36,740 research outputs found
Dimensionality Reduction for k-Means Clustering and Low Rank Approximation
We show how to approximate a data matrix with a much smaller
sketch that can be used to solve a general class of
constrained k-rank approximation problems to within error.
Importantly, this class of problems includes -means clustering and
unconstrained low rank approximation (i.e. principal component analysis). By
reducing data points to just dimensions, our methods generically
accelerate any exact, approximate, or heuristic algorithm for these ubiquitous
problems.
For -means dimensionality reduction, we provide relative
error results for many common sketching techniques, including random row
projection, column selection, and approximate SVD. For approximate principal
component analysis, we give a simple alternative to known algorithms that has
applications in the streaming setting. Additionally, we extend recent work on
column-based matrix reconstruction, giving column subsets that not only `cover'
a good subspace for \bv{A}, but can be used directly to compute this
subspace.
Finally, for -means clustering, we show how to achieve a
approximation by Johnson-Lindenstrauss projecting data points to just dimensions. This gives the first result that leverages the
specific structure of -means to achieve dimension independent of input size
and sublinear in
Differentially Private Mixture of Generative Neural Networks
Generative models are used in a wide range of applications building on large
amounts of contextually rich information. Due to possible privacy violations of
the individuals whose data is used to train these models, however, publishing
or sharing generative models is not always viable. In this paper, we present a
novel technique for privately releasing generative models and entire
high-dimensional datasets produced by these models. We model the generator
distribution of the training data with a mixture of generative neural
networks. These are trained together and collectively learn the generator
distribution of a dataset. Data is divided into clusters, using a novel
differentially private kernel -means, then each cluster is given to separate
generative neural networks, such as Restricted Boltzmann Machines or
Variational Autoencoders, which are trained only on their own cluster using
differentially private gradient descent. We evaluate our approach using the
MNIST dataset, as well as call detail records and transit datasets, showing
that it produces realistic synthetic samples, which can also be used to
accurately compute arbitrary number of counting queries.Comment: A shorter version of this paper appeared at the 17th IEEE
International Conference on Data Mining (ICDM 2017). This is the full
version, published in IEEE Transactions on Knowledge and Data Engineering
(TKDE
Relaxation dynamics of maximally clustered networks
We study the relaxation dynamics of fully clustered networks (maximal number
of triangles) to an unclustered state under two different edge dynamics---the
double-edge swap, corresponding to degree-preserving randomization of the
configuration model, and single edge replacement, corresponding to full
randomization of the Erd\H{o}s--R\'enyi random graph. We derive expressions for
the time evolution of the degree distribution, edge multiplicity distribution
and clustering coefficient. We show that under both dynamics networks undergo a
continuous phase transition in which a giant connected component is formed. We
calculate the position of the phase transition analytically using the
Erd\H{o}s--R\'enyi phenomenology
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