15,168 research outputs found
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
Distributed Clustering and Learning Over Networks
Distributed processing over networks relies on in-network processing and
cooperation among neighboring agents. Cooperation is beneficial when agents
share a common objective. However, in many applications agents may belong to
different clusters that pursue different objectives. Then, indiscriminate
cooperation will lead to undesired results. In this work, we propose an
adaptive clustering and learning scheme that allows agents to learn which
neighbors they should cooperate with and which other neighbors they should
ignore. In doing so, the resulting algorithm enables the agents to identify
their clusters and to attain improved learning and estimation accuracy over
networks. We carry out a detailed mean-square analysis and assess the error
probabilities of Types I and II, i.e., false alarm and mis-detection, for the
clustering mechanism. Among other results, we establish that these
probabilities decay exponentially with the step-sizes so that the probability
of correct clustering can be made arbitrarily close to one.Comment: 47 pages, 6 figure
Diffusion Variational Autoencoders
A standard Variational Autoencoder, with a Euclidean latent space, is
structurally incapable of capturing topological properties of certain datasets.
To remove topological obstructions, we introduce Diffusion Variational
Autoencoders with arbitrary manifolds as a latent space. A Diffusion
Variational Autoencoder uses transition kernels of Brownian motion on the
manifold. In particular, it uses properties of the Brownian motion to implement
the reparametrization trick and fast approximations to the KL divergence. We
show that the Diffusion Variational Autoencoder is capable of capturing
topological properties of synthetic datasets. Additionally, we train MNIST on
spheres, tori, projective spaces, SO(3), and a torus embedded in R3. Although a
natural dataset like MNIST does not have latent variables with a clear-cut
topological structure, training it on a manifold can still highlight
topological and geometrical properties.Comment: 10 pages, 8 figures Added an appendix with derivation of asymptotic
expansion of KL divergence for heat kernel on arbitrary Riemannian manifolds,
and an appendix with new experiments on binarized MNIST. Added a previously
missing factor in the asymptotic expansion of the heat kernel and corrected a
coefficient in asymptotic expansion KL divergence; further minor edit
Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks
Stochastic gradient descent (SGD) is widely believed to perform implicit
regularization when used to train deep neural networks, but the precise manner
in which this occurs has thus far been elusive. We prove that SGD minimizes an
average potential over the posterior distribution of weights along with an
entropic regularization term. This potential is however not the original loss
function in general. So SGD does perform variational inference, but for a
different loss than the one used to compute the gradients. Even more
surprisingly, SGD does not even converge in the classical sense: we show that
the most likely trajectories of SGD for deep networks do not behave like
Brownian motion around critical points. Instead, they resemble closed loops
with deterministic components. We prove that such "out-of-equilibrium" behavior
is a consequence of highly non-isotropic gradient noise in SGD; the covariance
matrix of mini-batch gradients for deep networks has a rank as small as 1% of
its dimension. We provide extensive empirical validation of these claims,
proven in the appendix
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