Quadratic Growth Research Articles

Limited Memory Online Gradient Descent for Kernelized Pairwise Learning with Dynamic Averaging

Pairwise learning, an important domain within machine learning, addresses loss functions defined on pairs of training examples, including those in metric learning and AUC maximization. Acknowledging the quadratic growth in computation complexity accompanying pairwise loss as the sample size grows, researchers have turned to online gradient descent (OGD) methods for enhanced scalability. Recently, an OGD algorithm emerged, employing gradient computation involving prior and most recent examples, a step that effectively reduces algorithmic complexity to O(T), with T being the number of received examples. This approach, however, confines itself to linear models while assuming the independence of example arrivals. We introduce a lightweight OGD algorithm that does not require the independence of examples and generalizes to kernel pairwise learning. Our algorithm builds the gradient based on a random example and a moving average representing the past data, which results in a sub-linear regret bound with a complexity of O(T). Furthermore, through the integration of O(√T logT) random Fourier features, the complexity of kernel calculations is effectively minimized. Several experiments with real-world datasets show that the proposed technique outperforms kernel and linear algorithms in offline and online scenarios.

Patterns and predictors of alcohol misuse trajectories from adolescence through early midlife.

We took a multilevel developmental contextual approach and characterized trajectories of alcohol misuse from adolescence through early midlife, examined genetic and environmental contributions to individual differences in those trajectories, and identified adolescent and young adult factors associated with change in alcohol misuse. Data were from two longitudinal population-based studies. FinnTwin16 is a study of Finnish twins assessed at 16, 17, 18, 25, and 35 years (N = 5659; 52% female; 32% monozygotic). The National Longitudinal Study of Adolescent to Adult Health (Add Health) is a study of adolescents from the United States, who were assessed at five time points from 1994 to 2018 (N = 18026; 50% female; 64% White, 21% Black, 4% Native American, 7% Asian, 9% Other race/ethnicity). Alcohol misuse was measured as frequency of intoxication in FinnTwin16 and frequency of binge drinking in Add Health. In both samples, trajectories of alcohol misuse were best described by a quadratic growth curve: Alcohol misuse increased across adolescence, peaked in young adulthood, and declined into early midlife. Individual differences in these trajectories were primarily explained by environmental factors. Several adolescent and young adult correlates were related to the course of alcohol misuse, including other substance use, physical and mental health, and parenthood.

On unitary algebras with graded involution of quadratic growth

Let F be a field of characteristic zero. By a ⁎-superalgebra we mean an algebra A with graded involution over F. Recently, algebras with graded involution have been extensively studied in PI-theory and the sequence of ⁎-graded codimensions {cngri(A)}n≥1 has been investigated by several authors. In this paper, we classify varieties generated by unitary ⁎-superalgebras having quadratic growth of ⁎-graded codimensions. As a result we obtain that a unitary ⁎-superalgebra with quadratic growth is T2⁎-equivalent to a finite direct sum of minimal unitary ⁎-superalgebras with at most quadratic growth, where at least one ⁎-superalgebra of this sum has quadratic growth. Furthermore, we provide a method to determine explicitly the factors of those direct sums.

Quadratic Growth of Out-of-Time-Ordered Correlators in Quantum Kicked Rotor Model.

We investigate both theoretically and numerically the dynamics of out-of-time-ordered correlators (OTOCs) in quantum resonance conditions for a kicked rotor model. We employ various operators to construct OTOCs in order to thoroughly quantify their commutation relation at different times, therefore unveiling the process of quantum scrambling. With the help of quantum resonance condition, we have deduced the exact expressions of quantum states during both forward evolution and time reversal, which enables us to establish the laws governing OTOCs' time dependence. We find interestingly that the OTOCs of different types increase in a quadratic function of time, breaking the freezing of quantum scrambling induced by the dynamical localization under non-resonance condition. The underlying mechanism is discovered, and the possible applications in quantum entanglement are discussed.

