Hypergraph Cuts with General Splitting FunctionsAustin Benson
The document discusses joint work on hypergraph cuts with Cornell researchers Nate Veldt and Jon Kleinberg. It presents at the SIAM MDS 2020 conference on Pattern Analysis for Networks and Network Generalizations. The work aims to generalize graph minimum s-t cuts to hypergraphs by minimizing a cut function subject to constraints.
Spectral embeddings and evolving networksAustin Benson
Spectral embeddings provide a fundamental approach for many machine learning tasks but are challenging for dynamic networks that change over time. The authors develop improved methods for maintaining spectral embeddings on evolving networks by (1) incrementally updating embeddings with small perturbations, (2) measuring convergence to speed computations, and (3) proposing a new dynamic graph model to test algorithms. They formalize these ideas with perturbation theory and algorithms that warm-start iterative solvers using the previous embedding.
This document summarizes a proposed assembly-free simulation method for additive manufacturing processes.
Current additive manufacturing simulation methods either use a "quiet" approach where all elements are assembled into a global stiffness matrix from the start, or an "inactive" approach where only deposited elements are assembled. Both have disadvantages like ill-conditioning or remeshing requirements.
The proposed method uses a voxel-based discretization where the workspace is meshed once into identical hexahedral elements. Material deposition is modeled by modifying element properties without reassembling matrices. This avoids remeshing and assembly, reducing memory needs. The method was demonstrated by simulating transient non-linear thermal behavior during laser deposition.
Identification of unknown parameters and prediction with hierarchical matrice...Alexander Litvinenko
We compare four numerical methods for the prediction of missing values in four different datasets.
These methods are 1) the hierarchical maximum likelihood estimation (H-MLE), and three machine learning (ML) methods, which include 2) k-nearest neighbors (kNN), 3) random forest, and 4) Deep Neural Network (DNN).
From the ML methods, the best results (for considered datasets) were obtained by the kNN method with three (or seven) neighbors.
On one dataset, the MLE method showed a smaller error than the kNN method, whereas, on another, the kNN method was better.
The MLE method requires a lot of linear algebra computations and works fine on almost all datasets. Its result can be improved by taking a smaller threshold and more accurate hierarchical matrix arithmetics. To our surprise, the well-known kNN method produces similar results as H-MLE and worked much faster.
This document discusses network analysis and measures of centrality and communicability in networks. It provides mathematical definitions and formulas for quantifying properties like betweenness centrality, clustering coefficient, communicability between nodes, and the number of walks and routes connecting nodes in a network. Examples of applying these metrics to real-world networks like social and biological networks are also mentioned.
Hypergraph Cuts with General Splitting FunctionsAustin Benson
The document discusses joint work on hypergraph cuts with Cornell researchers Nate Veldt and Jon Kleinberg. It presents at the SIAM MDS 2020 conference on Pattern Analysis for Networks and Network Generalizations. The work aims to generalize graph minimum s-t cuts to hypergraphs by minimizing a cut function subject to constraints.
Spectral embeddings and evolving networksAustin Benson
Spectral embeddings provide a fundamental approach for many machine learning tasks but are challenging for dynamic networks that change over time. The authors develop improved methods for maintaining spectral embeddings on evolving networks by (1) incrementally updating embeddings with small perturbations, (2) measuring convergence to speed computations, and (3) proposing a new dynamic graph model to test algorithms. They formalize these ideas with perturbation theory and algorithms that warm-start iterative solvers using the previous embedding.
This document summarizes a proposed assembly-free simulation method for additive manufacturing processes.
Current additive manufacturing simulation methods either use a "quiet" approach where all elements are assembled into a global stiffness matrix from the start, or an "inactive" approach where only deposited elements are assembled. Both have disadvantages like ill-conditioning or remeshing requirements.
The proposed method uses a voxel-based discretization where the workspace is meshed once into identical hexahedral elements. Material deposition is modeled by modifying element properties without reassembling matrices. This avoids remeshing and assembly, reducing memory needs. The method was demonstrated by simulating transient non-linear thermal behavior during laser deposition.
Identification of unknown parameters and prediction with hierarchical matrice...Alexander Litvinenko
We compare four numerical methods for the prediction of missing values in four different datasets.
These methods are 1) the hierarchical maximum likelihood estimation (H-MLE), and three machine learning (ML) methods, which include 2) k-nearest neighbors (kNN), 3) random forest, and 4) Deep Neural Network (DNN).
From the ML methods, the best results (for considered datasets) were obtained by the kNN method with three (or seven) neighbors.
