Similarities, Distances and ManifoldLearning

Prof. Richard C. Wilson

Dept. of Computer ScienceUniversity of York

Part I: Euclidean SpacePosition, Similarity and Distance

Manifold Learning in Euclidean space

Some famous techniques

Part II: Non-Euclidean ManifoldsAssessing Data

Nature and Properties of Manifolds

Data Manifolds

Learning some special types of manifolds

Part III: Advanced TechniquesMethods for intrinsically curved manifolds

Thanks to Edwin Hancock, Eliza Xu, Bob Duin for contributionsAnd support from the EU SIMBAD project

Part I: Euclidean Space

Position

The main arena for pattern recognition and machinelearning problems is vector space

– A set of n well defined features collected into a vector

ℝn

Also defined are addition of vectors andmultiplication by a scalarmultiplication by a scalar

Feature vector → position

Similarity

To make meaningful progress, we need a notion ofsimilarity, the inner product

• The inner-product ‹x,y› can be considered to be asimilarity between x and y

In Euclidean space, position, similarity are all neatly

i

ii yxyx,

In Euclidean space, position, similarity are all neatlyconnected

yxyxyx

yx

yx

,d

yx,i

ii

),((squared)Distance

Similarity

,Position

2

The Golden Trio

• In Euclidean space, the concepts of position, similarity anddistance are elegantly connected

PositionX

SimilarityK

DistanceD

Point position matrix

• In a normal manifold learning problem, we have a set ofsamples X={x1,x2,...,xm}

• These can be collected together in a matrix X

T

T

x

x

X

2

1

I use this convention, but othersmay write them vertically

Tmx

may write them vertically

Centring

A common and important operation is centring –moving the mean to the origin– Centred points behave better

is the mean matrix, so is the centredmatrix

m/JX m/JXX matrix

– J is the all-ones matrix

This can be done with C

– C is the centring matrix (and is symmetric C=CT)

CXXJIC / m

Position-Similarity

• The similarity matrix K is defined as

• From the definition of X, we simply get

• The Gram matrix is the similarity matrix ofthe centred points (from the definition of X)

PositionX

SimilarityK

jiijK xx ,

TXXK

the centred points (from the definition of X)

– i.e. a centring operation on K

• Kc is really a kernel matrix for the points(linear kernel)

CKCCCXXK TTc

Position-Similarity

• To go from K to X, we need to consider theeigendecomposition of K

• As long as we can take the square root of Λthen we can find X as

PositionX

SimilarityK

T

T

XXK

UUK

Λ

then we can find X as1/2ΛUX

Kernel embedding

kernel embedding

Finds a Euclidean manifold from object similarities

• Embeds a kernel matrix into a set of points in Euclideanspace (the points are automatically centred)

1/2ΛUX

TUUK Λ

space (the points are automatically centred)

• K must have no negative eigenvalues, i.e. it is a kernelmatrix (Mercer condition)

Similarity-Distance

SimilarityK

DistanceD

ijsijjjii

jijjii

jijiji

DKKK

d

,

2

2

,2,,

,),(

xxxxxx

xxxxxx

ijsijjjii DKKK ,2

• We can easily determine Ds from K

Similarity-Distance

What about finding K from Ds ?

Looking at the top equation, we might imagine that

K=-½ Ds is a suitable choice

ijjjiiijs KKKD 2,

• Not centred; the relationship is actually

CCDK s2

1

Classic MDS

• Classic Multidimensional Scaling embeds a (squared)distance matrix into Euclidean space

• Using what we have so far, the algorithm is simple

kernel theposeEigendecom Λ

kerneltheCompute2

1

KUU

CCDK

T

s

• This is MDS

kerneltheEmbedΛ1/2UX

PositionX

DistanceD

The Golden Trio

PositionX

Similarity

KernelEmbedding

MDS

SimilarityK

DistanceD

ijjjiiijs

s

KKKD 2

2

1

,

CCDK

Kernel methods

• A kernel is function k(i,j) which computes an inner-product

– But without needing to know the actual points (the space isimplicit)

• Using a kernel function we can directly compute K withoutknowing X

Position

jijik xx ,),(

PositionX

SimilarityK

DistanceD

Kernelfunction

Kernel methods

• The implied space may be very high dimensional, but atrue kernel will always produce a positive semidefinite Kand the implied space will be Euclidean

