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On the Computational Determination of the Third Focus of the Cartesian Oval 32

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On the Computational Determination of the Third Focus of the...

32nd Telecommunications Forum, TELFOR 2024

作者： Malesevic, Branko Rosic, Aleksandra Banjac, Bojan Micovic, Milos University of Belgrade School of Electrical Engineering Department of Applied Mathematics Bulevar kralja Aleksandra 73 Belgrade11000 Serbia University of Belgrade Faculty of Mathematics Department of Algebra and Mathematical Logic Studentski trg 16 Belgrade11000 Serbia University of Novi Sad Computer Graphics Chair Faculty of Technical Sciences Trg Dositeja Obradovica 16 Novi Sad21000 Serbia Intens Doo Bulevar Evrope 28 Novi Sad21000 Serbia School of Electrical Engineering University of Belgrade Science and Research Centre Department of Applied Mathematics Bulevar kralja Aleksandra 73 Belgrade11000 Serbia

ISBN: (纸本)9798350391053

In this paper, the correctness of the algebraic determination of the coordinate of the third focus of the Cartesian oval is examined and analysed. This analysis is necessary because the process includes solving systems with a large number of multivariate polynomial equations using the computer algebra system Maple. The paper also considers the difficulties and limitations of using computers in determining the coordinate of the third focus. © 2024 IEEE.

关键词： Polynomials

Finding Birkhoff Averages via Adaptive Filtering

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arXiv 2024年

作者： Ruth, Maximilian Bindel, David Center for Applied Mathematics Cornell University IthacaNY United States Department of Computer Science Cornell University IthacaNY United States

In many applications, one is interested in classifying trajectories of Hamiltonian systems as invariant tori, islands, or chaos. The convergence rate of ergodic Birkhoff averages can be used to categorize these regions, but many iterations of the return map are needed to implement this directly. Recently, it has been shown that a weighted Birkhoff average can be used to accelerate the convergence, resulting in a useful method for categorizing trajectories. In this paper, we show how a modified version the reduced rank extrapolation method (named Birkhoff RRE) can also be used to find optimal weights for the weighted average with a single linear least-squares solve. Using these, we classify trajectories with fewer iterations of the map than the standard weighted Birkhoff average. Furthermore, for the islands and invariant circles, a subsequent eigenvalue problem gives the number of islands and the rotation number. Using these numbers, we find Fourier parameterizations of invariant circles and islands. We show examples of Birkhoff RRE on the standard map and on magnetic field line *** Codes 37M25 © 2024, CC BY.

关键词： Adaptive filtering

A Resilient Distributed Algorithm for Solving Linear Equations

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A Resilient Distributed Algorithm for Solving Linear Equatio...

IEEE Conference on Decision and Control

作者： Jingxuan Zhu Alvaro Velasquez Ji Liu Department of Applied Mathematics and Statistics at Stony Brook University Department of Computer Science at University of Colorado Boulder Department of Electrical and Computer Engineering at Stony Brook University

This paper presents a resilient distributed algorithm for solving a system of linear algebraic equations over a multi-agent network in the presence of Byzantine agents capable of arbitrarily introducing untrustworthy information in communication. It is shown that the algorithm causes all non-Byzantine agents' states to converge to the same least squares solution exponentially fast, provided appropriate levels of graph redundancy and objective redundancy are established. An explicit convergence rate is also provided.

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FINDING PLANTED CLIQUES USING MARKOV CHAIN MONTE CARLO

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arXiv 2023年

作者： Gheissari, Reza Jagannath, Aukosh Xu, Yiming Department of Mathematics Northwestern University United States Department of Statistics and Actuarial Science Department of Applied Mathematics Cheriton School of Computer Science University of Waterloo Canada Department of Statistics and Actuarial Science University of Waterloo Canada

The planted clique problem is a paradigmatic model of statistical-to-computational gaps: the planted clique is information-theoretically detectable if its size k ≥ 2 log2 n but polynomial-time algorithms only exist for the recovery task when k = Ω(√n). By now, there are many simple and fast algorithms that succeed as soon as k = Ω(√n). Glaringly, however, no MCMC approach to the problem had been shown to work, including the Metropolis process on cliques studied by Jerrum since 1992. In fact, Chen, Mossel, and Zadik recently showed that any Metropolis process whose state space is the set of cliques fails to find any sub-linear sized planted clique in polynomial time if initialized naturally from the empty set. Here, we redeem MCMC performance for the planted clique problem by relaxing the state space to all vertex subsets and adding a corresponding energy penalty for missing edges. With that, we prove that energy-minimizing Markov chains (gradient descent and a low-temperature relaxation of it) succeed at recovering planted cliques of size k = Ω(√n) if initialized from the full graph. Importantly, initialized from the empty set, the relaxation still does not help the gradient descent find sub-linear planted cliques. We also demonstrate robustness of these Markov chain approaches under a natural contamination model. Copyright © 2023, The Authors. All rights reserved.

关键词： Temperature

Resilient Distributed Optimization*

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Resilient Distributed Optimization*

American Control Conference (ACC)

作者： Jingxuan Zhu Yixuan Lin Alvaro Velasquez Ji Liu Department of Applied Mathematics and Statistics Stony Brook University Department of Computer Science University of Colorado Boulder Department of Electrical and Computer Engineering Stony Brook University

This paper considers a distributed optimization problem in the presence of Byzantine agents capable of introducing untrustworthy information into the communication network. A resilient distributed subgradient algorithm is proposed based on graph redundancy and objective redundancy. It is shown that the algorithm causes all non-Byzantine agents’ states to asymptotically converge to the same optimal point under appropriate assumptions. A partial convergence rate result is also provided.

