1. Similarities, Distances and Manifold Learning Prof. Richard C. Wilson Dept. of Computer Science University of York

2. Background Typically objects are characterised by features –Face images –SIFT features –Object spectra –... If we measure n features → n-dimensional space The arena for our problem is an n-dimensional vector space

3. Background Example: Eigenfaces Raw pixel values: n by m gives nm features Feature space is space of all n by m images

4. Background The space of all face-like images is smaller than the space of all images Assumption is faces lie on a smaller manifold embedded in the global space All images Face images

5. Manifold: A space which locally looks Euclidean Manifold learning: Finding the manifold representing the objects we are interested in All objects should be on the manifold, non-objects outside

6. Part I: Euclidean Space Position, Similarity and Distance Manifold Learning in Euclidean space Some famous techniques Part II: Non-Euclidean Manifolds Assessing Data Nature and Properties of Manifolds Data Manifolds Learning some special types of manifolds Part III: Advanced Techniques Methods for intrinsically curved manifolds Thanks to Edwin Hancock, Eliza Xu, Bob Duin for contributions And support from the EU SIMBAD project

7. Part I: Euclidean Space

8. Position The main arena for pattern recognition and machine learning problems is vector space –A set of n well defined features collected into a vector ℝ n Also defined are addition of vectors and multiplication by a scalar Feature vector → position

9. Similarity To make meaningful progress, we need a notion of similarity Inner product The inner-product ‹x,y› can be considered to be a similarity between x and y

10. Induced norm The self-similarity ‹x,x› is the (square of) the ‘size’ of x and gives rise to the induced norm, of the length of x: Finally, the length of x allows the definition of a distance in our vector space as the length of the vector joining x and y Inner product also gets us distance

11. Euclidean space If we have a vector space for features, and the usual inner product, all three are connected:

12. non-Euclidean Inner Product If the inner-product has the form Then the vector space is Euclidean Note we recover all the expected stuff for Euclidean space, i.e. The inner-product doesn’t have to be like this; for example in Einstein’s special relativity, the inner-product of spacetime is

13. The Golden Trio In Euclidean space, the concepts of position, similarity and distance are elegantly connected Position X Similarity K Distance D

14. Point position matrix In a normal manifold learning problem, we have a set of samples X ={x 1,x 2,...,x m } These can be collected together in a matrix X I use this convention, but others may write them vertically

15. Centreing A common and important operation is centreing – moving the mean to the origin –Centred points behave better is the mean matrix, so is the centred matrix –J is the all-ones matrix This can be done with C –C is the centreing matrix (and is symmetric C=C T )

16. Position-Similarity The similarity matrix K is defined as From the definition of X, we simply get The Gram matrix is the similarity matrix of the centred points (from the definition of X) –i.e. a centring operation on K K c is really a kernel matrix for the points (linear kernel) Position X Similarity K

17. Position-Similarity To go from K to X, we need to consider the eigendecomposition of K As long as we can take the square root of Λ then we can find X as Position X Similarity K

18. Kernel embedding First manifold learning method – kernel embedding Finds a Euclidean manifold from object similarities Embeds a kernel matrix into a set of points in Euclidean space (the points are automatically centred) K must have no negative eigenvalues, i.e. it is a kernel matrix (Mercer condition)

19. Similarity-Distance Similarity K Distance D We can easily determine D s from K

20. Similarity-Distance What about finding K from D s ? Looking at the top equation, we might imagine that K=-½ D s is a suitable choice Not centred; the relationship is actually

21. Classic MDS Classic Multidimensional Scaling embeds a (squared) distance matrix into Euclidean space Using what we have so far, the algorithm is simple This is MDS Position X Distance D

22. The Golden Trio Position X Similarity K Distance D Kernel Embedding MDS

23. Kernel methods A kernel is function k(i,j) which computes an inner-product –But without needing to know the actual points (the space is implicit) Using a kernel function we can directly compute K without knowing X Position X Similarity K Distance D Kernel function

24. Kernel methods The implied space may be very high dimensional, but a true kernel will always produce a positive semidefinite K and 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 in the space spanned by the examples we are given Hence it can be written as a linear combination Look for α instead of u

25. Kernel PCA What about PCA? PCA solves the following problem Let’s kernelize:

26. Kernel PCA K 2 has the same eigenvectors as K, so the eigenvectors of PCA are the same as the eigenvectors of K The eigenvalues of PCA are related to the eigenvectors of K by Kernel PCA is a kernel embedding with an externally provided kernel matrix

27. Kernel PCA So kernel PCA gives the same solution as kernel embedding –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 Euclidean space Kernel embedding, MDS and PCA all give the same answer for a set of points in Euclidean space

