20,409 research outputs found
Unfolding Latent Tree Structures using 4th Order Tensors
Discovering the latent structure from many observed variables is an important
yet challenging learning task. Existing approaches for discovering latent
structures often require the unknown number of hidden states as an input. In
this paper, we propose a quartet based approach which is \emph{agnostic} to
this number. The key contribution is a novel rank characterization of the
tensor associated with the marginal distribution of a quartet. This
characterization allows us to design a \emph{nuclear norm} based test for
resolving quartet relations. We then use the quartet test as a subroutine in a
divide-and-conquer algorithm for recovering the latent tree structure. Under
mild conditions, the algorithm is consistent and its error probability decays
exponentially with increasing sample size. We demonstrate that the proposed
approach compares favorably to alternatives. In a real world stock dataset, it
also discovers meaningful groupings of variables, and produces a model that
fits the data better
Online Tensor Methods for Learning Latent Variable Models
We introduce an online tensor decomposition based approach for two latent
variable modeling problems namely, (1) community detection, in which we learn
the latent communities that the social actors in social networks belong to, and
(2) topic modeling, in which we infer hidden topics of text articles. We
consider decomposition of moment tensors using stochastic gradient descent. We
conduct optimization of multilinear operations in SGD and avoid directly
forming the tensors, to save computational and storage costs. We present
optimized algorithm in two platforms. Our GPU-based implementation exploits the
parallelism of SIMD architectures to allow for maximum speed-up by a careful
optimization of storage and data transfer, whereas our CPU-based implementation
uses efficient sparse matrix computations and is suitable for large sparse
datasets. For the community detection problem, we demonstrate accuracy and
computational efficiency on Facebook, Yelp and DBLP datasets, and for the topic
modeling problem, we also demonstrate good performance on the New York Times
dataset. We compare our results to the state-of-the-art algorithms such as the
variational method, and report a gain of accuracy and a gain of several orders
of magnitude in the execution time.Comment: JMLR 201
A Tensor-Based Dictionary Learning Approach to Tomographic Image Reconstruction
We consider tomographic reconstruction using priors in the form of a
dictionary learned from training images. The reconstruction has two stages:
first we construct a tensor dictionary prior from our training data, and then
we pose the reconstruction problem in terms of recovering the expansion
coefficients in that dictionary. Our approach differs from past approaches in
that a) we use a third-order tensor representation for our images and b) we
recast the reconstruction problem using the tensor formulation. The dictionary
learning problem is presented as a non-negative tensor factorization problem
with sparsity constraints. The reconstruction problem is formulated in a convex
optimization framework by looking for a solution with a sparse representation
in the tensor dictionary. Numerical results show that our tensor formulation
leads to very sparse representations of both the training images and the
reconstructions due to the ability of representing repeated features compactly
in the dictionary.Comment: 29 page
Smoothed Analysis in Unsupervised Learning via Decoupling
Smoothed analysis is a powerful paradigm in overcoming worst-case
intractability in unsupervised learning and high-dimensional data analysis.
While polynomial time smoothed analysis guarantees have been obtained for
worst-case intractable problems like tensor decompositions and learning
mixtures of Gaussians, such guarantees have been hard to obtain for several
other important problems in unsupervised learning. A core technical challenge
in analyzing algorithms is obtaining lower bounds on the least singular value
for random matrix ensembles with dependent entries, that are given by
low-degree polynomials of a few base underlying random variables.
In this work, we address this challenge by obtaining high-confidence lower
bounds on the least singular value of new classes of structured random matrix
ensembles of the above kind. We then use these bounds to design algorithms with
polynomial time smoothed analysis guarantees for the following three important
problems in unsupervised learning:
1. Robust subspace recovery, when the fraction of inliers in the
d-dimensional subspace is at least for any constant integer . This contrasts with the known
worst-case intractability when , and the previous smoothed
analysis result which needed (Hardt and Moitra, 2013).
2. Learning overcomplete hidden markov models, where the size of the state
space is any polynomial in the dimension of the observations. This gives the
first polynomial time guarantees for learning overcomplete HMMs in a smoothed
analysis model.
3. Higher order tensor decompositions, where we generalize the so-called
FOOBI algorithm of Cardoso to find order- rank-one tensors in a subspace.
This allows us to obtain polynomially robust decomposition algorithms for
'th order tensors with rank .Comment: 44 page
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