• Many (most?) PR algorithms can be kernelized– Made to use K rather than X or D

• The trick is to note that any interesting vector should lie inthe space spanned by the examples we are giventhe space spanned by the examples we are given

• Hence it can be written as a linear combination

• Look for α instead of u

αX

xxxuT

mm

2211

Kernel PCA

• What about PCA? PCA solves the following problem

• Let’s kernelize:

XuXu

Σuuu

u

u

TT

T

n

1minarg

minarg*

11

αKα

αXXXXα

αXXXαXXuXu

21

1

)()(11

T

TTT

TTTTTT

n

n

nn

Kernel PCA

• K2 has the same eigenvectors as K, so the eigenvectors ofPCA are the same as the eigenvectors of K

• The eigenvalues of PCA are related to the eigenvectors ofK by

2PCA

1K

n

• Kernel PCA is a kernel embedding with an externallyprovided kernel matrix

Kernel PCA

• So kernel PCA gives the same solution as kernelembedding– The eigenvalues are modified a bit

• They are essentially the same thing in Euclidean space

• MDS uses the kernel and kernel embedding

• MDS and PCA are essentially the same thing in Euclideanspacespace

• Kernel embedding, MDS and PCA all give the sameanswer for a set of points in Euclidean space

Some useful observations

• Your similarity matrix is Euclidean iff it has no negativeeigenvalues (i.e. it is a kernel matrix and PSD)

• By similar reasoning, your distance matrix is Euclidean iffthe similarity matrix derived from it is PSD

• If the feature space is small but the number of samples islarge, then the covariance matrix is small and it is better todo normal PCA (on the covariance matrix)do normal PCA (on the covariance matrix)

• If the feature space is large and the number of samples issmall, then the kernel matrix will be small and it is better todo kernel embedding

Part II: Non-Euclidean ManifoldsPart II: Non-Euclidean Manifolds

Non-linear data

• Much of the data in computer vision lies in a high-dimensional feature space but is constrained insome way

– The space of all images of a face is a subspace of thespace of all possible images

– The subspace is highly non-linear but low dimensional(described by a few parameters)(described by a few parameters)

Non-linear data

• This cannot be exploited by the linear subspace methodslike PCA– These assume that the subspace is a Euclidean space as well

• A classic example is the

‘swiss roll’ data:

‘Flat’ Manifolds• Fundamentally different types of data, for example:

• The embedding of this data into the high-dimensionalspace is highly curvedspace is highly curved– This is called extrinsic curvature, the curvature of the manifold

with respect to the embedding space

• Now imagine that this manifold was a piece of paper; youcould unroll the paper into a flat plane without distorting it– No intrinsic curvature, in fact it is homeomorphic to Euclidean

space

• This manifold is different:

• It must be stretched to map it onto a plane

– It has non-zero intrinsic curvature

Curved manifold

– It has non-zero intrinsic curvature

• A flatlander living on this manifold can tell that it is curved, forexample by measuring the ratio of the radius to the circumference of acircle

• In the first case, we might still hope to find Euclidean embedding

• We can never find a distortion free Euclidean embedding of the second(in the sense that the distances will always have errors)

Intrinsically Euclidean Manifolds

• We cannot use the previous methods on the second type ofmanifold, but there is still hope for the first

• The manifold is embedded in Euclidean space, butEuclidean distance is not the correct way to measuredistance

• The Euclidean distance ‘shortcuts’ the manifold

• The geodesic distance calculates the shortest path along themanifold

Geodesics

• The geodesic generalizes the concept of distance to curvedmanifolds– The shortest path joining two points which lies completely within

the manifold

• If we can correctly compute the geodesic distances, and themanifold is intrinsically flat, we should get Euclideandistances which we can plug into our Euclidean geometrymachine Position

X

SimilarityK

DistanceD

GeodesicDistances

ISOMAP

• ISOMAP is exactly such an algorithm

• Approximate geodesic distances are computed for the points from agraph

• Nearest neighbours graph

– For neighbours, Euclidean distance≈geodesic distances

– For non-neighbours, geodesic distance approximated by shortest distancein graph

• Once we have distances D, can use MDS to find Euclidean embedding

ISOMAP

• ISOMAP:

– Neighbourhood graph

– Shortest path algorithm

– MDS

• ISOMAP is distance-preserving – embeddeddistances should be close to geodesic distancesdistances should be close to geodesic distances

Laplacian Eigenmap

• The Laplacian Eigenmap is another graph-based method of embeddingnon-linear manifolds into Euclidean space