On one dataset, the MLE method showed a smaller error than the kNN method, whereas, on another, the kNN method was better.
The MLE method requires a lot of linear algebra computations and works fine on almost all datasets. Its result can be improved by taking a smaller threshold and more accurate hierarchical matrix arithmetics. To our surprise, the well-known kNN method produces similar results as H-MLE and worked much faster.
This document discusses network analysis and measures of centrality and communicability in networks. It provides mathematical definitions and formulas for quantifying properties like betweenness centrality, clustering coefficient, communicability between nodes, and the number of walks and routes connecting nodes in a network. Examples of applying these metrics to real-world networks like social and biological networks are also mentioned.
Computational Frameworks for Higher-order Network Data AnalysisAustin Benson
1. The document discusses computational frameworks for analyzing higher-order network data, where interactions can involve more than two nodes. Real-world systems often involve higher-order interactions that are reduced to pairwise connections.
2. The author presents several datasets involving higher-order interactions and shows that predicting the formation of new higher-order connections is similar to link prediction but considers groups of nodes rather than individual links. Structural properties like edge density and tie strength influence the likelihood of simplicial closure.
3. Models are proposed to score open simplices based on structural features and predict which will transition to closed simplices. Accounting for higher-order structure provides new insights beyond traditional network analysis of pairwise connections.
Higher-order link prediction and other hypergraph modelingAustin Benson
Higher-order link prediction and other hypergraph modeling can better model real-world systems composed of higher-order interactions that are often reduced to pairwise ones. Hypergraphs allow the modeling of interactions between more than two nodes, like groups of people collaborating, multiple recipients of emails, students gathering in groups, and drug compounds made of several substances.
Simplicial closure & higher-order link predictionAustin Benson
The document discusses higher-order link prediction in networks. It summarizes previous work representing higher-order interactions as tensors, hypergraphs, etc. It then proposes evaluating models of higher-order data using "higher-order link prediction" to predict which groups of more than two nodes will interact based on past data. The authors analyze dynamics of triadic closure in several real-world networks and propose methods to predict closure based on structural properties like edge weights.
Three hypergraph eigenvector centralitiesAustin Benson
Three hypergraph eigenvector centralities are proposed to measure the importance of nodes in complex systems modeled as hypergraphs. Hypergraphs generalize graphs by allowing edges to connect any number of nodes. The proposed centralities are adaptations of the standard graph eigenvector centrality to hypergraphs. They measure a node's centrality based on 1) the centralities of its neighbors, 2) being positive values, and 3) being the principal eigenvector of the hypergraph adjacency matrix.
The document discusses different models for how social networks grow over time, including preferential attachment and fitness models. It proposes using discrete choice theory as a way to model network growth, which allows incorporating covariates and flexible modeling. The approach is statistically rigorous and allows easy incorporation of new models and effects compared to traditional static network models.
Link prediction in networks with core-fringe structureAustin Benson
1. The document discusses link prediction in networks with a core-fringe structure. It examines how including connections from fringe nodes affects the performance of link prediction algorithms on the core nodes.
2. An experiment was conducted where a link prediction algorithm was run multiple times, each time including more fringe nodes and connections in order to measure the effect on link prediction accuracy for the core nodes.
3. The results showed that including more information from the fringe helped improve the link prediction performance on the core nodes.
This document discusses research on modeling and predicting higher-order interactions in networks beyond pairwise connections. The researchers collected datasets containing time-stamped groups or "simplices" of nodes and analyzed properties like triangle closure. They propose "higher-order link prediction" to predict which new simplices will form based on structural features like edge weights between nodes. Scoring functions were tested and averages of edge weights often performed well, differing from classical link prediction methods.
This document summarizes a talk about higher-order link prediction in networks. It discusses organizational principles of systems with higher-order interactions, how they evolve over time through simplicial closure events, and how insights can be used to create effective higher-order link prediction methods. Key points include that simplicial closure depends on the structure and strength of ties in the projected graph, and this closure process is similar for 3 and 4 nodes.
Random spatial network models for core-periphery structureAustin Benson
The document proposes a random spatial network model for generating networks with core-periphery structure. The model assigns each node u a weight e^θu and the probability of an edge between nodes u and v is proportional to e^θu + e^θv. This generates networks where high-weight nodes in the "core" have many connections and low-weight "periphery" nodes have few connections.
Random spatial network models for core-periphery structure.Austin Benson
The document proposes a random spatial network model for generating networks with core-periphery structure. The model assigns each node u a weight e^θu and the probability of an edge between nodes u and v is proportional to e^θu + e^θv. This leads to dense connections between high-weight core nodes and sparser connections between core and low-weight peripheral nodes.
