parameters: , related to the ratio
of the curvature of the current density at its maximum to the magnetic shear,
and , which now represents the separation of the resonance from the point
of maximum current density. The relation of our results to earlier studies of
tearing modes and to recent gyro-kinetic calculations of current driven
instabilities, is discussed, together with potential implications for the
stability of the tokamak pedestal.Comment: To appear in Plasma Physics and Controlled Fusio
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