Dictionary learning with large step gradient descent for sparse representations

This is the accepted version of an article published in Lecture Notes in Computer Science Volume 7191, 2012, pp 231-238. The final publication is available at link.springer.com http://www.springerlink.com/content/l1k4514765283618

Dirac Quantization of the Pais-Uhlenbeck Fourth Order Oscillator

As a model, the Pais-Uhlenbeck fourth order oscillator with equation of motion $d^4q/dt^4+(\omega_1^2+\omega_2^2)d^2q/dt^2 +\omega_1^2\omega_2^2 q=0$ is a quantum-mechanical prototype of a field theory containing both second and fourth order derivative terms. With its dynamical degrees of freedom obeying constraints due to the presence of higher order time derivatives, the model cannot be quantized canonically. We thus quantize it using the method of Dirac constraints to construct the correct quantum-mechanical Hamiltonian for the system, and find that the Hamiltonian diagonalizes in the positive and negative norm states that are characteristic of higher derivative field theories. However, we also find that the oscillator commutation relations become singular in the $\omega_1 \to \omega_2$ limit, a limit which corresponds to a prototype of a pure fourth order theory. Thus the particle content of the $\omega_1 =\omega_2$ theory cannot be inferred from that of the $\omega_1 \neq \omega_2$ theory; and in fact in the $\omega_1 \to \omega_2$ limit we find that all of the $\omega_1 \neq \omega_2$ negative norm states move off shell, with the spectrum of asymptotic in and out states of the equal frequency theory being found to be completely devoid of states with either negative energy or negative norm. As a byproduct of our work we find a Pais-Uhlenbeck analog of the zero energy theorem of Boulware, Horowitz and Strominger, and show how in the equal frequency Pais-Uhlenbeck theory the theorem can be transformed into a positive energy theorem instead.Comment: RevTeX4, 20 pages. Final version, to appear in Phys. Rev.

Minimax bounds for sparse PCA with noisy high-dimensional data

We study the problem of estimating the leading eigenvectors of a high-dimensional population covariance matrix based on independent Gaussian observations. We establish a lower bound on the minimax risk of estimators under the $l_2$ loss, in the joint limit as dimension and sample size increase to infinity, under various models of sparsity for the population eigenvectors. The lower bound on the risk points to the existence of different regimes of sparsity of the eigenvectors. We also propose a new method for estimating the eigenvectors by a two-stage coordinate selection scheme.Comment: 1 figur

Improved constraints on non-Newtonian forces at 10 microns

Several recent theories suggest that light moduli or particles in "large" extra dimensions could mediate macroscopic forces exceeding gravitational strength at length scales below a millimeter. Such new forces can be parameterized as a Yukawa-type correction to the Newtonian potential of strength $\alpha$ relative to gravity and range $\lambda$. To extend the search for such new physics we have improved our apparatus utilizing cryogenic micro-cantilevers capable of measuring attonewton forces, which now includes a switchable magnetic force for calibration. Our most recent experimental constraints on Yukawa-type deviations from Newtonian gravity are more than three times as stringent as our previously published results, and represent the best bound in the range of 5 - 15 microns, with a 95 percent confidence exclusion of forces with $|\alpha| > 14,000$ at $\lambda$ = 10 microns.Comment: 12 pages, 9 figures, accepted for publication in PRD. Minor changes, replaced and corrected Figs 4,5,

The approach to a superconductor-to-Bose-insulator transition in disordered films

Through a detailed study of scaling near the magnetic field-tuned superconductor-to-insulator transition in strongly disordered films, we find that results for a variety of materials can be collapsed onto a single phase diagram. The data display two clear branches, one with weak disorder and an intervening metallic phase, the other with strong disorder. Along the strongly disordered branch, the resistance at the critical point approaches $R_Q = h/4e^2$ and the scaling of the resistance is consistent with quantum percolation, and therefore with the predictions of the dirty boson model.Comment: 4 pages, 4 figure

High Resolution Polar Kerr Effect Measurements of Sr2RuO4: Evidence for Broken Time Reversal Symmetry in the Superconducting State

Polar Kerr effect in the spin-triplet superconductor Sr2RuO4 was measured with high precision using a Sagnac interferometer with a zero-area Sagnac loop. We observed non-zero Kerr rotations as big as 65 nanorad appearing below Tc in large domains. Our results imply a broken time reversal symmetry state in the superconducting state of Sr2RuO4, similar to He3-A

Entanglement, discord and the power of quantum computation

We show that the ability to create entanglement is necessary for execution of bipartite quantum gates even when they are applied to unentangled states and create no entanglement. Starting with a simple example we demonstrate that to execute such a gate bi-locally the local operations and classical communications (LOCC) should be supplemented by shared entanglement. Our results point to the changes in quantum discord, which is a measure of quantumness of correlations even in the absence of entanglement, as the indicator of failure of a LOCC implementation of the gates.Comment: Published version. More results are adde

Quantum discord and local demons

Quantum discord was proposed as a measure of the "quantumness" of correlations. There are at least three different discord-like quantities, two of which determine the difference between the efficiencies of a Szilard's engine under different sets of restrictions. The three discord measures vanish simulataneosly. We introduce an easy way to test for zero discord, relate it to the Cerf-Adami conditional entropy and show that there is no relation between the discord and the local disitnguishability.Comment: 7 pages, RevTeX. Some minor changes after comments from colleagues, some references added. Similar to published versio