IA Scholar Query: Implementing regularization implicitly via approximate eigenvector computation.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgMon, 26 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Delayed Geometric Discounts: An Alternative Criterion for Reinforcement Learning
https://scholar.archive.org/work/r2pq2vmzgzdbdi2mdkhgwm3xtm
The endeavor of artificial intelligence (AI) is to design autonomous agents capable of achieving complex tasks. Namely, reinforcement learning (RL) proposes a theoretical background to learn optimal behaviors. In practice, RL algorithms rely on geometric discounts to evaluate this optimality. Unfortunately, this does not cover decision processes where future returns are not exponentially less valuable. Depending on the problem, this limitation induces sample-inefficiency (as feed-backs are exponentially decayed) and requires additional curricula/exploration mechanisms (to deal with sparse, deceptive or adversarial rewards). In this paper, we tackle these issues by generalizing the discounted problem formulation with a family of delayed objective functions. We investigate the underlying RL problem to derive: 1) the optimal stationary solution and 2) an approximation of the optimal non-stationary control. The devised algorithms solved hard exploration problems on tabular environment and improved sample-efficiency on classic simulated robotics benchmarks.Firas Jarboui, Ahmed Akakziawork_r2pq2vmzgzdbdi2mdkhgwm3xtmMon, 26 Sep 2022 00:00:00 GMTTurbulence as Clebsch Confinement
https://scholar.archive.org/work/qrlmjshh65cfddfvw4x3lbhb44
We argue that in the strong turbulence phase, as opposed to the weak one, the Clebsch variables compactify to the sphere S_2 and are not observable as wave excitations like weak turbulence. Various topologically nontrivial configurations of this confined Clebsch field are responsible for vortex sheets. Stability equations (CVS) for closed vortex surfaces (bubbles of Clebsch field) are derived and investigated. The exact non-compact solution for the stable vortex sheet family is presented. Compact solutions are proven not to exist by De Lellis and Brué. Asymptotic conservation of anomalous dissipation on stable vortex surfaces in the turbulent limit is discovered. We derive an exact formula for this anomalous dissipation as a surface integral of the square of velocity gap times the square root of minus local normal strain. Topologically stable time-dependent solutions, which we call Kelvinons, are introduced. They have a conserved velocity circulation around static loop; this makes them responsible for asymptotic PDF tails of velocity circulation, perfectly matching numerical simulations. The loop equation for fluid dynamics is derived and studied. This equation is exactly equivalent to the Schrödinger equation in loop space, with viscosity ν playing the role of Planck's constant. Area law and the asymptotic scaling law for mean circulation at a large area are derived. The exact representation of the solution of the loop equation in terms of a singular stochastic equation for momentum loop trajectory is presented. Kelvinons are fixed points of the loop equation at turbulent limit ν→ 0. The Loop equation's linearity makes the PDF's general solution to be a superposition of Kelvinon solutions with different winding numbers.Alexander Migdalwork_qrlmjshh65cfddfvw4x3lbhb44Sun, 25 Sep 2022 00:00:00 GMTDeep Neural Networks for Visual Reasoning
https://scholar.archive.org/work/viks3f7ou5fkvegimdooztmaxi
Visual perception and language understanding are - fundamental components of human intelligence, enabling them to understand and reason about objects and their interactions. It is crucial for machines to have this capacity to reason using these two modalities to invent new robot-human collaborative systems. Recent advances in deep learning have built separate sophisticated representations of both visual scenes and languages. However, understanding the associations between the two modalities in a shared context for multimodal reasoning remains a challenge. Focusing on language and vision modalities, this thesis advances the understanding of how to exploit and use pivotal aspects of vision-and-language tasks with neural networks to support reasoning. We derive these understandings from a series of works, making a two-fold contribution: (i) effective mechanisms for content selection and construction of temporal relations from dynamic visual scenes in response to a linguistic query and preparing adequate knowledge for the reasoning process (ii) new frameworks to perform reasoning with neural networks by exploiting visual-linguistic associations, deduced either directly from data or guided by external priors.Thao Minh Lework_viks3f7ou5fkvegimdooztmaxiSat, 24 Sep 2022 00:00:00 GMTDynamic Cone-beam CT Reconstruction using Spatial and Temporal Implicit Neural Representation Learning (STINR)
