76 research outputs found
From Symmetry to Geometry: Tractable Nonconvex Problems
As science and engineering have become increasingly data-driven, the role of
optimization has expanded to touch almost every stage of the data analysis
pipeline, from the signal and data acquisition to modeling and prediction. The
optimization problems encountered in practice are often nonconvex. While
challenges vary from problem to problem, one common source of nonconvexity is
nonlinearity in the data or measurement model. Nonlinear models often exhibit
symmetries, creating complicated, nonconvex objective landscapes, with multiple
equivalent solutions. Nevertheless, simple methods (e.g., gradient descent)
often perform surprisingly well in practice.
The goal of this survey is to highlight a class of tractable nonconvex
problems, which can be understood through the lens of symmetries. These
problems exhibit a characteristic geometric structure: local minimizers are
symmetric copies of a single "ground truth" solution, while other critical
points occur at balanced superpositions of symmetric copies of the ground
truth, and exhibit negative curvature in directions that break the symmetry.
This structure enables efficient methods to obtain global minimizers. We
discuss examples of this phenomenon arising from a wide range of problems in
imaging, signal processing, and data analysis. We highlight the key role of
symmetry in shaping the objective landscape and discuss the different roles of
rotational and discrete symmetries. This area is rich with observed phenomena
and open problems; we close by highlighting directions for future research.Comment: review paper submitted to SIAM Review, 34 pages, 10 figure
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
A Method for Finding Structured Sparse Solutions to Non-negative Least Squares Problems with Applications
Demixing problems in many areas such as hyperspectral imaging and
differential optical absorption spectroscopy (DOAS) often require finding
sparse nonnegative linear combinations of dictionary elements that match
observed data. We show how aspects of these problems, such as misalignment of
DOAS references and uncertainty in hyperspectral endmembers, can be modeled by
expanding the dictionary with grouped elements and imposing a structured
sparsity assumption that the combinations within each group should be sparse or
even 1-sparse. If the dictionary is highly coherent, it is difficult to obtain
good solutions using convex or greedy methods, such as non-negative least
squares (NNLS) or orthogonal matching pursuit. We use penalties related to the
Hoyer measure, which is the ratio of the and norms, as sparsity
penalties to be added to the objective in NNLS-type models. For solving the
resulting nonconvex models, we propose a scaled gradient projection algorithm
that requires solving a sequence of strongly convex quadratic programs. We
discuss its close connections to convex splitting methods and difference of
convex programming. We also present promising numerical results for example
DOAS analysis and hyperspectral demixing problems.Comment: 38 pages, 14 figure
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