• As with ISOMAP, form a neighbourhood graph for the datapoints

• Find the graph Laplacian as follows

• The adjacency matrix A is

connectedareandif

2

jieAt

d

ij

ij

• The ‘degree’ matrix D is the diagonal matrix

• The normalized graph Laplacian is

otherwise0

Aij

j

ijii AD

2/12/1 ADDIL

Laplacian Eigenmap

• We find the Laplacian eigenmap embedding using theeigendecomposition of L

• The embedded positions are

• Similar to ISOMAP

TUUL

UDX 2/1• Similar to ISOMAP

– Structure preserving not distance preserving

Locally-Linear Embedding

• Locally-linear Embedding is another classic method which also beginswith a neighbourhood graph

• We make point i (in the original data) from a weighted sum of theneighbouring points

i j j

jiji W xx̂

• Wij is 0 for any point j not in the neighbourhood (and for i=j)

• We find the weights by minimising the reconstruction error

– Subject to the constrains that the weights are non-negative and sum to 1

• Gives a relatively simple closed-form solution

2|ˆ|min ii xx

j

ijij WW 1,0

Locally-Linear Embedding

• These weights encode how well a point j represents a pointi and can be interpreted as the adjacency between i and j

• A low dimensional embedding is found by then findingpoints to minimise the error

• In other words, we find a low-dimensional embeddingwhich preserves the adjacency relationships

j

jijii

ii W yyyy ˆ|ˆ|min 2

which preserves the adjacency relationships

• The solution to this embedding problem turns out to besimply the eigenvectors of the matrix M

• LLE is scale-free: the final points have the covariancematrix I– Unit scale

)()( WIWIM T

Comparison

• LLE might seem like quite a different process to the previous two, butactually very similar

• We can interpret the process as producing a kernel matrix followed byscale-free kernel embedding

UXUUΛK

WWWWJIK

T

TT

n

kk )1(

ISOMAP Lap. Eigenmap LLE

Representation Neighbourhoodgraph

Neighbourhoodgraph

Neighbourhoodgraph

Similarity matrix From geodesicdistances

Graph Laplacian Reconstructionweights

Embedding UDX 2/12/1 UX UX

Comparison

• ISOMAP is the only method which directly computes anduses the geodesic distances– The other two depend indirectly on the distances through local

structure

• LLE is scale-free, so the original distance scale is lost, butthe local structure is preserved

• Computing the necessary local dimensionality to find the• Computing the necessary local dimensionality to find thecorrect nearest neighbours is a problem for all suchmethods

Part II: Indefinite Similarities

Non-Euclidean data

• Data is Euclidean iff K is psd

• Unless you are using a kernel function, this is often nottrue

• Why does this happen?

What type of data do I have?

• Starting point: distance matrix

• However we do not know apriori if our measurements arerepresentable on a manifold– We will call them dissimilarities

• Our starting point to answer the question “What type ofdata do I have?” will be a matrix of dissimilarities Dbetween objectsbetween objects

• Types of dissimilarities– Euclidean (no intrinsic curvature)

– Non-Euclidean, metric (curved manifold)

– Non-metric (no point-like manifold representation)

Causes

• Example: Chicken pieces data

• Distance by alignment

• Global alignment of everything could find Euclideandistances

• Only local alignments are practical

Causes

Dissimilarities may also be non-metric

The data is metric if it obeys the metric conditions

1. Dij≥ 0 (nonegativity)

2. Dij= 0 iff i=j (identity of indiscernables)

3. Dij= Dji (symmetry)

4. D ≤D + D (triangle inequality)4. Dij≤Dik+ Dkj (triangle inequality)

Reasonable dissimilarites should meet 1&2

Causes

• Symmetry Dij= Dji

• May not be symmetric by definition

• Alignment: i→j may find a better solution thanj→i

Causes

• Triangle violations Dij≤Dik+ Dkj

• ‘Extended objects’

k

i j

0D

• Finally, noise in the measure of D can cause all of theseeffects

0

0

0

ij

kj

ik

D

D

D

Tests(1)

• Find the similarity matrix

• The data is Euclidean iff K is positive semidefinite (nonegative eigenvalues)– K is a kernel, explicit embedding from kernel embedding

CCDK s2

1

– K is a kernel, explicit embedding from kernel embedding

• We can then use K in a kernel algorithm

• Negative eigenfraction (NEF)