Simplicial closure & higher-order link predictionAustin Benson
This document discusses higher-order link prediction and simplicial closure as ways to analyze and model higher-order interactions in network data. It summarizes that networks can be viewed as weighted projected graphs where simplices "fill in" structures, and that new simplices and closed triangles tend to form through trajectories of nodes reaching "simplicial closure events". It proposes evaluating models of higher-order structure through higher-order link prediction, predicting the formation of new simplices.
Simplicial closure and simplicial diffusionsAustin Benson
This document summarizes research on modeling higher-order interactions in network data using simplicial complexes. It finds that most real-world network datasets exhibit a mixture of closed and open triangles, with the fraction varying by domain. A simple probabilistic model can account for this variation. The document proposes that groups of nodes go through trajectories of interactions until reaching a "simplicial closure event" where a new simplex is formed, analogous to triangle closure. It evaluates models' ability to predict such closures using a framework of "higher-order link prediction". Key indicators of closure are edge density and tie strength between nodes.
Sampling methods for counting temporal motifsAustin Benson
The document summarizes research on developing scalable algorithms for counting temporal network motifs in real-time from high-throughput temporal network data streams. It discusses existing methods being insufficient and the problem of not having algorithms that can analyze modern temporal network datasets at fine time scales and high frequencies. It also briefly introduces the idea of using parallel sampling to speed up motif counting algorithms and enable analysis of very large temporal networks.
The document discusses three perspectives on predicting sets of items rather than single items. It describes how sets are common in data such as team formations, medical codes, and online purchases. It then discusses three specific approaches to set prediction: (1) predicting which sets an individual will interact with based on their history, (2) modeling sequences of sets using a generative model, and (3) understanding characteristics of set-based data like subsets and repeats. Applications include prediction, analysis, and simulation.
1. The document discusses sequences of sets, which is a common data structure where each item in a sequence can be a set of elements rather than a single element. Examples of data that can be modeled as sequences of sets include email recipients, tags on questions, academic coauthors, and contacts.
2. The authors provide a generative model to capture important characteristics of sequences of sets, such as subsets and supersets of prior sets being common and recency bias in repeat behavior.
3. Applications of the model include predicting new sets, anomaly detection, understanding user behaviors, and simulation.
Simplicial closure and higher-order link prediction --- SIAMNS18Austin Benson
The document discusses higher-order link prediction, which aims to predict the formation of new groups or "simplices" containing more than two nodes, based on structural properties in timestamped simplex data from various domains. It finds that predicting the closure of open triangles (where a pair of nodes have interacted but not with the third) performs well, and that simply averaging the edge weights in a triangle is often a good predictor. Predicting new structures in communication, collaboration and proximity networks can provide insights beyond classical link prediction.
Simplicial closure and higher-order link prediction (SIAMNS18)Austin Benson
This document summarizes research on modeling and predicting the formation of higher-order relationships or interactions between nodes in network datasets. It introduces the concept of "simplicial closure" to describe how groups of nodes interact over time until forming a simplex or higher-order relationship. The researchers propose "higher-order link prediction" as a framework to evaluate models for predicting the formation of new simplices. They test various methods for scoring open triangles based on edge weights and other structural properties to predict which will become closed triangles. The results show these approaches can significantly outperform random prediction, with simply averaging edge weights often performing well.
Simplicial closure and higher-order link predictionAustin Benson
This document summarizes research on simplicial closure and higher-order link prediction in network science. It finds that groups of nodes often interact through complex trajectories before reaching "simplicial closure" where all nodes are jointly present in a simplex. Predicting these closed simplices is framed as a higher-order link prediction problem. Various score functions are proposed based on edge weights, node neighborhoods, and similarity measures. Scores combining local edge weight information consistently perform well, outperforming classical link prediction approaches. The results provide insights into higher-order structure and a framework for evaluating models of complex relational data.
Computational Frameworks for Higher-order Network Data AnalysisAustin Benson
1. The document discusses computational frameworks for analyzing higher-order network data, where interactions can involve more than two nodes. Real-world systems often involve higher-order interactions that are reduced to pairwise connections.
2. The author presents several datasets involving higher-order interactions and shows that predicting the formation of new higher-order connections is similar to link prediction but considers groups of nodes rather than individual links. Structural properties like edge density and tie strength influence the likelihood of simplicial closure.
