IA Scholar Query: SIAM Journal on Computing, Volume 6
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 29 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440DIGRAC: 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, named DIGRAC, 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_ey3lgxetkrgsrjjm4uxszkegimTue, 29 Nov 2022 00:00:00 GMT2019
https://scholar.archive.org/work/g6qfzbclcfe6pobzwry66zimou
On completion of this course, students will have knowledge in: • CO1.Basics of electrochemistry. Classical & modern batteries and fuel cells. CO2. Causes & effects of corrosion of metals and control of corrosion. Modification of surface properties of metals to develop resistance to corrosion, wear, tear, impact etc. by electroplating and electroless plating. CO3. Production & consumption of energy for industrialization of country and living standards of people. Utilization of solar energy for different useful forms of energy. CO4. Understanding Phase rule and instrumental techniques and its applications. CO5.Over viewing of synthesis, properties and applications of nanomaterials.BTECH.MECHwork_g6qfzbclcfe6pobzwry66zimouMon, 28 Nov 2022 00:00:00 GMTNumerical approximation of probabilistically weak and strong solutions of the stochastic total variation flow
https://scholar.archive.org/work/pdzbpla25vhlni74yph4iqw36q
We propose a fully practical numerical scheme for the simulation of the stochastic total variation flow (STVF). The approximation is based on a stable time-implicit finite element space-time approximation of a regularized STVF equation. The approximation also involves a finite dimensional discretization of the noise that makes the scheme fully implementable on physical hardware. We show that the proposed numerical scheme converges in law to a solution that is defined in the sense of stochastic variational inequalities (SVIs). Under strengthened assumptions the convergence can be show to holds even in probability. As a by product of our convergence analysis we provide a generalization of the concept of probabilistically weak solutions of stochastic partial differential equation (SPDEs) to the setting of SVIs. We also prove convergence of the numerical scheme to a probabilistically strong solution in probability if pathwise uniqueness holds. We perform numerical simulations to illustrate the behavior of the proposed numerical scheme {as well as its non-conforming variant} in the context of image denoising.Lubomir Banas, Martin Ondrejatwork_pdzbpla25vhlni74yph4iqw36qThu, 24 Nov 2022 00:00:00 GMTPerfect Sampling from Pairwise Comparisons
https://scholar.archive.org/work/vv2qscs2tja5tnfn4su7g56pty
In this work, we study how to efficiently obtain perfect samples from a discrete distribution 𝒟 given access only to pairwise comparisons of elements of its support. Specifically, we assume access to samples (x, S), where S is drawn from a distribution over sets 𝒬 (indicating the elements being compared), and x is drawn from the conditional distribution 𝒟_S (indicating the winner of the comparison) and aim to output a clean sample y distributed according to 𝒟. We mainly focus on the case of pairwise comparisons where all sets S have size 2. We design a Markov chain whose stationary distribution coincides with 𝒟 and give an algorithm to obtain exact samples using the technique of Coupling from the Past. However, the sample complexity of this algorithm depends on the structure of the distribution 𝒟 and can be even exponential in the support of 𝒟 in many natural scenarios. Our main contribution is to provide an efficient exact sampling algorithm whose complexity does not depend on the structure of 𝒟. To this end, we give a parametric Markov chain that mixes significantly faster given a good approximation to the stationary distribution. We can obtain such an approximation using an efficient learning from pairwise comparisons algorithm (Shah et al., JMLR 17, 2016). Our technique for speeding up sampling from a Markov chain whose stationary distribution is approximately known is simple, general and possibly of independent interest.Dimitris Fotakis, Alkis Kalavasis, Christos Tzamoswork_vv2qscs2tja5tnfn4su7g56ptyWed, 23 Nov 2022 00:00:00 GMTEfficient List-Decodable Regression using Batches
https://scholar.archive.org/work/n3g4c66bkzagzkcgxpvx6mzvr4
