A striking pathology of semidefinite programs (SDPs) is illustrated by a classical example of Khachiyan: feasible solutions in SDPs may need exponential space even to write down. Such exponential size solutions are the main obstacle to solve a long standing, fundamental open problem: can we decide feasibility of SDPs in polynomial time? The consensus seems that SDPs with large size solutions are rare. However, here we prove that they are actually quite common: a linear change of variables transforms every strictly feasible SDP into a Khachiyan type SDP, in which the leading variables are large. As to ``how large", that depends on the singularity degree of a dual problem. Further, we present some SDPs coming from sum-of-squares proofs, in which large solutions appear naturally, without any change of variables. We also partially answer the question: how do we represent such large solutions in polynomial space?
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Multiagent systems aim to accomplish highly complex learning tasks through decentralised consensus seeking dynamics and their use has garnered a great deal of attention in the signal processing and computational intelligence societies. This article examines the behaviour of multiagent networked systems with nonlinear filtering/learning dynamics. To this end, a general formulation for the actions of an agent in multiagent networked systems is presented and conditions for achieving a cohesive learning behaviour is given. Importantly, application of the so derived framework in distributed and federated learning scenarios are presented.
Quantum neural networks (QNNs) and quantum kernels stand as prominent figures in the realm of quantum machine learning, poised to leverage the nascent capabilities of near-term quantum computers to surmount classical machine learning challenges. Nonetheless, the training efficiency challenge poses a limitation on both QNNs and quantum kernels, curbing their efficacy when applied to extensive datasets. To confront this concern, we present a unified approach: coreset selection, aimed at expediting the training of QNNs and quantum kernels by distilling a judicious subset from the original training dataset. Furthermore, we analyze the generalization error bounds of QNNs and quantum kernels when trained on such coresets, unveiling the comparable performance with those training on the complete original dataset. Through systematic numerical simulations, we illuminate the potential of coreset selection in expediting tasks encompassing synthetic data classification, identification of quantum correlations, and quantum compiling. Our work offers a useful way to improve diverse quantum machine learning models with a theoretical guarantee while reducing the training cost.
ChatGPT is a powerful language model from OpenAI that is arguably able to comprehend and generate text. ChatGPT is expected to have a large impact on society, research, and education. An essential step to understand ChatGPT's expected impact is to study its domain-specific answering capabilities. Here, we perform a systematic empirical assessment of its abilities to answer questions across the natural science and engineering domains. We collected 594 questions from 198 faculty members across 5 faculties at Delft University of Technology. After collecting the answers from ChatGPT, the participants assessed the quality of the answers using a systematic scheme. Our results show that the answers from ChatGPT are on average perceived as ``mostly correct''. Two major trends are that the rating of the ChatGPT answers significantly decreases (i) as the complexity level of the question increases and (ii) as we evaluate skills beyond scientific knowledge, e.g., critical attitude.
In this work, we construct a fifth-order weighted essentially non-oscillatory (WENO) scheme with exponential approximation space for solving dispersive equations. A conservative third-order derivative formulation is developed directly using WENO spatial reconstruction procedure and third-order TVD Runge- Kutta scheme is used for the evaluation of time derivative. This exponential approximation space consists a tension parameter that may be optimized to fit the specific feature of the charecteristic data, yielding better results without spurious oscillations compared to the polynomial approximation space. A detailed formulation is presented for the the construction of conservative flux approximation, smoothness indicators, nonlinear weights and verified that the proposed scheme provides the required fifth convergence order. One and two-dimensional numerical examples are presented to support the theoretical claims.
Large Language Models (LLMs) have emerged as powerful generative Artificial Intelligence solutions which can be applied to several fields and areas of work. This paper presents results and reflection of an experiment done to use the model GPT 3.5-Turbo to emulate some aspects of an inductive Thematic Analysis. Previous research on this subject has largely worked on conducting deductive analysis. Thematic Analysis is a qualitative method for analysis commonly used in social sciences and it is based on interpretations made by the human analyst(s) and the identification of explicit and latent meanings in qualitative data. Attempting an analysis based on human interpretation with an LLM clearly is a provocation but also a way to learn something about how these systems can or cannot be used in qualitative research. The paper presents the motivations for attempting this emulation, it reflects on how the six steps to a Thematic Analysis proposed by Braun and Clarke can at least partially be reproduced with the LLM and it also reflects on what are the outputs produced by the model. The paper used two existing datasets of open access semi-structured interviews, previously analysed with Thematic Analysis by other researchers. It used the previously produced analysis (and the related themes) to compare with the results produced by the LLM. The results show that the model can infer at least partially some of the main Themes. The objective of the paper is not to replace human analysts in qualitative analysis but to learn if some elements of LLM data manipulation can to an extent be of support for qualitative research.
