Blind Two-Dimensional Super-Resolution and Its Performance Guarantee

In this work, we study the problem of identifying the parameters of a linear system from its response to multiple unknown input waveforms. We assume that the system response, which is the only given information, is a scaled superposition of time-delayed and frequency-shifted versions of the unknown waveforms. Such kind of problem is severely ill-posed and does not yield a unique solution without introducing further constraints. To fully characterize the linear system, we assume that the unknown waveforms lie in a common known low-dimensional subspace that satisfies certain randomness and concentration properties. Then, we develop a blind two-dimensional (2D) super-resolution framework that applies to a large number of applications such as radar imaging, image restoration, and indoor source localization. In this framework, we show that under a minimum separation condition between the time-frequency shifts, all the unknowns that characterize the linear system can be recovered precisely and with very high probability provided that a lower bound on the total number of the observed samples is satisfied. The proposed framework is based on 2D atomic norm minimization problem which is shown to be reformulated and solved efficiently via semidefinite programming. Simulation results that confirm the theoretical findings of the paper are provided

Convergence of Gradient Descent for Low-Rank Matrix Approximation

This paper provides a proof of global convergence of gradient search for low-rank matrix approximation. Such approximations have recently been of interest for large-scale problems, as well as for dictionary learning for sparse signal representations and matrix completion. The proof is based on the interpretation of the problem as an optimization on the Grassmann manifold and Fubiny-Study distance on this space

Wilson-Loop Characterization of Inversion-Symmetric Topological Insulators

The ground state of translationally-invariant insulators comprise bands which can assume topologically distinct structures. There are few known examples where this distinction is enforced by a point-group symmetry alone. In this paper we show that 1D and 2D insulators with the simplest point-group symmetry - inversion - have a $Z^{\geq}$ classification. In 2D, we identify a relative winding number that is solely protected by inversion symmetry. By analysis of Berry phases, we show that this invariant has similarities with the first Chern class (of time-reversal breaking insulators), but is more closely analogous to the $Z_2$ invariant (of time-reversal invariant insulators). Implications of our work are discussed in holonomy, the geometric-phase theory of polarization, the theory of maximally-localized Wannier functions, and in the entanglement spectrum.Comment: The updated version is accepted in Physical Review

Pseudogap and incommensurate magnetic fluctuations in YBa_2Cu_3O_{6. 6}

Unpolarized inelastic neutron scattering is used to study the temperature and wave vector dependence of the dynamical magnetic susceptibility, $\chi''(q,\omega)$, of a well characterized single crystal $YBa_2Cu_3O_{6.6}$ ($T_c=62.7$ K). We find that a pseudogap opens in the spin fluctuation spectrum at temperatures well above $T_c$. We speculate that the appearance of the low frequency incommensurate fluctuations is associated with the opening of the pseudogap. To within the error of the measurements, a gap in the spin fluctuation spectrum is found in the superconducting state.Comment: 6 pages, 3 ps figs, Proceedings of ICNS, Physica B, to be publishe

Observation of Magnetic Moments in the Superconducting State of YBa2_2Cu3_3O6.6_{6.6}

Neutron Scattering measurements for YBa$_2$Cu$_3$O$_{6.6}$ have identified small magnetic moments that increase in strength as the temperature is reduced below $T^\ast$ and further increase below $T_c$. An analysis of the data shows the moments are antiferromagnetic between the Cu-O planes with a correlation length of longer than 195 \AA in the $a$-$b$ plane and about 35 \AA along the c-axis. The origin of the moments is unknown, and their properties are discusssed both in terms of Cu spin magnetism and orbital bond currents.Comment: 9 pages, and 4 figure