3. Models are proposed to score open simplices based on structural features and predict which will transition to closed simplices. Accounting for higher-order structure provides new insights beyond traditional network analysis of pairwise connections.
Higher-order link prediction and other hypergraph modelingAustin Benson
Higher-order link prediction and other hypergraph modeling can better model real-world systems composed of higher-order interactions that are often reduced to pairwise ones. Hypergraphs allow the modeling of interactions between more than two nodes, like groups of people collaborating, multiple recipients of emails, students gathering in groups, and drug compounds made of several substances.
Simplicial closure & higher-order link predictionAustin Benson
The document discusses higher-order link prediction in networks. It summarizes previous work representing higher-order interactions as tensors, hypergraphs, etc. It then proposes evaluating models of higher-order data using "higher-order link prediction" to predict which groups of more than two nodes will interact based on past data. The authors analyze dynamics of triadic closure in several real-world networks and propose methods to predict closure based on structural properties like edge weights.
Three hypergraph eigenvector centralitiesAustin Benson
Three hypergraph eigenvector centralities are proposed to measure the importance of nodes in complex systems modeled as hypergraphs. Hypergraphs generalize graphs by allowing edges to connect any number of nodes. The proposed centralities are adaptations of the standard graph eigenvector centrality to hypergraphs. They measure a node's centrality based on 1) the centralities of its neighbors, 2) being positive values, and 3) being the principal eigenvector of the hypergraph adjacency matrix.
The document discusses different models for how social networks grow over time, including preferential attachment and fitness models. It proposes using discrete choice theory as a way to model network growth, which allows incorporating covariates and flexible modeling. The approach is statistically rigorous and allows easy incorporation of new models and effects compared to traditional static network models.
Link prediction in networks with core-fringe structureAustin Benson
1. The document discusses link prediction in networks with a core-fringe structure. It examines how including connections from fringe nodes affects the performance of link prediction algorithms on the core nodes.
2. An experiment was conducted where a link prediction algorithm was run multiple times, each time including more fringe nodes and connections in order to measure the effect on link prediction accuracy for the core nodes.
3. The results showed that including more information from the fringe helped improve the link prediction performance on the core nodes.
This document discusses research on modeling and predicting higher-order interactions in networks beyond pairwise connections. The researchers collected datasets containing time-stamped groups or "simplices" of nodes and analyzed properties like triangle closure. They propose "higher-order link prediction" to predict which new simplices will form based on structural features like edge weights between nodes. Scoring functions were tested and averages of edge weights often performed well, differing from classical link prediction methods.
This document summarizes a talk about higher-order link prediction in networks. It discusses organizational principles of systems with higher-order interactions, how they evolve over time through simplicial closure events, and how insights can be used to create effective higher-order link prediction methods. Key points include that simplicial closure depends on the structure and strength of ties in the projected graph, and this closure process is similar for 3 and 4 nodes.
Random spatial network models for core-periphery structureAustin Benson
The document proposes a random spatial network model for generating networks with core-periphery structure. The model assigns each node u a weight e^θu and the probability of an edge between nodes u and v is proportional to e^θu + e^θv. This generates networks where high-weight nodes in the "core" have many connections and low-weight "periphery" nodes have few connections.
Random spatial network models for core-periphery structure.Austin Benson
The document proposes a random spatial network model for generating networks with core-periphery structure. The model assigns each node u a weight e^θu and the probability of an edge between nodes u and v is proportional to e^θu + e^θv. This leads to dense connections between high-weight core nodes and sparser connections between core and low-weight peripheral nodes.
Simplicial closure & higher-order link predictionAustin Benson
This document discusses higher-order link prediction and simplicial closure as ways to analyze and model higher-order interactions in network data. It summarizes that networks can be viewed as weighted projected graphs where simplices "fill in" structures, and that new simplices and closed triangles tend to form through trajectories of nodes reaching "simplicial closure events". It proposes evaluating models of higher-order structure through higher-order link prediction, predicting the formation of new simplices.
Simplicial closure and simplicial diffusionsAustin Benson
This document summarizes research on modeling higher-order interactions in network data using simplicial complexes. It finds that most real-world network datasets exhibit a mixture of closed and open triangles, with the fraction varying by domain. A simple probabilistic model can account for this variation. The document proposes that groups of nodes go through trajectories of interactions until reaching a "simplicial closure event" where a new simplex is formed, analogous to triangle closure. It evaluates models' ability to predict such closures using a framework of "higher-order link prediction". Key indicators of closure are edge density and tie strength between nodes.