We begin the study of list-decodable linear regression using batches. In this setting only an α∈ (0,1] fraction of the batches are genuine. Each genuine batch contains ≥ n i.i.d. samples from a common unknown distribution and the remaining batches may contain arbitrary or even adversarial samples. We derive a polynomial time algorithm that for any n≥Ω̃(1/α) returns a list of size 𝒪(1/α^2) such that one of the items in the list is close to the true regression parameter. The algorithm requires only 𝒪̃(d/α^2) genuine batches and works under fairly general assumptions on the distribution. The results demonstrate the utility of batch structure, which allows for the first polynomial time algorithm for list-decodable regression, which may be impossible for the non-batch setting, as suggested by a recent SQ lower bound for the non-batch setting.Abhimanyu Das, Ayush Jain, Weihao Kong, Rajat Senwork_n3g4c66bkzagzkcgxpvx6mzvr4Wed, 23 Nov 2022 00:00:00 GMTProximal Stochastic Recursive Momentum Methods for Nonconvex Composite Decentralized Optimization
https://scholar.archive.org/work/gn6chyvf55fp7m3nru4urgyvzi
Consider a network of N decentralized computing agents collaboratively solving a nonconvex stochastic composite problem. In this work, we propose a single-loop algorithm, called DEEPSTORM, that achieves optimal sample complexity for this setting. Unlike double-loop algorithms that require a large batch size to compute the (stochastic) gradient once in a while, DEEPSTORM uses a small batch size, creating advantages in occasions such as streaming data and online learning. This is the first method achieving optimal sample complexity for decentralized nonconvex stochastic composite problems, requiring 𝒪(1) batch size. We conduct convergence analysis for DEEPSTORM with both constant and diminishing step sizes. Additionally, under proper initialization and a small enough desired solution error, we show that DEEPSTORM with a constant step size achieves a network-independent sample complexity, with an additional linear speed-up with respect to N over centralized methods. All codes are made available at .Gabriel Mancino-Ball, Shengnan Miao, Yangyang Xu, Jie Chenwork_gn6chyvf55fp7m3nru4urgyvziTue, 22 Nov 2022 00:00:00 GMTPreconditioners for computing multiple solutions in three-dimensional fluid topology optimization
https://scholar.archive.org/work/b2vjh3wg5rf7jk33qbrpiqyxvi
Topology optimization problems generally support multiple local minima, and real-world applications are typically three-dimensional. In previous work [I. P. A. Papadopoulos, P. E. Farrell, and T. M. Surowiec, Computing multiple solutions of topology optimization problems, SIAM Journal on Scientific Computing, (2021)], the authors developed the deflated barrier method, an algorithm that can systematically compute multiple solutions of topology optimization problems. In this work we develop preconditioners for the linear systems arising in the application of this method to Stokes flow, making it practical for use in three dimensions. In particular, we develop a nested block preconditioning approach which reduces the linear systems to solving two symmetric positive-definite matrices and an augmented momentum block. An augmented Lagrangian term is used to control the innermost Schur complement and we apply a geometric multigrid method with a kernel-capturing relaxation method for the augmented momentum block. We present multiple solutions in three-dimensional examples computed using the proposed iterative solver.Ioannis P. A. Papadopoulos, Patrick E. Farrellwork_b2vjh3wg5rf7jk33qbrpiqyxviTue, 22 Nov 2022 00:00:00 GMTAutomatic Detection of Horner Syndrome by Using Facial Images
https://scholar.archive.org/work/plvveonetzgcjpsigjnfi35xdu
Horner syndrome is a clinical constellation that presents with miosis, ptosis, and facial anhidrosis. It is important as a warning sign of the damaged oculosympathetic chain, potentially with serious causes. However, the diagnosis of Horner syndrome is operator dependent and subjective. This study aims to present an objective method that can recognize Horner sign from facial photos and verify its accuracy. A total of 173 images were collected, annotated, and divided into training and testing groups. Two types of classifiers were trained (two-stage classifier and one-stage classifier). The two-stage method utilized the MediaPipe face mesh to estimate the coordinates of landmarks and generate facial geometric features accordingly. Then, ten machine learning classifiers were trained based on this. The one-stage classifier was trained based on one of the latest algorithms, YOLO v5. The performance of the classifier