563 research outputs found
Comment on "Support Vector Machines with Applications"
Comment on ``Support Vector Machines with Applications'' [math.ST/0612817]Comment: Published at http://dx.doi.org/10.1214/088342306000000484 in the
Statistical Science (http://www.imstat.org/sts/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Counting coloured planar maps: differential equations
We address the enumeration of q-coloured planar maps counted bythe number of
edges and the number of monochromatic edges. We prove that the associated
generating function is differentially algebraic,that is, satisfies a
non-trivial polynomial differential equation withrespect to the edge variable.
We give explicitly a differential systemthat characterizes this series. We then
prove a similar result for planar triangulations, thus generalizing a result of
Tutte dealing with their proper q-colourings. Instatistical physics terms, we
solvethe q-state Potts model on random planar lattices. This work follows a
first paper by the same authors, where the generating functionwas proved to be
algebraic for certain values of q,including q=1, 2 and 3. It isknown to be
transcendental in general. In contrast, our differential system holds for an
indeterminate q.For certain special cases of combinatorial interest (four
colours; properq-colourings; maps equipped with a spanning forest), we derive
from this system, in the case of triangulations, an explicit differential
equation of order 2 defining the generating function. For general planar maps,
we also obtain a differential equation of order 3 for the four-colour case and
for the self-dual Potts model.Comment: 43 p
Local Rademacher complexities
We propose new bounds on the error of learning algorithms in terms of a
data-dependent notion of complexity. The estimates we establish give optimal
rates and are based on a local and empirical version of Rademacher averages, in
the sense that the Rademacher averages are computed from the data, on a subset
of functions with small empirical error. We present some applications to
classification and prediction with convex function classes, and with kernel
classes in particular.Comment: Published at http://dx.doi.org/10.1214/009053605000000282 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Precision-Recall Curves Using Information Divergence Frontiers
Despite the tremendous progress in the estimation of generative models, the
development of tools for diagnosing their failures and assessing their
performance has advanced at a much slower pace. Recent developments have
investigated metrics that quantify which parts of the true distribution is
modeled well, and, on the contrary, what the model fails to capture, akin to
precision and recall in information retrieval. In this paper, we present a
general evaluation framework for generative models that measures the trade-off
between precision and recall using R\'enyi divergences. Our framework provides
a novel perspective on existing techniques and extends them to more general
domains. As a key advantage, this formulation encompasses both continuous and
discrete models and allows for the design of efficient algorithms that do not
have to quantize the data. We further analyze the biases of the approximations
used in practice.Comment: Updated to the AISTATS 2020 versio
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