https://scholar.archive.org/work/mduct3vyfbbptew3gqidi2llbu
Objective: Dynamic cone-beam CT (CBCT) imaging is highly desired in image-guided radiation therapy to provide volumetric images with high spatial and temporal resolutions to enable applications including tumor motion tracking/prediction and intra-delivery dose calculation/accumulation. However, the dynamic CBCT reconstruction is a substantially challenging spatiotemporal inverse problem, due to the extremely limited projection sample available for each CBCT reconstruction (one projection for one CBCT volume). Approach: We developed a simultaneous spatial and temporal implicit neural representation (STINR) method for dynamic CBCT reconstruction. STINR mapped the unknown image and the evolution of its motion into spatial and temporal multi-layer perceptrons (MLPs), and iteratively optimized the neuron weighting of the MLPs via acquired projections to represent the dynamic CBCT series. In addition to the MLPs, we also introduced prior knowledge, in form of principal component analysis (PCA)-based patient-specific motion models, to reduce the complexity of the temporal INRs to address the ill-conditioned dynamic CBCT reconstruction problem. We used the extended cardiac torso (XCAT) phantom to simulate different lung motion/anatomy scenarios to evaluate STINR. The scenarios contain motion variations including motion baseline shifts, motion amplitude/frequency variations, and motion non-periodicity. The scenarios also contain inter-scan anatomical variations including tumor shrinkage and tumor position change. Main results: STINR shows consistently higher image reconstruction and motion tracking accuracy than a traditional PCA-based method and a polynomial-fitting based neural representation method. STINR tracks the lung tumor to an averaged center-of-mass error of <2 mm, with corresponding relative errors of reconstructed dynamic CBCTs <10%.You Zhang, Tielige Mengkework_mduct3vyfbbptew3gqidi2llbuFri, 23 Sep 2022 00:00:00 GMTEnsemble Kalman Methods: A Mean Field Perspective
https://scholar.archive.org/work/iyvmm22oongv3aylwgphbpuc3y
This paper provides a unifying mean field based framework for the derivation and analysis of ensemble Kalman methods. Both state estimation and parameter estimation problems are considered, and formulations in both discrete and continuous time are employed. For state estimation problems both the control and filtering approaches are studied; analogously, for parameter estimation (inverse) problems the optimization and Bayesian perspectives are both studied. The approach taken unifies a wide-ranging literature in the field, provides a framework for analysis of ensemble Kalman methods, and suggests open problems.Edoardo Calvello, Sebastian Reich, Andrew M. Stuartwork_iyvmm22oongv3aylwgphbpuc3yFri, 23 Sep 2022 00:00:00 GMTA singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations
https://scholar.archive.org/work/o5ofot3zyvggfee6zymipmq3ke
Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. The strategies employed to investigate their theoretical properties mainly rely on Euclidean geometry, but in the last years new approaches based on Riemannian geometry have been developed. Motivated by some open problems, we study a particular sequence of maps between manifolds, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the structures induced trough pullbacks on the other manifolds of the sequence and on some related quotients. In particular, we show that the pullbacks of the final Riemannian metric to any manifolds of the sequence is a degenerate Riemannian metric inducing a structure of pseudometric space, we show that the Kolmogorov quotient of this pseudometric space yields a smooth manifold, which is the base space of a particular vertical bundle. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between manifolds implementing neural networks of practical interest and we present some applications of the geometric framework we introduced in the first part of the paper.Alessandro Benfenati, Alessio Martawork_o5ofot3zyvggfee6zymipmq3keFri, 23 Sep 2022 00:00:00 GMTArtificial Intelligence and Advanced Materials
