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On Trivalent Logics, Compound Conditionals, and Probabilistic Deduction Theorems

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arXiv 2023年

作者： Gilio, Angelo Over, David E. Pfeifer, Niki Sanfilippo, Giuseppe Department of Basic and Applied Sciences for Engineering University of Rome "La Sapienza" Italy Department of Psychology Durham University United Kingdom Department of Philosophy University of Regensburg Germany Department of Mathematics and Computer Science University of Palermo Italy INdAM-GNAMPA Italy

In this paper we recall some results for conditional events, compound conditionals, conditional random quantities, p-consistency, and p-entailment. Then, we show the equivalence between bets on conditionals and conditional bets, by reviewing de Finetti’s trivalent analysis of conditionals. But our approach goes beyond de Finetti’s early trivalent logical analysis and is based on his later ideas, aiming to take his proposals to a higher level. We examine two recent articles that explore trivalent logics for conditionals and their definitions of logical validity and compare them with our approach to compound conditionals. We prove a Probabilistic Deduction Theorem for conditional events. After that, we study some probabilistic deduction theorems, by presenting several examples. We focus on iterated conditionals and the invalidity of the Import-Export principle in the light of our Probabilistic Deduction Theorem. We use the inference from a disjunction, A or B, to the conditional, if not-A then B, as an example to show the invalidity of the Import-Export principle. We also introduce a General Import-Export principle and we illustrate it by examining some p-valid inference rules of System P. Finally, we briefly discuss some related work relevant to AI. Copyright © 2023, The Authors. All rights reserved.

关键词： Artificial intelligence

Exact Recovery for System Identification with More Corrupt Data than Clean Data

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arXiv 2023年

作者： Yalcin, Baturalp Zhang, Haixiang Lavaei, Javad Arcak, Murat the Department of Industrial Engineering and Operations Research University of California BerkeleyCA94720 United States the Department of Mathematics University of California BerkeleyCA94720 United States the Department of Electrical Engineering and Computer Sciences University of California BerkeleyCA94720 United States

This paper investigates the system identification problem for linear discrete-time systems under adversaries and analyzes two lasso-type estimators. We examine both asymptotic and non-asymptotic properties of these estimators in two separate scenarios, corresponding to deterministic and stochastic models for the attack times. Since the samples collected from the system are correlated, the existing results on lasso are not applicable. We prove that when the system is stable and attacks are injected periodically, the sample complexity for exact recovery of the system dynamics is linear in terms of the dimension of the states. When adversarial attacks occur at each time instance with probability p, the required sample complexity for exact recovery scales polynomially in the dimension of the states and the probability p. This result implies almost sure convergence to the true system dynamics under the asymptotic regime. As a by-product, our estimators still learn the system correctly even when more than half of the data is compromised. We highlight that the attack vectors are allowed to be correlated with each other in this work, whereas we make some assumptions about the times at which the attacks happen. This paper provides the first mathematical guarantee in the literature on learning from correlated data for dynamical systems in the case when there is less clean data than corrupt *** Codes 62, 90, 93 © 2023, CC BY-NC-ND.

关键词： Linear systems

Phy-Diff: Physics-guided Hourglass Diffusion Model for Diffusion MRI Synthesis

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arXiv 2024年

作者： Zhang, Juanhua Yan, Ruodan Perelli, Alessandro Chen, Xi Li, Chao School of Science and Engineering University of Dundee United Kingdom Department of Applied Mathematics and Theoretical Physics University of Cambridge United Kingdom Department of Computer Science University of Bath United Kingdom Department of Clinical Neurosciences University of Cambridge United Kingdom School of Medicine University of Dundee United Kingdom

Diffusion MRI (dMRI) is an important neuroimaging technique with high acquisition costs. Deep learning approaches have been used to enhance dMRI and predict diffusion biomarkers through undersampled dMRI. To generate more comprehensive raw dMRI, generative adversarial network based methods are proposed to include b-values and b-vectors as conditions, but they are limited by unstable training and less desirable diversity. The emerging diffusion model (DM) promises to improve generative performance. However, it remains challenging to include essential information in conditioning DM for more relevant generation, i.e., the physical principles of dMRI and white matter tract structures. In this study, we propose a physics-guided diffusion model to generate high-quality dMRI. Our model introduces the physical principles of dMRI in the noise evolution in the diffusion process and introduces a query-based conditional mapping within the diffusion model. In addition, to enhance the anatomical fine details of the generation, we introduce the XTRACT atlas as a prior of white matter tracts by adopting an adapter technique. Our experiment results show that our method outperforms other state-of-the-art methods and has the potential to advance dMRI enhancement. © 2024, CC BY-NC-ND.