was evaluated by the diagnosis accuracy, sensitivity, and specificity. For the two-stage model, the MediaPipe successfully detected 92.2% of images in the testing group, and the Decision Tree Classifier presented the highest accuracy (0.790). The sensitivity and specificity of this classifier were 0.432 and 0.970, respectively. As for the one-stage classifier, the accuracy, sensitivity, and specificity were 0.65, 0.51, and 0.84, respectively. The results of this study proved the possibility of automatic detection of Horner syndrome from images. This tool could work as a second advisor for neurologists by reducing subjectivity and increasing accuracy in diagnosing Horner syndrome.Jingyuan Fan, Bengang Qin, Fanbin Gu, Zhaoyang Wang, Xiaolin Liu, Qingtang Zhu, Jiantao Yang, Mehdi Gheisariwork_plvveonetzgcjpsigjnfi35xduMon, 21 Nov 2022 00:00:00 GMTLinear Convergence of Natural Policy Gradient Methods with Log-Linear Policies
https://scholar.archive.org/work/fgxu4e4myncqlha2hgg5l2aeau
We consider infinite-horizon discounted Markov decision processes and study the convergence rates of the natural policy gradient (NPG) and the Q-NPG methods with the log-linear policy class. Using the compatible function approximation framework, both methods with log-linear policies can be written as inexact versions of the policy mirror descent (PMD) method. We show that both methods attain linear convergence rates and 𝒪̃(1/ϵ^2) sample complexities using a simple, non-adaptive geometrically increasing step size, without resorting to entropy or other strongly convex regularization. Lastly, as a byproduct, we obtain sublinear convergence rates for both methods with arbitrary constant step size.Rui Yuan, Simon S. Du, Robert M. Gower, Alessandro Lazaric, Lin Xiaowork_fgxu4e4myncqlha2hgg5l2aeauMon, 21 Nov 2022 00:00:00 GMTA priori error estimates of two fully discrete coupled schemes for Biot's consolidation model
https://scholar.archive.org/work/dtqachthwfgvzna2vkpbpibbk4
This paper concentrates on a priori error estimates of two fully discrete coupled schemes for Biot's consolidation model based on the three-field formulation introduced by Oyarzua et al. (SIAM Journal on Numerical Analysis, 2016). The spatial discretizations are based on the Taylor-Hood finite elements combined with Lagrange elements for the three primary variables. For time discretization, we consider two methods. One uses the backward Euler method, and the other applies a combination of the backward Euler and Crank-Nicolson methods. A priori error estimates show that the two schemes are unconditionally convergent with optimal error orders. Detailed numerical experiments are presented to validate the theoretical analysis.Huipeng Gu, Mingchao Cai, Jingzhi Li, Guoliang Juwork_dtqachthwfgvzna2vkpbpibbk4Sat, 19 Nov 2022 00:00:00 GMTA general sample complexity analysis of vanilla policy gradient
https://scholar.archive.org/work/bgnkgagrjbeenn64qgo5vqbc34
We adapt recent tools developed for the analysis of Stochastic Gradient Descent (SGD) in non-convex optimization to obtain convergence and sample complexity guarantees for the vanilla policy gradient (PG). Our only assumptions are that the expected return is smooth w.r.t. the policy parameters, that its H-step truncated gradient is close to the exact gradient, and a certain ABC assumption. This assumption requires the second moment of the estimated gradient to be bounded by A≥ 0 times the suboptimality gap, B ≥ 0 times the norm of the full batch gradient and an additive constant C ≥ 0, or any combination of aforementioned. We show that the ABC assumption is more general than the commonly used assumptions on the policy space to prove convergence to a stationary point. We provide a single convergence theorem that recovers the 𝒪(ϵ^-4) sample complexity of PG to a stationary point. Our results also affords greater flexibility in the choice of hyper parameters such as the step size and the batch size m, including the single trajectory case (i.e., m=1). When an additional relaxed weak gradient domination assumption is available, we establish a novel global optimum convergence theory of PG with 𝒪(ϵ^-3) sample complexity. We then instantiate our theorems in different settings, where we both recover existing results and obtain improved sample complexity, e.g., 𝒪(ϵ^-3) sample complexity for the convergence to the global optimum for Fisher-non-degenerated parametrized policies.Rui Yuan, Robert M. Gower, Alessandro Lazaricwork_bgnkgagrjbeenn64qgo5vqbc34Fri, 18 Nov 2022 00:00:00 GMTThe communication cost of security and privacy in federated frequency estimation