https://scholar.archive.org/work/tkf566mg6zf77a7xan6anloxvu
Artificial intelligence is gaining strength and materials science can both contribute to and profit from it. In a simultaneous progress race, new materials, systems and processes can be devised and optimized thanks to machine learning techniques and such progress can be turned into in-novative computing platforms. Future materials scientists will profit from understanding how machine learning can boost the conception of advanced materials. This review covers aspects of computation from the fundamentals to directions taken and repercussions produced by compu-tation to account for the origins, procedures and applications of artificial intelligence. Machine learning and its methods are reviewed to provide basic knowledge on its implementation and its potential. The materials and systems used to implement artificial intelligence with electric charges are finding serious competition from other information carrying and processing agents. The impact these techniques are having on the inception of new advanced materials is so deep that a new paradigm is developing where implicit knowledge is being mined to conceive materi-als and systems for functions instead of finding applications to found materials. How far this trend can be carried is hard to fathom as exemplified by the power to discover unheard of mate-rials or physical laws buried in data.Cefe Lópezwork_tkf566mg6zf77a7xan6anloxvuFri, 23 Sep 2022 00:00:00 GMTFrontiers, challenges, and solutions in modeling of swift heavy ion effects in materials
https://scholar.archive.org/work/kleq2g2lrnftbgdo2c5t6xb3bq
Since a few breakthroughs in the fundamental understanding of the effects of swift heavy ions (SHI) decelerating in the electronic stopping regime in the matter have been achieved in the last decade, it motivated us to review the state-of-the-art approaches in the modeling of SHI effects. The SHI track kinetics occurs via several well-separated stages: from attoseconds in ion-impact ionization depositing energy in a target, to femtoseconds of electron transport and hole cascades, to picoseconds of lattice excitation and response, to nanoseconds of atomic relaxation, and even longer macroscopic reaction. Each stage requires its own approaches for quantitative description. We discuss that understanding the links between the stages makes it possible to describe the entire track kinetics within a multiscale model without fitting procedures. The review focuses on the underlying physical mechanisms of each process, the dominant effects they produce, and the limitations of the existing approaches as well as various numerical techniques implementing these models. It provides an overview of ab-initio-based modeling of the evolution of the electronic properties; Monte Carlo simulations of nonequilibrium electronic transport; molecular dynamics modeling of atomic reaction on the surface and in the bulk; kinetic Mote Carlo of atomic defect kinetics; finite-difference methods of tracks interaction with chemical solvents describing etching kinetics. We outline the modern methods that couple these approaches into multiscale multidisciplinary models and point to their bottlenecks, strengths, and weaknesses. The analysis is accompanied by examples of important results improving the understanding of track formation in various materials. Summarizing the most recent advances in the field of the track formation process, the review delivers a comprehensive picture and detailed understanding of the phenomena.N. Medvedev, A.E. Volkov, R. Rymzhanov, F. Akhmetov, S. Gorbunov, R. Voronkov, P. Babaevwork_kleq2g2lrnftbgdo2c5t6xb3bqThu, 22 Sep 2022 00:00:00 GMTProximal Point Imitation Learning
https://scholar.archive.org/work/nkf7uknumrhjberzwfptwas64i