关键词： Magnetic resonance imaging

Robust personnel rostering: how accurate should absenteeism predictions be?

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arXiv 2024年

作者： Doneda, Martina Smet, Pieter Carello, Giuliana Lanzarone, Ettore Vanden Berghe, Greet Politecnico di Milano Department of Electronics Information and Bioengineering Milan Italy National Research Council Institute for Applied Mathematics and Information Technologies Milan Italy KU Leuven Department of Computer Science CODeS Gent Belgium University of Bergamo Department of Management Information and Production Engineering Dalmine Italy

Disruptions to personnel rosters caused by absenteeism often necessitate last-minute adjustments to the employees' working hours. A common strategy to mitigate the impact of such changes is to assign employees to reserve shifts: special on-call duties during which an employee can be called in to cover for an absent employee. To maximize roster robustness, we assume a predict-then-optimize approach that uses absence predictions from a machine learning model to schedule an adequate number of reserve shifts. In this paper we propose a methodology to evaluate the robustness of rosters generated by the predict-then-optimize approach, assuming the machine learning model will make predictions at a predetermined prediction performance level. Instead of training and testing machine learning models, our methodology simulates the predictions based on a characterization of model performance. We show how this methodology can be applied to identify the minimum performance level needed for the model to outperform simple non-data-driven robust rostering policies. In a computational study on a nurse rostering problem, we demonstrate how the predict-then-optimize approach outperforms non-data-driven policies under reasonable performance requirements, particularly when employees possess interchangeable skills. © 2024, CC BY.

关键词： Machine learning

Maximum Likelihood Estimation Based Complex-Valued Robust Chinese Remainder Theorem and Its Fast Algorithm

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arXiv 2025年

作者： Li, Xiaoping Sun, Shiyang Liao, Qunying Xia, Xiang-Gen School of Mathematical Science University of Electronic Science and Technology of China Chengdu611731 China Institute of Mathematics Science Sichuan Normal University Chengdu610068 China Department of Electrical and Computer Engineering University of Delaware NewarkDE19716 United States

Recently, a multi-channel self-reset analog-to-digital converter (ADC) system with complex-valued moduli has been proposed. This system enables the recovery of high dynamic range complex-valued bandlimited signals at low sampling rates via the Chinese remainder theorem (CRT). In this paper, we investigate complex-valued CRT (C-CRT) with erroneous remainders, where the errors follow wrapped complex Gaussian distributions. Based on the existing real-valued CRT utilizing maximum likelihood estimation (MLE), we propose a fast MLE-based C-CRT (MLE C-CRT). The proposed algorithm requires only 2L searches to obtain the optimal estimate of the common remainder, where L is the number of moduli. Once the common remainder is estimated, the complex number can be determined using the C-CRT. Furthermore, we obtain a necessary and sufficient condition for the fast MLE C-CRT to achieve robust estimation. Finally, we apply the proposed algorithm to ADCs. The results demonstrate that the proposed algorithm outperforms the existing methods. Copyright © 2025, The Authors. All rights reserved.

关键词： Maximum likelihood estimation

Almost Optimal Variance-Constrained Best Arm Identification

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arXiv 2022年

作者： Hou, Yunlong Tan, Vincent Y.F. Zhong, Zixin The Department of Mathematics National University of Singapore Singapore The Department of Electrical and Computer Engineering National University of Singapore Singapore The Department of Computing Science The University of Alberta Canada

We design and analyze Variance-Aware-Lower and Upper Confidence Bound (VA-LUCB), a parameter-free algorithm, for identifying the best arm under the fixed-confidence setup and under a stringent constraint that the variance of the chosen arm is strictly smaller than a given threshold. An upper bound on VA-LUCB's sample complexity is shown to be characterized by a fundamental variance-aware hardness quantity HVA. By proving an information-theoretic lower bound, we show that sample complexity of VA-LUCB is optimal up to a factor logarithmic in HVA. Extensive experiments corroborate the dependence of the sample complexity on the various terms in HVA. By comparing VA-LUCB's empirical performance to a close competitor RiskAverseUCB-BAI by David et al. [1], our experiments suggest that VA-LUCB has the lowest sample complexity for this class of risk-constrained best arm identification problems, especially for the riskiest instances. Copyright © 2022, The Authors. All rights reserved.