Quadratic Growth and Linear Convergence of a DCA Method for Quartic Minimization over the Sphere

Quadratic Growth and Linear Convergence of a DCA Method for Quartic Minimization over the Sphere

Wellposedness of anticipated BSDEs with quadratic growth and unbounded terminal value

Wellposedness of anticipated BSDEs with quadratic growth and unbounded terminal value

Degree growth of lattice equations defined on a 3x3 stencil

We study complexity in terms of degree growth of one-component lattice equations defined on a $3\times 3$ stencil. The equations include two in Hirota bilinear form and the Boussinesq equations of regular, modified and Schwarzian type. Initial values are given on a staircase or on a corner configuration and depend linearly or rationally on a special variable, for example $f_{n,m}=\alpha_{n,m}z+\beta_{n,m}$, in which case we count the degree in $z$ of the iterates. Known integrable cases have linear growth if only one initial values contains $z$, and quadratic growth if all initial values contain $z$. Even a small deformation of an integrable equation changes the degree growth from polynomial to exponential, because the deformation will change factorization properties and thereby prevent cancellations.

Torelli-type theorems for gravitational instantons with quadratic volume growth

Torelli-type theorems for gravitational instantons with quadratic volume growth

On quasilinear parabolic systems and FBSDEs of quadratic growth

On quasilinear parabolic systems and FBSDEs of quadratic growth

Macroscopic effects in generation of attosecond XUV pulses via high-order frequency mixing in gases and plasma

We theoretically study the generation of attosecond XUV pulses via high-order frequency mixing (HFM) of two intense generating fields, and compare this process with the more common high-order harmonic generation (HHG) process. We calculate the macroscopic XUV signal by numerically integrating the 1D propagation equation coupled with the 3D time-dependent Schrödinger equation. We analytically find the length scales which limit the quadratic growth of the HFM macroscopic signal with propagation length. Compared to HHG these length scales are much longer for a group of HFM components, with orders defined by the frequencies of the generating fields. This results in a higher HFM macroscopic signal despite the microscopic response being lower than for HHG. In our numerical simulations in a gas target, the intensity of the HFM signal is several times higher than that for HHG, while for a plasma target the HFM generation efficiency is up to three orders of magnitude higher than for HHG. The HFM with relatively long generating pulses provides very narrow XUV lines ( δω/ω=4.6×10−4 ) with well-defined frequencies, thus allowing for a simple extension of optical frequency standards to the XUV range. Finally, we show that the group of HFM components effectively generated due to macroscopic effects provides a train of attosecond pulses such that the carrier-envelope phase of an individual attosecond pulse can be easily controlled by tuning the phase of one of the generating fields.

A stochastic maximum principle for forward–backward stochastic control systems with quadratic generators and sample-wise constraints

This paper examines the stochastic maximum principle (SMP) for a forward–backward stochastic control system where the backward state equation is characterized by the backward stochastic differential equation (BSDE) with quadratic growth and the forward state at the terminal time is constrained in a convex set with probability one. With the help of the theory of BSDEs with quadratic growth and the bounded mean oscillation (BMO) martingales, we employ the terminal perturbation approach and Ekeland’s variational principle to obtain a dynamic stochastic maximum principle. The main result has a wide range of applications in mathematical finance and we investigate a robust recursive utility maximization problem with bankruptcy prohibition as an example.

Static Blame for gradual typing

Abstract Gradual typing integrates static and dynamic typing by introducing a dynamic type and a consistency relation. A problem of gradual type systems is that dynamic types can easily hide erroneous data flows since consistency relations are not transitive. Therefore, a more rigorous static check is required to reveal these hidden data flows statically. However, in order to preserve the expressiveness of gradually typed languages, static checks for gradually typed languages cannot simply reject programs with potentially erroneous data flows. By contrast, a more reasonable request is to show how these data flows can affect the execution of the program. In this paper, we propose and formalize Static Blame, a framework that can reveal hidden data flows for gradually typed programs and establish the correspondence between static-time data flows and runtime behavior. With this correspondence, we build a classification of potential errors detected from hidden data flows and formally characterize the possible impact of potential errors in each category on program execution, without simply rejecting the whole program. We implemented Static Blame on Grift, an academic gradually typed language, and evaluated the effectiveness of Static Blame by mutation analysis to verify our theoretical results. Our findings revealed that Static Blame exhibits a notable level of precision and recall in detecting type-related bugs. Furthermore, we conducted a manual classification to elucidate the reasons behind instances of failure. We also evaluated the performance of Static Blame, showing a quadratic growth in run time as program size increases.