This work develops new algorithms with rigorous efficiency guarantees for infinite horizon imitation learning (IL) with linear function approximation without restrictive coherence assumptions. We begin with the minimax formulation of the problem and then outline how to leverage classical tools from optimization, in particular, the proximal-point method (PPM) and dual smoothing, for online and offline IL, respectively. Thanks to PPM, we avoid nested policy evaluation and cost updates for online IL appearing in the prior literature. In particular, we do away with the conventional alternating updates by the optimization of a single convex and smooth objective over both cost and Q-functions. When solved inexactly, we relate the optimization errors to the suboptimality of the recovered policy. As an added bonus, by re-interpreting PPM as dual smoothing with the expert policy as a center point, we also obtain an offline IL algorithm enjoying theoretical guarantees in terms of required expert trajectories. Finally, we achieve convincing empirical performance for both linear and neural network function approximation.Luca Viano and Angeliki Kamoutsi and Gergely Neu and Igor Krawczuk and Volkan Cevherwork_nkf7uknumrhjberzwfptwas64iThu, 22 Sep 2022 00:00:00 GMTMatrix factorisation and the interpretation of geodesic distance
https://scholar.archive.org/work/aifzqflr45ddxjchkyf4hiosra
Given a graph or similarity matrix, we consider the problem of recovering a notion of true distance between the nodes, and so their true positions. We show that this can be accomplished in two steps: matrix factorisation, followed by nonlinear dimension reduction. This combination is effective because the point cloud obtained in the first step lives close to a manifold in which latent distance is encoded as geodesic distance. Hence, a nonlinear dimension reduction tool, approximating geodesic distance, can recover the latent positions, up to a simple transformation. We give a detailed account of the case where spectral embedding is used, followed by Isomap, and provide encouraging experimental evidence for other combinations of techniques.Nick Whiteley, Annie Gray, Patrick Rubin-Delanchywork_aifzqflr45ddxjchkyf4hiosraThu, 22 Sep 2022 00:00:00 GMTDeveloping, Evaluating and Scaling Learning Agents in Multi-Agent Environments
https://scholar.archive.org/work/r2sl3udro5fhjn6ofsxl5lqhaq
The Game Theory & Multi-Agent team at DeepMind studies several aspects of multi-agent learning ranging from computing approximations to fundamental concepts in game theory to simulating social dilemmas in rich spatial environments and training 3-d humanoids in difficult team coordination tasks. A signature aim of our group is to use the resources and expertise made available to us at DeepMind in deep reinforcement learning to explore multi-agent systems in complex environments and use these benchmarks to advance our understanding. Here, we summarise the recent work of our team and present a taxonomy that we feel highlights many important open challenges in multi-agent research.Ian Gemp, Thomas Anthony, Yoram Bachrach, Avishkar Bhoopchand, Kalesha Bullard, Jerome Connor, Vibhavari Dasagi, Bart De Vylder, Edgar Duenez-Guzman, Romuald Elie, Richard Everett, Daniel Hennes, Edward Hughes, Mina Khan, Marc Lanctot, Kate Larson, Guy Lever, Siqi Liu, Luke Marris, Kevin R. McKee, Paul Muller, Julien Perolat, Florian Strub, Andrea Tacchetti, Eugene Tarassov, Zhe Wang, Karl Tuylswork_r2sl3udro5fhjn6ofsxl5lqhaqThu, 22 Sep 2022 00:00:00 GMTScaled σ-functionals for the Kohn-Sham correlation energy with scaling functions from the homogeneous electron gas
https://scholar.archive.org/work/csw3ckg7lzarpemvireq3uz2wa
The recently introduced σ-functionals constitute a new type of functionals for the Kohn-Sham (KS) correlation energy. σ-Functionals are based on the adiabatic-connection fluctuation-dissipation theorem, are computationally closely related to the well-known direct random phase approximation (dRPA), and are formally rooted in many-body perturbation theory along the adiabatic connection. In σ-functionals, the function of the eigenvalues σ of the Kohn-Sham response matrix that enters the coupling constant and frequency integration in the dRPA is replaced by another function optimized with the help of reference sets of atomization, reaction, transition state, and non-covalent interaction energies. σ-Functionals are highly accurate and yield chemical accuracy of 1 kcal/mol in reaction or transition state energies, in main group chemistry. A shortcoming of σ-functionals is their inability to accurately describe processes involving a change of the electron number, such as ionizations or electron attachments. This problem is attributed to unphysical self-interactions caused by the neglect of the exchange kernel in the dRPA and σ-functionals. Here, we tackle this problem by introducing a frequency- and σ-dependent scaling of the eigenvalues σ of the KS response function that models the effect of the exchange kernel. The