28. Some useful observations Your similarity matrix is Euclidean iff it has no negative eigenvalues (i.e. it is a kernel matrix and PSD) By similar reasoning, your distance matrix is Euclidean iff the similarity matrix derived from it is PSD If the feature space is small but the number of samples is large, then the covariance matrix is small and it is better to do normal PCA (on the covariance matrix) If the feature space is large and the number of samples is small, then the kernel matrix will be small and it is better to do kernel embedding

29. Part II: Non-Euclidean Manifolds

30. Non-linear data Much of the data in computer vision lies in a high- dimensional feature space but is constrained in some way –The space of all images of a face is a subspace of the space of all possible images –The subspace is highly non-linear but low dimensional (described by a few parameters)

31. Non-linear data This cannot be exploited by the linear subspace methods like PCA –These assume that the subspace is a Euclidean space as well A classic example is the ‘swiss roll’ data:

32. ‘Flat’ Manifolds Fundamentally different types of data, for example: The embedding of this data into the high-dimensional space 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; you could unroll the paper into a flat plane without distorting it –No intrinsic curvature, in fact it is homeomorphic to Euclidean space

33. This manifold is different: It must be stretched to map it onto a plane –It has non-zero intrinsic curvature A flatlander living on this manifold can tell that it is curved, for example by measuring the ratio of the radius to the circumference of a circle 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) Curved manifold

34. Intrinsically Euclidean Manifolds We cannot use the previous methods on the second type of manifold, but there is still hope for the first The manifold is embedded in Euclidean space, but Euclidean distance is not the correct way to measure distance The Euclidean distance ‘shortcuts’ the manifold The geodesic distance calculates the shortest path along the manifold

35. Geodesics The geodesic generalizes the concept of distance to curved manifolds –The shortest path joining two points which lies completely within the manifold If we can correctly compute the geodesic distances, and the manifold is intrinsically flat, we should get Euclidean distances which we can plug into our Euclidean geometry machine Position X Similarity K Distance D Geodesic Distances

36. ISOMAP ISOMAP is exactly such an algorithm Approximate geodesic distances are computed for the points from a graph Nearest neighbours graph –For neighbours, Euclidean distance≈geodesic distances –For non-neighbours, geodesic distance approximated by shortest distance in graph Once we have distances D, can use MDS to find Euclidean embedding

37. ISOMAP ISOMAP: –Neighbourhood graph –Shortest path algorithm –MDS ISOMAP is distance-preserving – embedded distances should be close to geodesic distances

38. Laplacian Eigenmap The Laplacian Eigenmap is another graph-based method of embedding non-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 The ‘degree’ matrix D is the diagonal matrix The normalized graph Laplacian is

39. Laplacian Eigenmap We find the Laplacian eigenmap embedding using the eigendecomposition of L The embedded positions are Similar to ISOMAP –Structure preserving not distance preserving

40. Locally-Linear Embedding Locally-linear Embedding is another classic method which also begins with a neighbourhood graph We make point i (in the original data) from a weighted sum of the neighbouring points W ij 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 i j

41. Locally-Linear Embedding These weights encode how well a point j represents a point i and can be interpreted as the adjacency between i and j A low dimensional embedding is found by then finding points to minimise the error In other words, we find a low-dimensional embedding which preserves the adjacency relationships The solution to this embedding problem turns out to be simply the eigenvectors of the matrix M LLE is scale-free: the final points have the covariance matrix I –Unit scale

42. Comparison LLE might seem like quite a different process to the previous two, but actually very similar We can interpret the process as producing a kernel matrix followed by scale-free kernel embedding ISOMAPLap. EigenmapLLE RepresentationNeighbourhood graph Similarity matrixFrom geodesic distances Graph LaplacianReconstruction weights Embedding

43. Comparison ISOMAP is the only method which directly computes and uses 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, but the local structure is preserved Computing the necessary local dimensionality to find the correct nearest neighbours is a problem for all such methods

45. Non-Euclidean data Data is Euclidean iff K is psd Unless you are using a kernel function, this is often not true Why does this happen?

46. What type of data do I have? Starting point: distance matrix However we do not know apriori if our measurements are representable on a manifold –We will call them dissimilarities Our starting point to answer the question “What type of data do I have?” will be a matrix of dissimilarities D between objects Types of dissimilarities –Euclidean (no intrinsic curvature) –Non-Euclidean, metric (curved manifold) –Non-metric (no point-like manifold representation)

47. Causes Example: Chicken pieces data Distance by alignment Global alignment of everything could find Euclidean distances Only local alignments are practical

48. Causes Dissimilarities may also be non-metric The data is metric if it obeys the metric conditions 1. D ij ≥ 0 (nonegativity) 2. D ij = 0 iff i=j (identity of indiscernables) 3. D ij = D ji (symmetry) 4. D ij ≤D ik + D kj (triangle inequality) Reasonable dissimilarites should meet 1&2