Well-Posedness of Backward Stochastic Differential Equations with Jumps and Irregular Coefficients

In this paper, we focus on investigating the well-posedness of backward stochastic differential equations with jumps (BSDEJs) driven by irregular coefficients. We establish new results regarding the existence and uniqueness of solutions for a specific class of singular BSDEJs. Unlike previous studies, our approach considers terminal data that are square-integrable, eliminating the need for them to be necessarily bounded. The generators in our study encompass a standard drift, a signed measure across the entire real line, and the local time of the unknown process. This broadens the scope to include BSDEJs with quadratic growth in the Brownian component and exponential growth concerning the jump noise. The key methodology involves establishing Krylov-type estimates for a subset of solutions to irregular BSDEJs and subsequently proving the Tanaka-Krylov formula. Additionally, we employ a space transformation technique to simplify the initial BSDEJs, leading to a standard form without singular terms. We also provide various examples and special cases, shedding light on BSDEJs with irregular drift coefficients and contributing to new findings in the field.

DOES LIVING WITH A FAMILY AFFECT SOCIAL CONTACT/PARTICIPATION? VARIATIONS ACROSS BIRTH COHORTS AND GENDER IN JAPAN

Abstract Changing family structures and values might have affected social relationships outside the household in a different way for men and women. This study clarified birth cohort and gender differences among older Japanese in terms of (a) trajectories of social contact and participation and (b) effects of living with a family on them. Data were obtained from ten waves of the National Survey of the Japanese Elderly (1987-2021) for 5,855 individuals who commenced participation by 2012. Quadratic growth models were examined as a function of age at time t (≥63 years) for the frequency of contact with children living apart, with friends and neighbors, and participation in community groups, whose intercept and slope were predicted by gender and birth cohorts (C1:1902-13, C2:1914-25, C3:1926-37, C4:1938-49). Whether effects of being married and co-residing with a child would differ by gender and cohorts was also examined, controlling for socioeconomic and health status. The results showed that the level of contact/participation at age 75 increased in women in recent cohorts. The frequency for friends and groups declined with age after the mid-70s across cohorts. C4 showed a steeper decline in the child/friend contact, indicating the COVID-19 influence. Living with a family generally impeded social contact, except that married men contacted non-coresident children more frequently than their unmarried counterparts. The negative effect of living with a child was greater in women and the earliest birth cohort. Thus, social isolation could matter not only for older adults living alone, but also for those living with a family.

Performance and wake interaction between two aligned vertical axis turbines

Performance and wake interaction between two aligned vertical turbines are investigated by solving the three-dimensional Navier-Stokes equations with the k−ω SST IDDES turbulence model. Specifically, the impact of longitudinal spacing between the upstream and downstream turbines is examined. The numerical results show that as the longitudinal distance increases, the performance of the downstream turbine shows a quadratic growth. This is phenomenon is strongly correlated to the wake recovery of the upstream turbine. Based on the velocity and turbulence distribution, the wake of the upstream turbine could be divided into three distinct zones. However, the zone characterized by low axial velocity and turbulence level is not observed in the wake of the downstream turbine. This difference primarily arises from the high turbulence present in the incoming wake from the upstream turbine, combined with the turbulence generated by the downstream turbine. Interestingly, regardless of the longitudinal distances between the turbines, the downstream turbine displays similar wake patterns for different streamwise distances between turbines. This consistency is attributed to the influence of the upstream turbine's wake on the stability of the downstream turbine's wake. The downstream turbine is suggested to be positioned in the zone where the upstream turbine's wake has fully developed.

Option Pricing and Portfolio Optimization under a Multi-Asset Jump-Diffusion Model with Systemic Risk

We explore a multi-asset jump-diffusion pricing model, combining a systemic risk asset with several conditionally independent ordinary assets. Our approach allows for analyzing and modeling a portfolio that integrates high-activity security, such as an exchange trading fund (ETF) tracking a major market index (e.g., S&P500), along with several low-activity securities infrequently traded on financial markets. The model retains tractability even as the number of securities increases. The proposed framework allows for constructing models with common and asset-specific jumps with normally or exponentially distributed sizes. One of the main features of the model is the possibility of estimating parameters for each asset price process individually. We present the conditional maximum likelihood estimation (MLE) method for fitting asset price processes to empirical data. For the case with common jumps only, we derive a closed-form solution to the conditional MLE method for ordinary assets that works even if the data are incomplete and asynchronous. Alternatively, to find risk-neutral parameters, the least-square method calibrates the model to option values. The number of parameters grows linearly in the number of assets compared to the quadratic growth through the correlation matrix, which is typical for many other multi-asset models. We delve into the properties of the proposed model, its parameter estimation using the MLE method and least-squares technique, the evaluation of VaR and CVaR metrics, the identification of optimal portfolios, and the pricing of European-style basket options. We propose a Laplace-transform-based approach to computing Value at Risk (VaR) and conditional VaR (also known as the expected shortfall) of portfolio returns. Additionally, European-style basket options written on the extreme and average stock prices or returns can be evaluated semi-analytically. For numerical demonstration, we examine a combination of the SPDR S&P 500 ETF (as a systemic risk asset) with eight ordinary assets representing diverse industries. Using historical assets and options prices, we estimate the real-world and risk-neutral parameters of the model with common jumps, construct several optimal portfolios, and evaluate various basket options with the eight assets. The results affirm the robustness and efficiency of the estimation and evaluation methodologies. Computational results are compared with Monte Carlo estimates.