https://scholar.archive.org/work/2vd3z3yevzbkldfe7wcfgkuswe
We consider the federated frequency estimation problem, where each user holds a private item X_i from a size-d domain and a server aims to estimate the empirical frequency (i.e., histogram) of n items with n ≪ d. Without any security and privacy considerations, each user can communicate its item to the server by using log d bits. A naive application of secure aggregation protocols would, however, require dlog n bits per user. Can we reduce the communication needed for secure aggregation, and does security come with a fundamental cost in communication? In this paper, we develop an information-theoretic model for secure aggregation that allows us to characterize the fundamental cost of security and privacy in terms of communication. We show that with security (and without privacy) Ω( n log d ) bits per user are necessary and sufficient to allow the server to compute the frequency distribution. This is significantly smaller than the dlog n bits per user needed by the naive scheme, but significantly higher than the log d bits per user needed without security. To achieve differential privacy, we construct a linear scheme based on a noisy sketch which locally perturbs the data and does not require a trusted server (a.k.a. distributed differential privacy). We analyze this scheme under ℓ_2 and ℓ_∞ loss. By using our information-theoretic framework, we show that the scheme achieves the optimal accuracy-privacy trade-off with optimal communication cost, while matching the performance in the centralized case where data is stored in the central server.Wei-Ning Chen, Ayfer Özgür, Graham Cormode, Akash Bharadwajwork_2vd3z3yevzbkldfe7wcfgkusweFri, 18 Nov 2022 00:00:00 GMTAdaptive Reduced Basis Methods for Multiscale Problems and Large-scale PDE-constrained Optimization
https://scholar.archive.org/work/43ykt52yc5b4tduxemv42ohgxy
This thesis presents recent advances in model order reduction methods with the primary aim to construct online-efficient reduced surrogate models for parameterized multiscale phenomena and accelerate large-scale PDE-constrained parameter optimization methods. In particular, we present several different adaptive RB approaches that can be used in an error-aware trust-region framework for progressive construction of a surrogate model used during a certified outer optimization loop. In addition, we elaborate on several different enhancements for the trust-region reduced basis (TR-RB) algorithm and generalize it for parameter constraints. Thanks to the a posteriori error estimation of the reduced model, the resulting algorithm can be considered certified with respect to the high-fidelity model. Moreover, we use the first-optimize-then-discretize approach in order to take maximum advantage of the underlying optimality system of the problem. In the first part of this thesis, the theory is based on global RB techniques that use an accurate FEM discretization as the high-fidelity model. In the second part, we focus on localized model order reduction methods and develop a novel online efficient reduced model for the localized orthogonal decomposition (LOD) multiscale method. The reduced model is internally based on a two-scale formulation of the LOD and, in particular, is independent of the coarse and fine discretization of the LOD. The last part of this thesis is devoted to combining both results on TR-RB methods and localized RB approaches for the LOD. To this end, we present an algorithm that uses adaptive localized reduced basis methods in the framework of a trust-region localized reduced basis (TR-LRB) algorithm. The basic ideas from the TR-RB are followed, but FEM evaluations of the involved systems are entirely avoided.Tim Keilwork_43ykt52yc5b4tduxemv42ohgxyThu, 17 Nov 2022 00:00:00 GMTEfficient Gradient Flows in Sliced-Wasserstein Space
https://scholar.archive.org/work/u7hdgrxk75ceropnsoroqosite
