The document discusses exact repair problems with multiple sources in distributed storage systems. It presents the problem definition for (n,k,d) exact repair with multiple sources. It then provides an example of a 2-source (3,2,2) exact repair problem and outlines techniques for computing achievable rate regions using polyhedral bounds and a projection algorithm. The techniques are implemented in software that can be used to obtain rate regions and optimal codes for coded distributed storage networks.
Full Communication in a Wireless Sensor Network by Merging Blocks of a Key Pr...cscpconf
Wireless Sensor Networks (WSN) are constraint by the limited resources available to its
constituting sensors. Thus the use of public-key cryptographyduring message exchange gets
forbidden. One has to invoke symmetric key techniques. This leads to key distribution in the
sensors which in itself is a major challenge. Again due to resource constraints, Key
Predistrubution (KPD) methods are preferred to other distribution techniques. It requires
predistribution of keys in nodes prior to deployment and establishing immediately once
deployed. However there are certain weaknesses in various existing KPD schemes. For
instance, often it is not guaranteed that any given pair of nodes communicate directly. This
leads one to revert to multi-hop communication involving intermediate sensor nodes resulting
in increased cost of communication. In this work a key predistribution technique using ReedSolomon
codes is considered which is faced with the above weakness. The authors suggests a
novel technique of merging certain number of sensors into blocks ensuring that the blocks
have full connectivity amongst themselves. Here the blocks are chosen in such a way that it
ensures no intra-node communication. Further this approach improves both time and space
complexity of the system
Performance Comparison of Cluster based and Threshold based Algorithms for De...Eswar Publications
In mobile ad-hoc networks (MANET), the movement of the nodes may quickly change the networks topology resulting in the increase of the overhead message in topology maintenance. The nodes communicate with each other by exchanging the hello packet and constructing the neighbor list at each node. MANET is vulnerable to attacks such as black hole attack, gray hole attack, worm hole attack and sybil attack. A black hole attack makes a serious impact on routing, packet delivery ratio, throughput, and end to end delay of packets. In this paper, the performance comparison of clustering based and threshold based algorithms for detection and prevention of
cooperative in MANETs is examined. In this study every node is monitored by its own cluster head (CH), while server (SV) monitors the entire network by channel overhearing method. Server computes the trust value based on sent and receive count of packets of the receiver node. It is implemented using AODV routing protocol in the NS2 simulations. The results are obtained by comparing the performance of clustering based and threshold based methods by varying the concentration of black hole nodes and are analyzed in terms of throughput,
packet delivery ratio. The results demonstrate that the threshold based method outperforms the clustering based method in terms of throughput, packet delivery ratio and end to end delay.
Robust Watermarking Technique using 2D Logistic Map and Elliptic Curve Crypto...idescitation
Copyright protection is a vital issue in modern day’s data transmission over
internet. For copyright protection, watermarking technique is extensively used. In this
paper, we have proposed a robust watermarking scheme using 2D Logistic map and elliptic
curve cryptosystem (ECC) in the DWT domain. The combined encryption has been taken to
enhance the security of the watermark before the embedding phase. The PSNR value shows
the difference between original cover and embedded cover is minimal. Similarly, NC values
show the robustness and resistance capability of the proposed technique from the common
attacks such as scaling, Gaussian noise etc. Thus, this combined version of 2D Logistic map
and Elliptic curve cryptosystem can be used in case of higher security requirement of the
watermark signal.
Adjacency Decomposition Method: Breaking up problemsJayant Apte, PhD
Adjacency decomposition method breaks up a large polyhedral representation conversion problem into several smaller representation conversion problems. Given a group G acting on the set of rays, the smaller problems are that of finding G-in-equivalent neighbors of a given extreme ray.
Rate regions provide fundamental limits on storage and transfer of information in networks in a multi-source multi-sink network setting. We formulate three enumeration problems related to rate region computation and propose algorithms to solve them.
