Near-Optimal Algorithms for Differentially-Private Principal Components

Principal components analysis (PCA) is a standard tool for identifying good low-dimensional approximations to data in high dimension. Many data sets of interest contain private or sensitive information about individuals. Algorithms which operate on such data should be sensitive to the privacy risks in publishing their outputs. Differential privacy is a framework for developing tradeoffs between privacy and the utility of these outputs. In this paper we investigate the theory and empirical performance of differentially private approximations to PCA and propose a new method which explicitly optimizes the utility of the output. We show that the sample complexity of the proposed method differs from the existing procedure in the scaling with the data dimension, and that our method is nearly optimal in terms of this scaling. We furthermore illustrate our results, showing that on real data there is a large performance gap between the existing method and our method.Comment: 37 pages, 8 figures; final version to appear in the Journal of Machine Learning Research, preliminary version was at NIPS 201

Evidence for two spin-glass transitions with magnetoelastic and magnetoelectric couplings in the multiferroic (Bi1−x_{1-x}Bax_x)(Fe1−x_{1-x}Tix_x)O3_3 system

For disordered Heisenberg systems with small single ion anisotropy, two spin glass transitions below the long range ordered phase transition temperature has been predicted theoretically for compositions close to the percolation threshold. Experimental verification of these predictions is still controversial for conventional spin glasses. We show that multiferroic spin glass systems can provide a unique platform for verifying these theoretical predictions via a study of change in magnetoelastic and magnetoelectric couplings, obtained from an analysis of diffraction data, at the spin glass transition temperatures. Results of macroscopic and microscopic (x-ray and neutron scattering) measurements are presented on disordered BiFeO3, a canonical Heisenberg system with small single ion anisotropy, which reveal appearance of two spin glass phases SG1 and SG2 in coexistence with the LRO phase below the A-T and G-T lines. It is shown that the temperature dependence of the integrated intensity of the antiferromagnetic peak shows dips with respect to the Brillouin function behaviour around the SG1 and SG2 transition temperatures. The ferroelectric polarisation changes significantly at the two spin glass transition temperatures. These results, obtained using microscopic techniques, clearly demonstrate that the SG1 and SG2 transitions occur on the same magnetic sublattice and are intrinsic to the system. We also construct a phase diagram showing all the magnetic phases in BF-xBT system. While our results on the two spin glass transitions support the theoretical predictions, it also raises several open questions which need to be addressed by revisiting the existing theories of spin glass transitions by taking into account the effect of magnetoelastic and magnetoelectric couplings as well as electromagnons.Comment: 59 pages 21 figure

Decay of Spin-One Particle into Two Photons in Presence of Uniform External Magnetic Field

Yang's theorem states that an initial J=1 state cannot decay into two photons. Because of this result some reactions relating to elementary particles or atomic transitions can be ruled out. The theorem is not valid in the presence of background electric or magnetic fields. In this work we show that the decay of a J=1 particle into two photons is permitted by Bose symmetry and rotational invariance when the background of the decay process is not pure vacuum but contains an external classical magnetic/electric field. We also discuss constraints on these amplitudes from {\bf CP} invariance.Comment: Tex fil

Disclination-mediated thermo-optical response in nematic glass sheets

Nematic solids respond strongly to changes in ambient heat or light, significantly differently parallel and perpendicular to the director. This phenomenon is well characterized for uniform director fields, but not for defect textures. We analyze the elastic ground states of a nematic glass in the membrane approximation as a function of temperature for some disclination defects with an eye towards reversibly inducing three-dimensional shapes from flat sheets of material, at the nano-scale all the way to macroscopic objects, including non-developable surfaces. The latter offers a new paradigm to actuation via switchable stretch in thin systems.Comment: Specific results for spiral defects now added. References to Witten, Mahadevan and Ben Amar now added

Stability of Lending Rate Stickiness: A Case Study of India

The paper postulates that in an environment of continuous financial reforms, the lending rate stickiness in an economy could be changing over the period. The financial reforms (of which deregulation of interest rates formed a major part) during the 1990s and the early 2000s and the changing role attributed to different policy rates during the reforms make India an interesting case study. The paper finds evidence of diminishing lending rate stickiness in case of India. During the major part of the study, Indian policymakers used the discount rate for policy signaling. The paper observes that as a result, the long-term rates like the lending rates did not react sufficiently to the changes in the short-term rates (e.g., repo rate) in this period unless the discount rate was also changed. Such behavior changed when policymakers started to use short-term rates like repo rates for policy signaling. Results in this paper suggest that when the impacts are added together, a change of 100 basis points in all policy rates towards the end of the reference period could change the lending rate in India almost by similar magnitude. These findings help to reconcile some of the contrasting findings on lending rate stickiness in case of India. Among possible factors still responsible for lending rate stickiness, the study identifies inelastic credit demand in India as an important factor. From policymaking perspective, however, it is postulated that as demand for personal and housing loans in India are likely to increase in future due to demographic factor, it is likely that such increase could tend to increase inflexibility in loan rates.Lending Rate Stickiness, Discount Rate Addiction, Monetary Policy Transmission

The Linearly Independent Non Orthogonal yet Energy Preserving (LINOEP) vectors

It is well known that, in any inner product space, a set of linearly independent (LI) vectors can be transformed to a set of orthogonal vectors, spanning the same space, by the Gram-Schmidt Orthogonalization Method (GSOM). In this paper, we propose a transformation from a set of LI vectors to a set of LI non orthogonal yet energy (square of the norm) preserving (LINOEP) vectors in an inner product space and we refer it as LINOEP method. We also show that there are various solutions to preserve the square of the norm.Comment: 6 pages, 2 figure

Magnonic spin-transfer torque MRAM with low power, high speed, and error-free switching

A new class of spin-transfer torque magnetic random access memory (STT-MRAM) is discussed, in which writing is achieved using thermally initiated magnonic current pulses as an alternative to conventional electric current pulses. The magnonic pulses are used to destabilize the magnetic free layer from its initial direction, and are followed immediately by a bipolar electric current exerting conventional spin-transfer torque on the free layer. The combination of thermal and electric currents greatly reduces switching errors, and simultaneously reduces the electric switching current density by more than an order of magnitude as compared to conventional STT-MRAM. The energy efficiency of several possible electro-thermal circuit designs have been analyzed numerically. As compared to STT-MRAM with perpendicular magnetic anisotropy, magnonic STT-MRAM reduces the overall switching energy by almost 80%. Furthermore, the lower electric current density allows the use of thicker tunnel barriers, which should result in higher tunneling magneto-resistance and improved tunnel barrier reliability. The combination of lower power, improved reliability, higher integration density, and larger read margin make magnonic STT-MRAM a promising choice for future non-volatile storage.Comment: 9 Pages, 11 Figure