DOCSLIB.ORG

Complex-Valued Deep Learning with Applications to Magnetic Resonance Image Synthesis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Complex-Valued Deep Learning with Applications to Magnetic Resonance Image Synthesis Complex-valued Deep Learning with Applications to Magnetic Resonance Image Synthesis Patrick Virtue Electrical Engineering and Computer Sciences University of California at Berkeley Technical Report No. UCB/EECS-2019-126 http://www2.eecs.berkeley.edu/Pubs/TechRpts/2019/EECS-2019-126.html August 16, 2019 Copyright © 2019, by the author(s). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. Complex-valued Deep Learning with Applications to Magnetic Resonance Image Synthesis By Patrick M. Virtue A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering – Electrical Engineering and Computer Sciences in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Michael Lustig, Co-chair Professor Stella X. Yu, Co-chair Professor Trevor Darrell Professor Bruno Olshausen Summer 2019 Complex-valued Deep Learning with Applications to Magnetic Resonance Image Synthesis Copyright 2019 by Patrick M. Virtue 1 Abstract Complex-valued Deep Learning with Applications to Magnetic Resonance Image Synthesis by Patrick M. Virtue Doctor of Philosophy in Engineering – Electrical Engineering and Computer Sciences University of California, Berkeley Professor Michael Lustig, Co-chair Professor Stella X. Yu, Co-chair Magnetic resonance imaging (MRI) has the ability to produce a series of images that each have different visual contrast between tissues, allowing clinicians to qualitatively assess pathologies that may be visible in one contrast-weighted image but not others. Unfortu- nately, these standard contrast-weighted images do not contain quantitative values, produc- ing challenges for post-processing, assessment, and longitudinal studies. MR fingerprinting is a recent technique that produces quantitative tissue maps from a single pseudorandom acquisition, but it relies on computationally heavy nearest neighbor algorithms to solve the associated nonlinear inverse problem. In this dissertation, we present our deep learning methods to speed up quantitative MR fingerprinting and synthesize the standard contrast- weighted images directly from the same MR fingerprinting scan. Adapting deep learning methodologies to MR image synthesis presents two specific chal- lenges: 1) complex-valued data and 2) the presence of noise while undersampling. MRI signals are inherently complex-valued, as they are measurements of rotating mag- netization within the body. However, modern neural networks are not designed to support complex values. As an example, the pervasive ReLU activation function is undefined for complex numbers. This limitation curtails the impact of deep learning for complex data applications, such as MRI, radio frequency modulation identification, and target recogni- tion in synthetic-aperture radar images. In this dissertation, we discuss the motivation for complex-valued networks, the changes that we have made to implement complex backpropa- gation, and our new complex cardioid activation function that made it possible to outperform real-valued networks for MR fingerprinting image synthesis. In Fourier-based medical imaging, undersampling results in an underdetermined system, in which a linear reconstruction will exhibit artifacts. Another consequence is lower signal- to-noise ratio (SNR) because of fewer acquired measurements. The coupled effects of low SNR and underdetermined system during reconstruction makes it difficult to model the 2 signal and analyze image reconstruction algorithms. We demonstrate that neural networks trained only with a Gaussian noise model fail to process in vivo MR fingerprinting data, while our proposed empirical noise model allows neural networks to successfully synthesize quantitative images. Additionally, to better understand the impact of noise on undersampled imaging systems, we present an image quality prediction process that reconstructs fully sampled, fully determined data with noise added to simulate the SNR loss induced by a given undersampling pattern. The resulting prediction image empirically shows the effects of noise in undersampled image reconstruction without any effect from an underdetermined system, allowing MR pulse sequence and reconstruction developers to determine if low SNR, rather than the underdetermined system, is the limiting factor for a successful reconstruction. i For Caitlin and Molly ii Contents List of Figuresv List of Tables xiii Acknowledgments xiv 1 Introduction1 1.1 MRI Overview...................................2 1.2 Complex-valued Deep Learning.........................3 1.3 Deep Learning for MR Fingerprinting......................4 1.4 Noise in Undersampled Reconstruction.....................5 1.5 Organization and Previously Published Work..................6 2 Complex-valued Deep Learning7 2.1 Complex numbers.................................8 2.2 Motivation.....................................9 2.2.1 Where Do Complex Values Occur in Deep Learning Applications?..9 2.2.2 Why Not Use a Real Network with Two Channels?.......... 11 2.2.3 Why Not Directly Extend Real-valued Layers to Complex?...... 14 2.3 Complex Calculus Background.......................... 17 2.3.1 Complex-differentiable.......................... 17 2.3.2 Wirtinger/CR Calculus.......................... 18 2.3.3 Gradient Descent............................. 20 2.4 Complex-valued Neural Networks........................ 21 2.4.1 Backpropagation with Real-valued Loss................. 23 2.4.2 Hybrid Real/Complex Networks..................... 