5,573 research outputs found
A fast semi-direct least squares algorithm for hierarchically block separable matrices
We present a fast algorithm for linear least squares problems governed by
hierarchically block separable (HBS) matrices. Such matrices are generally
dense but data-sparse and can describe many important operators including those
derived from asymptotically smooth radial kernels that are not too oscillatory.
The algorithm is based on a recursive skeletonization procedure that exposes
this sparsity and solves the dense least squares problem as a larger,
equality-constrained, sparse one. It relies on a sparse QR factorization
coupled with iterative weighted least squares methods. In essence, our scheme
consists of a direct component, comprised of matrix compression and
factorization, followed by an iterative component to enforce certain equality
constraints. At most two iterations are typically required for problems that
are not too ill-conditioned. For an HBS matrix with
having bounded off-diagonal block rank, the algorithm has optimal complexity. If the rank increases with the spatial dimension as is
common for operators that are singular at the origin, then this becomes
in 1D, in 2D, and
in 3D. We illustrate the performance of the method on
both over- and underdetermined systems in a variety of settings, with an
emphasis on radial basis function approximation and efficient updating and
downdating.Comment: 24 pages, 8 figures, 6 tables; to appear in SIAM J. Matrix Anal. App
Identifiability for Blind Source Separation of Multiple Finite Alphabet Linear Mixtures
We give under weak assumptions a complete combinatorial characterization of
identifiability for linear mixtures of finite alphabet sources, with unknown
mixing weights and unknown source signals, but known alphabet. This is based on
a detailed treatment of the case of a single linear mixture. Notably, our
identifiability analysis applies also to the case of unknown number of sources.
We provide sufficient and necessary conditions for identifiability and give a
simple sufficient criterion together with an explicit construction to determine
the weights and the source signals for deterministic data by taking advantage
of the hierarchical structure within the possible mixture values. We show that
the probability of identifiability is related to the distribution of a hitting
time and converges exponentially fast to one when the underlying sources come
from a discrete Markov process. Finally, we explore our theoretical results in
a simulation study. Our work extends and clarifies the scope of scenarios for
which blind source separation becomes meaningful
- …