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Mirror Games Against an Open Book Player

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arXiv 2022年

作者： Magen, Roey Naor, Moni Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot Israel

Mirror games were invented by Garg and Schneider (ITCS 2019). Alice and Bob take turns (with Alice s) in declaring numbers from the set {1, 2, . . ., 2n}. If a player picks a number that was previously played, that player loses the game and the other player wins. If all numbers are declared without repetition, the result is a draw. Bob has a simple mirror strategy that assures he won’t lose the game and requires no memory. On the other hand, Garg and Schneider showed that every deterministic Alice requires memory of size that is proportional to n in order to secure a draw. Regarding probabilistic strategies, previous work showed that assuming Alice has access to a secret random perfect matching over {1, 2, . . ., 2n} allows her to achieve a draw in the game w.p. at least 1 − n1 and using only polylog bits of memory. We show that the requirement for secret bits is crucial: for an ‘open book’ Alice with no secrets (Bob knows her memory but not future coin flips) and memory of at most n/4c bits for any c ≥ 2, there is a Bob that wins w.p. close to 1 − 2−c/2 Copyright © 2022, The Authors. All rights reserved.

关键词： Mirrors

Debiasing Synthetic Data Generated by Deep Generative Models

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arXiv 2024年

作者： Decruyenaere, Alexander Dehaene, Heidelinde Rabaey, Paloma Polet, Christiaan Decruyenaere, Johan Demeester, Thomas Vansteelandt, Stijn Ghent University Hospital SYNDARA Belgium Ghent University imec Belgium Ghent University Department of Applied Mathematics Computer Science and Statistics Belgium

While synthetic data hold great promise for privacy protection, their statistical analysis poses significant challenges that necessitate innovative solutions. The use of deep generative models (DGMs) for synthetic data generation is known to induce considerable bias and imprecision into synthetic data analyses, compromising their inferential utility as opposed to original data analyses. This bias and uncertainty can be substantial enough to impede statistical convergence rates, even in seemingly straightforward analyses like mean calculation. The standard errors of such estimators then exhibit slower shrinkage with sample size than the typical 1 over root-n rate. This complicates fundamental calculations like p-values and confidence intervals, with no straightforward remedy currently available. In response to these challenges, we propose a new strategy that targets synthetic data created by DGMs for specific data analyses. Drawing insights from debiased and targeted machine learning, our approach accounts for biases, enhances convergence rates, and facilitates the calculation of estimators with easily approximated large sample variances. We exemplify our proposal through a simulation study on toy data and two case studies on real-world data, highlighting the importance of tailoring DGMs for targeted data analysis. This debiasing strategy contributes to advancing the reliability and applicability of synthetic data in statistical inference. © 2024, CC BY.

关键词： Differential privacy

A Penny for Your (visual) Thoughts: Self-Supervised Reconstruction of Natural Movies from Brain Activity

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arXiv 2022年

作者： Kupershmidt, Ganit Beliy, Roman Gaziv, Guy Irani, Michal Department of Computer Science and Applied Mathematics The Weizmann Institute of Science Rehovot Israel

Reconstructing natural videos from fMRI brain recordings is very challenging, for two main reasons: (i) As fMRI data acquisition is difficult, we only have a limited amount of supervised samples, which is not enough to cover the huge space of natural videos;and (ii) The temporal resolution of fMRI recordings is much lower than the frame rate of natural videos. In this paper, we propose a self-supervised approach for natural-movie reconstruction. By employing cycle-consistency over Encoding-Decoding natural videos, we can: (i) exploit the full framerate of the training videos, and not be limited only to clips that correspond to fMRI recordings;(ii) exploit massive amounts of external natural videos which the subjects never saw inside the fMRI machine. These enable increasing the applicable training data by several orders of magnitude, introducing natural video priors to the decoding network, as well as temporal coherence. Our approach significantly outperforms competing methods, since those train only on the limited supervised data. We further introduce a new and simple temporal prior of natural videos, which – when folded into our fMRI decoder – further allows us to reconstruct videos at a higher frame-rate (HFR) of up to x8 of the original fMRI sample rate. © 2022, CC BY-NC-SA.

关键词： Data acquisition

MinePlanner: A Benchmark for Long-Horizon Planning in Large Minecraft Worlds

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arXiv 2023年

作者： Hill, William Liu, Ireton De Mello Koch, Anita Harvey, Damion Kumar, Nishanth Konidaris, George James, Steven School of Computer Science and Applied Mathematics University of the Witwatersrand South Africa Department of Computer Science Brown University United States MIT Computer Science and Artificial Intelligence Laboratory United States

We propose a new benchmark for planning tasks based on the Minecraft game. Our benchmark contains 45 tasks overall, but also provides support for creating both propositional and numeric instances of new Minecraft tasks automatically. We benchmark numeric and propositional planning systems on these tasks, with results demonstrating that state-of-the-art planners are currently incapable of dealing with many of the challenges advanced by our new benchmark, such as scaling to instances with thousands of objects. Based on these results, we identify areas of improvement for future planners. Our framework is made available at https://***/IretonLiu/mine-pddl/. Copyright © 2023, The Authors. All rights reserved.

关键词： Machine learning