Minimizing functionals in the space of probability distributions can be done with Wasserstein gradient flows. To solve them numerically, a possible approach is to rely on the Jordan-Kinderlehrer-Otto (JKO) scheme which is analogous to the proximal scheme in Euclidean spaces. However, it requires solving a nested optimization problem at each iteration, and is known for its computational challenges, especially in high dimension. To alleviate it, very recent works propose to approximate the JKO scheme leveraging Brenier's theorem, and using gradients of Input Convex Neural Networks to parameterize the density (JKO-ICNN). However, this method comes with a high computational cost and stability issues. Instead, this work proposes to use gradient flows in the space of probability measures endowed with the sliced-Wasserstein (SW) distance. We argue that this method is more flexible than JKO-ICNN, since SW enjoys a closed-form differentiable approximation. Thus, the density at each step can be parameterized by any generative model which alleviates the computational burden and makes it tractable in higher dimensions.Clément Bonet, Nicolas Courty, François Septier, Lucas Drumetzwork_u7hdgrxk75ceropnsoroqositeTue, 15 Nov 2022 00:00:00 GMTDeep Learning for Recommender System: Toward Structures and Sequences
https://scholar.archive.org/work/pku32vnwsne7fnkathcrkotzam
Recommender systems are nowadays everywhere to help users to explore online new resources, items, contents and etc. Along the booming of recommender systems, it is the urgent desire to develop applicable algorithms and frameworks to enhance the performance of recommendation results, where deep learning methods serve the best. In this thesis, I investigate two type of information that existing recommender systems are able to leverage, \textit{i.e.} the structural and sequential information. The former is focused in the interactions between users and items, where we construct graphs to characterize the underlying patterns. The latter incorporate interactions in chronological orders, which can thus reflect their sequential correlations. I present two papers associated with the structural information, including how to design spectral graph convolution for cross-domain recommendation and how to resolve basket recommendation with graph neural networks. Moreover, I introduce two papers associated with the sequential information, including augmenting sequential recommendation via pseudo-prior items and contrastive self-supervised learning for sequential recommendation. Additionally, I discuss how to simultaneously leverage the structure information and sequential information via the temporal collaborative transformer over a temporal graph. Finally, I conclude this thesis by summarizing deep learning techniques towards structures and sequences.Zhiwei Liuwork_pku32vnwsne7fnkathcrkotzamTue, 08 Nov 2022 00:00:00 GMTDimples, jets and self-similarity in nonlinear capillary waves
https://scholar.archive.org/work/ipgbcnily5bqzojvrobzcarlhy
Numerical studies of dimple and jet formation from a collapsing cavity often model the initial cavity shape as a truncated sphere, mimicking a bursting bubble. In this study, we present a minimal model containing only nonlinear inertial and capillary forces, which produces dimples and jets from a collapsing capillary wave trough. The trough in our simulation develops from a smooth initial perturbation, chosen to be an eigenmode to the linearised ${O}(\epsilon )$ problem ( $\epsilon$ is the non-dimensional amplitude). We explain the physical mechanism of dimple formation and demonstrate that, for moderate $\epsilon$ , the sharp dimple seen in simulations is well captured by the weakly nonlinear ${O}(\epsilon ^3)$ theory developed here. For $\epsilon \gg 1$ the regime is strongly nonlinear, spreading surface energy into many modes, and the precursor dimple now develops into a sharply rising jet. Here, simulations reveal a novel localised window (in space and time) where the jet evolves self-similarly following inviscid (Keller & Miksis, SIAM J. Appl. Maths, vol. 43, issue 2, 1983, pp. 268–277) scales. We develop an analogy of this regime to a self-similar solution of the first kind, for linearised capillary waves. Our first-principles study demonstrates that, at sufficiently small scales, dimples and jets can form from radial inward focusing of capillary waves, and the formation of this may be described by a relatively simple model employing (nonlinear) inertial and capillary effects. Viscosity and gravity can, however, significantly influence the focusing process, either intensifying the singularity or weakening it (Walls et al., Phys. Rev. E, vol. 92, issue 2, 2015, 021002; Gordillo & Rodríguez-Rodríguez, J. Fluid Mech., vol. 867, 2019, pp. 556–571). This leads, in particular, to critical values of Ohnesorge and Bond numbers, which cannot be obtained from our minimal model.Lohit Kayal, Saswata Basak, Ratul Dasguptawork_ipgbcnily5bqzojvrobzcarlhyTue, 08 Nov 2022 00:00:00 GMTHypermedia online publishing : the transformation of the scholarly journal