Sampling methods for counting temporal motifsAustin Benson
The document summarizes research on developing scalable algorithms for counting temporal network motifs in real-time from high-throughput temporal network data streams. It discusses existing methods being insufficient and the problem of not having algorithms that can analyze modern temporal network datasets at fine time scales and high frequencies. It also briefly introduces the idea of using parallel sampling to speed up motif counting algorithms and enable analysis of very large temporal networks.
The document discusses three perspectives on predicting sets of items rather than single items. It describes how sets are common in data such as team formations, medical codes, and online purchases. It then discusses three specific approaches to set prediction: (1) predicting which sets an individual will interact with based on their history, (2) modeling sequences of sets using a generative model, and (3) understanding characteristics of set-based data like subsets and repeats. Applications include prediction, analysis, and simulation.
1. The document discusses sequences of sets, which is a common data structure where each item in a sequence can be a set of elements rather than a single element. Examples of data that can be modeled as sequences of sets include email recipients, tags on questions, academic coauthors, and contacts.
2. The authors provide a generative model to capture important characteristics of sequences of sets, such as subsets and supersets of prior sets being common and recency bias in repeat behavior.
3. Applications of the model include predicting new sets, anomaly detection, understanding user behaviors, and simulation.
Simplicial closure and higher-order link prediction --- SIAMNS18Austin Benson
The document discusses higher-order link prediction, which aims to predict the formation of new groups or "simplices" containing more than two nodes, based on structural properties in timestamped simplex data from various domains. It finds that predicting the closure of open triangles (where a pair of nodes have interacted but not with the third) performs well, and that simply averaging the edge weights in a triangle is often a good predictor. Predicting new structures in communication, collaboration and proximity networks can provide insights beyond classical link prediction.
Simplicial closure and higher-order link prediction (SIAMNS18)Austin Benson
This document summarizes research on modeling and predicting the formation of higher-order relationships or interactions between nodes in network datasets. It introduces the concept of "simplicial closure" to describe how groups of nodes interact over time until forming a simplex or higher-order relationship. The researchers propose "higher-order link prediction" as a framework to evaluate models for predicting the formation of new simplices. They test various methods for scoring open triangles based on edge weights and other structural properties to predict which will become closed triangles. The results show these approaches can significantly outperform random prediction, with simply averaging edge weights often performing well.
Simplicial closure and higher-order link predictionAustin Benson
This document summarizes research on simplicial closure and higher-order link prediction in network science. It finds that groups of nodes often interact through complex trajectories before reaching "simplicial closure" where all nodes are jointly present in a simplex. Predicting these closed simplices is framed as a higher-order link prediction problem. Various score functions are proposed based on edge weights, node neighborhoods, and similarity measures. Scores combining local edge weight information consistently perform well, outperforming classical link prediction approaches. The results provide insights into higher-order structure and a framework for evaluating models of complex relational data.
STATATHON: Unleashing the Power of Statistics in a 48-Hour Knowledge Extravag...sameer shah
"Join us for STATATHON, a dynamic 2-day event dedicated to exploring statistical knowledge and its real-world applications. From theory to practice, participants engage in intensive learning sessions, workshops, and challenges, fostering a deeper understanding of statistical methodologies and their significance in various fields."
Build applications with generative AI on Google CloudMárton Kodok
We will explore Vertex AI - Model Garden powered experiences, we are going to learn more about the integration of these generative AI APIs. We are going to see in action what the Gemini family of generative models are for developers to build and deploy AI-driven applications. Vertex AI includes a suite of foundation models, these are referred to as the PaLM and Gemini family of generative ai models, and they come in different versions. We are going to cover how to use via API to: - execute prompts in text and chat - cover multimodal use cases with image prompts. - finetune and distill to improve knowledge domains - run function calls with foundation models to optimize them for specific tasks. At the end of the session, developers will understand how to innovate with generative AI and develop apps using the generative ai industry trends.
Open Source Contributions to Postgres: The Basics POSETTE 2024ElizabethGarrettChri
Postgres is the most advanced open-source database in the world and it's supported by a community, not a single company. So how does this work? How does code actually get into Postgres? I recently had a patch submitted and committed and I want to share what I learned in that process. I’ll give you an overview of Postgres versions and how the underlying project codebase functions. I’ll also show you the process for submitting a patch and getting that tested and committed.
The Ipsos - AI - Monitor 2024 Report.pdfSocial Samosa
According to Ipsos AI Monitor's 2024 report, 65% Indians said that products and services using AI have profoundly changed their daily life in the past 3-5 years.
Global Situational Awareness of A.I. and where its headedvikram sood
You can see the future first in San Francisco.
