Deformed Kazhdan-Lusztig elements and Macdonald polynomials

We introduce deformations of Kazhdan-Lusztig elements and specialised nonsymmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of a maximal parabolic subalgebra of the Hecke algebra. We give explicit integral formula for these polynomials, and explicitly describe the transition matrices between classes of polynomials. We further develop a combinatorial interpretation of homogeneous evaluations using an expansion in terms of Schubert polynomials in the deformation parameters.Comment: major revision, 29 pages, 22 eps figure

Tracking the effects of interactions on spinons in gapless Heisenberg chains

We consider the effects of interactions on spinon excitations in Heisenberg spin-1/2 chains. We compute the exact two-spinon part of the longitudinal structure factor of the infinite chain in zero field for all values of anisotropy in the gapless antiferromagnetic regime, via an exact algebraic approach. Our results allow us to quantitatively describe the behavior of these fundamental excitations throughout the observable continuum, for cases ranging from free to fully coupled chains, thereby explicitly mapping the effects of "turning on the interactions" in a strongly correlated system

Efficient, noise-tolerant, and private learning via boosting

We introduce a simple framework for designing private boosting algorithms. We give natural conditions under which these algorithms are differentially private, efficient, and noise-tolerant PAC learners. To demonstrate our framework, we use it to construct noise-tolerant and private PAC learners for large-margin halfspaces whose sample complexity does not depend on the dimension. We give two sample complexity bounds for our large-margin halfspace learner. One bound is based only on differential privacy, and uses this guarantee as an asset for ensuring generalization. This first bound illustrates a general methodology for obtaining PAC learners from privacy, which may be of independent interest. The second bound uses standard techniques from the theory of large-margin classification (the fat-shattering dimension) to match the best known sample complexity for differentially private learning of large-margin halfspaces, while additionally tolerating random label noise.https://arxiv.org/pdf/2002.01100.pd

Securing Approximate Homomorphic Encryption Using Differential Privacy

Recent work of Li and Micciancio (Eurocrypt 2021) has shown that the traditional formulation of indistinguishability under chosen plaintext attack (INDCPA) is not adequate to capture the security of approximate homomorphic encryption against passive adversaries, and identified a stronger INDCPA^D security definition (INDCPA with decryption oracles) as the appropriate security target for approximate encryption schemes. We show how to any approximate homomorphic encryption scheme achieving the weak INDCPA security definition, into one which is provably INDCPA^D secure, offering strong guarantees against realistic passive attacks. The method works by post-processing the output of the decryption function with a mechanism satisfying an appropriate notion of differential privacy (DP), adding an amount of noise tailored to the worst-case error growth of the homomorphic computation. We apply these results to the approximate homomorphic encryption scheme of Cheon, Kim, Kim, and Song (CKKS, Asiacrypt 2017), proving that adding Gaussian noise to the output of CKKS decryption suffices to achieve INDCPA^D security. We precisely quantify how much Gaussian noise must be added by proving nearly matching upper and lower bounds, showing that one cannot hope to significantly reduce the amount of noise added in this post-processing step. As an additional contribution, we present and use a finer-grained definition of bit security that distinguishes between a computational security parameter (c) and a statistical one (s). Based on our upper and lower bounds, we propose parameters for the counter-measures recently adopted by open-source libraries implementing CKKS. Lastly, we investigate the plausible claim that smaller DP noise parameters might suffice to achieve INDCPA^D-security for schemes supporting more accurate (dynamic, key dependent) estimates of ciphertext noise during decryption. Perhaps surprisingly, we show that this claim is false, and that DP mechanisms with noise parameters tailored to the error present in a given ciphertext, rather than worst-case error, are vulnerable to INDCPA^D attacks

Stability is Stable: Connections between Replicability, Privacy, and Adaptive Generalization