A general a posteriori error analysis applies to five lowest-order finite element methods for two fourth-order semi-linear problems with trilinear non-linearity and a general source. A quasi-optimal smoother extends the source term to the discrete trial space, and more importantly, modifies the trilinear term in the stream-function vorticity formulation of the incompressible 2D Navier-Stokes and the von K\'{a}rm\'{a}n equations. This enables the first efficient and reliable a posteriori error estimates for the 2D Navier-Stokes equations in the stream-function vorticity formulation for Morley, two discontinuous Galerkin, $C^0$ interior penalty, and WOPSIP discretizations with piecewise quadratic polynomials.
The accurate representation of precipitation in Earth system models (ESMs) is crucial for reliable projections of the ecological and socioeconomic impacts in response to anthropogenic global warming. The complex cross-scale interactions of processes that produce precipitation are challenging to model, however, inducing potentially strong biases in ESM fields, especially regarding extremes. State-of-the-art bias correction methods only address errors in the simulated frequency distributions locally at every individual grid cell. Improving unrealistic spatial patterns of the ESM output, which would require spatial context, has not been possible so far. Here, we show that a post-processing method based on physically constrained generative adversarial networks (cGANs) can correct biases of a state-of-the-art, CMIP6-class ESM both in local frequency distributions and in the spatial patterns at once. While our method improves local frequency distributions equally well as gold-standard bias-adjustment frameworks, it strongly outperforms any existing methods in the correction of spatial patterns, especially in terms of the characteristic spatial intermittency of precipitation extremes.
Mediation analysis is widely used in health science research to evaluate the extent to which an intermediate variable explains an observed exposure-outcome relationship. However, the validity of analysis can be compromised when the exposure is measured with error. Motivated by the Health Professionals Follow-up Study (HPFS), we investigate the impact of exposure measurement error on assessing mediation with a survival outcome, based on the Cox proportional hazards outcome model. When the outcome is rare and there is no exposure-mediator interaction, we show that the uncorrected estimators of the natural indirect and direct effects can be biased into either direction, but the uncorrected estimator of the mediation proportion is approximately unbiased as long as the measurement error is not large or the mediator-exposure association is not strong. We develop ordinary regression calibration and risk set regression calibration approaches to correct the exposure measurement error-induced bias when estimating mediation effects and allowing for an exposure-mediator interaction in the Cox outcome model. The proposed approaches require a validation study to characterize the measurement error process. We apply the proposed approaches to the HPFS (1986-2016) to evaluate extent to which reduced body mass index mediates the protective effect of vigorous physical activity on the risk of cardiovascular diseases, and compare the finite-sample properties of the proposed estimators via simulations.
We present a novel computational model for the dynamics of alveolar recruitment/derecruitment (RD), which reproduces the underlying characteristics typically observed in injured lungs. The basic idea is a pressure- and time-dependent variation of the stress-free reference volume in reduced dimensional viscoelastic elements representing the acinar tissue. We choose a variable reference volume triggered by critical opening and closing pressures in a time-dependent manner from a straightforward mechanical point of view. In the case of (partially and progressively) collapsing alveolar structures, the volume available for expansion during breathing reduces and vice versa, eventually enabling consideration of alveolar collapse and reopening in our model. We further introduce a method for patient-specific determination of the underlying critical parameters of the new alveolar RD dynamics when integrated into the tissue elements, referred to as terminal units, of a spatially resolved physics-based lung model that simulates the human respiratory system in an anatomically correct manner. Relevant patient-specific parameters of the terminal units are herein determined based on medical image data and the macromechanical behavior of the lung during artificial ventilation. We test the whole modeling approach for a real-life scenario by applying it to the clinical data of a mechanically ventilated patient. The generated lung model is capable of reproducing clinical measurements such as tidal volume and pleural pressure during various ventilation maneuvers. We conclude that this new model is an important step toward personalized treatment of ARDS patients by considering potentially harmful mechanisms - such as cyclic RD and overdistension - and might help in the development of relevant protective ventilation strategies to reduce ventilator-induced lung injury (VILI).
Deep learning is usually described as an experiment-driven field under continuous criticizes of lacking theoretical foundations. This problem has been partially fixed by a large volume of literature which has so far not been well organized. This paper reviews and organizes the recent advances in deep learning theory. The literature is categorized in six groups: (1) complexity and capacity-based approaches for analyzing the generalizability of deep learning; (2) stochastic differential equations and their dynamic systems for modelling stochastic gradient descent and its variants, which characterize the optimization and generalization of deep learning, partially inspired by Bayesian inference; (3) the geometrical structures of the loss landscape that drives the trajectories of the dynamic systems; (4) the roles of over-parameterization of deep neural networks from both positive and negative perspectives; (5) theoretical foundations of several special structures in network architectures; and (6) the increasingly intensive concerns in ethics and security and their relationships with generalizability.
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