ambiguity and uncertainty - ucla statisticssczhu/courses/ucla/stat_232b/handouts/ch6_… ·...

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Part 3: Computing Multiple Solutions from MCMC

Chapter 6 Advanced Algorithms for Inference in Gibbs fields

Computing multiple distinct solution is of great importance for preserving ambiguities and uncertainty. We introduce two algorithms

1, C4: Clustering with Cooperative and Competitive Constraints. in J. Porway, PAMI 2011

2, K-adventure’s algorithm in DDMCMC.

Stat 232B-CS276B Stat computing and inference. © S.C. Zhu

in Z. Tu, PAMI 2002

Ambiguity and uncertainty

Ambiguities : multiple distinct solutionsp(x)

Uncertainty : a range of variants

Both increase the entropy of the perception (posterior probability):


H(W) | I = entropy (p(W | I )

When H(W) | I is too large, we say W is imperceptible. Thus we seek for a simpleand low-dimensional perception W_. Taught in stat232A.

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Rationale for computing multiple solutions

Computing multiple solutions to avoid premature commitments and to preserve the intrinsic ambiguities.


Two criteria for designing “effective algorithms”:

1, Fast convergence to global optimal solutions: “not greedy”

Rationale for computing multiple solutions (cont.)

g g p g y

But this is not enough, especially when the probability has multiple modes.

2, Exploring diverse solutions (liquidate, fast mixing) : “not stubborn”

1 0 2 2 0 1 0 2 1 … is more dynamic than 1 1 1 2 2 2 0 0 0


And thus the former can better represent the underlying distributions by a shorter sequence.

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Sampling Probabilities with Multiple Modes

The two criteria for MCMC design: 1, Short “burn-in” period --- The MC reaches the equilibrium fast 2, Fast “mixing rate” --- The MC states are less correlated in time


p(X)

Sometimes, to design ambiguities is to create abstract art.

Rationale for computing multiple solutions (cont.)


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Example: creating abstract art from picture


M.T. Zhao, “Abstract Painting with Interactive Control of Perceptual Entropy,” ACM Trans. on Applied Perception, 2013.

Example of computing multiple solutions in vision

a. Input image b. Segmented texture regions c. synthesis by texture models


d. curve processes + bkgd region e. synthesis by curve models

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Ambiguities are ubiquitous in images

Example of google earth image in aerial image understanding


Local interpretations are often strongly coupled !

They form “clusters” and the search algorithms often get stuck.

Revisit: Swendsen-Wang and cluster sampling

(a) Augment with auxiliary bonding variable U along edges E:

)(

,

1),();,(1

)( ts xxtst

tss exxxx

ZXp

Ising model:

}}10{{ XtU }}1,0{;,,{ stst uXtsuU

(c) Select a connected component V0 and update its nodes’ labels.state A state B

V2V2

(b) Turn edges “on” (ust = +1) or “off” (ust = 0) probabilistically.


V0V0

V1V1

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Potts Model with Negative Edges (“frustrated” system)

)()(1

)(X

Split our edge set E into E+ and E- to represent positive and negative interactions:

)(1)(1

,,

),(,),(

),(),()(

tsts xxts

xxts

tEji

stEts

s

exxexx

xxxxZ

Xp

Form components similarly to Swendsen-Wang.


Components now consist of positively connected sub-components connected by negative interactions.

Experiments on Negative Edge Ising Model

Created “checkerboard” constraint problem.p


Initial State Solution 1 Solution 2

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CCCP: Composite Connected Components


Edge probabilities for clustering cccp’s

For positive edge, the edge variable follows a Bernoulli probability

1

01)|(

10

01)|(

1

01)|(

ij

jiij

ij

ij

jiij

ij

ij

jiij

uxxup

u

uxxup

u

uxxup

p y

ue ~ Bernoulli(qe1(xs=xt))

For negative edge, it follows

ue ~ Bernoulli(qe1(xs xt))


10

01)|(

1

ij

ij

jiij

ijjiij

u

uxxup

u

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Mixing rate

Solution states over time for Ising model (two states)

Sta

teS

tate

SW

C4


TimeC4 has- Low burn-in time (converges quickly).- High mixing rate (samples remaining unbiased over short run).

Simulating the Potts model

Energy vs. time

SW


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Representation: Candidacy Graphs

We formulate a candidacy graph representation, as it can represents1, MRF and CRF structures2, Soft and hard constraints.3, Positive (collaborative) and negative (competitive) edges.


Flattening the Candidacy Graphs


This becomes a graph coloring problem: each color is a solution.

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Ex: Candidacy Graph for Necker Cube


Some labels/edges omitted for clarity

Application I: Line Drawing

Solution1 Solution2

Solution 2

Solution 1 Solution 2Solution 1


C4 finds both solutions, swaps corner labels continuously.