25 2.4.3 Complex Activation Functions...................... 28 2.4.4 Well-defined Layers............................ 35 2.4.5 Software.................................. 39 2.5 Discussion..................................... 41 Contents iii 3 Deep Learning for MR Fingerprinting 43 3.1 Background for MR Fingerprinting....................... 44 3.1.1 Measuring Net Magnetization...................... 44 3.1.2 Scanner Parameters............................ 45 3.1.3 Tissue Parameters............................ 46 3.1.4 Modeling the MR System........................ 47 3.1.5 Parameter Mapping and Contrast-weighted Imaging.......... 47 3.1.6 MR Fingerprinting............................ 49 3.2 Complex-valued Deep Learning......................... 51 3.2.1 Methods.................................. 52 3.2.2 Results................................... 54 3.2.3 Discussion................................. 56 3.3 MRF Training Data Synthesis.......................... 59 3.3.1 Introduction................................ 59 3.3.2 Methods.................................. 63 3.3.3 Results................................... 68 3.3.4 Discussion................................. 71 3.4 Direct Contrast Synthesis............................ 73 3.4.1 Methods.................................. 74 3.4.2 Results................................... 80 3.4.3 Discussions and conclusions....................... 83 4 Empirical Effect of Gaussian Noise in Undersampled MRI Reconstruction 87 4.1 Introduction.................................... 87 4.1.1 Measurement Time and SNR...................... 91 4.1.2 Undersampling and Expected Measurement Time........... 92 4.1.3 Image Quality Prediction......................... 93 4.1.4 Weighted Least Squares Reconstruction................. 94 4.2 Methodology................................... 96 4.2.1 Effect of Measurement Time Distribution................ 96 4.2.2 Effects of Measurement Noise and Undersampling Rate........ 98 4.2.3 Image Quality Prediction......................... 98 4.2.4 Regularization Parameter........................ 100 4.3 Results....................................... 100 4.3.1 Effect of Measurement Time Distribution................ 101 4.3.2 Effects of Measurement Noise and Undersampling Rate........ 103 4.3.3 Image Quality Prediction......................... 104 4.3.4 Regularization Parameter........................ 106 4.4 Discussion..................................... 106 Bibliography 109 Contents iv A Chapter 2 Derivations 118 A.1 Complex Normalization.............................. 118 A.2 Phase....................................... 119 A.3 Separable Sigmoid................................ 120 A.4 Siglog....................................... 122 A.5 iGaussian..................................... 123 A.6 Cardioid...................................... 124 A.7 Batch Normalization............................... 125 A.8 Euclidean Loss.................................. 128 B Chapter 4 Derivations 132 B.1 Variance of Added Gaussian Noise........................ 132 B.2 Weighted Least Squares............................. 132 v List of Figures 1.1 Illustration of how a complex-valued neural network could learn a unique trans- form function, despite an under-specified training dataset. Even though the train- ing data lie on a single line, left, the complex identity theorem implies that this is sufficient to define a unique complex-valued holomorphic transform function, cen- ter [1]. However, there is no such theorem when considering real-valued transform functions, right....................................3 1.2 Our proposed direct contrast synthesis (DCS) uses a trained neural network to transform the MR fingerprinting
Load more

Recommended publications
  • Emerging Topics in Internet Technology: a Complex Networks Approach
    1 Emerging Topics in Internet Technology: A Complex Networks Approach Bisma S. Khan 1, Muaz A. Niazi *,2 COMSATS Institute of Information Technology, Islamabad, Pakistan Abstract —Communication networks, in general, and internet technology, in particular, is a fast-evolving area of research. While it is important to keep track of emerging trends in this domain, it is such a fast-growing area that it can be very difficult to keep track of literature. The problem is compounded by the fast-growing number of citation databases. While other databases are gradually indexing a large set of reliable content, currently the Web of Science represents one of the most highly valued databases. Research indexed in this database is known to highlight key advancements in any domain. In this paper, we present a Complex Network-based analytical approach to analyze recent data from the Web of Science in communication networks. Taking bibliographic records from the recent period of 2014 to 2017 , we model and analyze complex scientometric networks. Using bibliometric coupling applied over complex citation data we present answers to co-citation patterns of documents, co-occurrence patterns of terms, as well as the most influential articles, among others, We also present key pivot points and intellectual turning points. Complex network analysis of the data demonstrates a considerably high level of interest in two key clusters labeled descriptively as “social networks” and “computer networks”. In addition, key themes in highly cited literature were clearly identified as “communication networks,” “social networks,” and “complex networks”. Index Terms —CiteSpace, Communication Networks, Network Science, Complex Networks, Visualization, Data science.