• Between 0 and 0.5– 0 for Euclidean similarities

i

i

i

0

NEF

Tests(2)

3. Dij= Dji (symmetry)

– Mean, maximum asymmetry

– Easy to check by looking at pairs

4. Dij≤Dik+ Dkj (triangle inequality)

– Number, maximum violation

– Check these for your data (3rd involves checking all– Check these for your data (3rd involves checking alltriples – possibly expensive)

• Metric data is embeddable on a (curved)Reimannian manifold

Determining the causes

• The negative eigenvalues

-5

-4

-3

-2

-1

0

Eig

env

alu

e

Noise

-2.5

-2

-1.5

-1

-0.5

0

Eig

env

alu

e

Extended Objects

-7

-6

-5

-4

-3.5

-3

-25

-20

-15

-10

-5

0

Eig

env

alu

e

Spherical manifold

Corrections

• If the data is non-metric or non-Euclidean, we can ‘correct it’

• Symmetry violations

– Average

– For min-cost distances may be moreappropriate

• Triangle violations

– Constant offset

– This will also remove non-Euclidean behaviour for large enough c

)(2

1jiijjiij DDDD

),min( jiijjiij DDDD

)( jicDD ijij

– This will also remove non-Euclidean behaviour for large enough c

• Euclidean violations

– Discard negative eigenvalues

– Even when the violations are caused by noise, some information is stilllost

• There are many other approaches*

* “On Euclidean corrections for non-Euclidean dissimilarities”, Duin, Pekalska, Harol,Lee and Bunke, S+SSPR 08

Part III: Techniques for non-Euclidean EmbeddingsPart III: Techniques for non-Euclidean Embeddings

Known Manifolds

• Sometimes we have data which lies on a known but non-Euclidean manifold

• Examples in Computer Vision– Surface normals

– Rotation matrices

– Flow tensors (DT-MRI)

• This is not Manifold Learning, as we already know what• This is not Manifold Learning, as we already know whatthe manifold is

• What tools do we need to be able to process data like this?– As before, distances are the key

Example: 2D direction

Direction of an edge in an image, encoded as a unit vector

1x

2x

x

The average of the direction vector isn’t even a direction vector (not unitlength), let alone the correct ‘average’ direction

The normal definition of mean is not correct

– Because the manifold is curved

i

in

xx1

Tangent space

• The tangent space (TP) is the Euclidean space which isparallel to the manifold(M) at a particular point (P)

M

T

P

• The tangent space is a very useful tool because it isEuclidean

TP

Exponential Map

• Exponential map:

• ExpP maps a point X on the tangent plane onto a point A onthe manifold– P is the centre of the mapping and is at the origin on the tangent

space

XA

MT

P

PP

Exp

:Exp

space

– The mapping is one-to-one in a local region of P

– The most important property of the mapping is that the distances tothe centre P are preserved

– The geodesic distance on the manifold equals the Euclidean distance on thetangent plane (for distances to the centre only)

),(),( PAdPXd MTP

Exponential map

• The log map goes the other way, from manifold to tangentplane

MX

TM

P

pP

Log

:Log

Exponential Map

• Example on the circle: Embed the circle in the complexplane

• The manifold representing the circle is a complex numberwith magnitude 1 and can be written x+iy=exp(i)

ImPieP

Re

• In this case it turns out that the map is related to the normalexp and log functions

M

TP PieP

PAi

i

P

P

A

e

ei

P

AiAX

log

logLog

X

AieA APAP

P

iii

iXPXA

exp)(expexp

expExp

Intrinsic mean

• The mean of a set of samples is usually defined as the sumof the samples divided by the number– This is only true in Euclidean space

• A more general formula

• Minimises the distances from the mean to the samples

i

igd ),(minarg 2 xxxx

• Minimises the distances from the mean to the samples(equivalent in Euclidean space)

Intrinsic mean

• We can compute this intrinsic mean using the exponentialmap

• If we knew what the mean was, then we can use the meanas the centre of a map

• From the properties of the Exp-map, the distances are the

iMi AX Log

• From the properties of the Exp-map, the distances are thesame

• So the mean on the tangent plane is equal to the mean onthe manifold

),(),( MAdMXd igie

Intrinsic mean

• Start with a guess at the mean and move towards correctanswer

• This gives us the following algorithm– Guess at a mean M0

1. Map on to tangent plane using Mi

2. Compute the mean on the tangent plane to get new estimate Mi+1

1

iiMMk A

nM

kkLog

1Exp1

Intrinsic Mean

• For many manifolds, this procedure will converge to theintrinsic mean– Convergence not always guaranteed

