2,784 research outputs found
An Exponential Lower Bound on the Complexity of Regularization Paths
For a variety of regularized optimization problems in machine learning,
algorithms computing the entire solution path have been developed recently.
Most of these methods are quadratic programs that are parameterized by a single
parameter, as for example the Support Vector Machine (SVM). Solution path
algorithms do not only compute the solution for one particular value of the
regularization parameter but the entire path of solutions, making the selection
of an optimal parameter much easier.
It has been assumed that these piecewise linear solution paths have only
linear complexity, i.e. linearly many bends. We prove that for the support
vector machine this complexity can be exponential in the number of training
points in the worst case. More strongly, we construct a single instance of n
input points in d dimensions for an SVM such that at least \Theta(2^{n/2}) =
\Theta(2^d) many distinct subsets of support vectors occur as the
regularization parameter changes.Comment: Journal version, 28 Pages, 5 Figure
Slime mould computes planar shapes
Computing a polygon defining a set of planar points is a classical problem of
modern computational geometry. In laboratory experiments we demonstrate that a
concave hull, a connected alpha-shape without holes, of a finite planar set is
approximated by slime mould Physarum polycephalum. We represent planar points
with sources of long-distance attractants and short-distance repellents and
inoculate a piece of plasmodium outside the data set. The plasmodium moves
towards the data and envelops it by pronounced protoplasmic tubes
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