Full Communication in a Wireless Sensor Network by Merging Blocks of a Key Pr...cscpconf
Wireless Sensor Networks (WSN) are constraint by the limited resources available to its
constituting sensors. Thus the use of public-key cryptographyduring message exchange gets
forbidden. One has to invoke symmetric key techniques. This leads to key distribution in the
sensors which in itself is a major challenge. Again due to resource constraints, Key
Predistrubution (KPD) methods are preferred to other distribution techniques. It requires
predistribution of keys in nodes prior to deployment and establishing immediately once
deployed. However there are certain weaknesses in various existing KPD schemes. For
instance, often it is not guaranteed that any given pair of nodes communicate directly. This
leads one to revert to multi-hop communication involving intermediate sensor nodes resulting
in increased cost of communication. In this work a key predistribution technique using ReedSolomon
codes is considered which is faced with the above weakness. The authors suggests a
novel technique of merging certain number of sensors into blocks ensuring that the blocks
have full connectivity amongst themselves. Here the blocks are chosen in such a way that it
ensures no intra-node communication. Further this approach improves both time and space
complexity of the system
Performance Comparison of Cluster based and Threshold based Algorithms for De...Eswar Publications
In mobile ad-hoc networks (MANET), the movement of the nodes may quickly change the networks topology resulting in the increase of the overhead message in topology maintenance. The nodes communicate with each other by exchanging the hello packet and constructing the neighbor list at each node. MANET is vulnerable to attacks such as black hole attack, gray hole attack, worm hole attack and sybil attack. A black hole attack makes a serious impact on routing, packet delivery ratio, throughput, and end to end delay of packets. In this paper, the performance comparison of clustering based and threshold based algorithms for detection and prevention of
cooperative in MANETs is examined. In this study every node is monitored by its own cluster head (CH), while server (SV) monitors the entire network by channel overhearing method. Server computes the trust value based on sent and receive count of packets of the receiver node. It is implemented using AODV routing protocol in the NS2 simulations. The results are obtained by comparing the performance of clustering based and threshold based methods by varying the concentration of black hole nodes and are analyzed in terms of throughput,
packet delivery ratio. The results demonstrate that the threshold based method outperforms the clustering based method in terms of throughput, packet delivery ratio and end to end delay.
Robust Watermarking Technique using 2D Logistic Map and Elliptic Curve Crypto...idescitation
Copyright protection is a vital issue in modern day’s data transmission over
internet. For copyright protection, watermarking technique is extensively used. In this
paper, we have proposed a robust watermarking scheme using 2D Logistic map and elliptic
curve cryptosystem (ECC) in the DWT domain. The combined encryption has been taken to
enhance the security of the watermark before the embedding phase. The PSNR value shows
the difference between original cover and embedded cover is minimal. Similarly, NC values
show the robustness and resistance capability of the proposed technique from the common
attacks such as scaling, Gaussian noise etc. Thus, this combined version of 2D Logistic map
and Elliptic curve cryptosystem can be used in case of higher security requirement of the
watermark signal.
Adjacency Decomposition Method: Breaking up problemsJayant Apte, PhD
Adjacency decomposition method breaks up a large polyhedral representation conversion problem into several smaller representation conversion problems. Given a group G acting on the set of rays, the smaller problems are that of finding G-in-equivalent neighbors of a given extreme ray.
Rate regions provide fundamental limits on storage and transfer of information in networks in a multi-source multi-sink network setting. We formulate three enumeration problems related to rate region computation and propose algorithms to solve them.
Entropic Inequalities and marginal problems (Fritz and Chavez) Jayant Apte, PhD
Aforementioned paper discusses 'marginal problem': Given distributions on certain subsets of N random variables, determine whether exists a joint distribution that marginalizes to given subset distributions.
I go over ways of including pictures in LaTeX documents. I cover distinction between image formats viz. bitmaps and vector graphics. Finally I demonstrate a few tools to create vector graphics such as Inkscape, PSTricks and LatexDraw and a simple way of including LaTeX in your presentations i.e. TexMaths equations editor.
Network Coding for Distributed Storage Systems(Group Meeting Talk)Jayant Apte, PhD
Reviews work of Koetter et al. and Dimakis et al.
The former provides an algebraic framework for linear network coding. The latter reduces the so called repair problem to single-source multicast network-coding problem and shows that there is a tradeoff between amount of data stored in a distributed sturage system and amount of data transfer required to repair the system if a node(hard-drive) fails.
