[PDF] - Lifted Message Passing Rorschach Test K. Kersting 2 Lifted PDF Document

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GRAPHBOT 2010. IROS WORKSHOP ON PROBABILISTIC GRAPHICAL MODELS IN ROBOTICS Taipei, Taiwan, October 22. 2010

Lifted Message Passing

Kristian Kersting

Babak Ahmadi Sriraam Natarajan Fabian Hadiji Scott Sanner Youssef El Massaoudi

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K. Kersting

Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Rorschach Test

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Etzioni’s Rorschach Test for Computer Scientists

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Moore’s Law?

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Storage Capacity?

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Number of Facebook Users?

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Number of Scientific Publications?

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Number of Web Pages?

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Number of Actions?

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Computing 2020: Science in an Exponential World How to deal with millions of images ? How to accumulate general knowledge automatically from the Web ? How to deal with billions of shared users’ perceptions stored at massive scale ? How to realize the vision of social search? How to realize a roboter outofthebox?

“The amount of scientific data is doubling every year”

[Szalay,Gray; 440, 413414 (23 March 2006) ]

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Machine Learning in an Exponential World

Machine Learning = Data + Model

Most effort has gone into the modeling part How much can the data itself help us to solve a problem?

[Fergus et al. PAMI 30(11) 2008; Halevy et al., IEEE Intelligent Systems, 24 2009]

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K. Kersting

Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Spectrally Hashed Logistic Regression

[Behley, K, Schulz, Steinhage, Cremers IROS10]

Classes: Car Vegetation Ground Building

Ground Truth Our Approach

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K. Kersting

Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010 Real world is structured in terms of objects and relations Relational knowledge can reveal additional correlations between

variables of interest . Abstraction allows one to compactly model general knowledge and to move to complex inference

Machine Learning in an Exponential World

Machine Learning = Data + Model ML = Structured Data + Model + Reasoning

Most effort has gone into the modeling part How much can the data itself help us to solve a problem?

[Fergus et al. PAMI 30(11) 2008; Halevy et al., IEEE Intelligent Systems, 24 2009]

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

http://www.cs.washington.edu/research/textrunner/ Object Object Relation Uncertainty

“Programs will consume, combine, and correlate everything in the universe of structured information and help users reason over it.”

[S. Parastatidis et al., CACM Vol. 52(12):3337 ]

[Etzioni et al. ACL08]

No complex inference (yet) !

TextRunner: (Turing, born in, London) + WordNet: (London, part of, England) + Rule: ‘born in’ is transitive thru ‘part of’ Conclusion: (Turing, born in, )

So, how do computer systems deal with uncertainty, objects, and relations? How do we realize the vision of a world:wide:mind?

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

(First:order) Logic handles Complexity

atomic propositional firstorder/relational

Many types of entities Relations between them Arbitrary knowledge

19th C 5th C B.C. Explicit enumeration

daugtherof(cecily,john) daugtherof(lily,tom) M

E.g., rules of chess (which is a tiny problem): 1 page in firstorder logic, ~100000 pages in propositional logic, ~100000000000000000000000000000000000000 pages as atomicstate model Logic true/false

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Probability handles Uncertainty

Logic true/false Probability atomic propositional firstorder/relational

Sensor noise Human error Inconsistencies Unpredictability

5th C B.C. 19th C 17th C 20th C

Many types of entities Relations between them Arbitrary knowledge

Explicit enumeration

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Will Traditional AI Scale ?

Logic true/false Probability atomic propositional firstorder/relational

Sensor noise Human error Inconsistencies Unpredictability

5th C B.C. 19th C 17th C 20th C

Many types of entities Relations between them Arbitrary knowledge

Explicit enumeration “Scaling up the environment will inevitably overtax the resources of the current AI architecture.”

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Statistical Relational Learning / AI (StarAI*)

M unifies logical and statistical AI, M solid formal foundations, M is of interest to many communities.