    [Show full text]
  • Complex Networks in Computer Vision
    Complex networks in computer vision Pavel Voronin Shape to Network [Backes 2009], [Backes 2010a], [Backes 2010b] Weights + thresholds Feature vectors Features = characteris>cs of thresholded undirected binary networks: degree or joint degree (avg / min / max), avg path length, clustering coefficient, etc. Mul>scale Fractal Dimension Properes • Only uses distances => rotaon invariant • Weights normalized by max distance => scale invariant • Uses pixels not curve elements => robust to noise and outliers => applicable to skeletons, mul>ple contours Face recogni>on [Goncalves 2010], [Tang 2012a] Mul>ple binarizaon thresholds Texture to Network [Chalumeau 2008], [Backes 2010c], [Backes 2013] Thresholds + features Features: degree, hierarchical degree (avg / min / max) Invariance, robustness Graph structure analysis [Tang 2012b] Graph to Network Thresholds + descriptors Degree, joint degree, clustering-distance Saliency Saccade eye movements + different fixaon >me = saliency map => model them as walks in networks [Harel 2006], [Costa 2007], [Gopalakrishnan 2010], [Pal 2010], [Kim 2013] Network construc>on Nodes • pixels • segments • blocks Edges • local (neighbourhood) • global (most similar) • both Edge weights = (distance in feature space) / (distance in image space) Features: • relave intensity • entropy of local orientaons • compactness of local colour Random walks Eigenvector centrality == staonary distribu>on for Markov chain == expected >me a walker spends in the node Random walk with restart (RWR) If we have prior info on node importance,
    [Show full text]
  • Analysis of Statistical and Structural Properties of Complex Networks with Random Networks
    Appl. Math. Inf. Sci. 11, No. 1, 137-146 (2017) 137 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/110116 Analysis of Statistical and Structural Properties of Complex networks with Random Networks S. Vairachilai1,∗, M. K. Kavitha Devi2 and M. Raja 1 1 Department of CSE, Faculty of Science and Technology, IFHE Hyderabad, Telangana -501203, India. 2 Department of CSE, ThiagarajarCollege of Engineering, Madurai,Tamil Nadu-625015, India. Received: 2 Sep. 2016, Revised: 25 Oct. 2016, Accepted: 29 Oct. 2016 Published online: 1 Jan. 2017 Abstract: Random graphs are extensive, in addition, it is used in several functional areas of research, particularly in the field of complex networks. The study of complex networks is a useful and active research areas in science, such as electrical power grids and telecommunication networks, collaboration and citation networks of scientists,protein interaction networks, World-Wide Web and Internet Social networks, etc. A social network is a graph in which n vertices and m edges are selected at random, the vertices represent people and the edges represent relationships between them. In network analysis, the number of properties is defined and studied in the literature to identify the important vertex in a network. Recent studies have focused on statistical and structural properties such as diameter, small world effect, clustering coefficient, centrality measure, modularity, community structure in social networks like Facebook, YouTube, Twitter, etc. In this paper, we first provide a brief introduction to the complex network properties. We then discuss the complex network properties with values expected for random graphs.