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Line drawing interpretation

+ ++ + ++ - -


+

++

+

+ +++

++

-

+-

+ +++

+

-+

-+

-

- ---

-+

Application II: CRF for Building Detection

- Learn a CRF for building detection1.

- CRF labels each window of image as {0,1} .


1S. Kumar, M. Hebert, “Man-Made Structure Detection in Natural Images using a Causal Multiscale Random Field”, CVPR, 2003.

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Comparison: CRF for Building Detection


LBP = Loopy Belief Propagation ICM = Iterated Conditional Modes

Comparison: CRF Building Detection

Algorithm FP (per image) Detection Rate (%)

Loopy BP 23.18 0.451

Swendsen‐Wang 156.05 0.468

C4 47.12 0.696

ICM 61.78 0.697


Note: Couldn’t recreate Kumar’s exact results, so compared our CRF with his inference method (ICM) and ours (C4).

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Application III: Aerial Image Segmentation

-Detect objects in an aerial image.

- MANY false positives.

- Have C4 determine which ones best explain the image by turning detected objects “on” and “off”.

Car

Car

Car

Parking Lot

Tree Car

Car

Car

Parking Lot

Tree

(a) (b)

Car

Car

Car

Parking Lot

Tree

(c)


Car

Building

Car

Car

BuildingCCP1

CCP2

Positive Edges

Negative Edges

In Current Explanation

Not In Current Explanation

Car

Car

BuildingCCP1

CCP2

Results: Aerial Image SegmentationTreesRoadsCarsRoofsParking Lots


Initial Bottom-up Detections C4-Selected Subset

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Results: Aerial Image Segmentation

P-R curve for original detections vs. C4 pruned detections.

Original

C4

Example of C4’s benefits: Misinterpreted


Misinterpreted section of image is later updated by large composite move. (b)(a)

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Problem with “Flat”-- C4

The “love triangles”:create inconsistent clusters

B

C

Ears

Eye

Head

Love triangle

A

C


Eye

Beak Nosee.g. Love triangles in the duck/rabbit illusion.

Hierarchical C4

The candidacy graph so far represent pairwise edges, high-order relations arerepresented by extended candidacy graphs


System will now flip between duck and rabbit without love triangle issue.

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Elephant Illusion

Elephant

Hierarchical part model Layered representation of hierarchy


Leg Pair, Head, Back

Leg, Trunk

Line

Composing the Bottom Layer

Trunk compatibilitiesp

Leg compatibilities


Elephant


Leg, Trunk

Line

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Part Binding for Next Layer

Trunk bindingsTrunk bindings

Leg bindings


Elephant


Leg, Trunk

Line

Continue Sampling/Binding for each layer

Leg Pair compatibilities

Leg Pair bindings


Elephant


Leg, Trunk

Line

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Top level bindings and sampling

Elephantcompatibilities


Elephant


Leg, Trunk

Line

1, A candidacy graph representation incorporates

MRF and CRF structures

Summary

Soft and hard constraints.Positive (collaborative) and negative (competitive) edges.

hierarchic compositional structures for high-order relations.

2, An algorithm with large composite moves

Fast convergence to global solutions


Fast convergence to global solutionsEffective mixing between solutions

Relaxation labeling Gibbs sampling SW Cut C4

(Rosenfeld/Zucker/Hammel 1978) (Geman/ Geman 1984) (Barbu/Zhu 2005) (Porway/ Zhu 2009)

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General MCMC Design

Design Strategy I Design Strategy II

InputI

1. Proposal Q(x,y) 2. Transition K(x,y)3. Initial prob.

MC=(, K, )

x1, x2, …., xn ~ (x) Solutions))(ˆ||)((minarg xxKL **

2*1 ,...,, Kxxx


Computing Multiple Solutions

To faithfully preserve the posterior probability p(W|I),

We compute a set of weighted scene particles {W1, W2, …, WM},

A mathematical principle:


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Computing Multiple Solutions

To faithfully preserve the posterior probability p(W|I),

We compute a set of weighted scene particles {W1, W2, …, WM},

A mathematical principle:


Pursuit of Multiple Solutions

The Kullback-Leibler divergence can be computed if we assumemixture of Gaussian distributions.

--- a simple fact: the KL-divergence of two Gaussians is the signal-to-noise rati


a simple fact: the KL divergence of two Gaussians is the signal to noise rati

Intuition: S includes global maximum, local modes, apart from each other.

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A k-adventurer algorithm


Preserving Distinct Particles


x1 x2 x3 x4

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An Example of Keeping Multiple Solutions

An example of illustration:

1

23 4


Preserving Distinct Particles

min D(q || q1)

An example of illustration:min | q – q2 |


A model p with 50 particles two approximate models q with 6 particles

q q1 q2

ambiguity and uncertainty - ucla statisticssczhu/courses/ucla/stat_232b/handouts/ch6_… ·...

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