Como aplicar a Laudato Si e vivenciar sua espiritualidadeAfonso Murad (FAJE)
Material para padres, religiosos/as e agentes de pastoral, visando incorporar na ação pastoral o cuidado com o nosso planeta. Contém também links para estudo e sugestões de atividades.
Esquema para discussão, visando colaborar no trabalho das secretarias paroquiais, na perspectiva da Evangelii Gaudium: Igreja em saída e conversão pastoral.
Entropic Inequalities and marginal problems (Fritz and Chavez) Jayant Apte, PhD
Aforementioned paper discusses 'marginal problem': Given distributions on certain subsets of N random variables, determine whether exists a joint distribution that marginalizes to given subset distributions.
I go over ways of including pictures in LaTeX documents. I cover distinction between image formats viz. bitmaps and vector graphics. Finally I demonstrate a few tools to create vector graphics such as Inkscape, PSTricks and LatexDraw and a simple way of including LaTeX in your presentations i.e. TexMaths equations editor.
Network Coding for Distributed Storage Systems(Group Meeting Talk)Jayant Apte, PhD
Reviews work of Koetter et al. and Dimakis et al.
The former provides an algebraic framework for linear network coding. The latter reduces the so called repair problem to single-source multicast network-coding problem and shows that there is a tradeoff between amount of data stored in a distributed sturage system and amount of data transfer required to repair the system if a node(hard-drive) fails.
Como aplicar a Laudato Si e vivenciar sua espiritualidadeAfonso Murad (FAJE)
Material para padres, religiosos/as e agentes de pastoral, visando incorporar na ação pastoral o cuidado com o nosso planeta. Contém também links para estudo e sugestões de atividades.
Esquema para discussão, visando colaborar no trabalho das secretarias paroquiais, na perspectiva da Evangelii Gaudium: Igreja em saída e conversão pastoral.
We propose an efficient algorithmic framework for time domain circuit simulation using exponential integrators. This work addresses several critical issues exposed by previous matrix exponential based circuit simulation research, and makes it capable of simulating stiff nonlinear circuit system at a large scale. In this framework, the system’s nonlinearity is treated with exponential Rosenbrock-Euler formulation. The matrix exponential and vector product is computed using invert Krylov subspace method. Our proposed method has several distinguished advantages over conventional formulations (e.g., the well-known backward Euler with Newton-Raphson method). The matrix factorization is performed only for the conductance/resistance matrix G, without being performed for the combinations of the capacitance/inductance matrix C and matrix G, which are used in traditional implicit formulations. Furthermore, due to the explicit nature of our formulation, we do not need to repeat LU decompositions when adjusting the length of time steps for error controls. Our algorithm is better suited to solving tightly coupled post-layout circuits in the pursuit for full-chip simulation. Our experimental results validate the advantages of our framework.
Segmenting Medical MRI via Recurrent Decoding CellSeunghyun Hwang
Review : Segmenting Medical MRI via Recurrent Decoding Cell
- by Seunghyun Hwang (Yonsei University, Severance Hospital, Center for Clinical Data Science)
https://imatge.upc.edu/web/publications/impact-visual-saliency-prediction-image-classification
This thesis introduces an architecture to improve the accuracy of a Convolutional Neural Network trained for image classification using visual saliency predictions from the original images. In this thesis the accuracy of a Convolutional Neural Network (CNN) trained for classification has been improved using saliency maps from the original images. The network had an AlexNet architecture and was trained using 1.2 million images from the Imagenet dataset. Two methods had been explored in order to exploit the information from the visual saliency predictions. The first methodologies implemented applied the saliency maps directly to the existing layers of the CNN, which in some cases were already trained for classification and in other they were initialized with random weights. In the second methodology the information from the saliency maps was merged from a new branch, trained at the same time as the initial CNN. In order to speed up the training of the networks the experiments were implemented using images reduced to 128x128. With this sizes the proposed model achieves 12.39% increase in Top-1 accuracy performance with respect to the original CNN, and additionally reduces the number of parameters needed compared to AlexNet. Regarding the original size images 227x227 a model that increases 1.72% Top-1 accuracy is proposed. To accelerate the training process of the network the images have been reduced. The methodology that provides the higher improvement in accuracy will be implemented using the original size of the images. The results will be compared to those obtained from the network trained only with the original images. All the methodologies proposed are implemented in a network previously trained for classification. Additionally the most successful methodologies will be implemented in the training of a network. The results will provide information about the best way to add saliency maps to improve the accuracy.