The notion of replicable algorithms was introduced in Impagliazzo et al. [STOC '22] to describe randomized algorithms that are stable under the resampling of their inputs. More precisely, a replicable algorithm gives the same output with high probability when its randomness is fixed and it is run on a new i.i.d. sample drawn from the same distribution. Using replicable algorithms for data analysis can facilitate the verification of published results by ensuring that the results of an analysis will be the same with high probability, even when that analysis is performed on a new data set. In this work, we establish new connections and separations between replicability and standard notions of algorithmic stability. In particular, we give sample-efficient algorithmic reductions between perfect generalization, approximate differential privacy, and replicability for a broad class of statistical problems. Conversely, we show any such equivalence must break down computationally: there exist statistical problems that are easy under differential privacy, but that cannot be solved replicably without breaking public-key cryptography. Furthermore, these results are tight: our reductions are statistically optimal, and we show that any computational separation between DP and replicability must imply the existence of one-way functions. Our statistical reductions give a new algorithmic framework for translating between notions of stability, which we instantiate to answer several open questions in replicability and privacy. This includes giving sample-efficient replicable algorithms for various PAC learning, distribution estimation, and distribution testing problems, algorithmic amplification of $\delta$ in approximate DP, conversions from item-level to user-level privacy, and the existence of private agnostic-to-realizable learning reductions under structured distributions.Comment: STOC 2023, minor typos fixe

Effect of skin tone on the accuracy of the estimation of arterial oxygen saturation by pulse oximetry: a systematic review

BACKGROUND: Pulse oximetry-derived oxygen saturation (SpO2) is an estimate of true arterial oxygen saturation (SaO2). The aim of this review was to evaluate available evidence determining the effect of skin tone on the ability of pulse oximeters to accurately estimate SaO2. METHODS: Published literature was screened to identify clinical and non-clinical studies enrolling adults and children when SpO2 was compared with a paired co-oximetry SaO2 value. We searched literature databases from their inception to March 20, 2023. Risk of bias (RoB) was assessed using the QUADAS-2 tool. Certainty of assessment was evaluated using the GRADE tool. RESULTS: Forty-four studies were selected reporting on at least 222 644 participants (6121 of whom were children) and 733 722 paired SpO2-SaO2 measurements. Methodologies included laboratory studies, prospective clinical, and retrospective clinical studies. A high RoB was detected in 64% of studies and there was considerable heterogeneity in study design, data analysis, and reporting metrics. Only 11 (25%) studies measured skin tone in 2353 (1.1%) participants; the remainder reported participant ethnicity: 68 930 (31.0%) participants were of non-White ethnicity or had non-light skin tones. The majority of studies reported overestimation of SaO2 by pulse oximetry in participants with darker skin tones or from ethnicities assumed to have darker skin tones. Several studies reported no inaccuracy related to skin tone. Meta-analysis of the data was not possible. CONCLUSIONS: Pulse oximetry can overestimate true SaO2 in people with darker skin tones. The clinical relevance of this bias remains unclear, but its magnitude is likely to be greater when SaO2 is lower. SYSTEMATIC REVIEW PROTOCOL: International Prospective Register of Systematic Reviews (PROSPERO): CRD42023390723

Relaxation rate of the reverse biased asymmetric exclusion process

We compute the exact relaxation rate of the partially asymmetric exclusion process with open boundaries, with boundary rates opposing the preferred direction of flow in the bulk. This reverse bias introduces a length scale in the system, at which we find a crossover between exponential and algebraic relaxation on the coexistence line. Our results follow from a careful analysis of the Bethe ansatz root structure.Comment: 22 pages, 12 figure

Complex WKB Analysis of a PT Symmetric Eigenvalue Problem

The spectra of a particular class of PT symmetric eigenvalue problems has previously been studied, and found to have an extremely rich structure. In this paper we present an explanation for these spectral properties in terms of quantisation conditions obtained from the complex WKB method. In particular, we consider the relation of the quantisation conditions to the reality and positivity properties of the eigenvalues. The methods are also used to examine further the pattern of eigenvalue degeneracies observed by Dorey et al. in [1,2].Comment: 22 pages, 13 figures. Added references, minor revision