34,483 research outputs found
Semivariogram methods for modeling Whittle-Mat\'ern priors in Bayesian inverse problems
We present a new technique, based on semivariogram methodology, for obtaining
point estimates for use in prior modeling for solving Bayesian inverse
problems. This method requires a connection between Gaussian processes with
covariance operators defined by the Mat\'ern covariance function and Gaussian
processes with precision (inverse-covariance) operators defined by the Green's
functions of a class of elliptic stochastic partial differential equations
(SPDEs). We present a detailed mathematical description of this connection. We
will show that there is an equivalence between these two Gaussian processes
when the domain is infinite -- for us, -- which breaks down when
the domain is finite due to the effect of boundary conditions on Green's
functions of PDEs. We show how this connection can be re-established using
extended domains. We then introduce the semivariogram method for estimating the
Mat\'ern covariance parameters, which specify the Gaussian prior needed for
stabilizing the inverse problem. Results are extended from the isotropic case
to the anisotropic case where the correlation length in one direction is larger
than another. Finally, we consider the situation where the correlation length
is spatially dependent rather than constant. We implement each method in
two-dimensional image inpainting test cases to show that it works on practical
examples
Matrix completion with queries
In many applications, e.g., recommender systems and traffic monitoring, the
data comes in the form of a matrix that is only partially observed and low
rank. A fundamental data-analysis task for these datasets is matrix completion,
where the goal is to accurately infer the entries missing from the matrix. Even
when the data satisfies the low-rank assumption, classical matrix-completion
methods may output completions with significant error -- in that the
reconstructed matrix differs significantly from the true underlying matrix.
Often, this is due to the fact that the information contained in the observed
entries is insufficient. In this work, we address this problem by proposing an
active version of matrix completion, where queries can be made to the true
underlying matrix. Subsequently, we design Order&Extend, which is the first
algorithm to unify a matrix-completion approach and a querying strategy into a
single algorithm. Order&Extend is able identify and alleviate insufficient
information by judiciously querying a small number of additional entries. In an
extensive experimental evaluation on real-world datasets, we demonstrate that
our algorithm is efficient and is able to accurately reconstruct the true
matrix while asking only a small number of queries.Comment: Proceedings of the 21th ACM SIGKDD International Conference on
Knowledge Discovery and Data Minin
Solving systems of phaseless equations via Kaczmarz methods: A proof of concept study
We study the Kaczmarz methods for solving systems of quadratic equations,
i.e., the generalized phase retrieval problem. The methods extend the Kaczmarz
methods for solving systems of linear equations by integrating a phase
selection heuristic in each iteration and overall have the same per iteration
computational complexity. Extensive empirical performance comparisons establish
the computational advantages of the Kaczmarz methods over other
state-of-the-art phase retrieval algorithms both in terms of the number of
measurements needed for successful recovery and in terms of computation time.
Preliminary convergence analysis is presented for the randomized Kaczmarz
methods
Performance Analysis of Sparse Recovery Based on Constrained Minimal Singular Values
The stability of sparse signal reconstruction is investigated in this paper.
We design efficient algorithms to verify the sufficient condition for unique
sparse recovery. One of our algorithm produces comparable results with
the state-of-the-art technique and performs orders of magnitude faster. We show
that the -constrained minimal singular value (-CMSV) of the
measurement matrix determines, in a very concise manner, the recovery
performance of -based algorithms such as the Basis Pursuit, the Dantzig
selector, and the LASSO estimator. Compared with performance analysis involving
the Restricted Isometry Constant, the arguments in this paper are much less
complicated and provide more intuition on the stability of sparse signal
recovery. We show also that, with high probability, the subgaussian ensemble
generates measurement matrices with -CMSVs bounded away from zero, as
long as the number of measurements is relatively large. To compute the
-CMSV and its lower bound, we design two algorithms based on the
interior point algorithm and the semi-definite relaxation
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