    [Show full text]
  • Arxiv:2006.02870V1 [Cs.SI] 4 Jun 2020
    The why, how, and when of representations for complex systems Leo Torres Ann S. Blevins [email protected] [email protected] Network Science Institute, Department of Bioengineering, Northeastern University University of Pennsylvania Danielle S. Bassett Tina Eliassi-Rad [email protected] [email protected] Department of Bioengineering, Network Science Institute and University of Pennsylvania Khoury College of Computer Sciences, Northeastern University June 5, 2020 arXiv:2006.02870v1 [cs.SI] 4 Jun 2020 1 Contents 1 Introduction 4 1.1 Definitions . .5 2 Dependencies by the system, for the system 6 2.1 Subset dependencies . .7 2.2 Temporal dependencies . .8 2.3 Spatial dependencies . 10 2.4 External sources of dependencies . 11 3 Formal representations of complex systems 12 3.1 Graphs . 13 3.2 Simplicial Complexes . 13 3.3 Hypergraphs . 15 3.4 Variations . 15 3.5 Encoding system dependencies . 18 4 Mathematical relationships between formalisms 21 5 Methods suitable for each representation 24 5.1 Methods for graphs . 24 5.2 Methods for simplicial complexes . 25 5.3 Methods for hypergraphs . 27 5.4 Methods and dependencies . 28 6 Examples 29 6.1 Coauthorship . 29 6.2 Email communications . 32 7 Applications 35 8 Discussion and Conclusion 36 9 Acknowledgments 38 10 Citation diversity statement 38 2 Abstract Complex systems thinking is applied to a wide variety of domains, from neuroscience to computer science and economics. The wide variety of implementations has resulted in two key challenges: the progenation of many domain-specific strategies that are seldom revisited or questioned, and the siloing of ideas within a domain due to inconsistency of complex systems language.
    [Show full text]
  • Complex Design Networks: Structure and Dynamics
    Complex Design Networks: Structure and Dynamics Dan Braha New England Complex Systems Institute, Cambridge, MA 02138, USA University of Massachusetts, Dartmouth, MA 02747, USA Email: [email protected] Why was the $6 billion FAA air traffic control project scrapped? How could the 1977 New York City blackout occur? Why do large scale engineering systems or technology projects fail? How do engineering changes and errors propagate, and how is that related to epidemics and earthquakes? In this paper we demonstrate how the rapidly expanding science of complex design networks could provide answers to these intriguing questions. We review key concepts, focusing on non-trivial topological features that often occur in real-world large-scale product design and development networks; and the remarkable interplay between these structural features and the dynamics of design rework and errors, network robustness and resilience, and design leverage via effective resource allocation. We anticipate that the empirical and theoretical insights gained by modeling real-world large-scale product design and development systems as self-organizing complex networks will turn out to be the standard framework of a genuine science of design. 1. Introduction: the road to complex networks in engineering design Large-scale product design and development is often a distributed process, which involves an intricate set of interconnected tasks carried out in an iterative manner by hundreds of designers (see Fig. 1), and is fundamental to the creation of complex man- made systems [1—16]. This complex network of interactions and coupling is at the heart of large-scale project failures as well as of large-scale engineering and software system failures (see Table 1).
    [Show full text]
  • Optimizing the Topology of Complex Neural Networks
    Optimising the topology of complex neural networks Fei Jiang1;2, Hugues Berry1, and Marc Schoenauer2 1Project-team Alchemy and 2Project-team TAO, INRIA Futurs, Parc Club Orsay Universite, 91893 Orsay Cedex France, [email protected] Summary. In this paper, we study instances of complex neural networks, i.e. neural networks with complex topologies. We use Self-Organizing Map neural networks whose neighbourhood relationships are defined by a complex network, to classify handwritten digits. We show that topology has a small impact on performance and robustness to neuron failures, at least at long learning times. Performance may however be increased (by almost 10%) by artificial evolution of the network topology. In our experimental conditions, the evolved networks are more random than their parents, but display a more heterogeneous degree distribution. 1 Introduction The connectivity structure of complex networks (i.e. their topology) is a cru- cial determinant of information transfer in large networks (internet, social networks, metabolic networks...) [1]. Hence, the computation made by com- plex neural networks, i.e. neural networks with complex connectivity struc- ture, could be dependent on their topology. For instance, recent studies have shown that introducing a small-world topology in a multilayer perceptron increases its performance [13, 2]. However, other studies have inspected the performance of Hopfield [5, 10, 9, 14] or Echo state networks [3] with small- world or scale-free topologies and reported more contrasted results. Using artificial evolutionary algorithms to modify the topology of neu- ral networks so as to optimise their performance has become widespread in the artificial neural networks community for several years [11, 16].
