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Fast and Provable algorithms for Learning Two-Layer Polynomial Neural Networks

IEEE TRANSACTIONS ON SIGNAL PROCESSING 2019年第13期67卷 3361-3371页

作者： Soltani, Mohammadreza Hegde, Chinmay Iowa State Univ Elect & Comp Engn Dept Ames IA 50010 USA

In this paper, we bridge the problem of (provably) learning shallow neural networks with the well-studied problem of low-rank matrix estimation. In particular, we consider two-layer networks with quadratic activations, and focus on the under-parameterized regime where the number of neurons in the hidden layer is smaller than the dimension of the input. Our main approach is to "lift" the learning problem into a higher dimension, which enables us to borrow algorithmic techniques from low-rank matrix estimation. Using this intuition, we propose three novel, non-convex training algorithms. We support our algorithms with rigorous theoretical analysis, and show that all three enjoy a linear convergence, fast running time per iteration, and near-optimal sample complexity. Finally, we complement our theoretical results with numerical experiments.

关键词： Deep learning low-rank estimation approximate algorithms sample complexity rank-one projection

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Optimizing Number, Placement, and Backhaul Connectivity of Multi-UAV Networks

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IEEE INTERNET OF THINGS JOURNAL 2022年第21期9卷 21548-21560页

作者： Sabzehali, Javad Shah, Vijay K. Fan, Qiang Choudhury, Biplav Liu, Lingjia Reed, Jeffrey H. Virginia Tech Wireless VT Blacksburg VA 24061 USA Virginia Tech Bradley Dept Elect & Comp Engn Blacksburg VA 24061 USA George Mason Univ Dept Cybersecur Engn Fairfax VA 22030 USA Qualcomm San Jose CA 95110 USA

Multi unmanned aerial vehicle (UAV) network is a promising solution to providing wireless coverage to ground users in challenging rural areas (such as Internet of Things (IoT) devices in farmlands), where the traditional cellular networks are sparse or unavailable. A key challenge in such networks is the 3-D placement of all UAV base stations (BSs) such that the formed multi-UAV network: 1) utilizes a minimum number of UAVs while ensuring-2) backhaul connectivity directly (or via other UAVs) to the nearby terrestrial BS;and 3) wireless coverage to all ground users in the area of operation. This joint backhaul-and-coverage-aware drone deployment (BoaRD) problem is largely unaddressed in the literature and, thus, is the focus of this article. We first formulate the BoaRD problem as integer linear programming (ILP). However, the problem is NP-hard and, therefore, we propose a low complexity algorithm with a provable performance guarantee to solve the problem efficiently. Our simulation study shows that the Proposed algorithm performs very close to that of the Optimal algorithm (solved using ILP solver) for smaller scenarios, where the area size and the number of users are relatively small. For larger scenarios, where the area size and the number of users are relatively large, the proposed algorithm greatly outperforms the baseline approaches-Backhaul-aware Greedy and random algorithm, respectively, by up to 17% and 95% in utilizing fewer UAVs while ensuring 100% ground-user coverage and backhaul connectivity for all deployed UAVs across all considered simulation setting.

关键词： Wireless communication Base stations Autonomous aerial vehicles Optimization Internet of Things Three-dimensional displays Atmospheric modeling approximate algorithms backhaul graph theory multi-UAV networks unmanned aerial vehicles (UAVs)

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A survey on algorithms for mining frequent itemsets over data streams

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KNOWLEDGE AND INFORMATION SYSTEMS 2008年第1期16卷 1-27页

作者： Cheng, James Ke, Yiping Ng, Wilfred Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Kowloon Hong Kong Peoples R China

The increasing prominence of data streams arising in a wide range of advanced applications such as fraud detection and trend learning has led to the study of online mining of frequent itemsets (FIs). Unlike mining static databases, mining data streams poses many new challenges. In addition to the one-scan nature, the unbounded memory requirement and the high data arrival rate of data streams, the combinatorial explosion of itemsets exacerbates the mining task. The high complexity of the FI mining problem hinders the application of the stream mining techniques. We recognize that a critical review of existing techniques is needed in order to design and develop efficient mining algorithms and data structures that are able to match the processing rate of the mining with the high arrival rate of data streams. Within a unifying set of notations and terminologies, we describe in this paper the efforts and main techniques for mining data streams and present a comprehensive survey of a number of the state-of-the-art algorithms on mining frequent itemsets over data streams. We classify the stream-mining techniques into two categories based on the window model that they adopt in order to provide insights into how and why the techniques are useful. Then, we further analyze the algorithms according to whether they are exact or approximate and, for approximate approaches, whether they are false-positive or false-negative. We also discuss various interesting issues, including the merits and limitations in existing research and substantive areas for future research.