Over the past year, the talk of the town has shifted from $10 billion compute clusters to $100 billion clusters to trillion-dollar clusters. Every six months another zero is added to the boardroom plans. Behind the scenes, there’s a fierce scramble to secure every power contract still available for the rest of the decade, every voltage transformer that can possibly be procured. American big business is gearing up to pour trillions of dollars into a long-unseen mobilization of American industrial might. By the end of the decade, American electricity production will have grown tens of percent; from the shale fields of Pennsylvania to the solar farms of Nevada, hundreds of millions of GPUs will hum.
The AGI race has begun. We are building machines that can think and reason. By 2025/26, these machines will outpace college graduates. By the end of the decade, they will be smarter than you or I; we will have superintelligence, in the true sense of the word. Along the way, national security forces not seen in half a century will be un-leashed, and before long, The Project will be on. If we’re lucky, we’ll be in an all-out race with the CCP; if we’re unlucky, an all-out war.
Everyone is now talking about AI, but few have the faintest glimmer of what is about to hit them. Nvidia analysts still think 2024 might be close to the peak. Mainstream pundits are stuck on the wilful blindness of “it’s just predicting the next word”. They see only hype and business-as-usual; at most they entertain another internet-scale technological change.
Before long, the world will wake up. But right now, there are perhaps a few hundred people, most of them in San Francisco and the AI labs, that have situational awareness. Through whatever peculiar forces of fate, I have found myself amongst them. A few years ago, these people were derided as crazy—but they trusted the trendlines, which allowed them to correctly predict the AI advances of the past few years. Whether these people are also right about the next few years remains to be seen. But these are very smart people—the smartest people I have ever met—and they are the ones building this technology. Perhaps they will be an odd footnote in history, or perhaps they will go down in history like Szilard and Oppenheimer and Teller. If they are seeing the future even close to correctly, we are in for a wild ride.
Let me tell you what we see.
4th Modern Marketing Reckoner by MMA Global India & Group M: 60+ experts on W...Social Samosa
The Modern Marketing Reckoner (MMR) is a comprehensive resource packed with POVs from 60+ industry leaders on how AI is transforming the 4 key pillars of marketing – product, place, price and promotions.
Beyond the Basics of A/B Tests: Highly Innovative Experimentation Tactics You...Aggregage
This webinar will explore cutting-edge, less familiar but powerful experimentation methodologies which address well-known limitations of standard A/B Testing. Designed for data and product leaders, this session aims to inspire the embrace of innovative approaches and provide insights into the frontiers of experimentation!
ViewShift: Hassle-free Dynamic Policy Enforcement for Every Data LakeWalaa Eldin Moustafa
Dynamic policy enforcement is becoming an increasingly important topic in today’s world where data privacy and compliance is a top priority for companies, individuals, and regulators alike. In these slides, we discuss how LinkedIn implements a powerful dynamic policy enforcement engine, called ViewShift, and integrates it within its data lake. We show the query engine architecture and how catalog implementations can automatically route table resolutions to compliance-enforcing SQL views. Such views have a set of very interesting properties: (1) They are auto-generated from declarative data annotations. (2) They respect user-level consent and preferences (3) They are context-aware, encoding a different set of transformations for different use cases (4) They are portable; while the SQL logic is only implemented in one SQL dialect, it is accessible in all engines.
#SQL #Views #Privacy #Compliance #DataLake
ViewShift: Hassle-free Dynamic Policy Enforcement for Every Data Lake
Hypergraph Cuts with General Splitting Functions (JMM)
1. 1
Joint work with
Nate Veldt & Jon Kleinberg (Cornell)
Hypergraph Cuts with General Splitting Functions
Austin R. Benson · Cornell University
AMS Special Session on Applied Combinatorial Methods
Joint Mathematics Meetings · January 9, 2021
2. Graph minimum s-t cuts are fundamental.
2
minimizeS⇢V cut(S)
subject to s 2 S, t /2 S.<latexit 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1 3
2 4
5
6
7
8
s
t
• Maximum flow / min s-t cut [Ford,Fulkerson,Dantzig 1950s]
• Computer vision [Bokykov-Kolmogorov 01; Kolmogorov-Zabih 04]
• Densest subgraph [Goldberg 84; Shang+ 18]
• First graph-based semi-supervised learning algorithms [Blum-Chawla 01]
• Local graph clustering [Andersen-Lang 08; Oreccchia-Zhu 14; Veldt+ 16]
Also see any undergraduate algorithms class
poly-time algorithms!