    [Show full text]
  • Applications of Network Science to Education Research: Quantifying Knowledge and the Development of Expertise Through Network Analysis
    education sciences Commentary Applications of Network Science to Education Research: Quantifying Knowledge and the Development of Expertise through Network Analysis Cynthia S. Q. Siew Department of Psychology, National University of Singapore, Singapore 117570, Singapore; [email protected] Received: 31 January 2020; Accepted: 3 April 2020; Published: 8 April 2020 Abstract: A fundamental goal of education is to inspire and instill deep, meaningful, and long-lasting conceptual change within the knowledge landscapes of students. This commentary posits that the tools of network science could be useful in helping educators achieve this goal in two ways. First, methods from cognitive psychology and network science could be helpful in quantifying and analyzing the structure of students’ knowledge of a given discipline as a knowledge network of interconnected concepts. Second, network science methods could be relevant for investigating the developmental trajectories of knowledge structures by quantifying structural change in knowledge networks, and potentially inform instructional design in order to optimize the acquisition of meaningful knowledge as the student progresses from being a novice to an expert in the subject. This commentary provides a brief introduction to common network science measures and suggests how they might be relevant for shedding light on the cognitive processes that underlie learning and retrieval, and discusses ways in which generative network growth models could inform pedagogical strategies to enable meaningful long-term conceptual change and knowledge development among students. Keywords: education; network science; knowledge; learning; expertise; development; conceptual representations 1. Introduction Cognitive scientists have had a long-standing interest in quantifying cognitive structures and representations [1,2]. The empirical evidence from cognitive psychology and psycholinguistics demonstrates that the structure of cognitive systems influences the psychological processes that operate within them.
    [Show full text]
  • Reverse Image Search for Scientific Data Within and Beyond the Visible
    Journal, Vol. XXI, No. 1, 1-5, 2013 Additional note Reverse Image Search for Scientific Data within and beyond the Visible Spectrum Flavio H. D. Araujo1,2,3,4,a, Romuere R. V. Silva1,2,3,4,a, Fatima N. S. Medeiros3,Dilworth D. Parkinson2, Alexander Hexemer2, Claudia M. Carneiro5, Daniela M. Ushizima1,2,* Abstract The explosion in the rate, quality and diversity of image acquisition instruments has propelled the development of expert systems to organize and query image collections more efficiently. Recommendation systems that handle scientific images are rare, particularly if records lack metadata. This paper introduces new strategies to enable fast searches and image ranking from large pictorial datasets with or without labels. The main contribution is the development of pyCBIR, a deep neural network software to search scientific images by content. This tool exploits convolutional layers with locality sensitivity hashing for querying images across domains through a user-friendly interface. Our results report image searches over databases ranging from thousands to millions of samples. We test pyCBIR search capabilities using three convNets against four scientific datasets, including samples from cell microscopy, microtomography, atomic diffraction patterns, and materials photographs to demonstrate 95% accurate recommendations in most cases. Furthermore, all scientific data collections are released. Keywords Reverse image search — Content-based image retrieval — Scientific image recommendation — Convolutional neural network 1University of California, Berkeley, CA, USA 2Lawrence Berkeley National Laboratory, Berkeley, CA, USA 3Federal University of Ceara,´ Fortaleza, CE, Brazil 4Federal University of Piau´ı, Picos, PI, Brazil 5Federal University of Ouro Preto, Ouro Preto, MG, Brazil *Corresponding author: [email protected] aContributed equally to this work.
    [Show full text]
  • Immunization for Complex Network Based on the Effective Degree Of
    Immunization for complex network based on the effective degree of vertex Ke Hu and Yi Tang* Department of Physics and Institute of Modern Physics, Xiangtan University, Xiangtan 411105, Hunan, China The basic idea of many effective immunization strategies is first to rank the importance of vertices according to the degrees of vertices and then remove the vertices from highest importance to lowest until the network becomes disconnected. Here we define the effective degrees of vertex, i.e., the number of its connections linking to un-immunized nodes in current network during the immunization procedure, to rank the importance of vertex, and modify these strategies by using the effective degrees of vertices. Simulations on both the scale-free network models with various degree correlations and two real networks have revealed that the immunization strategies based on the effective degrees are often more effective than those based on the degrees in the initial network. Keywords: Immunization, epidemic spreading, scale-free network, effective degree PACS number(s): 89.75.Hc, 87.23.Ge 1. Introduction The epidemic dynamic on complex network is largely benefited from the need of understanding the spreading of both computer viruses and human sexual diseases[1-3], which has recently become a subject of *Corresponding author. Email address: [email protected] 1 intensive investigations.[4-15] Since most of real networks involving the Internet and the web of sexual contacts appear to be of large heterogeneity in the connectivity properties,[16-19] epidemic modeling deals with the complex systems with heterogeneous connectivity to approach a more realistic situation. This highly heterogeneous topology of the networks is mainly reflected in the small average path lengths among any two nodes (small-world property)[20], and in a power law connectivity distribution (scale-free property)[19].
