22,974 research outputs found
Faster Inversion and Other Black Box Matrix Computations Using Efficient Block Projections
Block projections have been used, in [Eberly et al. 2006], to obtain an
efficient algorithm to find solutions for sparse systems of linear equations. A
bound of softO(n^(2.5)) machine operations is obtained assuming that the input
matrix can be multiplied by a vector with constant-sized entries in softO(n)
machine operations. Unfortunately, the correctness of this algorithm depends on
the existence of efficient block projections, and this has been conjectured. In
this paper we establish the correctness of the algorithm from [Eberly et al.
2006] by proving the existence of efficient block projections over sufficiently
large fields. We demonstrate the usefulness of these projections by deriving
improved bounds for the cost of several matrix problems, considering, in
particular, ``sparse'' matrices that can be be multiplied by a vector using
softO(n) field operations. We show how to compute the inverse of a sparse
matrix over a field F using an expected number of softO(n^(2.27)) operations in
F. A basis for the null space of a sparse matrix, and a certification of its
rank, are obtained at the same cost. An application to Kaltofen and Villard's
Baby-Steps/Giant-Steps algorithms for the determinant and Smith Form of an
integer matrix yields algorithms requiring softO(n^(2.66)) machine operations.
The derived algorithms are all probabilistic of the Las Vegas type
Sparse Recovery from Combined Fusion Frame Measurements
Sparse representations have emerged as a powerful tool in signal and
information processing, culminated by the success of new acquisition and
processing techniques such as Compressed Sensing (CS). Fusion frames are very
rich new signal representation methods that use collections of subspaces
instead of vectors to represent signals. This work combines these exciting
fields to introduce a new sparsity model for fusion frames. Signals that are
sparse under the new model can be compressively sampled and uniquely
reconstructed in ways similar to sparse signals using standard CS. The
combination provides a promising new set of mathematical tools and signal
models useful in a variety of applications. With the new model, a sparse signal
has energy in very few of the subspaces of the fusion frame, although it does
not need to be sparse within each of the subspaces it occupies. This sparsity
model is captured using a mixed l1/l2 norm for fusion frames.
A signal sparse in a fusion frame can be sampled using very few random
projections and exactly reconstructed using a convex optimization that
minimizes this mixed l1/l2 norm. The provided sampling conditions generalize
coherence and RIP conditions used in standard CS theory. It is demonstrated
that they are sufficient to guarantee sparse recovery of any signal sparse in
our model. Moreover, a probabilistic analysis is provided using a stochastic
model on the sparse signal that shows that under very mild conditions the
probability of recovery failure decays exponentially with increasing dimension
of the subspaces
Sparse Estimation with the Swept Approximated Message-Passing Algorithm
Approximate Message Passing (AMP) has been shown to be a superior method for
inference problems, such as the recovery of signals from sets of noisy,
lower-dimensionality measurements, both in terms of reconstruction accuracy and
in computational efficiency. However, AMP suffers from serious convergence
issues in contexts that do not exactly match its assumptions. We propose a new
approach to stabilizing AMP in these contexts by applying AMP updates to
individual coefficients rather than in parallel. Our results show that this
change to the AMP iteration can provide theoretically expected, but hitherto
unobtainable, performance for problems on which the standard AMP iteration
diverges. Additionally, we find that the computational costs of this swept
coefficient update scheme is not unduly burdensome, allowing it to be applied
efficiently to signals of large dimensionality.Comment: 11 pages, 3 figures, implementation available at
https://github.com/eric-tramel/SwAMP-Dem
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