3. Real-world systems have“higher-order”interactions.
3
Physical proximity
• nodes are students
• People gather in groups
linear-algebra discrete-mathematics
math-software
combinatorics
category-theory
logic
terminology
algebraic-graph-theory
combinatorial-designs
hypergraphs
graph-theory
cayley-graphs
group-theory
finite-groups
Categorical information
• nodes are tags
• groups of tags applied to info
(same question on mathoverflow.com)
Networks beyond pairwise interactions: structure and dynamics. Battiston et al., 2020.
The why, how, and when of representations for complex systems. Torres et al., 2020.
Commerce
• nodes are products
• hyperedges are students
in the same class
4. We can model“higher-order”interactions
with hypergraphs.
4
H = (V, E), edge e 2 E is a subset of V (e ⇢ V)<latexit 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1 2
3
4
5
V = {1, 2, 3, 4, 5}
E = {{1, 2, 3}, {2, 4, 5}}<latexit 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5. 5
1. What is a hypergraph minimum s-t cut?
2. If we know what they are, can we find them efficiently?
3. If we can find them efficiently, what can we use them for?
We should have a foundation for
hypergraph minimum s-t cuts,but…
6. What is a hypergraph minimum s-t cut?
6
s
t
Should we treat the 2/2 split
differently from the 1/3 split?
Historically, no. [Lawler 73,Ihler+ 93]
More recently, yes.
[Li-Milenkovic 17,Veldt-Benson-Kleinberg 20]
1 3
2 4
5
6
7
8
s
t
There is only one way to
split an edge (1/1).
7. We model hypergraph cuts with splitting functions.
7
s
t
Given a cut defined by S,
we incur penalty of
at each hyperedge e.
Hypergraph minimum s-t cut problem.
Cardinality-Based splitting functions.
S<latexit 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cutH(S) = f (2) + f (1)<latexit 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sha1_base64="JdV0NHpso/GwwYvqd/CeIvys+E4=">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</latexit>
<latexit sha1_base64="6FFH4JtCJ1Fb69WjugRCayyF5vI=">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</latexit>
we(e S)
<latexit sha1_base64="QjrhfsKLxaK/82LxnRurhFCzCik=">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</latexit>
minimizeS⇢V
P
e2E we(e S) ⌘ cutH(S)
subject to s 2 S, t /2 S.
<latexit sha1_base64="vCSQ5hxLftoc4zdzUNdXcsthqGM=">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</latexit>
Non-negativity we(A) 0.
Non-split ignoring we(e) = we(;) = 0.
C-B we(A) = f (min(|A|, |Ae|)).
9. Cardinality-based splitting functions are easy to specify.
9
minimizeS⇢V
P
e2E we(e S) ⌘ cutH(S)
subject to s 2 S, t /2 S.<latexit 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s
t
One extra scaling DOF, so set w1 = 1. Specify w2, ... , wbr/2c.<latexit 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cutH(S) = f (2) + f (1) = w2 + 1<latexit 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Only need to specify f(1), f(2), …, f(⌊r / 2⌋), where r = max hyperedge size.
Just scalars. f(i) = wi.
Cardinality-based splitting functions.
<latexit sha1_base64="vCSQ5hxLftoc4zdzUNdXcsthqGM=">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</latexit>
Non-negativity we(A) 0.
Non-split ignoring we(e) = we(;) = 0.
C-B we(A) = f (min(|A|, |Ae|)).
10. Cardinality-based splitting functions are easy to specify.
10
Just need to specify w2, ... , wbr/2c and assume w1 = 1.<latexit 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r = 2 (graphs) r = 3 (3-uniform hypergraph)
“Only one way to split a triangle”
[Benson+ 16; Li-Milenkovic 17; Yin+ 17]
s
t
s
t
s
t
r = 4 w2 = 0.5 solution w2 = 1.5 solution w3 = 1.5 solution
11. 1.0 1.25 1.5 1.75 2.0
fusion- systems
topological- stacks
graph- invariants
adjacency- matrix
signed- graph
gorenstein
cohen- macaulay
topological- k- theory
difference- sets
pushforward
regular- rings
graph- connectivity
block- matrices
directed- graphs
eulerian- path
central- extensions
group- extensions
semidirect- product
wreath- product
graded- algebras
supergeometry
geometric- complexity
soliton- theory
matrix- congruences
teichmueller- theory
superalgebra
string- theory
riemann- surfaces
group- cohomology
dglas
celestial- mechanics
s- seed = symplectic- linear- algebra
t- seed = bernoulli- numbers
Different weights lead to different min cuts in practice.
