228 research outputs found
Representation Learning for Clustering: A Statistical Framework
We address the problem of communicating domain knowledge from a user to the
designer of a clustering algorithm. We propose a protocol in which the user
provides a clustering of a relatively small random sample of a data set. The
algorithm designer then uses that sample to come up with a data representation
under which -means clustering results in a clustering (of the full data set)
that is aligned with the user's clustering. We provide a formal statistical
model for analyzing the sample complexity of learning a clustering
representation with this paradigm. We then introduce a notion of capacity of a
class of possible representations, in the spirit of the VC-dimension, showing
that classes of representations that have finite such dimension can be
successfully learned with sample size error bounds, and end our discussion with
an analysis of that dimension for classes of representations induced by linear
embeddings.Comment: To be published in Proceedings of UAI 201
Sample-Efficient Learning of Mixtures
We consider PAC learning of probability distributions (a.k.a. density
estimation), where we are given an i.i.d. sample generated from an unknown
target distribution, and want to output a distribution that is close to the
target in total variation distance. Let be an arbitrary class of
probability distributions, and let denote the class of
-mixtures of elements of . Assuming the existence of a method
for learning with sample complexity ,
we provide a method for learning with sample complexity
. Our mixture
learning algorithm has the property that, if the -learner is
proper/agnostic, then the -learner would be proper/agnostic as
well.
This general result enables us to improve the best known sample complexity
upper bounds for a variety of important mixture classes. First, we show that
the class of mixtures of axis-aligned Gaussians in is
PAC-learnable in the agnostic setting with
samples, which is tight in and up to logarithmic factors. Second, we
show that the class of mixtures of Gaussians in is
PAC-learnable in the agnostic setting with sample complexity
, which improves the previous known
bounds of and
in its dependence on and . Finally,
we show that the class of mixtures of log-concave distributions over
is PAC-learnable using
samples.Comment: A bug from the previous version, which appeared in AAAI 2018
proceedings, is fixed. 18 page
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