53,090 research outputs found
Spectral Condition Numbers of Orthogonal Projections and Full Rank Linear Least Squares Residuals
A simple formula is proved to be a tight estimate for the condition number of
the full rank linear least squares residual with respect to the matrix of least
squares coefficients and scaled 2-norms. The tight estimate reveals that the
condition number depends on three quantities, two of which can cause
ill-conditioning. The numerical linear algebra literature presents several
estimates of various instances of these condition numbers. All the prior values
exceed the formula introduced here, sometimes by large factors.Comment: 15 pages, 1 figure, 2 table
Sparse seismic imaging using variable projection
We consider an important class of signal processing problems where the signal
of interest is known to be sparse, and can be recovered from data given
auxiliary information about how the data was generated. For example, a sparse
Green's function may be recovered from seismic experimental data using sparsity
optimization when the source signature is known. Unfortunately, in practice
this information is often missing, and must be recovered from data along with
the signal using deconvolution techniques.
In this paper, we present a novel methodology to simultaneously solve for the
sparse signal and auxiliary parameters using a recently proposed variable
projection technique. Our main contribution is to combine variable projection
with sparsity promoting optimization, obtaining an efficient algorithm for
large-scale sparse deconvolution problems. We demonstrate the algorithm on a
seismic imaging example.Comment: 5 pages, 4 figure
A multiscattering series for impedance tomography in layered media
We introduce an inversion algorithm for tomographic images of layered media. The algorithm is based on a multiscattering series expansion of the Green function that, unlike the Born series, converges unconditionally. Our inversion algorithm obtains images of the medium that improves iteratively as we use more and more terms in the multiscattering series. We present the derivation of the multiscattering series, formulate the inversion algorithm and demonstrate its performance through numerical experiments
Perturbed Datasets Methods for Hypothesis Testing and Structure of Corresponding Confidence Sets
Hypothesis testing methods that do not rely on exact distribution assumptions
have been emerging lately. The method of sign-perturbed sums (SPS) is capable
of characterizing confidence regions with exact confidence levels for linear
regression and linear dynamical systems parameter estimation problems if the
noise distribution is symmetric. This paper describes a general family of
hypothesis testing methods that have an exact user chosen confidence level
based on finite sample count and without relying on an assumed noise
distribution. It is shown that the SPS method belongs to this family and we
provide another hypothesis test for the case where the symmetry assumption is
replaced with exchangeability. In the case of linear regression problems it is
shown that the confidence regions are connected, bounded and possibly
non-convex sets in both cases. To highlight the importance of understanding the
structure of confidence regions corresponding to such hypothesis tests it is
shown that confidence sets for linear dynamical systems parameter estimates
generated using the SPS method can have non-connected parts, which have far
reaching consequences
- …