scaling factors are determined with the help of the homogeneous electron gas. The resulting scaled σ-functionals retain the accuracy of their unscaled parent functionals but in addition yield very accurate ionization potentials and electron affinities. Moreover, atomization and total energies are found to be exceptionally accurate. Scaled σ-functionals are computationally highly efficient like their unscaled counterparts.Jannis Erhard, Steffen Fauser, Egor Trushin, Andreas Görlingwork_csw3ckg7lzarpemvireq3uz2waWed, 21 Sep 2022 00:00:00 GMTA Riemann–Hilbert approach to the perturbation theory for orthogonal polynomials: Applications to numerical linear algebra and random matrix theory
https://scholar.archive.org/work/kos63w7dxbfbfpcti37kuffmoq
We establish a new perturbation theory for orthogonal polynomials using a Riemann--Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the orthogonal polynomials with respect to two measures can be effectively compared using the difference of their Stieltjes transforms on a suitably chosen contour. Moreover, when two measures are close and satisfy some regularity conditions, we use the theta functions of a hyperelliptic Riemann surface to derive explicit and accurate expansion formulae for the perturbed orthogonal polynomials. In contrast to other approaches, a key strength of the methodology is that estimates can remain valid as the degree of the polynomial grows. The results are applied to analyze several numerical algorithms from linear algebra, including the Lanczos tridiagonalization procedure, the Cholesky factorization and the conjugate gradient algorithm. As a case study, we investigate these algorithms applied to a general spiked sample covariance matrix model by considering the eigenvector empirical spectral distribution and its limits. For the first time, we give precise estimates on the output of the algorithms, applied to this wide class of random matrices, as the number of iterations diverges. In this setting, beyond the first order expansion, we also derive a new mesoscopic central limit theorem for the associated orthogonal polynomials and other quantities relevant to numerical algorithms.Xiucai Ding, Thomas Trogdonwork_kos63w7dxbfbfpcti37kuffmoqWed, 21 Sep 2022 00:00:00 GMTA probabilistic deep learning model of inter-fraction anatomical variations in radiotherapy
https://scholar.archive.org/work/4oqx4nr46nhwnfiyb6dxr2azoi
In radiotherapy, the internal movement of organs between treatment sessions causes errors in the final radiation dose delivery. Motion models can be used to simulate motion patterns and assess anatomical robustness before delivery. Traditionally, such models are based on principal component analysis (PCA) and are either patient-specific (requiring several scans per patient) or population-based, applying the same deformations to all patients. We present a hybrid approach which, based on population data, allows to predict patient-specific inter-fraction variations for an individual patient. We propose a deep learning probabilistic framework that generates deformation vector fields (DVFs) warping a patient's planning computed tomography (CT) into possible patient-specific anatomies. This daily anatomy model (DAM) uses few random variables capturing groups of correlated movements. Given a new planning CT, DAM estimates the joint distribution over the variables, with each sample from the distribution corresponding to a different deformation. We train our model using dataset of 312 CT pairs from 38 prostate cancer patients. For 2 additional patients (22 CTs), we compute the contour overlap between real and generated images, and compare the sampled and ground truth distributions of volume and center of mass changes. With a DICE score of 0.86 and a distance between prostate contours of 1.09 mm, DAM matches and improves upon PCA-based models. The distribution overlap further indicates that DAM's sampled movements match the range and frequency of clinically observed daily changes on repeat CTs. Conditioned only on a planning CT and contours of a new patient without any pre-processing, DAM can accurately predict CTs seen during following treatment sessions, which can be used for anatomically robust treatment planning and robustness evaluation against inter-fraction anatomical changes.Oscar Pastor-Serrano, Steven Habraken, Mischa Hoogeman, Danny Lathouwers, Dennis Schaart, Yusuke Nomura, Lei Xing, Zoltán Perkówork_4oqx4nr46nhwnfiyb6dxr2azoiTue, 20 Sep 2022 00:00:00 GMTSynaptic balancing: A biologically plausible local learning rule that provably increases neural network noise robustness without sacrificing task performance
