8,234 research outputs found
Computational Complexity versus Statistical Performance on Sparse Recovery Problems
We show that several classical quantities controlling compressed sensing
performance directly match classical parameters controlling algorithmic
complexity. We first describe linearly convergent restart schemes on
first-order methods solving a broad range of compressed sensing problems, where
sharpness at the optimum controls convergence speed. We show that for sparse
recovery problems, this sharpness can be written as a condition number, given
by the ratio between true signal sparsity and the largest signal size that can
be recovered by the observation matrix. In a similar vein, Renegar's condition
number is a data-driven complexity measure for convex programs, generalizing
classical condition numbers for linear systems. We show that for a broad class
of compressed sensing problems, the worst case value of this algorithmic
complexity measure taken over all signals matches the restricted singular value
of the observation matrix which controls robust recovery performance. Overall,
this means in both cases that, in compressed sensing problems, a single
parameter directly controls both computational complexity and recovery
performance. Numerical experiments illustrate these points using several
classical algorithms.Comment: Final version, to appear in information and Inferenc
PEAR: PEriodic And fixed Rank separation for fast fMRI
In functional MRI (fMRI), faster acquisition via undersampling of data can
improve the spatial-temporal resolution trade-off and increase statistical
robustness through increased degrees-of-freedom. High quality reconstruction of
fMRI data from undersampled measurements requires proper modeling of the data.
We present an fMRI reconstruction approach based on modeling the fMRI signal as
a sum of periodic and fixed rank components, for improved reconstruction from
undersampled measurements. We decompose the fMRI signal into a component which
a has fixed rank and a component consisting of a sum of periodic signals which
is sparse in the temporal Fourier domain. Data reconstruction is performed by
solving a constrained problem that enforces a fixed, moderate rank on one of
the components, and a limited number of temporal frequencies on the other. Our
approach is coined PEAR - PEriodic And fixed Rank separation for fast fMRI.
Experimental results include purely synthetic simulation, a simulation with
real timecourses and retrospective undersampling of a real fMRI dataset.
Evaluation was performed both quantitatively and visually versus ground truth,
comparing PEAR to two additional recent methods for fMRI reconstruction from
undersampled measurements. Results demonstrate PEAR's improvement in estimating
the timecourses and activation maps versus the methods compared against at
acceleration ratios of R=8,16 (for simulated data) and R=6.66,10 (for real
data). PEAR results in reconstruction with higher fidelity than when using a
fixed-rank based model or a conventional Low-rank+Sparse algorithm. We have
shown that splitting the functional information between the components leads to
better modeling of fMRI, over state-of-the-art methods
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