2,333 research outputs found
A new robust and efficient estimator for ill-conditioned linear inverse problems with outliers
Solving a linear inverse problem may include difficulties such as the presence of outliers and a mixing matrix with a large condition number. In such cases a regularized robust estimator is needed. We propose a new tau-type regularized robust estimator that is simultaneously highly robust against outliers, highly efficient in the presence of purely Gaussian noise, and also stable when the mixing matrix has a large condition number. We also propose an algorithm to compute the estimates, based on a regularized iterative reweighted least squares algorithm. A basic and a fast version of the algorithm are given. Finally, we test the performance of the proposed approach using numerical experiments and compare it with other estimators. Our estimator provides superior robustness, even up to 40% of outliers, while at the same time performing quite close to the optimal maximum likelihood estimator in the outlier-free case
Variational Downscaling, Fusion and Assimilation of Hydrometeorological States via Regularized Estimation
Improved estimation of hydrometeorological states from down-sampled
observations and background model forecasts in a noisy environment, has been a
subject of growing research in the past decades. Here, we introduce a unified
framework that ties together the problems of downscaling, data fusion and data
assimilation as ill-posed inverse problems. This framework seeks solutions
beyond the classic least squares estimation paradigms by imposing proper
regularization, which are constraints consistent with the degree of smoothness
and probabilistic structure of the underlying state. We review relevant
regularization methods in derivative space and extend classic formulations of
the aforementioned problems with particular emphasis on hydrologic and
atmospheric applications. Informed by the statistical characteristics of the
state variable of interest, the central results of the paper suggest that
proper regularization can lead to a more accurate and stable recovery of the
true state and hence more skillful forecasts. In particular, using the Tikhonov
and Huber regularization in the derivative space, the promise of the proposed
framework is demonstrated in static downscaling and fusion of synthetic
multi-sensor precipitation data, while a data assimilation numerical experiment
is presented using the heat equation in a variational setting
Significance Regression: Robust Regression for Collinear Data
This paper examines robust linear multivariable regression from collinear data. A brief review of M-estimators discusses the strengths of this approach for tolerating outliers and/or perturbations in the error distributions. The review reveals that M-estimation may be unreliable if the data exhibit collinearity. Next, significance regression (SR) is discussed. SR is a successful method for treating collinearity but is not robust. A new significance regression algorithm for the weighted-least-squares error criterion (SR-WLS) is developed. Using the weights computed via M-estimation with the SR-WLS algorithm yields an effective method that robustly mollifies collinearity problems. Numerical examples illustrate the main points
Vector Approximate Message Passing for the Generalized Linear Model
The generalized linear model (GLM), where a random vector is
observed through a noisy, possibly nonlinear, function of a linear transform
output , arises in a range of applications such
as robust regression, binary classification, quantized compressed sensing,
phase retrieval, photon-limited imaging, and inference from neural spike
trains. When is large and i.i.d. Gaussian, the generalized
approximate message passing (GAMP) algorithm is an efficient means of MAP or
marginal inference, and its performance can be rigorously characterized by a
scalar state evolution. For general , though, GAMP can
misbehave. Damping and sequential-updating help to robustify GAMP, but their
effects are limited. Recently, a "vector AMP" (VAMP) algorithm was proposed for
additive white Gaussian noise channels. VAMP extends AMP's guarantees from
i.i.d. Gaussian to the larger class of rotationally invariant
. In this paper, we show how VAMP can be extended to the GLM.
Numerical experiments show that the proposed GLM-VAMP is much more robust to
ill-conditioning in than damped GAMP
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