    [Show full text]
  • Complex Networks in Climate Science: Progress, Opportunities and Challenges
    COMPLEX NETWORKS IN CLIMATE SCIENCE: PROGRESS, OPPORTUNITIES AND CHALLENGES KARSTEN STEINHAEUSER1,2, NITESH V. CHAWLA1, AND AUROOP R. GANGULY2 Abstract. Networks have been used to describe and model a wide range of complex systems, both natural as well as man-made. One particularly interesting application in the earth sciences is the use of complex networks to represent and study the global climate system. In this paper, we motivate this general approach, explain the basic methodology, report on the state of the art (including our contributions), and outline open questions and opportunities for future research. 1. Introduction Datasets and systems that can be represented as interaction networks (or graphs), broadly defined as any collection of interrelated objects or entities, have received considerable attention both from a theoretical viewpoint [1, 2, 6, 8, 13, 31] as well as various application domains; examples include the analysis of social networks [30], chemical interactions between proteins [26], the behavior of financial markets [12], and many others. Recently, the study of complex networks – that is, networks which exhibit non-trivial topological properties – has permeated numerous fields and disciplines spanning the physical, social, and computational sciences. So why do networks enjoy such broad appeal? Briefly, it is their ability to serve at once as a data representation, as an analysis framework, and as a visualization tool. The analytic capabilities in particular are quite powerful, as networks can uncover structure and patterns at multiple scales, ranging from local properties to global phenomena, and thus help better understand the characteristics of complex systems. We focus on one particular application of networks in the earth sciences, namely, the construction and analysis of climate networks [25].
    [Show full text]
  • Complex Network Theory
    Complex Network Theory Lecture 1: Networks: what they are and what they are good for Instructor: S. Mehdi Vahidipour ([email protected]) Spring 2018 Thanks A. Rezvanian L. Adamic , A. Barabasi, J. Leskovec Feb., 2018 Outline Overview of class topics Course overview Syllabus References Evaluation First topic Types of networks Network models Implications of network structure Next class: Basic definitions & metrics Complex Network Theory, S. Mehdi Vahidipour, Spring 2018. 2 Course overview Ref: Newman, Mark. Networks: an introduction. Oxford University Press, 2010. Goals Introduce the basic mathematical tools to understand the fundamentals of complex networks. Provide the skills that are needed to perform basic analyses of such networks. Means Study fundamental concepts from graph theory and random networks. Lots of exercises in proving properties of various well-known networks Practice the use of network analysis tools Complex Network Theory, S. Mehdi Vahidipour, Spring 2018. 3 Topics Complex network theory Complex network characteristics Complex network models (Erdős–Rényi, watts-strogatz, Barabási– Albert) Network metrics, centralities, measures & algorithms Structural and behavioral analysis (static & dynamic) Subgraphs, cliques and Motifs analysis & algorithms Social network analysis Spectral and random walks techniques on networks Community structure discovery and analysis in complex networks Diffusion models on networks Cascading failures in real networks Network visualizations Supplementary
    [Show full text]
  • Neural Network FAQ, Part 1 of 7
    Neural Network FAQ, part 1 of 7: Introduction Archive-name: ai-faq/neural-nets/part1 Last-modified: 2002-05-17 URL: ftp://ftp.sas.com/pub/neural/FAQ.html Maintainer: [email protected] (Warren S. Sarle) Copyright 1997, 1998, 1999, 2000, 2001, 2002 by Warren S. Sarle, Cary, NC, USA. --------------------------------------------------------------- Additions, corrections, or improvements are always welcome. Anybody who is willing to contribute any information, please email me; if it is relevant, I will incorporate it. The monthly posting departs around the 28th of every month. --------------------------------------------------------------- This is the first of seven parts of a monthly posting to the Usenet newsgroup comp.ai.neural-nets (as well as comp.answers and news.answers, where it should be findable at any time). Its purpose is to provide basic information for individuals who are new to the field of neural networks or who are just beginning to read this group. It will help to avoid lengthy discussion of questions that often arise for beginners. SO, PLEASE, SEARCH THIS POSTING FIRST IF YOU HAVE A QUESTION and DON'T POST ANSWERS TO FAQs: POINT THE ASKER TO THIS POSTING The latest version of the FAQ is available as a hypertext document, readable by any WWW (World Wide Web) browser such as Netscape, under the URL: ftp://ftp.sas.com/pub/neural/FAQ.html. If you are reading the version of the FAQ posted in comp.ai.neural-nets, be sure to view it with a monospace font such as Courier. If you view it with a proportional font, tables and formulas will be mangled.
    [Show full text]