https://scholar.archive.org/work/3uibrtkhlncuncjpeghzg3cjpq
We introduce a novel, biologically plausible local learning rule that provably increases the robustness of neural dynamics to noise in nonlinear recurrent neural networks with homogeneous nonlinearities. Our learning rule achieves higher noise robustness without sacrificing performance on the task and without requiring any knowledge of the particular task. The plasticity dynamics-an integrable dynamical system operating on the weights of the network-maintains a multiplicity of conserved quantities, most notably the network's entire temporal map of input to output trajectories. The outcome of our learning rule is a synaptic balancing between the incoming and outgoing synapses of every neuron. This synaptic balancing rule is consistent with many known aspects of experimentally observed heterosynaptic plasticity, and moreover makes new experimentally testable predictions relating plasticity at the incoming and outgoing synapses of individual neurons. Overall, this work provides a novel, practical local learning rule that exactly preserves overall network function and, in doing so, provides new conceptual bridges between the disparate worlds of the neurobiology of heterosynaptic plasticity, the engineering of regularized noise-robust networks, and the mathematics of integrable Lax dynamical systems.Christopher H Stock, Sarah E Harvey, Samuel A Ocko, Surya Ganguliwork_3uibrtkhlncuncjpeghzg3cjpqMon, 19 Sep 2022 00:00:00 GMTDIGRAC: Digraph Clustering Based on Flow Imbalance
https://scholar.archive.org/work/ey3lgxetkrgsrjjm4uxszkegim
Node clustering is a powerful tool in the analysis of networks. We introduce a graph neural network framework to obtain node embeddings for directed networks in a self-supervised manner, including a novel probabilistic imbalance loss, which can be used for network clustering. Here, we propose directed flow imbalance measures, which are tightly related to directionality, to reveal clusters in the network even when there is no density difference between clusters. In contrast to standard approaches in the literature, in this paper, directionality is not treated as a nuisance, but rather contains the main signal. DIGRAC optimizes directed flow imbalance for clustering without requiring label supervision, unlike existing graph neural network methods, and can naturally incorporate node features, unlike existing spectral methods. Extensive experimental results on synthetic data, in the form of directed stochastic block models, and real-world data at different scales, demonstrate that our method, based on flow imbalance, attains state-of-the-art results on directed graph clustering when compared against 10 state-of-the-art methods from the literature, for a wide range of noise and sparsity levels, graph structures and topologies, and even outperforms supervised methods.Yixuan He and Gesine Reinert and Mihai Cucuringuwork_ey3lgxetkrgsrjjm4uxszkegimSun, 18 Sep 2022 00:00:00 GMTApproximation results for Gradient Descent trained Shallow Neural Networks in 1d
https://scholar.archive.org/work/2hldwrv4bvaffcd2riocbmgnpq
Two aspects of neural networks that have been extensively studied in the recent literature are their function approximation properties and their training by gradient descent methods. The approximation problem seeks accurate approximations with a minimal number of weights. In most of the current literature these weights are fully or partially hand-crafted, showing the capabilities of neural networks but not necessarily their practical performance. In contrast, optimization theory for neural networks heavily relies on an abundance of weights in over-parametrized regimes. This paper balances these two demands and provides an approximation result for shallow networks in 1d with non-convex weight optimization by gradient descent. We consider finite width networks and infinite sample limits, which is the typical setup in approximation theory. Technically, this problem is not over-parametrized, however, some form of redundancy reappears as a loss in approximation rate compared to best possible rates.R. Gentile, G. Welperwork_2hldwrv4bvaffcd2riocbmgnpqSat, 17 Sep 2022 00:00:00 GMTWhen could NISQ algorithms start to create value in discrete manufacturing ?