Let‘s deal with uncertainty, objects, and relations jointly

A

Robotics CV Search Planning SAT Probability Statistics Logic Graphs Trees Learning

Natural domain modeling:
bjects, properties,

relations

Compact, natural models
Properties of entities can

depend on properties of related entities

Generalization over a

variety of situations

(*)First StarAI workshop at AAAI10;cochaired with S. Russell, L. Kaelbling, A.Halevy, S. Natarajan, and L. Milhalkova

The study and design of intelligent agents that act in noisy worlds composed of

bjects and relations among

the objects

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Pros of SRL / StarAI

Relations can reveal additional

correlations. Abstraction allows for

generalization and compactness SRL/StarAI techniques have the potential to lay the foundations of next generation AI systems

+++

Better performance Better understanding of domains Growth path for machine learning, artificial intelligence, and robotics

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K. Kersting

Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

For example, we can A

M learn probabilistic relational models automatically from

millions of interrelated objects

M generate optimal plans and learn to act optimally in uncertain

environments involving millions of objects and relations among them

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Markov Logic Networks

( )

) , ( _ ) , ( ) , ( , ) ( ) ( ) , ( _ , ) ( ) ( _ ) ( _ ) ( ) , ( y x author co p y author p x author p y x y smart x smart y x author co y x p accepted p quality high x p quality high x smart p x author x ⇒ ∧ ∃ ∀ ⇔ ⇒ ∀ ⇒ ∀ ⇒ ∧ ∀

∞ 2 . 1 1 . 1 5 . 1

Suppose we have constants: alice, bob and p1

smart(bob) smart(alice) high_quality(p1) author(p1,alice) author(p1,bob) accepted(p1) co_author(bob,alice) co_author(alice,bob) co_author(alice,alice) co_author(bob,bob)

[Richardson, Domingos MLJ 62(12): 107136, 2006]

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Relational Exploration [Lange, Toussaint, K ECML:PKDD10]

Simulated robot manipulation domain (realistic physics engine, > 2^100 states)
Robot starts from zero knowledge and actively generates training trajectories

fully grounded network!!

Goal: pile objects

Placeholders (X,Y A) allow one to generalize experience

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Pros and Cons of SRL / StarAI

Relations can reveal additional

correlations. Abstraction allows for

generalization and compactness Yes, SRL/StarAI is challenging but knowing

ne of its ingredients is half the battle

SRL/StarAI techniques have the potential to lay the foundations of next generation AI systems

+++

Better performance Better understanding of domains Growth path for machine learning, artificial intelligence, and robotics

:::

Learning is much harder Inference becomes a crucial issue Greater complexity for user

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

So, can we do better?

Inference in firstorder logic is not „ground“
M it is lifted, i.e., it never „touches“ the ground

talk Yes, we can and it is exactly the focus of my talk Resulting lifted approaches are often

Faster
More compact

... and provide more structure for optimization

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Distributions can naturally be represented as factor graphs

There is an edge between a circle and a box if the variable is in

the domain/scope of the factor

unnormalized !

Random variable Factor resp. potential

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Factor Graphs from Graphical Models

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Variable Elimination Sum out nonquery variables

ne by one

A

popular start series attends(p1) attends(p2) attends(p)

φ1(pop, att(p1)) φ2(att(p1), ser)

∑

att(p1)

φ′(pop, ser)

Time is linear in number of invitees

Inviting people to a workshop Does not exploit symmetries encoded in the structure of the model

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

[Poole IJCAI03; de Salvo Braz et al. IJCAI05, Milch,Zettlemoyer, Haims,K, Kaelbling AAAI08]

First:Order Variable Elimination

∀X. φ1(popular, attends(X)) ∀X. φ2(attends(X), series)

A

popular start series attends(p1) attends(p2) attends(p)

!"

