Lecture 26: MDS / Canonical Forms COMPSCI/MATH 290-04 Chris Tralie, - - PowerPoint PPT Presentation

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Lecture 26: MDS / Canonical Forms COMPSCI/MATH 290-04 Chris Tralie, - - PowerPoint PPT Presentation

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Lecture 26: MDS / Canonical Forms COMPSCI/MATH 290-04 Chris Tralie, Duke University 4/19/2016 COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms Announcements Group Assignment 3 Final Deadline Tuesday 4/26 Guest Lecture Thursday

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Lecture 26: MDS / Canonical Forms

COMPSCI/MATH 290-04

Chris Tralie, Duke University

4/19/2016

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Announcements

⊲ Group Assignment 3 Final Deadline Tuesday 4/26 ⊲ Guest Lecture Thursday ⊲ No office hours Thursday

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Spin Images

Why did they all look so boring and unlike the objects in question?

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Spin Images

I made a mistake on the assignment! First principal axis is vertical axis in image

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Table of Contents

◮ Multidimensional Scaling ⊲ Canonical Forms

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Academic Majors Distances: Your Choices

Multidimensional Scaling down to R2

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  • 0.2
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0.1 0.2 0.3 0.4

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0.1 0.2 0.3 Art History English Math CS ECE Philosophy

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Academic Majors Distances: Chris’s Choices

Multidimensional Scaling down to R2

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0.1 0.2 0.3 0.4

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0.1 0.2 0.3 Art History English Math CS ECE Philosophy

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Multidimensional Scaling

⊲ Given an N × N symmetric discrete similarity matrix D (i.e. Dij = Dji) ⊲ Given a Euclidean dimension K Find a point cloud X ∈ RN×K so that Dij ≈

  • K
  • k=1

(X[i, k] − X[j, k])2

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Multidimensional Scaling

⊲ Given an N × N symmetric discrete similarity matrix D (i.e. Dij = Dji) ⊲ Given a Euclidean dimension K Find a point cloud X ∈ RN×K so that Dij ≈

  • K
  • k=1

(X[i, k] − X[j, k])2 In other words, find a point cloud in Euclidean K-space that best approximates the distances

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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MDS: Euclidean Dimension Reduction

Dij ≈

  • K
  • k=1

(X[i, k] − X[j, k])2 What if Dij comes from a Euclidena space of dimension d > k? Can we solve this using something else we learned in the course?

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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MDS: Euclidean Dimension Reduction

Dij ≈

  • K
  • k=1

(X[i, k] − X[j, k])2 What if Dij comes from a Euclidena space of dimension d > k? Can we solve this using something else we learned in the course? This is equivalent to PCA!! If we let k = d, then we can represent distances exactly

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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MDS: Non-Euclidean Space Reduction

Can we always find a point cloud that satisfies a given D by making k arbitrarily high?

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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MDS: Non-Euclidean Space Reduction

Can we always find a point cloud that satisfies a given D by making k arbitrarily high? Assume sphere of radius 2/π with points in the following configuration:

1 2 3 4

v1 v2 v3 v4 v1 v2 v3 v4

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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MDS: Non-Euclidean Space Reduction

Can we always find a point cloud that satisfies a given D by making k arbitrarily high? Assume sphere of radius 2/π with points in the following configuration:

1 2 3 4

v1 v2 v3 v4 v1 2 1 1 v2 2 1 1 v3 1 1 1 v4 1 1 1

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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MDS: Non-Euclidean Space Reduction

Assume sphere of radius 2/π with points in the following configuration:

1 2 3 4

v1 v2 v3 v4 v1 2 1 1 v2 2 1 1 v3 1 1 1 v4 1 1 1

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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MDS: Non-Euclidean Space Reduction

Assume sphere of radius 2/π with points in the following configuration:

1 2 3 4

v1 v2 v3 v4 v1 2 1 1 v2 2 1 1 v3 1 1 1 v4 1 1 1

v1 v3 v2

1 1 2

v1, v3, v2 along a line

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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MDS: Non-Euclidean Space Reduction

Assume sphere of radius 2/π with points in the following configuration:

1 2 3 4

v1 v2 v3 v4 v1 2 1 1 v2 2 1 1 v3 1 1 1 v4 1 1 1

v1 v3 v2

1 1 2

v1, v3, v2 along a line

v1 v4 v2

1 1 2

v1, v4, v2 also along line!

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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MDS: Non-Euclidean Space Reduction

v1 v3 v2

1 1 2

v1, v3, v2 along a line

v1 v4 v2

1 1 2

v1, v4, v2 also along line! This implies that v4 and v3 must collapse to the same point in any Euclidean space. ⊲ In other words, distances along the sphere cannot be perfectly realized using a Euclidean space of any finite dimension!

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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MDS: Non-Euclidean Space Reduction

v1 v3 v2

1 1 2

v1, v3, v2 along a line

v1 v4 v2

1 1 2

v1, v4, v2 also along line! This implies that v4 and v3 must collapse to the same point in any Euclidean space. ⊲ In other words, distances along the sphere cannot be perfectly realized using a Euclidean space of any finite dimension! ⊲ (But let’s do our best and see what we come up with)

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Table of Contents

◮ Multidimensional Scaling ⊲ Canonical Forms

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Nonrigid Shape Alignment

How do I align these two camels??

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Geodesic Distances

Geodesic distances are invariant to isometries (aka bending without stretching)

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Geodesic Distances

Geodesic distances are invariant to isometries (aka bending without stretching) What if we try to apply MDS to the distance matrix we get from geodesic distances?

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Face Example

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Face Example

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Face Example

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms

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Face Example

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More Examples

Bronstein

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Lots More Examples

Elad Kimmel 2001: “Bending Invariant Representations for Surfaces”

COMPSCI/MATH 290-04 Lecture 26: MDS / Canonical Forms