Input Sparsity and Hardness for Robust Subspace Approximation

In the subspace approximation problem, we seek a k-dimensional subspace F of R^d that minimizes the sum of p-th powers of Euclidean distances to a given set of n points a_1, ..., a_n in R^d, for p >= 1. More generally than minimizing sum_i dist(a_i,F)^p,we may wish to minimize sum_i M(dist(a_i,F)) for some loss function M(), for example, M-Estimators, which include the Huber and Tukey loss functions. Such subspaces provide alternatives to the singular value decomposition (SVD), which is the p=2 case, finding such an F that minimizes the sum of squares of distances. For p in [1,2), and for typical M-Estimators, the minimizing $F$ gives a solution that is more robust to outliers than that provided by the SVD. We give several algorithmic and hardness results for these robust subspace approximation problems. We think of the n points as forming an n x d matrix A, and letting nnz(A) denote the number of non-zero entries of A. Our results hold for p in [1,2). We use poly(n) to denote n^{O(1)} as n -> infty. We obtain: (1) For minimizing sum_i dist(a_i,F)^p, we give an algorithm running in O(nnz(A) + (n+d)poly(k/eps) + exp(poly(k/eps))), (2) we show that the problem of minimizing sum_i dist(a_i, F)^p is NP-hard, even to output a (1+1/poly(d))-approximation, answering a question of Kannan and Vempala, and complementing prior results which held for p >2, (3) For loss functions for a wide class of M-Estimators, we give a problem-size reduction: for a parameter K=(log n)^{O(log k)}, our reduction takes O(nnz(A) log n + (n+d) poly(K/eps)) time to reduce the problem to a constrained version involving matrices whose dimensions are poly(K eps^{-1} log n). We also give bicriteria solutions, (4) Our techniques lead to the first O(nnz(A) + poly(d/eps)) time algorithms for (1+eps)-approximate regression for a wide class of convex M-Estimators.Comment: paper appeared in FOCS, 201

Majorana braiding with thermal noise

We investigate the self-correcting properties of a network of Majorana wires, in the form of a trijunction, in contact with a parity-preserving thermal environment. As opposed to the case where Majorana bound states (MBSs) are immobile, braiding MBSs within a trijunction introduces dangerous error processes that we identify. Such errors prevent the lifetime of the memory from increasing with the size of the system. We confirm our predictions with Monte Carlo simulations. Our findings put a restriction on the degree of self-correction of this specific quantum computing architecture.Comment: 6 pages, 3 figures, long version: arXiv: 1507.0089

Stochastic Master Equation Analysis of Optimized Three-Qubit Nondemolition Parity Measurement

We analyze a direct parity measurement of the state of three superconducting qubits in circuit quantum electrodynamics. The parity is inferred from a homodyne measurement of the reflected/transmitted microwave radiation and the measurement is direct in the sense that the parity is measured without the need for any quantum circuit operations or for ancilla qubits. Qubits are coupled to two resonant cavity modes, allowing the steady state of the emitted radiation to satisfy the necessary conditions to act as a pointer state for the parity. However, the transient dynamics violates these conditions and we analyze this detrimental effect and show that it can be overcome in the limit of weak measurement signal. Our analysis shows that, with a moderate degree of post-selection, it is possible to achieve post-measurement states with fidelity of order 95%. We believe that this type of measurement could serve as a benchmark for future error-correction protocols in a scalable architecture

No well-defined remnant Fermi surface in Sr2CuO2Cl2

In angle-resolved photoelectron spectra of the antiferromagnetic insulators Ca2CuO2Cl2 and Sr2CuO2Cl2 a sharp drop of the spectral intensity of the lowest-lying band is observed along a line in k space equivalent to the Fermi surface of the optimally doped high-temperature superconductors. This was interpreted as a signature of the existence of a remnant Fermi surface in the insulating phase of the high-temperature superconductors. In this paper it is shown that the drop of the spectral intensity is not related to the spectral function but is a consequence of the electron-photon matrix elementComment: 4 pages, 3 figure

Galactic R Coronae Borealis Stars: The C-2 Swan Bands, The Carbon Problem, And The C-12/C-13 Ratio

Observed spectra of R Coronae Borealis (RCB) and hydrogen-deficient carbon (HdC) stars are analyzed by synthesizing the C-2 Swan bands (1, 0), (0, 0), and (0, 1) using our detailed line list and the Uppsala model atmospheres. The (0, 1) and (0, 0) C-2 bands are used to derive the C-12 abundance, and the (1, 0) (CC)-C-12-C-13 band to determine the C-12/C-13 ratios. The carbon abundance derived from the C-2 Swan bands is about the same for the adopted models constructed with different carbon abundances over the range 8.5 (C/He = 0.1%) to 10.5 (C/He = 10%). Carbon abundances derived from C I lines are about a factor of four lower than the carbon abundance of the adopted model atmosphere over the same C/He interval, as reported by Asplund et al., who dubbed the mismatch between adopted and derived C abundance as the "carbon problem." In principle, the carbon abundances obtained from C-2 Swan bands and that assumed for the model atmosphere can be equated for a particular choice of C/He that varies from star to star. Then, the carbon problem for C-2 bands is eliminated. However, such C/He ratios are in general less than those of the extreme helium stars, the seemingly natural relatives to the RCB and HdC stars. A more likely solution to the C-2 carbon problem may lie in a modification of the model atmosphere's temperature structure. The derived carbon abundances and the C-12/C-13 ratios are discussed in light of the double degenerate and the final flash scenarios.Robert A. Welch Foundation of Houston, TX F-634McDonald Observator

Heartbeat Anomaly Detection using Adversarial Oversampling

Cardiovascular diseases are one of the most common causes of death in the world. Prevention, knowledge of previous cases in the family, and early detection is the best strategy to reduce this fact. Different machine learning approaches to automatic diagnostic are being proposed to this task. As in most health problems, the imbalance between examples and classes is predominant in this problem and affects the performance of the automated solution. In this paper, we address the classification of heartbeats images in different cardiovascular diseases. We propose a two-dimensional Convolutional Neural Network for classification after using a InfoGAN architecture for generating synthetic images to unbalanced classes. We call this proposal Adversarial Oversampling and compare it with the classical oversampling methods as SMOTE, ADASYN, and RandomOversampling. The results show that the proposed approach improves the classifier performance for the minority classes without harming the performance in the balanced classes

Quantum channels in nonlinear optical processes

Quantum electrodynamics furnishes a new type of representation for the characterisation of nonlinear optical processes. The treatment elicits the detailed role and interplay of specific quantum channels, information that is not afforded by other methods. Following an illustrative application to the case of Rayleigh scattering, the method is applied to second and third harmonic generation. Derivations are given of parameters that quantify the various quantum channels and their interferences; the results are illustrated graphically. With given examples, it is shown in some systems that optical nonlinearity owes its origin to an isolated channel, or a small group of channels. © 2009 World Scientific Publishing Company