62 research outputs found
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
Enrichment of Turbulence Field Using Wavelets
This thesis is composed of two parts. The first part presents a new turbulence generation method based on stochastic wavelets and tests various properties of the generated turbulence field in both the homogeneous and inhomogeneous cases. Numerical results indicate that turbulence fields can be generated with much smaller bases in comparison to synthetic Fourier methods while maintaining comparable accuracy. Adaptive generation of inhomogeneous turbulence is achieved by a scale reduction algorithm, which greatly reduces the computational cost and practically introduces no error. The generating formula proposed in this research could be adjusted to generate fully inhomogeneous and anisotropic turbulence with given RANS data under a divergence-free constraint, which was not achieved previously in similar research. Numerical examples show that the generated homogeneous and inhomogeneous turbulence are in good agreement with the input data and theoretical results. The second part presents a framework of solving turbulence deconvolution problems using optimization techniques on Riemannian manifolds. A filtered velocity field was deconvoluted without any information of the filter. The deconvolution results shows high accuracy compared with the original velocity field. The computational cost of the optimization problem was largely reduced using wavelet representation while still maintaining high accuracy. Utilization of divergence-free wavelets ensures the incompressible property of deconvolution results, which was barely achieved in previous research
Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective
We study a general matrix optimization problem with a fixed-rank positive
semidefinite (PSD) constraint. We perform the Burer-Monteiro factorization and
consider a particular Riemannian quotient geometry in a search space that has a
total space equipped with the Euclidean metric. When the original objective f
satisfies standard restricted strong convexity and smoothness properties, we
characterize the global landscape of the factorized objective under the
Riemannian quotient geometry. We show the entire search space can be divided
into three regions: (R1) the region near the target parameter of interest,
where the factorized objective is geodesically strongly convex and smooth; (R2)
the region containing neighborhoods of all strict saddle points; (R3) the
remaining regions, where the factorized objective has a large gradient. To our
best knowledge, this is the first global landscape analysis of the
Burer-Monteiro factorized objective under the Riemannian quotient geometry. Our
results provide a fully geometric explanation for the superior performance of
vanilla gradient descent under the Burer-Monteiro factorization. When f
satisfies a weaker restricted strict convexity property, we show there exists a
neighborhood near local minimizers such that the factorized objective is
geodesically convex. To prove our results we provide a comprehensive landscape
analysis of a matrix factorization problem with a least squares objective,
which serves as a critical bridge. Our conclusions are also based on a result
of independent interest stating that the geodesic ball centered at Y with a
radius 1/3 of the least singular value of Y is a geodesically convex set under
the Riemannian quotient geometry, which as a corollary, also implies a
quantitative bound of the convexity radius in the Bures-Wasserstein space. The
convexity radius obtained is sharp up to constants.Comment: The abstract is shortened to meet the arXiv submission requiremen
Approximate matrix and tensor diagonalization by unitary transformations: convergence of Jacobi-type algorithms
We propose a gradient-based Jacobi algorithm for a class of maximization
problems on the unitary group, with a focus on approximate diagonalization of
complex matrices and tensors by unitary transformations. We provide weak
convergence results, and prove local linear convergence of this algorithm.The
convergence results also apply to the case of real-valued tensors
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