More investment in Research and Development for better Education in the future?Dhafer Malouche
I presented this talk the 7th Edition of the CEAFE. We show in this talk whether R&D efforts affect education performance in small classes. Merging two datasets collected from the PISA
studies and the World Development Indicators and using Learning Bayesian Networks, we prove the existence of a statistical causal relationship between investment in R&D of a country and its education performance (PISA scores). We also prove that the effect of R&D on Education is long term as a country has to invest at least 10 years before beginning to improve the level of young pupils.
The increased need for unattended authentication in multiple scenarios has motivated a wide deployment of biometric systems in the last few years. This has in turn led to the disclosure of security concerns specifically related to biometric systems. Among them, presentation attacks (PAs, i.e., attempts to log into the system with a fake biometric characteristic or presentation attack instrument) pose a severe threat to the security of the system: any person could eventually fabricate or order a gummy finger or face mask to impersonate someone else. In this context, we present a novel fingerprint presentation attack detection (PAD) scheme based on i) a new capture device able to acquire images within the short wave infrared (SWIR) spectrum, and ii) an in-depth analysis of several state-of-the art techniques based on both handcrafted and deep learning features. The approach is evaluated on a database comprising over 4700 samples, stemming from 562 different subjects and 35 different presentation attack instrument (PAI) species. The results show the soundness of the proposed approach with a detection equal error rate (D-EER) as low as 1.35% even in a realistic scenario where five different PAI species are considered only for testing purposes (i.e., unknown attacks).
https://telecombcn-dl.github.io/2018-dlcv/
Deep learning technologies are at the core of the current revolution in artificial intelligence for multimedia data analysis. The convergence of large-scale annotated datasets and affordable GPU hardware has allowed the training of neural networks for data analysis tasks which were previously addressed with hand-crafted features. Architectures such as convolutional neural networks, recurrent neural networks and Q-nets for reinforcement learning have shaped a brand new scenario in signal processing. This course will cover the basic principles and applications of deep learning to computer vision problems, such as image classification, object detection or image captioning.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Kubernetes & AI - Beauty and the Beast !?! @KCD Istanbul 2024Tobias Schneck
As AI technology is pushing into IT I was wondering myself, as an “infrastructure container kubernetes guy”, how get this fancy AI technology get managed from an infrastructure operational view? Is it possible to apply our lovely cloud native principals as well? What benefit’s both technologies could bring to each other?
Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
Speakers:
Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
Neuro-symbolic is not enough, we need neuro-*semantic*
Exact Repair problems with multiple sources: CISS 2014
1. CISS 2014, Princeton NJ 1
Exact Repair Problems with Multiple
Sources
Jayant Apte*, Congduan Li,
John MacLaren Walsh, Steven Weber
ECE Dept. Drexel University
2. CISS 2014, Princeton NJ 2
Outline
● Problem Definition
● Computer assisted proofs: General Structure
● Polyhedral bounds on
● Polyhedral computation interpretation of rate
region computation
● A projection technique for computing
achievable rate region
3. CISS 2014, Princeton NJ 3
Outline
● Problem Definition
● Computer assisted proofs: General structure
● Polyhedral bounds on
● Polyhedral computation interpretation of rate
region computation
● A projection technique for computing
achievable rate region
15. CISS 2014, Princeton NJ 15
Outline
● Problem Definition
● Computer assisted proofs: General structure
● Polyhedral bounds on
● Polyhedral computation interpretation of rate
region computation
● A projection technique for computing
achievable rate region
20. CISS 2014, Princeton NJ 20
Computer assisted converse
Inequalities obtained as an
implication of linear Shannon-type,
non-Shannon-type, non-linear
