7,117 research outputs found
Smoothed Analysis of the Condition Number Under Low-Rank Perturbations
Let be an arbitrary by matrix of rank . We study the
condition number of plus a \emph{low-rank} perturbation where
are by random Gaussian matrices. Under some necessary assumptions, it
is shown that is unlikely to have a large condition number. The main
advantages of this kind of perturbation over the well-studied dense Gaussian
perturbation, where every entry is independently perturbed, is the cost
to store and the increase in time complexity for performing the
matrix-vector multiplication . This improves the space
and time complexity increase required by a dense perturbation, which is
especially burdensome if is originally sparse. Our results also extend to
the case where and have rank larger than and to symmetric and
complex settings. We also give an application to linear systems solving and
perform some numerical experiments. Lastly, barriers in applying low-rank noise
to other problems studied in the smoothed analysis framework are discussed
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
Smoothed Analysis of Tensor Decompositions
Low rank tensor decompositions are a powerful tool for learning generative
models, and uniqueness results give them a significant advantage over matrix
decomposition methods. However, tensors pose significant algorithmic challenges
and tensors analogs of much of the matrix algebra toolkit are unlikely to exist
because of hardness results. Efficient decomposition in the overcomplete case
(where rank exceeds dimension) is particularly challenging. We introduce a
smoothed analysis model for studying these questions and develop an efficient
algorithm for tensor decomposition in the highly overcomplete case (rank
polynomial in the dimension). In this setting, we show that our algorithm is
robust to inverse polynomial error -- a crucial property for applications in
learning since we are only allowed a polynomial number of samples. While
algorithms are known for exact tensor decomposition in some overcomplete
settings, our main contribution is in analyzing their stability in the
framework of smoothed analysis.
Our main technical contribution is to show that tensor products of perturbed
vectors are linearly independent in a robust sense (i.e. the associated matrix
has singular values that are at least an inverse polynomial). This key result
paves the way for applying tensor methods to learning problems in the smoothed
setting. In particular, we use it to obtain results for learning multi-view
models and mixtures of axis-aligned Gaussians where there are many more
"components" than dimensions. The assumption here is that the model is not
adversarially chosen, formalized by a perturbation of model parameters. We
believe this an appealing way to analyze realistic instances of learning
problems, since this framework allows us to overcome many of the usual
limitations of using tensor methods.Comment: 32 pages (including appendix
Semi-blind Eigen-analyses of Recombination Histories Using CMB Data
Cosmological parameter measurements from CMB experiments such as Planck,
ACTpol, SPTpol and other high resolution follow-ons fundamentally rely on the
accuracy of the assumed recombination model, or one with well prescribed
uncertainties. Deviations from the standard recombination history might suggest
new particle physics or modified atomic physics. Here we treat possible
perturbative fluctuations in the free electron fraction, \Xe(z), by a
semi-blind expansion in densely-packed modes in redshift. From these we
construct parameter eigenmodes, which we rank order so that the lowest modes
provide the most power to probe the \Xe(z) with CMB measurements. Since the
eigenmodes are effectively weighed by the fiducial \Xe history, they are
localized around the differential visibility peak, allowing for an excellent
probe of hydrogen recombination, but a weaker probe of the higher redshift
helium recombination and the lower redshift highly neutral freeze-out tail. We
use an information-based criterion to truncate the mode hierarchy, and show
that with even a few modes the method goes a long way towards morphing a
fiducial older {\sc Recfast} into the new and improved {\sc
CosmoRec} and {\sc HyRec} in the hydrogen recombination
regime, though not well in the helium regime. Without such a correction, the
derived cosmic parameters are biased. We discuss an iterative approach for
updating the eigenmodes to further hone in on if large
deviations are indeed found. We also introduce control parameters that
downweight the attention on the visibility peak structure, e.g., focusing the
eigenmode probes more strongly on the \Xe (z) freeze-out tail, as would be
appropriate when looking for the \Xe signature of annihilating or decaying
elementary particles.Comment: 28 pages, 26 Fig
Polynomial-time Tensor Decompositions with Sum-of-Squares
We give new algorithms based on the sum-of-squares method for tensor
decomposition. Our results improve the best known running times from
quasi-polynomial to polynomial for several problems, including decomposing
random overcomplete 3-tensors and learning overcomplete dictionaries with
constant relative sparsity. We also give the first robust analysis for
decomposing overcomplete 4-tensors in the smoothed analysis model. A key
ingredient of our analysis is to establish small spectral gaps in moment
matrices derived from solutions to sum-of-squares relaxations. To enable this
analysis we augment sum-of-squares relaxations with spectral analogs of maximum
entropy constraints.Comment: to appear in FOCS 201
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