https://scholar.archive.org/work/m5jceljgm5c4bklrzdbxdjqjpq
This thesis looks at the impact of the technologies of networking and hypermedia on the scholarly journal. It does so in five main sections. The first section, Overview and Theory, begins by outlining the aims of the study and examining prior related work. Next it defines the three main theoretical perspectives that inform the research (a constructuralist ecology of communication, punctuated equilibrium, and a genre-based framework for new media) as well as considering and rejecting a number of lternatives. The second section. Publishing and Technology, first places the scholarly journal in its historical context and then identifies the stakeholders in the scholarly journal ecology. It then looks at the range of technology developments over the last twenty years that have the potential to be applied to scholarly comunication. The third section. Potentials and Responses, looks at the ways in which both publishing functions and stakeholder roles could be transformed and at some of the pressures for such a transformation. It then considers some of the responses that have developed because of these pressures and the potentials of the available technologies. The fourth section. Surveys and Case Studies, presents evidence gathered in this thesis project about users and libraries as key stakeholders. The survey is designed to gather evidence from users about their access to technology, use of electronic publishing, and attitudes to electronic journals. The library case studies look at leading edge examples of libraries who are actively facilitating electronic publishing. The final section. Interpretations and Conclusions, takes the results of all the research activities and discusses them in the context of possible transformations of the roles and practices of stakeholders and the form and function of journals. Evidence from each of the theoretical perspectives, research literature, survey and case studies is brought to bear on each transformation. The concluding chapter discusses the future of the journal as artefa [...]Andrew Edward Treloarwork_m5jceljgm5c4bklrzdbxdjqjpqMon, 07 Nov 2022 00:00:00 GMTSketches, metrics and fast algorithms
https://scholar.archive.org/work/q2ke5ehfxbhdzeqzosxoqh756q
As it has become easier and cheaper to collect big datasets in the last few decades, designing efficient and low-cost algorithms for these datasets has attracted unprecedented attention. However, in most applications, even storing datasets as acquired has become extremely costly and inefficient, which motivates the study of sublinear algorithms. This thesis focuses on studying two fundamental graph problems in the sublinear regime. Furthermore, it presents a fast kernel density estimation algorithm and data structure. The first part of this thesis focuses on graph spectral sparsification in dynamic streams. Our algorithm achieves almost optimal runtime and space simultaneously in a single pass. Our method is based on a novel bucketing scheme that enables us to recover high effective resistance edges faster. This contribution presents a novel approach to the effective resistance embedding of the graph, using locality-sensitive hash functions, with possible further future applications. The second part of this thesis presents spanner construction results in the dynamic streams and the simultaneous communication models. First, we show how one can construct a O(n 2/3 )-spanner using the above-mentioned almost-optimal single-pass spectral sparsifier, resulting in the first single-pass algorithm for non-trivial spanner construction in the literature. Then, we generalize this result to constructing O(n 2/3(1−α) )-spanners using O(n 1+α ) space for any α ∈ [0, 1], providing a smooth trade-off between distortion and memory complexity. Moreover, we study the simultaneous communication model and propose a novel protocol with low per player information. Also, we show how one can leverage more rounds of communication in this setting to achieve better distortion guarantees. Finally, in the third part of this thesis, we study the kernel density estimation problem. In this problem, given a kernel function, an input dataset imposes a kernel density on the space. The goal is to design fast and memory-efficient data structures that can output