• Other statistics and probability distributions on manifoldsare problematic.– Can hypothesis a normal distribution on tangent plane, but

distortions inevitabledistortions inevitable

Some useful manifolds and maps

• Some useful manifolds and exponential maps

• Directional vectors (surface normals etc.)

map)(Log)cos(sin

1,

pax

aa

• a, p unit vectors, x lies in an (n-1)D space

map)(Expsin

cos xpa

Some useful manifolds and maps

• Symmetric positive definite matrices (covariance, flowtensors etc)

map)(Expexp

map)(Loglog

00,

21

21

21

21

21

21

21

21

PXPPPA

PAPPPX

uAuuA

T

• A is symmetric positive definite, X is just symmetric

• log is the matrix log defined as a generalized matrixfunction

map)(Expexp PXPPPA

Some useful manifolds and maps

• Orthogonal matrices (rotation matrices, eigenvectormatrices)

map)(Expexp

map)(Loglog

I,

XPA

APX

AAA

T

T

• A orthogonal, X antisymmetric (X+XT=0)

• These are the matrix exp and log functions as before

• In fact there are multiple solutions to the matrix log– Only one is the required real antisymmetric matrix; not easy to find

– Rest are complex

Embedding on Sn

• On S2 (surface of a sphere in 3D) the followingparameterisation is well known

• The distance between two points (the length of thegeodesic) is

Trrr )cos,sinsin,cossin( x

rd coscossinsincos 1

xyd

x

y

yxxyyxij rd coscossinsincos

More Spherical Geometry

• But on a sphere, the distance is the highlighted arc-length– Much neater to use inner-product

– And works in any number of dimensions

2

1

2

,cos

coscos,

rrrd

rxy

xyxy

xyxy

yx

yx

xyrθ

xyθ

x

y

– And works in any number of dimensions

Spherical Embedding

• Say we had the distances between some objects (dij),measured on the surface of a [hyper]sphere of dimension n

• The sphere (and objects) can be embedded into an n+1dimensional space– Let X be the matrix of point positions

• Z=XXT is a kernel matrix

• But Z xx ,• But

• And

• We can compute Z from D and find the sphericalembedding!

jiijZ xx ,

r

drZ

rrd

ij

jiij

xy

cos,

,cos

2

2

1

xx

yx

Spherical Embedding

• But wait, we don’t know what r is!

• The distances D are non-Euclidean, and if we use thewrong radius, Z is not a kernel matrix– Negative eigenvalues

• Use this to find the radius– Choose r to minimise the negative eigenvalues

)(minarg* rZr or

Example: Texture Mapping

• As an alternative to unwrapping object onto a plane andtexture-mapping the plane

• Embed onto a sphere and texture-map the sphere

Plane Sphere

Backup slidesBackup slides

Laplacian and related processes

• As well as embedding objects onto manifolds, we canmodel many interesting processes on manifolds

• Example: the way ‘heat’ flows across a manifold can bevery informative

•

equationheat2udt

du

2 isitspaceEuclidean3DinandLaplaciantheis•

• On a sphere it is

2

2

2

2

2

2

2 isitspaceEuclidean3DinandLaplaciantheis

zyx

sin

sin

1

sin

122

2

22 rr

Heat flow

• Heat flow allows us to do interesting things on a manifold

• Smoothing: Heat flow is a diffusion process (will smooththe data)

• Characterising the manifold (heat content, heat kernelcoefficients...)

• The Laplacian depends on the geometry of the manifold• The Laplacian depends on the geometry of the manifold– We may not know this

– It may be hard to calculate explicitly

• Graph Laplacian

Graph Laplacian

• Given a set of datapoints on the manifold, describe them bya graph– Vertices are datapoints, edges are adjacency relation

• Adjacency matrix (for example)

2

2 )/exp(

ij

ij

ijd

dA

• Then the graph Laplacian is

• The graph Laplacian is a discrete approximation of themanifold Laplacian

ijd

j

ijii AVAVL

Heat Kernel

• Using the graph Laplacian, we can easily implement heat-flow methods on the manifold using the heat-kernel

kernelheat)exp(

equationheat

t

dt

d

LH

Luu

• Can diffuse a function on the manifold by

Hff '