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Topology in Manifold Learning

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Simplex. An n-simplex is the convex hull of (n 1) (independent) points. 0-simplex. 1-simplex ... Any face of a simplex in S is also in S ... – PowerPoint PPT presentation

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Title: Topology in Manifold Learning


1
Topology in Manifold Learning
  • Jonathan Huang
  • Presented at misc-read, 11.22.06

2
Bibliography
  • Simultaneous Inference of View and Body Pose
    Using Torus Manifolds Chan-Su Lee and Ahmed
    Elgammal The 18th International Conference on
    Pattern Recognition (ICPR), Hong Kong, August
    21-24, 2006
  • Finding the Homology of Submanifolds with High
    Confidence from Random Samples. P. Niyogi, S.
    Smale, and S. Weinberger to appear, Discrete and
    Computational Geometry, 2006.
  • On the Local Behavior of Spaces of Natural
    Images G. Carlsson, T. Ishkhanov, V. de Silva,
    and A. Zomorodian, /preprint/, May 31, 2006.
  • Computing Persistent Homology A. Zomorodian and
    G. Carlsson, Discrete and Computational Geometry,
    33 (2), pp. 247-274, 2005.

3
Outline
  • The Role of Topology in Manifold Learning
  • Constrained Topology Manifold Learning
  • Topology Basics
  • Learning a Topology from Noisy Data
  • Statistical Approach
  • Multi-scale Approach

4
The ISOMAP Algorithm
  • The ISOMAP algorithm
  • Compute pairwise distances for some point cloud
  • Choose an embedding dimension, d (d2 in this
    example)
  • Run the following code in matlab
  • gtgt options.dims 2
  • gtgt Y, R, E Isomap(DistanceMatrix, options)

ISOMAP
5
Selecting the Dimensionality
  • Problem How do we choose embedding dimension?
  • Several solutions (maybe some non-NIPS solutions
    too)
  • Brand (NIPS 2003)
  • Kegl (NIPS 2003)
  • Levina and Bickel (NIPS 2005)
  • Raginsky and Lazebnik (NIPS 2006)

Dimensionality Estimation
d2
6
Why none of these are actually solutions
Dimensionality Estimation
d2
ISOMAP
(samples from a sphere)
7
Moral
  • Manifold Learning is hard if you dont take
    topology into account!

8
Outline
  • The Role of Topology in Manifold Learning
  • Constrained Topology Manifold Learning
  • Topology Basics
  • Learning a Topology from Noisy Data
  • Statistical Approaches
  • Multi-scale Approaches

9
Manifold Learning with Known Topology
  • Images of a periodic gait from varying viewpoints
  • What is the intrinsic topology of this set of
    images?

Body Pose
View angle (0-330)
10
Learning a Mapping from a Torus
  • The product of two periodic spaces is a torus!
  • Use kernel methods to learn a map to/from a Torus

11
Some Results
12
Outline
  • The Role of Topology in Manifold Learning
  • Constrained Topology Manifold Learning
  • Topology Basics
  • Learning a Topology from Noisy Data
  • Statistical Approaches
  • Multiresolution Approaches

13
What is Topology?
  • Popular answer Its the branch of math that
    cant tell the difference between a coffee cup
    and a donut

14
What is Topology?
  • Topology cares about how a space is connected
  • It does not care about distances
  • We will define a very general class of
    topological spaces the simplicial complexes

15
Simplex
  • An n-simplex is the convex hull of (n1)
    (independent) points

0-simplex
1-simplex
2-simplex
16
Simplicial Complex
  • A simplicial complex is a finite collection of
    simplices S such that
  • Any face of a simplex in S is also in S
  • The intersection of two simplices in S is either
    empty or a face for both simplices
  • The dimension of S is the maximum dimension over
    all simplices in S

17
Boundary Maps
  • Graphs are one-dimensional simplicial complexes
  • Define a boundary matrix M1 with rows
    corresponding to vertices and columns
    corresponding to edges
  • M1(v,e) 0 if vertex v is not part of edge e
  • M1(v,e) -1 if vertex v is the first vertex of
    edge e
  • M1(v,e) 1 if vertex v is the second vertex of
    edge e

a
ab
ac
b
c
bc
18
Boundary Maps
  • Example
  • The boundary of an edge is its two vertices

19
Boundary Maps
  • Example
  • The boundary of the loop is empty

20
Boundary Maps
  • In general, the dim(Nullspace(M)) is the number
    of different loops in the graph
  • And (vertices)-Rank(M) is the number of
    connected components

21
Betti Numbers
  • For a simplicial complex S, we can define a
    boundary matrix Mk at each dimension k of S
  • Define the kth betti number to be
  • ?k Dim(Nullspace(Mk-1))-Dim(Complement of
    column space(Mk))

22
Betti Numbers
  • For an object in 3d space
  • ?0 is the number of connected components
  • ?1 is the number of tunnels or handles
  • ?2 is the number of voids

Point
Circle
Torus
23
Outline
  • The Role of Topology in Manifold Learning
  • Constrained Topology Manifold Learning
  • Topology Basics
  • Learning a Topology from Noisy Data
  • Statistical Approaches
  • Multi-scale Approaches

24
Problem Formulation
  • Given x1,x2,,xn, i.i.d samples from a manifold.
    What are the betti numbers of the manifold?

25
An Well Known Algorithm
  • Two Steps
  • First put an ?-ball around each point
  • Compute the betti numbers of the union of these
    balls using the Nerve Complex

26
The Nerve Complex
  • Given a collection of balls, U1,U2, in Euclidean
    space, what is the topology of their union?
  • Construct the Nerve Complex
  • For each ball, add a vertex
  • Add a k-simplex whenever k1 balls have nonempty
    intersection

27
Nerve Complex
  • The Nerve Lemma states that the original space ?
    and the Nerve Complex have the same Betti
    numbers!
  • Example

A B
B
A
A
B
A C
B C
C
C
28
A PAC-bound
  • With high probability and enough samples, we can
    recover the true Betti numbers!
  • The number of samples depends on the volume of
    the manifold and its condition number (how close
    it gets to itself)
  • (Niyogi, Smale, Weinberger, 2006)
  • (but how does one choose ??)

29
Outline
  • The Role of Topology in Manifold Learning
  • Constrained Topology Manifold Learning
  • Topology Basics
  • Learning a Topology from Noisy Data
  • Statistical Approaches
  • Multi-scale Approaches

30
Another Example
Spiral or Torus???
31
Topological Persistence
  • Topological Persistence looks at topology from
    all scales at once
  • Surprisingly, its not much harder to compute the
    betti numbers at every scale!

32
Filtered Complex
  • A Filtered Complex is an increasing sequence of
    simplicial complexes (Ct ? Ct1)
  • t is called the filtration index

t0
t1
t3
t2
33
Barcodes - Example
34
Barcodes
  • Barcodes represent betti numbers as a function of
    filtration index
  • Intuitively, Barcodes measure the lifetime of
    topological features
  • The Persistence Algorithm provides a way to
    compute barcodes efficiently

35
Persistence Algorithm
  • Empirically, the algorithm works in linear time
  • Worst case complexity bound O(m3)
  • Where m is the of simplices
  • Pros
  • Persistence distinguishes local features from
    global features
  • Applies to learning manifold topology from noisy
    data
  • Cons
  • No real probabilistic semantics

36
Conclusion
  • Learning a topology is not hopeless. So
  • The next time you decide to learn a manifold,
    take a moment to contemplate the underlying
    topology!

37
Thank You!
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