Infinite Horizon Irregular Quadratic BSDE and Applications to Quadratic PDE and Epidemic Models with Singular Coefficients

In an infinite time horizon, we focused on examining the well-posedness of problems for a particular category of Backward Stochastic Differential Equations having quadratic growth (QBSDEs) with terminal conditions that are merely square integrable and generators that are measurable. Our approach employs a Zvonkin-type transformation in conjunction with the Itô–Krylov’s formula. We applied our findings to derive probabilistic representation of a particular set of Partial Differential Equations par have quadratic growth in the gradient (QPDEs) characterized by coefficients that are measurable and almost surely continuous. Additionally, we explored a stochastic control optimization problem related to an epidemic model, interpreting it as an infinite time horizon QBSDE with a measurable and integrable drifts.

Structural indicator synergy for mitigating extreme urban heat island effects in industrial city: Simulation and verification based on machine learning

Industrial areas emit large amounts of extremely high temperatures and aggravate the extreme urban heat island (E-UHI) effect, which has become a non-negligible environmental issue worldwide. Due to the lack of in-depth understanding of the relationship between E-UHI and surface structural synergy, few studies explored the causes and mitigation measures of E-UHI. Therefore, this study accordingly defined the E-UHI effect at a finer scale based on the thermal characteristics and human thermal perceptions, and a multi-indicator’ method based on machine learning was proposed to quantify internal causes and collaborative mitigation measures. The results revealed that the average E-UHI scale shows a quadratic growth in Wu’an. Artificial heat from production space (PS) is the main cause of E-UHI formation. The synergy between green space (GS), PS and shape of industrial land determines the development of E-UHI. Although both the decrease of PS and the increase of GS are ways to mitigate high land surface temperature, when PS is higher than 50%, the cooling effect of GS is not obvious. When GS is less than 2.5%, the adjustment of PS is invalid for improving the thermal environment. For mitigating E-UHI, a factory with high PS needs to increase its shape index. The synergy between the structural indicators is a novel model to solve the shortcoming of adjusting a single factor in mitigating complex E-UHI. The results provide theoretical guidance for mitigating E-UHIs.

Invariant representation for generators of general time interval quadratic BSDEs under stochastic growth conditions

This paper is devoted to proving a general invariant representation theorem for generators of general time interval backward stochastic differential equations, where the generator g has a quadratic growth in the unknown variable z and satisfies some stochastic growth conditions in the unknown variable y. This unifies and strengthens some known results. A natural and innovative idea is used to prove the representation theorem.

Processing Speed throughout Primary School Education: Evidence from a Cross-Country Longitudinal Study.

This cross-country four-year longitudinal study investigated the development of processing speed throughout primary school education. The analyses were conducted on data accumulated from 441 pupils in grades from 1 to 4 (aged 6.42 to 11.85 years) in Kyrgyzstan and Russia. Mixed effects growth modeling was applied to estimate average and individual growth trajectories for processing speed in two cross-country samples. Latent class growth modeling was conducted to describe various types of growth trajectories for processing speed and to compare the distribution of the types within the analyzed samples. According to the results, processing speed significantly increases across primary school years. The trajectory is described by nonlinear changes with most dynamic growth between grades 1 and 2, which slows down until grade 4. No significant cross-country differences were found in the initial score of processing speed or developmental changes in processing speed across primary school years. The development of processing speed is described by a model including three quadratic growth types but this minimally differs. It is concluded that in both samples, the development of processing speed may be characterized by homogeneity, with the most intensive growth from grade 1 to grade 2 and subsequent linear improvement until grade 4.