1,618 research outputs found
Estimating a bivariate linear relationship
Solutions of the bivariate, linear errors-in-variables estimation problem
with unspecified errors are expected to be invariant under interchange and
scaling of the coordinates. The appealing model of normally distributed true
values and errors is unidentified without additional information. I propose a
prior density that incorporates the fact that the slope and variance parameters
together determine the covariance matrix of the unobserved true values but is
otherwise diffuse. The marginal posterior density of the slope is invariant to
interchange and scaling of the coordinates and depends on the data only through
the sample correlation coefficient and ratio of standard deviations. It covers
the interval between the two ordinary least squares estimates but diminishes
rapidly outside of it. I introduce the R package leiv for computing the
posterior density, and I apply it to examples in astronomy and method
comparison.Comment: 27 pages, 7 figure
The condition number of join decompositions
The join set of a finite collection of smooth embedded submanifolds of a
mutual vector space is defined as their Minkowski sum. Join decompositions
generalize some ubiquitous decompositions in multilinear algebra, namely tensor
rank, Waring, partially symmetric rank and block term decompositions. This
paper examines the numerical sensitivity of join decompositions to
perturbations; specifically, we consider the condition number for general join
decompositions. It is characterized as a distance to a set of ill-posed points
in a supplementary product of Grassmannians. We prove that this condition
number can be computed efficiently as the smallest singular value of an
auxiliary matrix. For some special join sets, we characterized the behavior of
sequences in the join set converging to the latter's boundary points. Finally,
we specialize our discussion to the tensor rank and Waring decompositions and
provide several numerical experiments confirming the key results
The average condition number of most tensor rank decomposition problems is infinite
The tensor rank decomposition, or canonical polyadic decomposition, is the
decomposition of a tensor into a sum of rank-1 tensors. The condition number of
the tensor rank decomposition measures the sensitivity of the rank-1 summands
with respect to structured perturbations. Those are perturbations preserving
the rank of the tensor that is decomposed. On the other hand, the angular
condition number measures the perturbations of the rank-1 summands up to
scaling.
We show for random rank-2 tensors with Gaussian density that the expected
value of the condition number is infinite. Under some mild additional
assumption, we show that the same is true for most higher ranks as
well. In fact, as the dimensions of the tensor tend to infinity, asymptotically
all ranks are covered by our analysis. On the contrary, we show that rank-2
Gaussian tensors have finite expected angular condition number.
Our results underline the high computational complexity of computing tensor
rank decompositions. We discuss consequences of our results for algorithm
design and for testing algorithms that compute the CPD. Finally, we supply
numerical experiments
A Sensitivity Matrix Methodology for Inverse Problem Formulation
We propose an algorithm to select parameter subset combinations that can be estimated using an ordinary least-squares (OLS) inverse problem formulation with a given data set. First, the algorithm selects the parameter combinations that correspond to sensitivity matrices with full rank. Second, the algorithm involves uncertainty quantification by using the inverse of the Fisher Information Matrix. Nominal values of parameters are used to construct synthetic data sets, and explore the effects of removing certain parameters from those to be estimated using OLS procedures. We quantify these effects in a score for a vector parameter defined using the norm of the vector of standard errors for components of estimates divided by the estimates. In some cases the method leads to reduction of the standard error for a parameter to less than 1% of the estimate
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