non-Shannon type inequalities and
network constraints
24. CISS 2014, Princeton NJ 24
Outline
● Problem Definition
● Computer assisted proofs: General structure
● Polyhedral bounds on
● Polyhedral computation interpretation of rate
region computation
● A projection technique for computing
achievable rate region
25. CISS 2014, Princeton NJ 25
● Closure of set of all 'entropic' vectors
arising from N-variable probability
distributions
3-D rendition of
26. CISS 2014, Princeton NJ 26
● Closure of set of all 'entropic' vectors
arising from N-variable probability
distributions
● Each entropic vector is formed by
stacking entropies of subsets of N
random variables
3-D rendition of
27. CISS 2014, Princeton NJ 27
● Closure of set of all 'entropic' vectors
arising from N-variable probability
distributions
● Each entropic vector is formed by
stacking entropies of subsets of N
random variables
● Cone:
3-D rendition of
28. CISS 2014, Princeton NJ 28
● Cannot be expressed as intersection
of finite number of linear inequalities
for N>3
● For N=4, existence of single nonlinear
● non-Shannon inequality(necessary and
sufficient) is known [Liu & Walsh 2014]
● Additionally, several hundred linear
non-Shannon inequalities are known
[DFZ 2011, Csirmaz 2013]
3-D rendition of
50. CISS 2014, Princeton NJ 50
Polyhedral bounds on rate region
● Using polyhedral inner/outer bound on yields
polyhedral inner/outer bounds on rate region
● Lemma 1: Inner bounds on rate region
computed using or are achievable
using linear codes
51. CISS 2014, Princeton NJ 51
Outline
● Problem Definition
● Computer assisted proofs: General structure
● Polyhedral bounds on
– Shannon (Outer) bound
– Matroid (Inner) bound(s)
– Subspace (Inner) bounds
● Polyhedral computation interpretation of rate region
computation
● A projection technique for computing achievable rate region
53. CISS 2014, Princeton NJ 53
Network Coding constraints
● Consider a type 1 or type 2 constraint H
● In general, computing extreme rays of given H and
extreme rays of is equivalent to an iteration of Double
Description Method of polyhedral representation conversion
● Lemma 2 [Li et al. 2013]: An extreme ray of is an extreme
ray of if it is contained in the hyperplane corresponding
to H
● Hence, simple membership check suffices to find extreme rays of
57. CISS 2014, Princeton NJ 57
A projection technique for computing
achievable rate region
58. CISS 2014, Princeton NJ 58
A projection technique for computing
achievable rate region
59. CISS 2014, Princeton NJ 59
A projection technique for computing
achievable rate region
60. CISS 2014, Princeton NJ 60
Polyhedral projection via chm
● chm is an implementation of polyhedral projection
algorithm called Convex Hull Method by Jayant Apte*
● chmlib v0.x is available at:
http://www.ece.drexel.edu/walsh/aspitrg/software.html
● Rational arithmetic using FLINT: Fast Library for
Number Theory
● Rational LP solver based on qsopt
61. CISS 2014, Princeton NJ 61
Polyhedral projection via chm
● Has been used for
– The current work
– Computer assisted converse proofs of rate regions of Multilevel
Diversity Coding Systems(a special case of multi-source network
coding)
– Finding non-Shannon Information Inequalities via Generalized
Copy Lemma of Csirmaz
● Can be used for
– Finding necessary conditions for non-contexuality of small
marginal scenarios(Quantum Information)
65. CISS 2014, Princeton NJ 65
References
● X. Yan, R.W. Yeung, and Zhen Zhang. An implicit characterization of the achievable rate region for acyclic
multisource multisink network coding. Information Theory, IEEE Transactions on, 58(9):5625–5639, 2012.
● Dougherty, Randall, Chris Freiling, and Kenneth Zeger. "Non-Shannon information inequalities in four
random variables." arXiv preprint arXiv:1104.3602 (2011).
● Csirmaz, László. "Information inequalities for four variables." CEU (2013).
● Yunshu Liu and John M. Walsh, "Only One Nonlinear Non-Shannon Inequality is Necessary for Four
Variables", submitted to IEEE Int. Symp. Information Theory (ISIT2014)
● Congduan Li, J. Apte, J.M. Walsh, and S. Weber. A new computational approach for determining rate regions
and optimal codes for coded networks. In Network Coding (NetCod), 2013 International Symposium on,
pages 1–6, 2013.
● Congduan Li, John MacLaren Walsh, Steven Weber. Matroid bounds on the region of entropic vectors. In
51th Annual Allerton Conference on Communication, Control and Computing, October 2013.