关键词： frequent itemsets stream mining window models approximate algorithms

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1.0957-approximation algorithm for random MAX-3SAT

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RAIRO-OPERATIONS RESEARCH 2007年第1期41卷 95-103页

作者： Fernandez de la Vega, Wenceslas Karpinski, Marek Univ Paris Sud CNRS Orsay France Yale Univ Dept Comp Sci New Haven CT 06520 USA Univ Bonn Dept Comp Sci D-5300 Bonn Germany Isaac Newton Inst Math Sci Cambridge England

We prove that MAX-3SAT can be approximated in polynomial time within a factor 1.0957 on random instances.

关键词： random satisfiability approximate algorithms

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PF-OLA: a high-performance framework for parallel online aggregation

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DISTRIBUTED AND PARALLEL DATABASES 2014年第3期32卷 337-375页

作者： Qin, Chengjie Rusu, Florin Univ Calif Merced CA 95343 USA

Online aggregation provides estimates to the final result of a computation during the actual processing. The user can stop the computation as soon as the estimate is accurate enough, typically early in the execution. This allows for the interactive data exploration of the largest datasets. In this paper we introduce the first framework for parallel online aggregation in which the estimation virtually does not incur any overhead on top of the actual execution. We define a generic interface to express any estimation model that abstracts completely the execution details. We design a novel estimator specifically targeted at parallel online aggregation. When executed by the framework over a massive 8 TB TPC-H instance, the estimator provides accurate confidence bounds early in the execution even when the cardinality of the final result is seven orders of magnitude smaller than the dataset size and without incurring overhead.

关键词： Sampling Online estimation Parallel processing approximate algorithms User-defined aggregates (UDA)

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Finding top-k elements in data streams

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INFORMATION SCIENCES 2010年第24期180卷 4958-4974页

作者： Homem, Nuno Carvalho, Joao Paulo INESC ID TULisbon Inst Super Tecn P-1000029 Lisbon Portugal

Identifying the most frequent elements in a data stream is a well known and difficult problem. Identifying the most frequent elements for each individual, especially in very large populations, is even harder. The use of fast and small memory footprint algorithms is paramount when the number of individuals is very large. In many situations such analysis needs to be performed and kept up to date in near real time. Fortunately, approximate answers are usually adequate when dealing with this problem. This paper presents a new and innovative algorithm that addresses this problem by merging the commonly used counter-based and sketch-based techniques for top-k identification. The algorithm provides the top-k list of elements, their frequency and an error estimate for each frequency value. It also provides strong guarantees on the error estimate, order of elements and inclusion of elements in the list depending on their real frequency. Additionally the algorithm provides stochastic bounds on the error and expected error estimates. Telecommunications customer's behavior and voice call data is used to present concrete results obtained with this algorithm and to illustrate improvements over previously existing algorithms. (C) 2010 Elsevier Inc. All rights reserved.

关键词： approximate algorithms Top-k algorithms Most frequent Estimation Data stream frequencies

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Price of dependence: stochastic submodular maximization with dependent items

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JOURNAL OF COMBINATORIAL OPTIMIZATION 2020年第2期39卷 305-314页

作者： Tang, Shaojie Univ Texas Dallas Richardson TX 75083 USA

In this paper, we study the stochastic submodular maximization problem with dependent items subject to downward-closed and prefix-closed constraints. The input of our problem is a finite set of items, and each item is in a particular state from a set of possible states. After picking an item, we are able to observe its state. We assume a monotone and submodular utility function over items and states, and our objective is to select a group of items adaptively so as to maximize the expected utility. Previous studies on stochastic submodular maximization often assume that items' states are independent, however, this assumption may not hold in general. This motivates us to study the stochastic submodular maximization problem with dependent items. We first introduce the concept of degree of independence to capture the degree to which one item's state is dependent on others'. Then we propose a non-adaptive policy that approximates the optimal adaptive policy within a factor of alpha(1 - e(-kappa/2+kappa/18m2) - kappa+2/3m kappa where the value of alpha is depending on the type of constraints, e.g., alpha = 1 for matroid constraint, kappa is the degree of independence, e.g., kappa = 1 for independent items, and m is the number of items. We also analyze the adaptivity gap, i.e., the ratio of the values of best adaptive policy and best non-adaptive policy, of our problem with prefix-closed constraints.