https://scholar.archive.org/work/t4k7yrg27bh5dg5phuhygeqlci
Are quantum advantages in discrete manufacturing achievable in the near term? As manufacturing-relevant NISQ algorithms, we identified Quantum Annealing (QA) and the Quantum Approximate Optimization Algorithm (QAOA) for combinatorial optimization as well as Derivative Quantum Circuits (DQC) for solving non-linear PDEs. While there is evidence for QAOA's outperformance, this requires post-NISQ circuit depths. In the case of QA, there is up to now no unquestionable evidence for advantage compared to classical computation. Yet different protocols could lead to finding such instances. Together with a well-chosen quantum feature map, DQC are a promising concept. Further investigations for higher dimensional problems and improvements in training could follow.Oxana Shayawork_t4k7yrg27bh5dg5phuhygeqlciSat, 17 Sep 2022 00:00:00 GMTOn Weighted Graph Sparsification by Linear Sketching
https://scholar.archive.org/work/dpgdgz4hpvbuvpxjz3ey7lrw2a
A seminal work of [Ahn-Guha-McGregor, PODS'12] showed that one can compute a cut sparsifier of an unweighted undirected graph by taking a near-linear number of linear measurements on the graph. Subsequent works also studied computing other graph sparsifiers using linear sketching, and obtained near-linear upper bounds for spectral sparsifiers [Kapralov-Lee-Musco-Musco-Sidford, FOCS'14] and first non-trivial upper bounds for spanners [Filtser-Kapralov-Nouri, SODA'21]. All these linear sketching algorithms, however, only work on unweighted graphs. In this paper, we initiate the study of weighted graph sparsification by linear sketching by investigating a natural class of linear sketches that we call incidence sketches, in which each measurement is a linear combination of the weights of edges incident on a single vertex. Our results are: 1. Weighted cut sparsification: We give an algorithm that computes a (1 + ϵ)-cut sparsifier using Õ(n ϵ^-3) linear measurements, which is nearly optimal. 2. Weighted spectral sparsification: We give an algorithm that computes a (1 + ϵ)-spectral sparsifier using Õ(n^6/5ϵ^-4) linear measurements. Complementing our algorithm, we then prove a superlinear lower bound of Ω(n^21/20-o(1)) measurements for computing some O(1)-spectral sparsifier using incidence sketches. 3. Weighted spanner computation: We focus on graphs whose largest/smallest edge weights differ by an O(1) factor, and prove that, for incidence sketches, the upper bounds obtained by [Filtser-Kapralov-Nouri, SODA'21] are optimal up to an n^o(1) factor.Yu Chen, Sanjeev Khanna, Huan Liwork_dpgdgz4hpvbuvpxjz3ey7lrw2aFri, 16 Sep 2022 00:00:00 GMTFinding unstable periodic orbits for nonlinear dynamical systems using polynomial optimisation
https://scholar.archive.org/work/ofdrlw4v6rf7zf6ylretgfua74
Computing unstable periodic orbits (UPOs) for systems governed by ordinary differential equations (ODEs) is a fundamental problem in the study of nonlinear dynamical systems that exhibit chaotic dynamics. Success of any existing method to compute UPOs relies on the availability of very good initial guesses for both the UPO and its time period. This thesis presents a computational framework for computing UPOs that are extremal, in the sense that they optimise the infinite-time average of a certain observable. Constituting this framework are two novel techniques. The first is a method to localise extremal UPOs for polynomial ODE systems that does not rely on numerical integration. The UPO search procedure relies on polynomial optimisation to construct nonnegative polynomials whose sublevel sets approximately localise parts of the extremal periodic orbit. Points inside the relevant sublevel sets can then be computed efficiently through direct nonlinear optimisation. Such points provide good initial conditions for UPO computations with existing algorithms. The second technique involves the addition of a control term to the original polynomial ODE system to reduce the instability of the extremal UPO, and, in some cases, to provably stabilise it. This control methodology produces a family of controlled systems parametrised by a control amplitude, to which existing UPO-finding algorithms are often more easily applied. The practical potential of these techniques is demonstrated by applying them to find extremal UPOs for a nine-dimensional model of sinusoidally forced shear flow, an extended version of the Lorenz system, and two different three-dimensional chaotic ODE systems. Extensions of the framework to non-polynomial and Hamiltonian ODE systems are also discussed.Mayur Venkatram Lakshmi, Sergei Chernyshenko, Engineering And Physical Sciences Research Council (EPSRC)work_ofdrlw4v6rf7zf6ylretgfua74Fri, 16 Sep 2022 00:00:00 GMT