Based on logically parameterized factors

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

First:Order Variable Elimination Idea: Sum out all attends(X) variables at once

∀X. φ1(popular, attends(X)) ∀X. φ2(attends(X), series)

A

popular start series attends(p1) attends(p2) attends(p) [Poole IJCAI03; de Salvo Braz et al. IJCAI05, Milch,Zettlemoyer, Haims,K, Kaelbling AAAI08]

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K. Kersting

Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

First:Order Variable Elimination Idea: Sum out all attends(X) variables at once

∀X. φ2(attends(X), series) ∀X. φ′(popular, series) φ′(popular, series)

A

popular start series attends(p1) attends(p2) attends(p)

Time is constant in

This exploits symmetries encoded in the structure of the model

∀X. φ1(popular, attends(X))

[Poole IJCAI03; de Salvo Braz et al. IJCAI05, Milch,Zettlemoyer, Haims,K, Kaelbling AAAI08]

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Results: Competing Workshops

50 100 150 200 50 100 150 200 Time (ms) Number of Invitees VE FOVE CFOVE

1

These exact inference approaches are

rather complex

so far do not easily scale to realistic

domains,

and hence have only been applied to

rather small artificial problems

What about approximate inference approaches?

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

How do you spend your spare time?

YouTube like media portals have changed the way users access media content in the Internet Every day, millions of people visit social media sites such as Flickr, YouTube, and Jumpcut, among

thers, to share their photos and videos, M

while others enjoy themselves by searching, watching, commenting, and rating the photos and videos; what your friends like will bear great significance for you.

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

How do you efficiently broadcast information?

Quite similar to the task allocation problem

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Content Distribution using Stochastic Policies

[Bickson et al. WDAS04]

Large (distributed) networks, so Bickson et al. propose to use (loopy) belief propagation

Add an edge potential between any two nodes which can take a part from a common neighboring node.

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

The Sum:Product Algorithm aka Belief Propagation

Iterative process in which neighboring variables “talk” to each
ther, passing messages such as:

“I (variable belong in these states with various likelihoodsM”

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Loopy Belief Propagation

After enough iterations, this series of conversations is likely to

converge to a consensus that determines the marginal probabilities of all the variables.

SumProduct/BP

(1) update messages until convergence (2) compute single node marginals

Variants exist for solving

SAT problems, systems of linear equations, matching problems and

for abitrary distributions (based on sampling)

A lot of shared factors! Can we make use of this? Can we speed up belief propagation in this case?

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Yes, we can: Lifted (Loopy) Belief Propagation

[Singla, Domingos AAAI08, K, Ahmadi, Natarajan UAI09]

Counting shared factors can result in great efficiency gains for (loopy) belief propagation Shared factors appear more often than you think in relevant real world problems

identical

http://www:kd.iai.uni:bonn.de/index.php?page=software_details&id=16

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Social Network Analysis

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Social Network Analysis

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Social Network Analysis

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Step 1: Compression

http://www:kd.iai.uni:bonn.de/index.php?page=software_details&id=16

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

http://www:kd.iai.uni:bonn.de/index.php?page=software_details&id=16

Step 2: Modified Belief Propagation

Only difference* : counts c(f,x)

How often do elements of superfactor F send messages to elements of supernode X

*actually also depends on the position at which the element appears in the factor

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Lifted Factored Frontier

20 people over 10 time steps. Max number of friends 5. Cancer never observed. Time step randomly selected.

Satisfied by Lifted Message Passing?

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Lifted Satisfiability

[K, Ahmadi, Natarajan UAI09, Hadiji, K, Ahmadi StarAI10]

BPCount, Warning and survey propagation can also be lifted
Enables lifted treatment of both prob. and det. knowledge

Similar, we can lift linear and quadratic assignment solvers based on message passing algorithms (min:sum)!

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Content Distribution (Gnutella): Lifted BP vs. BP

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Message Errors to the Rescue!

