30,957 research outputs found
Scalable Subspace Methods for Derivative-Free Nonlinear Least-Squares Optimization
We introduce a general framework for large-scale model-based derivative-free
optimization based on iterative minimization within random subspaces. We
present a probabilistic worst-case complexity analysis for our method, where in
particular we prove high-probability bounds on the number of iterations before
a given optimality is achieved. This framework is specialized to nonlinear
least-squares problems, with a model-based framework based on the Gauss-Newton
method. This method achieves scalability by constructing local linear
interpolation models to approximate the Jacobian, and computes new steps at
each iteration in a subspace with user-determined dimension. We then describe a
practical implementation of this framework, which we call DFBGN. We outline
efficient techniques for selecting the interpolation points and search
subspace, yielding an implementation that has a low per-iteration linear
algebra cost (linear in the problem dimension) while also achieving fast
objective decrease as measured by evaluations. Extensive numerical results
demonstrate that DFBGN has improved scalability, yielding strong performance on
large-scale nonlinear least-squares problems
Bayesian uncertainty quantification in linear models for diffusion MRI
Diffusion MRI (dMRI) is a valuable tool in the assessment of tissue
microstructure. By fitting a model to the dMRI signal it is possible to derive
various quantitative features. Several of the most popular dMRI signal models
are expansions in an appropriately chosen basis, where the coefficients are
determined using some variation of least-squares. However, such approaches lack
any notion of uncertainty, which could be valuable in e.g. group analyses. In
this work, we use a probabilistic interpretation of linear least-squares
methods to recast popular dMRI models as Bayesian ones. This makes it possible
to quantify the uncertainty of any derived quantity. In particular, for
quantities that are affine functions of the coefficients, the posterior
distribution can be expressed in closed-form. We simulated measurements from
single- and double-tensor models where the correct values of several quantities
are known, to validate that the theoretically derived quantiles agree with
those observed empirically. We included results from residual bootstrap for
comparison and found good agreement. The validation employed several different
models: Diffusion Tensor Imaging (DTI), Mean Apparent Propagator MRI (MAP-MRI)
and Constrained Spherical Deconvolution (CSD). We also used in vivo data to
visualize maps of quantitative features and corresponding uncertainties, and to
show how our approach can be used in a group analysis to downweight subjects
with high uncertainty. In summary, we convert successful linear models for dMRI
signal estimation to probabilistic models, capable of accurate uncertainty
quantification.Comment: Added results from a group analysis and a comparison with residual
bootstra
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