关键词： Stochastic optimization Dependent variables approximate algorithms Submodular maximization

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Distance bounds of ε-points on hypersurfaces

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THEORETICAL COMPUTER SCIENCE 2006年第1-3期359卷 344-368页

作者： Perez-Diaz, Sonia Sendra, Juana Sendra, J. Rafael Univ Alcala de Henares Dept Matemat E-28871 Alcala De Henares Spain Univ Carlos III Madrid Dept Matemat E-28911 Madrid Spain

epsilon-Points were introduced by the authors (see [S. Perez-Diaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic curves by lines, Theoret. Comput. Sci. 315(2-3) (2004) 627-650 (Special issue);S. Perez-Diaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic surfaces by lines, Comput. Aided Geom. Design 22(2) (2005) 147-181;S. Perez-Diaz, J.R. Sendra, J. Sendra, Distance properties of epsilon-points on algebraic curves, in: Series Mathematics and Visualization, Computational Methods for Algebraic Spline Surfaces, Springer, Berlin, 2005, pp. 45-61]) as a generalization of the notion of approximate root of a univariate polynomial. The notion of epsilon-point of an algebraic hypersurface is quite intuitive. It essentially consists in a point such that when substituted in the implicit equation of the hypersurface gives values of small module. Intuition says that an epsilon-point of a hypersurface is a point close to it. In this paper, we formally analyze this assertion giving bounds of the distance of the epsilon-point to the hypersurface. For this purpose, we introduce the notions of height, depth and weight of an epsilon-point. The height and the depth control when the distance bounds are valid, while the weight is involved in the bounds. (C) 2006 Elsevier B.V. All rights reserved.

关键词： epsilon-points distance bounds hypersurfaces approximate algorithms

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Probabilistic Support Prediction: Fast Frequent Itemset Mining in Dense Data

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IEEE ACCESS 2024年 12卷 39330-39350页

作者： Sadeequllah, Muhammad Rauf, Azhar Rehman, Saif Ur Alnazzawi, Noha Univ Peshawar Dept Comp Sci Peshawar 25120 Pakistan Yanbu Ind Coll Comp Sci & Engn Dept Yanbu 46452 Saudi Arabia

Frequent itemset mining (FIM) is a highly resource-demanding data-mining task fundamental to numerous data-mining applications. Support calculation is a frequently performed computation-intensive operation of FIM algorithms, whereas storing transactional data is memory-intensive. The FIM is even more resource-hungry for dense data than for sparse data. The rapidly growing size of datasets further exacerbates this situation and necessitates the design of out-of-the-box highly efficient solutions. This paper proposes a novel approach to frequent itemset mining for dense datasets. This approach, after the initial stage, does not use transactional data, which makes it memory efficient. It also replaces processing-intensive support calculations with efficient support predictions, which are probabilistic and need no transactional data. To predict the support of an itemset, it only needs the support of its subsets. However, this technique works only for itemsets of size three or higher. We also propose an FIM algorithm ProbBF, which incorporates this technique. The ProbBF discards transactional data after it uses it to calculate frequent one and two-size itemsets. For the itemsets of size k, where k >= 3 , ProbBF uses the proposed probabilistic technique to predict their support. It is considered frequent if the predicted support is greater than a given threshold. Our experiments show that ProbBF is efficient in both time and space against state-of-the-art FIM algorithms that use transactional data. The experiments also show that ProbBF can successfully generate the majority of the frequent itemsets on real-world datasets. Since ProbBF is probabilistic, some loss in quality is inevitable.

关键词： Association rule data mining frequent itemsets mining support-count approximation approximate algorithms transaction databases

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The labeled perfect matching in bipartite graphs

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INFORMATION PROCESSING LETTERS 2005年第3期96卷 81-88页

作者： Monnot, J Univ Paris 09 LAMSADE CNRS F-75775 Paris France

in this paper, we deal with both the complexity and the approximability of the labeled perfect matching problem in bipartite graphs. Given a simple graph G = (V, E) with vertical bar V vertical bar = 2n vertices such that E contains a perfect matching (of size n), together with a color (or label) function L: E -> {c(1),..., C-q}, the labeled perfect matching problem consists in finding a perfect matching on G that uses a minimum or a maximum number of colors. (c) 2005 Elsevier B.V. All rights reserved.

关键词： labeled matching bipartite graphs NP-complete approximate algorithms

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