approximations to the kernel density at queried points. This thesis presents a data structure based on the classical near neighbor search and localitysensitive hashing techniques that improves or matches the query time and space complexity for radial kernels considered in the literature. The approach is based on an implementation of (approximate) importance sampling for each distance range and then using near neighbor search algorithms to recover points from these distance ranges. Later, we show how to improve the runtime, using recent advances in the data-dependent near neighbor search data structures, for a class of radial kernels that includes the Gaussian kernel.Navid Nouriwork_q2ke5ehfxbhdzeqzosxoqh756qMon, 07 Nov 2022 00:00:00 GMTLower Bounds for the Convergence of Tensor Power Iteration on Random Overcomplete Models
https://scholar.archive.org/work/czlpwjl64jfv7pavfu225talom
Tensor decomposition serves as a powerful primitive in statistics and machine learning. In this paper, we focus on using power iteration to decompose an overcomplete random tensor. Past work studying the properties of tensor power iteration either requires a non-trivial data-independent initialization, or is restricted to the undercomplete regime. Moreover, several papers implicitly suggest that logarithmically many iterations (in terms of the input dimension) are sufficient for the power method to recover one of the tensor components. In this paper, we analyze the dynamics of tensor power iteration from random initialization in the overcomplete regime. Surprisingly, we show that polynomially many steps are necessary for convergence of tensor power iteration to any of the true component, which refutes the previous conjecture. On the other hand, our numerical experiments suggest that tensor power iteration successfully recovers tensor components for a broad range of parameters, despite that it takes at least polynomially many steps to converge. To further complement our empirical evidence, we prove that a popular objective function for tensor decomposition is strictly increasing along the power iteration path. Our proof is based on the Gaussian conditioning technique, which has been applied to analyze the approximate message passing (AMP) algorithm. The major ingredient of our argument is a conditioning lemma that allows us to generalize AMP-type analysis to non-proportional limit and polynomially many iterations of the power method.Yuchen Wu, Kangjie Zhouwork_czlpwjl64jfv7pavfu225talomMon, 07 Nov 2022 00:00:00 GMTTutorial and Practice in Linear Programming: Optimization Problems in Supply Chain and Transport Logistics
https://scholar.archive.org/work/xcm5wjx3lvcyvbfkpehqjxkzpu
This tutorial is an andragogical guide for students and practitioners seeking to understand the fundamentals and practice of linear programming. The exercises demonstrate how to solve classical optimization problems with an emphasis on spatial analysis in supply chain management and transport logistics. All exercises display the Python programs and optimization libraries used to solve them. The first chapter introduces key concepts in linear programming and contributes a new cognitive framework to help students and practitioners set up each optimization problem. The cognitive framework organizes the decision variables, constraints, the objective function, and variable bounds in a format for direct application to optimization software. The second chapter introduces two types of mobility optimization problems (shortest path in a network and minimum cost tour) in the context of delivery and service planning logistics. The third chapter introduces four types of spatial optimization problems (neighborhood coverage, flow capturing, zone heterogeneity, service coverage) and contributes a workflow to visualize the optimized solutions in maps. The workflow creates decision variables from maps by using the free geographic information systems (GIS) programs QGIS and GeoDA. The fourth chapter introduces three types of spatial logistical problems (spatial distribution, flow maximization, warehouse location optimization) and demonstrates how to scale the cognitive framework in software to reach solutions. The final chapter summarizes lessons learned and provides insights about how students and practitioners can modify the Phyton programs and GIS workflows to solve their own optimization problem and visualize the results.Raj Bridgelallwork_xcm5wjx3lvcyvbfkpehqjxkzpuFri, 04 Nov 2022 00:00:00 GMT