267 research outputs found
Efficient Orthogonal Tensor Decomposition, with an Application to Latent Variable Model Learning
Decomposing tensors into orthogonal factors is a well-known task in
statistics, machine learning, and signal processing. We study orthogonal outer
product decompositions where the factors in the summands in the decomposition
are required to be orthogonal across summands, by relating this orthogonal
decomposition to the singular value decompositions of the flattenings. We show
that it is a non-trivial assumption for a tensor to have such an orthogonal
decomposition, and we show that it is unique (up to natural symmetries) in case
it exists, in which case we also demonstrate how it can be efficiently and
reliably obtained by a sequence of singular value decompositions. We
demonstrate how the factoring algorithm can be applied for parameter
identification in latent variable and mixture models
Fourier PCA and Robust Tensor Decomposition
Fourier PCA is Principal Component Analysis of a matrix obtained from higher
order derivatives of the logarithm of the Fourier transform of a
distribution.We make this method algorithmic by developing a tensor
decomposition method for a pair of tensors sharing the same vectors in rank-
decompositions. Our main application is the first provably polynomial-time
algorithm for underdetermined ICA, i.e., learning an matrix
from observations where is drawn from an unknown product
distribution with arbitrary non-Gaussian components. The number of component
distributions can be arbitrarily higher than the dimension and the
columns of only need to satisfy a natural and efficiently verifiable
nondegeneracy condition. As a second application, we give an alternative
algorithm for learning mixtures of spherical Gaussians with linearly
independent means. These results also hold in the presence of Gaussian noise.Comment: Extensively revised; details added; minor errors corrected;
exposition improve
Tensor decompositions for learning latent variable models
This work considers a computationally and statistically efficient parameter
estimation method for a wide class of latent variable models---including
Gaussian mixture models, hidden Markov models, and latent Dirichlet
allocation---which exploits a certain tensor structure in their low-order
observable moments (typically, of second- and third-order). Specifically,
parameter estimation is reduced to the problem of extracting a certain
(orthogonal) decomposition of a symmetric tensor derived from the moments; this
decomposition can be viewed as a natural generalization of the singular value
decomposition for matrices. Although tensor decompositions are generally
intractable to compute, the decomposition of these specially structured tensors
can be efficiently obtained by a variety of approaches, including power
iterations and maximization approaches (similar to the case of matrices). A
detailed analysis of a robust tensor power method is provided, establishing an
analogue of Wedin's perturbation theorem for the singular vectors of matrices.
This implies a robust and computationally tractable estimation approach for
several popular latent variable models
Effective criteria for specific identifiability of tensors and forms
In applications where the tensor rank decomposition arises, one often relies
on its identifiability properties for interpreting the individual rank-
terms appearing in the decomposition. Several criteria for identifiability have
been proposed in the literature, however few results exist on how frequently
they are satisfied. We propose to call a criterion effective if it is satisfied
on a dense, open subset of the smallest semi-algebraic set enclosing the set of
rank- tensors. We analyze the effectiveness of Kruskal's criterion when it
is combined with reshaping. It is proved that this criterion is effective for
both real and complex tensors in its entire range of applicability, which is
usually much smaller than the smallest typical rank. Our proof explains when
reshaping-based algorithms for computing tensor rank decompositions may be
expected to recover the decomposition. Specializing the analysis to symmetric
tensors or forms reveals that the reshaped Kruskal criterion may even be
effective up to the smallest typical rank for some third, fourth and sixth
order symmetric tensors of small dimension as well as for binary forms of
degree at least three. We extended this result to symmetric tensors by analyzing the Hilbert function, resulting in a
criterion for symmetric identifiability that is effective up to symmetric rank
, which is optimal.Comment: 31 pages, 2 Macaulay2 code
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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