17,311 research outputs found
Maximum Likelihood for Matrices with Rank Constraints
Maximum likelihood estimation is a fundamental optimization problem in
statistics. We study this problem on manifolds of matrices with bounded rank.
These represent mixtures of distributions of two independent discrete random
variables. We determine the maximum likelihood degree for a range of
determinantal varieties, and we apply numerical algebraic geometry to compute
all critical points of their likelihood functions. This led to the discovery of
maximum likelihood duality between matrices of complementary ranks, a result
proved subsequently by Draisma and Rodriguez.Comment: 22 pages, 1 figur
Discretization of Parametrizable Signal Manifolds
Transformation-invariant analysis of signals often requires the computation
of the distance from a test pattern to a transformation manifold. In
particular, the estimation of the distances between a transformed query signal
and several transformation manifolds representing different classes provides
essential information for the classification of the signal. In many
applications the computation of the exact distance to the manifold is costly,
whereas an efficient practical solution is the approximation of the manifold
distance with the aid of a manifold grid. In this paper, we consider a setting
with transformation manifolds of known parameterization. We first present an
algorithm for the selection of samples from a single manifold that permits to
minimize the average error in the manifold distance estimation. Then we propose
a method for the joint discretization of multiple manifolds that represent
different signal classes, where we optimize the transformation-invariant
classification accuracy yielded by the discrete manifold representation.
Experimental results show that sampling each manifold individually by
minimizing the manifold distance estimation error outperforms baseline sampling
solutions with respect to registration and classification accuracy. Performing
an additional joint optimization on all samples improves the classification
performance further. Moreover, given a fixed total number of samples to be
selected from all manifolds, an asymmetric distribution of samples to different
manifolds depending on their geometric structures may also increase the
classification accuracy in comparison with the equal distribution of samples
Blind Demixing for Low-Latency Communication
In the next generation wireless networks, lowlatency communication is
critical to support emerging diversified applications, e.g., Tactile Internet
and Virtual Reality. In this paper, a novel blind demixing approach is
developed to reduce the channel signaling overhead, thereby supporting
low-latency communication. Specifically, we develop a low-rank approach to
recover the original information only based on a single observed vector without
any channel estimation. Unfortunately, this problem turns out to be a highly
intractable non-convex optimization problem due to the multiple non-convex
rankone constraints. To address the unique challenges, the quotient manifold
geometry of product of complex asymmetric rankone matrices is exploited by
equivalently reformulating original complex asymmetric matrices to the
Hermitian positive semidefinite matrices. We further generalize the geometric
concepts of the complex product manifolds via element-wise extension of the
geometric concepts of the individual manifolds. A scalable Riemannian
trust-region algorithm is then developed to solve the blind demixing problem
efficiently with fast convergence rates and low iteration cost. Numerical
results will demonstrate the algorithmic advantages and admirable performance
of the proposed algorithm compared with the state-of-art methods.Comment: 14 pages, accepted by IEEE Transaction on Wireless Communicatio
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