4,394 research outputs found
Inference and Uncertainty Quantification for Noisy Matrix Completion
Noisy matrix completion aims at estimating a low-rank matrix given only
partial and corrupted entries. Despite substantial progress in designing
efficient estimation algorithms, it remains largely unclear how to assess the
uncertainty of the obtained estimates and how to perform statistical inference
on the unknown matrix (e.g.~constructing a valid and short confidence interval
for an unseen entry).
This paper takes a step towards inference and uncertainty quantification for
noisy matrix completion. We develop a simple procedure to compensate for the
bias of the widely used convex and nonconvex estimators. The resulting
de-biased estimators admit nearly precise non-asymptotic distributional
characterizations, which in turn enable optimal construction of confidence
intervals\,/\,regions for, say, the missing entries and the low-rank factors.
Our inferential procedures do not rely on sample splitting, thus avoiding
unnecessary loss of data efficiency. As a byproduct, we obtain a sharp
characterization of the estimation accuracy of our de-biased estimators, which,
to the best of our knowledge, are the first tractable algorithms that provably
achieve full statistical efficiency (including the preconstant). The analysis
herein is built upon the intimate link between convex and nonconvex
optimization --- an appealing feature recently discovered by
\cite{chen2019noisy}.Comment: published at Proceedings of the National Academy of Sciences Nov
2019, 116 (46) 22931-2293
A closed-form solution to estimate uncertainty in non-rigid structure from motion
Semi-Definite Programming (SDP) with low-rank prior has been widely applied
in Non-Rigid Structure from Motion (NRSfM). Based on a low-rank constraint, it
avoids the inherent ambiguity of basis number selection in conventional
base-shape or base-trajectory methods. Despite the efficiency in deformable
shape reconstruction, it remains unclear how to assess the uncertainty of the
recovered shape from the SDP process. In this paper, we present a statistical
inference on the element-wise uncertainty quantification of the estimated
deforming 3D shape points in the case of the exact low-rank SDP problem. A
closed-form uncertainty quantification method is proposed and tested. Moreover,
we extend the exact low-rank uncertainty quantification to the approximate
low-rank scenario with a numerical optimal rank selection method, which enables
solving practical application in SDP based NRSfM scenario. The proposed method
provides an independent module to the SDP method and only requires the
statistic information of the input 2D tracked points. Extensive experiments
prove that the output 3D points have identical normal distribution to the 2D
trackings, the proposed method and quantify the uncertainty accurately, and
supports that it has desirable effects on routinely SDP low-rank based NRSfM
solver.Comment: 9 pages, 2 figure
Maximum-a-posteriori estimation with Bayesian confidence regions
Solutions to inverse problems that are ill-conditioned or ill-posed may have
significant intrinsic uncertainty. Unfortunately, analysing and quantifying
this uncertainty is very challenging, particularly in high-dimensional
problems. As a result, while most modern mathematical imaging methods produce
impressive point estimation results, they are generally unable to quantify the
uncertainty in the solutions delivered. This paper presents a new general
methodology for approximating Bayesian high-posterior-density credibility
regions in inverse problems that are convex and potentially very
high-dimensional. The approximations are derived by using recent concentration
of measure results related to information theory for log-concave random
vectors. A remarkable property of the approximations is that they can be
computed very efficiently, even in large-scale problems, by using standard
convex optimisation techniques. In particular, they are available as a
by-product in problems solved by maximum-a-posteriori estimation. The
approximations also have favourable theoretical properties, namely they
outer-bound the true high-posterior-density credibility regions, and they are
stable with respect to model dimension. The proposed methodology is illustrated
on two high-dimensional imaging inverse problems related to tomographic
reconstruction and sparse deconvolution, where the approximations are used to
perform Bayesian hypothesis tests and explore the uncertainty about the
solutions, and where proximal Markov chain Monte Carlo algorithms are used as
benchmark to compute exact credible regions and measure the approximation
error
Conformalized matrix completion
Matrix completion aims to estimate missing entries in a data matrix, using
the assumption of a low-complexity structure (e.g., low rank) so that
imputation is possible. While many effective estimation algorithms exist in the
literature, uncertainty quantification for this problem has proved to be
challenging, and existing methods are extremely sensitive to model
misspecification. In this work, we propose a distribution-free method for
predictive inference in the matrix completion problem. Our method adapts the
framework of conformal prediction, which provides confidence intervals with
guaranteed distribution-free validity in the setting of regression, to the
problem of matrix completion. Our resulting method, conformalized matrix
completion (cmc), offers provable predictive coverage regardless of the
accuracy of the low-rank model. Empirical results on simulated and real data
demonstrate that cmc is robust to model misspecification while matching the
performance of existing model-based methods when the model is correct.Comment: accepted to 37th Conference on Neural Information Processing Systems
(NeurIPS 2023
Bayesian Framework for Simultaneous Registration and Estimation of Noisy, Sparse and Fragmented Functional Data
Mathematical and Physical Sciences: 3rd Place (The Ohio State University Edward F. Hayes Graduate Research Forum)In many applications, smooth processes generate data that is recorded under a variety of observation regimes, such as dense sampling and sparse or fragmented observations that are often contaminated with error. The statistical goal of registering and estimating the individual underlying functions from discrete observations has thus far been mainly approached sequentially without formal uncertainty propagation, or in an application-specific manner by pooling information across subjects. We propose a unified Bayesian framework for simultaneous registration and estimation, which is flexible enough to accommodate inference on individual functions under general observation regimes. Our ability to do this relies on the specification of strongly informative prior models over the amplitude component of function variability. We provide two strategies for this critical choice: a data-driven approach that defines an empirical basis for the amplitude subspace based on available training data, and a shape-restricted approach when the relative location and number of local extrema is well-understood. The proposed methods build on the elastic functional data analysis framework to separately model amplitude and phase variability inherent in functional data. We emphasize the importance of uncertainty quantification and visualization of these two components as they provide complementary information about the estimated functions. We validate the proposed framework using simulation studies, and real applications to estimation of fractional anisotropy profiles based on diffusion tensor imaging measurements, growth velocity functions and bone mineral density curves.No embarg
Expectation Propagation for Nonlinear Inverse Problems -- with an Application to Electrical Impedance Tomography
In this paper, we study a fast approximate inference method based on
expectation propagation for exploring the posterior probability distribution
arising from the Bayesian formulation of nonlinear inverse problems. It is
capable of efficiently delivering reliable estimates of the posterior mean and
covariance, thereby providing an inverse solution together with quantified
uncertainties. Some theoretical properties of the iterative algorithm are
discussed, and the efficient implementation for an important class of problems
of projection type is described. The method is illustrated with one typical
nonlinear inverse problem, electrical impedance tomography with complete
electrode model, under sparsity constraints. Numerical results for real
experimental data are presented, and compared with that by Markov chain Monte
Carlo. The results indicate that the method is accurate and computationally
very efficient.Comment: Journal of Computational Physics, to appea
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