7,594 research outputs found
Bounding Embeddings of VC Classes into Maximum Classes
One of the earliest conjectures in computational learning theory-the Sample
Compression conjecture-asserts that concept classes (equivalently set systems)
admit compression schemes of size linear in their VC dimension. To-date this
statement is known to be true for maximum classes---those that possess maximum
cardinality for their VC dimension. The most promising approach to positively
resolving the conjecture is by embedding general VC classes into maximum
classes without super-linear increase to their VC dimensions, as such
embeddings would extend the known compression schemes to all VC classes. We
show that maximum classes can be characterised by a local-connectivity property
of the graph obtained by viewing the class as a cubical complex. This geometric
characterisation of maximum VC classes is applied to prove a negative embedding
result which demonstrates VC-d classes that cannot be embedded in any maximum
class of VC dimension lower than 2d. On the other hand, we show that every VC-d
class C embeds in a VC-(d+D) maximum class where D is the deficiency of C,
i.e., the difference between the cardinalities of a maximum VC-d class and of
C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible
results on embedding into maximum classes. For some special classes of Boolean
functions, relationships with maximum classes are investigated. Finally we give
a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum
classes for smallest k.Comment: 22 pages, 2 figure
Bifurcation diagram and pattern formation in superconducting wires with electric currents
We examine the behavior of a one-dimensional superconducting wire exposed to
an applied electric current. We use the time-dependent Ginzburg-Landau model to
describe the system and retain temperature and applied current as parameters.
Through a combination of spectral analysis, asymptotics and canonical numerical
computation, we divide this two-dimensional parameter space into a number of
regions. In some of them only the normal state or a stationary state or an
oscillatory state are stable, while in some of them two states are stable. One
of the most interesting features of the analysis is the evident collision of
real eigenvalues of the associated PT-symmetric linearization, leading as it
does to the emergence of complex elements of the spectrum. In particular this
provides an explanation to the emergence of a stable oscillatory state. We show
that part of the bifurcation diagram and many of the emerging patterns are
directly controlled by this spectrum, while other patterns arise due to
nonlinear interaction of the leading eigenfunctions
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