11
1.00 1.25 1.50 1.75 2.00
0.7
0.8
0.9
1.0
JaccardSimilarity
12. 12
1. What is a hypergraph minimum s-t cut?
2. If we know what they are, can we find them efficiently?
3. If we can find them efficiently, what can we use them for?
We should have a foundation for
hypergraph minimum s-t cuts,but…
13. We solve hypergraph cut problems with graph reductions.
13
1/21/2
1/2
1
1
1
1
∞
∞ ∞
∞
∞∞
Gadgets (expansions) model a hyperedge with a small graph.
clique expansion star expansion Lawler gadget [1973]hyperedge
In a graph reduction, we first replace all hyperedges with graph gadgets...
s
t
s
t
s
t
s
t
… then solve the (min s-t cut) problem exactly on the graph,
and finally convert the solution to a hypergraph solution.
14. b
We made a new gadget for C-B splitting functions.
14
This gadget models min(|A|, |eA|, b).
Theorem [Veldt-Benson-Kleinberg 20a]. Nonnegative linear combinations of the
C-B gadget can model any submodular cardinality-based splitting function.
See also Graph Cuts for Minimizing Robust Higher Order Potentials,Kohli et al.,2008.
<latexit sha1_base64="beQz4cdyY+p8N+9L01TDcNAiwcQ=">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</latexit>
C-B we(A) = f (min(|A|, |eA|)).
(F is submodular on X if F(A B) + F(A [ B) F(A) + F(B) for any A, B ✓ X.)<latexit 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15. 15
Theorem [Veldt-Benson-Kleinberg 20a]. The hypergraph min s-t cut problem
with a cardinality-based splitting function is graph-reducible (via gadgets)
if and only if the splitting function is submodular.
Cardinality-based splitting functions.
s
t
S<latexit 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cutH(S) = f (2) + f (1)<latexit 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Submodularity is key to efficient algorithms.
What happens when the splitting function isn’t submodular?
Is there some other efficient algorithm?
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Non-negativity we(A) 0.
Non-split ignoring we(e) = we(;) = 0.
C-B we(A) = f (min(|A|, |Ae|)).
16. 16
Unlike graph min s-t cut,
hypergraph min s-t cut can be NP-hard.
w1 = 1
0 1 2 w2
??
Reducible/Submodular
NP-hard
Unknown
Hard Reducible
w3
3
2.5
2
1.5
1
0.5 1 1.5 2 2.5 w2
0.5
w2
w3
w4
4
3
2
1
0
1
1.5
2
2.5
1
2
3
max hyperedge size 4 or 5 max hyperedge size 6 or 7 max hyperedge size 8 or 9
Theorem [Veldt-Benson-Kleinberg 20]. For C-B splitting functions,
Open Question: For 4-uniform hypergraphs, is there an efficient algorithm
to find the minimum s-t cut with no 2-2 splits (w1 = 1, w2 = ∞).
s
t
cutH(S) = f (2) + f (1)
= w2 + 1<latexit 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17. 17
1. What is a hypergraph minimum s-t cut?
2. If we know what they are, can we find them efficiently?
3. If we can find them efficiently, what can we use them for?
We should have a foundation for
hypergraph minimum s-t cuts,but…
18. G = (V,E) is a graph.
R ⊆ V (Reference or seed set).
Finds a “good” cluster S “near” R.
18
Background.Local clustering has been studied
extensively in graphs,but not much in hypergraphs.
19. Rewards high
overlap with R.
Penalizes nodes
outside R.
R(S) =
cut(S)
vol(S R) "vol(S ¯R)
Max Flow.Quot.Imp.(Lang,Rao,2004)
Flow-Improve (Andersen,Lang 2008)
Local-Improve (Orecchia,Allen-Zhou 2014)
SimpleLocal (Veldt,Gleich,Mahoney 2016)
FlowSeed (Veldt,Klymko,Gleich 2019)
Great survey paper! (Fountoulakis et al.2020)
19
Background.Flow-based methods minimize a
localized variant of conductance.
Rewards
contained clusters
vol(T) = sum of
degrees in T.
minimize
node sets S
FAST ALGORITHMS FOR
EXACT MINIMIZATION!
21. We generalize local flow-based techniques to
the hypergraph setting.
21
• We introduce localized hypergraph conductance
• We can minimize it exactly with our hypergraph min s-t cuts framework
• Strongly-local runtime! (Only depends on size of seed set)
• Normalized cut improvement guarantees The analysis provides even new
guarantees for the graph case!