关键词： Stochastic systems

Ranks of elliptic curves and deep neural networks

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arXiv 2022年

作者： Kazalicki, Matija Vlah, Domagoj Department of Mathematics University of Zagreb Bijenička Cesta 30 Zagreb10000 Croatia Department of Applied Mathematics Faculty of Electrical Engineering and Computing University of Zagreb Unska 3 Zagreb10000 Croatia

Determining the rank of an elliptic curve E/Q is a hard problem, and in some applications (e.g. when searching for curves of high rank) one has to rely on heuristics aimed at estimating the analytic rank (which is equal to the rank under the Birch and Swinnerton-Dyer conjecture). In this paper, we develop rank classification heuristics modeled by deep convolutional neural networks (CNN). Similarly to widely used Mestre-Nagao sums, it takes as an input the conductor of E and a sequence of normalized ap-s (where ap = p+1−#E(Fp) if p is a prime of good reduction) in some range (p k for k = 3,4,5), and tries to predict rank (or detect curves of "high" rank). The model has been trained and tested on two datasets: the LMFDB and a custom dataset consisting of elliptic curves with trivial torsion, conductor up to 1030, and rank up to 10. For comparison, eight simple neural network models of Mestre-Nagao sums have also been developed. Experiments showed that CNN performed better than Mestre-Nagao sums on the LMFDB dataset (interestingly neural network that took as an input all Mestre-Nagao sums performed much better than each sum individually), while they were roughly equal on custom made dataset. © 2022, CC BY.

关键词： Deep neural networks

Inverse design of 3D-printable metalenses with complementary dispersion for terahertz imaging

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arXiv 2025年

作者： Chen, Mo Chan, Ka Fai Hammond, Alec M. Chan, Chi Hou Johnson, Steven G. Department of Mathematics Massachusetts Institute of Technology CambridgeMA02139 United States State Key Laboratory of Terahertz and Millimeter Waves CityU Hong Kong School of Electrical and Computer Engineering Georgia Institute of Technology AtlantaGA30308 United States

This study formulates a volumetric inverse-design methodology to generate a pair of complementary focusing metalenses for terahertz imaging: the two lenses exhibit equal and opposite shifts in focal length with frequency. An asymmetry arises, where we find a focal length that decreases with frequency to be more challenging to achieve (without material dispersion) given fabrication constraints, but it is still possible. We employ topology optimization, coupled with manufacturing constraints, to explore fully freeform designs compatible with 3D printing. Formulating an optimization problem that quantifies the goal of maximal complementary focal shifts, while remaining differentiable and tractable, requires a carefully selected sequence of constraints and approximations. Copyright © 2025, The Authors. All rights reserved.

关键词： Shape optimization

Off-policy estimation of linear functionals: Non-asymptotic theory for semi-parametric efficiency

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arXiv 2022年

作者： Mou, Wenlong Wainwright, Martin J. Bartlett, Peter L. Department of Electrical Engineering and Computer Sciences United States Department of Statistics UC Berkeley United States Department of Mathematics Massachusetts Institute of Technology United States

The problem of estimating a linear functional based on observational data is canonical in both the causal inference and bandit literatures. We analyze a broad class of two-stage procedures that first estimate the treatment effect function, and then use this quantity to estimate the linear functional. We prove non-asymptotic upper bounds on the mean-squared error of such procedures: these bounds reveal that in order to obtain non-asymptotically optimal procedures, the error in estimating the treatment effect should be minimized in a certain weighted L2-norm. We analyze a two-stage procedure based on constrained regression in this weighted norm, and establish its instance-dependent optimality in finite samples via matching non-asymptotic local minimax lower bounds. These results show that the optimal non-asymptotic risk, in addition to depending on the asymptotically efficient variance, depends on the weighted norm distance between the true outcome function and its approximation by the richest function class supported by the sample size. Copyright © 2022, The Authors. All rights reserved.

关键词： Optimization