Make use of decaying message errors already at lifting time Make use of decaying message errors already at lifting time

Ihler et al. 05: BP message errors decay along paths LBP may spuriously assume some nodes send and

receive different messages and, hence, produce pessimistic lifted network

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Make use of decaying message errors already at lifting time Make use of decaying message errors already at lifting time

Ihler et al. 05: BP message errors decay along paths LBP may spuriously assume some nodes send and

receive different messages and, hence, produce pessimistic lifted network

Informed Lifted Belief Propagation

[El Massaoudi, K, Ahmadi, Hadiji AAAI10]

Informed LBP: (1) Alternate Lifting and Modified BP (2) Cluster according to actually computed messages

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Social Networks / Lifted Content Distribution

1 file, Gnutella snapshort

10876 nodes and 39994 edges iLBP 4.3 mio mess.< BP 5.8 mio mess.< LBP 6.4 mio mess.

On a different network:

iLBP 1.9mio < LBP 2.9mio < BP 5.8mio

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Workhorse of robotics: Kalman Filter

An efficient iterative algorithm to estimate the state of a discrete:time controlled process that is governed by a linear stochastic difference eq. Why should robotics care about all this? Why should robotics care about all this?

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Workhorse of robotics: Kalman Filter

An efficient iterative algorithm to estimate the state of a discrete:time controlled process that is governed by a linear stochastic difference eq. 1. Prediction: 2. 3. 4. Correction: 5. 6. 7. 8. Return Σ

t t t t t

u B A + =

−1

t

T t t t t

R A A + Σ = Σ

−1 1

) (

−

+ Σ Σ =

t T t t t T t t t

Q C C C K ) (

t t t t t t

C z K

−

+ =

t t t t

C K I Σ − = Σ ) (

Matrix inversion can be computed efficiently in a distributed fashion using the Gaussian Belief Propagation algorithm [Shental et al. ISIT08]

∏ ∏

=

∝

j} {i, ij K 1 i i

) , ( ) ( ) (

j i i

x x x x p ψ φ ) exp( ) , ( ) 2 / exp( ) (

ij 2 i j ij i j i i ii i i i

x A x x x x A x b x − = − = ψ φ

A x b ⋅ =

Lifted Kalman Filter [ongoing work]

So, let‘s run Lifted Gaussian Belief Propagation

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Solving System of Linear Equations using GBP

T

X X X           =                     − − 5 . 5 . 1 . 55 . 45 . 45 . 55 .

3 2 1

) 82 . 1 , 91 . ( ) 55 . , 55 . / 5 . (

1 1

) ) = ≈

−

φ ) 82 . 1 , 91 . (

2

) ≈ φ ) , (

3

∞ ≈ ) φ

) 45 . exp( ) , (

2 1 2 1 12

x x x x = ψ ) 45 . exp( ) , (

1 2 1 2 21

x x x x = ψ

[Shental et al. ISIT08]

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Matrix Inversion using GBP

A

I1 =E1 E1

A

I2 = E2

A IN

IN=EN EN

A

1

A I

− =

Factor Graph Part 1 Factor Graph Part 2 Factor Graph Part N

A

Gaussian BP

k k

A I e ⋅ =

I1 I2 IN IN

A

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Lifted Matrix Inversion

A

I1 =E1 E1

A

I2 = E2

A IN

IN=EN EN

A

Super Factor Graph Factor Graph Part 1 Factor Graph Part 2 Factor Graph Part N

A

Lifted Factor Graph

Lifting

I

Modified Gaussian BP

k k

A I e ⋅ =

1

A I

− =

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

First Results

dtx q1 r1 std

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

Conclusions

StarAI ≥ Objects&Relations + Probabilities + Machine Learning
It covers the whole AI spectrum

Lifted SAT, Relational (PO)MDPs, Robotics, M

Lifted/efficient reasoning crucial to StarAI

Exploit symmetries revealed by (relational) model

More tasks and applications!
Lifted inference for arbitrary distributions?
Lifted linear programs?
Is there a relational cortex of a robot?

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Lifted Message Passing GRAPHBOT@ ROS 2010 Taipei, Taiwan, October 22, 2010

How to solve commonsense reasoning How to solve Natural Language Processing How to solve robotics? How to solve Vision Domain Knowledge Inference and Learning Language Knowledge Inference and Learning Domain & Robot Knowledge Inference and Learning Objects & Optics Knowledge Inference and Learning