Differentially Private Regression for Discrete-Time Survival Analysis

In survival analysis, regression models are used to understand the effects of explanatory variables (e.g., age, sex, weight, etc.) to the survival probability. However, for sensitive survival data such as medical data, there are serious concerns about the privacy of individuals in the data set when medical data is used to fit the regression models. The closest work addressing such privacy concerns is the work on Cox regression which linearly projects the original data to a lower dimensional space. However, the weakness of this approach is that there is no formal privacy guarantee for such projection. In this work, we aim to propose solutions for the regression problem in survival analysis with the protection of differential privacy which is a golden standard of privacy protection in data privacy research. To this end, we extend the Output Perturbation and Objective Perturbation approaches which are originally proposed to protect differential privacy for the Empirical Risk Minimization (ERM) problems. In addition, we also propose a novel sampling approach based on the Markov Chain Monte Carlo (MCMC) method to practically guarantee differential privacy with better accuracy. We show that our proposed approaches achieve good accuracy as compared to the non-private results while guaranteeing differential privacy for individuals in the private data set.Comment: 19 pages, CIKM1

From quantum circuits to adiabatic algorithms

This paper explores several aspects of the adiabatic quantum computation model. We first show a way that directly maps any arbitrary circuit in the standard quantum computing model to an adiabatic algorithm of the same depth. Specifically, we look for a smooth time-dependent Hamiltonian whose unique ground state slowly changes from the initial state of the circuit to its final state. Since this construction requires in general an n-local Hamiltonian, we will study whether approximation is possible using previous results on ground state entanglement and perturbation theory. Finally we will point out how the adiabatic model can be relaxed in various ways to allow for 2-local partially adiabatic algorithms as well as 2-local holonomic quantum algorithms.Comment: Version accepted by and to appear in Phys. Rev.

Finite Generation of Canonical Ring by Analytic Method

In the 80th birthday conference for Professor LU Qikeng in June 2006 I gave a talk on the analytic approach to the finite generation of the canonical ring for a compact complex algebraic manifold of general type. This article is my contribution to the proceedings of that conference from my talk. In this article I give an overview of the analytic proof and focus on explaining how the analytic method handles the problem of infinite number of interminable blow-ups in the intuitive approach to prove the finite generation of the canonical ring. The proceedings of the LU Qikeng conference will appear as Issue No. 4 of Volume 51 of Science in China Series A: Mathematics (www.springer.com/math/applications/journal/11425)

Effective algebraic degeneracy

We prove that any nonconstant entire holomorphic curve from the complex line C into a projective algebraic hypersurface X = X^n in P^{n+1}(C) of arbitrary dimension n (at least 2) must be algebraically degenerate provided X is generic if its degree d = deg(X) satisfies the effective lower bound: d larger than or equal to n^{{(n+1)}^{n+5}}

Low-momentum ring diagrams of neutron matter at and near the unitary limit

We study neutron matter at and near the unitary limit using a low-momentum ring diagram approach. By slightly tuning the meson-exchange CD-Bonn potential, neutron-neutron potentials with various $^1S_0$ scattering lengths such as $a_s=-12070fm$ and $+21fm$ are constructed. Such potentials are renormalized with rigorous procedures to give the corresponding $a_s$-equivalent low-momentum potentials $V_{low-k}$, with which the low-momentum particle-particle hole-hole ring diagrams are summed up to all orders, giving the ground state energy $E_0$ of neutron matter for various scattering lengths. At the limit of $a_s\to \pm \infty$, our calculated ratio of $E_0$ to that of the non-interacting case is found remarkably close to a constant of 0.44 over a wide range of Fermi-momenta. This result reveals an universality that is well consistent with the recent experimental and Monte-Carlo computational study on low-density cold Fermi gas at the unitary limit. The overall behavior of this ratio obtained with various scattering lengths is presented and discussed. Ring-diagram results obtained with $V_{low-k}$ and those with $G$-matrix interactions are compared.Comment: 9 pages, 7 figure

Banach Analytic Sets and a Non-Linear Version of the Levi Extension Theorem

We prove a certain non-linear version of the Levi extension theorem for meromorphic functions. This means that the meromorphic function in question is supposed to be extendable along a sequence of complex curves, which are arbitrary, not necessarily straight lines. Moreover, these curves are not supposed to belong to any finite dimensional analytic family. The conclusion of our theorem is that nevertheless the function in question meromorphically extends along an (infinite dimensional) analytic family of complex curves and its domain of existence is a pinched domain filled in by this analytic family.Comment: 19 pages, This is the final version with significant corrections and improvements. To appear in Arkiv f\"or matemati

On the cohomology of pseudoeffective line bundles

The goal of this survey is to present various results concerning the cohomology of pseudoeffective line bundles on compact K{\"a}hler manifolds, and related properties of their multiplier ideal sheaves. In case the curvature is strictly positive, the prototype is the well known Nadel vanishing theorem, which is itself a generalized analytic version of the fundamental Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested here in the case where the curvature is merely semipositive in the sense of currents, and the base manifold is not necessarily projective. In this situation, one can still obtain interesting information on cohomology, e.g. a Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his PhD thesis defended in Grenoble, obtained a general K{\"a}hler vanishing theorem that depends on the concept of numerical dimension of a given pseudoeffective line bundle. The proof of these results depends in a crucial way on a general approximation result for closed (1,1)-currents, based on the use of Bergman kernels, and the related intersection theory of currents. Another important ingredient is the recent proof by Guan and Zhou of the strong openness conjecture. As an application, we discuss a structure theorem for compact K{\"a}hler threefolds without nontrivial subvarieties, following a joint work with F.Campana and M.Verbitsky. We hope that these notes will serve as a useful guide to the more detailed and more technical papers in the literature; in some cases, we provide here substantially simplified proofs and unifying viewpoints.Comment: 39 pages. This survey is a written account of a lecture given at the Abel Symposium, Trondheim, July 201