49. Causes Symmetry D ij = D ji May not be symmetric by definition Alignment: i→j may find a better solution than j→i

50. Causes Triangle violations D ij ≤D ik + D kj ‘Extended objects’ Finally, noise in the measure of D can cause all of these effects k ij

51. Tests(1) Find the similarity matrix The data is Euclidean iff K is positive semidefinite (no negative eigenvalues) –K is a kernel, explicit embedding from kernel embedding We can then use K in a kernel algorithm

52. Tests(2) Negative eigenfraction (NEF) Between 0 and 0.5

53. Tests(3) 1. D ij ≥ 0 (nonegativity) 2. D ij = 0 iff i=j (identity of indiscernables) 3. D ij = D ji (symmetry) 4. D ij ≤D ik + D kj (triangle inequality) –Check these for your data (3 rd involves checking all triples) –Metric data is embeddable on a (curved) Reimannian manifold

54. Corrections If the data is non-metric or non-Euclidean, we can ‘correct it’ Symmetry violations –Average –For min-cost distances may be more appropriate Triangle violations –Constant offset –This will also remove non-Euclidean behaviour for large enough c Euclidean violations –Discard negative eigenvalues There are many other approaches * * “ On Euclidean corrections for non-Euclidean dissimilarities”, Duin, Pekalska, Harol, Lee and Bunke, S+SSPR 08

55. Part III: Advanced techniques for non-Euclidean Embeddings

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

57. Example: 2D direction Direction of an edge in an image, encoded as a unit vector The average of the direction vector isn’t even a direction vector (not unit length), let alone the correct ‘average’ direction The normal definition of mean is not correct –Because the manifold is curved

58. Tangent space The tangent space (T P ) is the Euclidean space which is parallel to the manifold(M) at a particular point (P) The tangent space is a very useful tool because it is Euclidean M TPTP P

59. Exponential Map Exponential map: Exp P maps a point X on the tangent plane onto a point A on the manifold –P is the centre of the mapping and is at the origin on the tangent space –The mapping is one-to-one in a local region of P –The most important property of the mapping is that the distances to the centre P are preserved –The geodesic distance on the manifold equals the Euclidean distance on the tangent plane (for distances to the centre only)

60. Exponential map The log map goes the other way, from manifold to tangent plane

61. Exponential Map Example on the circle: Embed the circle in the complex plane The manifold representing the circle is a complex number with magnitude 1 and can be written x+iy=exp(i  ) Re Im

62. In this case it turns out that the map is related to the normal exp and log functions M TPTP

63. Intrinsic mean The mean of a set of samples is usually defined as the sum of 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 (equivalent in Euclidean space)

64. Intrinsic mean We can compute this intrinsic mean using the exponential map If we knew what the mean was, then we can use the mean as the centre of a map From the properties of the Exp-map, the distances are the same So the mean on the tangent plane is equal to the mean on the manifold

65. Intrinsic mean Start with a guess at the mean and move towards correct answer This gives us the following algorithm –Guess at a mean M 0 1.Map on to tangent plane using M i 2.Compute the mean on the tangent plane to get new estimate M i+1

66. Intrinsic Mean For many manifolds, this procedure will converge to the intrinsic mean –Convergence not always guaranteed Other statistics and probability distributions on manifolds are problematic. –Can hypothesis a normal distribution on tangent plane, but distortions inevitable

67. Some useful manifolds and maps Some useful manifolds and exponential maps Directional vectors (surface normals etc.) a, p unit vectors, x lies in an (n-1)D space

68. Some useful manifolds and maps Symmetric positive definite matrices (covariance, flow tensors etc) A is symmetric positive definite, X is just symmetric log is the matrix log defined as a generalized matrix function

69. Some useful manifolds and maps Orthogonal matrices (rotation matrices, eigenvector matrices) A orthogonal, X antisymmetric (X+X T =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

71. Embedding on S n On S 2 (surface of a sphere in 3D) the following parameterisation is well known The distance between two points (the length of the geodesic) is

72. 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

73. Spherical Embedding Say we had the distances between some objects (d ij ), measured on the surface of a [hyper]sphere of dimension n The sphere (and objects) can be embedded into an n+1 dimensional space –Let X be the matrix of point positions Z=XX T is a kernel matrix But And We can compute Z from D and find the spherical embedding!

74. Spherical Embedding But wait, we don’t know what r is! The distances D are non-Euclidean, and if we use the wrong radius, Z is not a kernel matrix –Negative eigenvalues Use this to find the radius –Choose r to minimise the negative eigenvalues

75. Example: Texture Mapping As an alternative to unwrapping object onto a plane and texture-mapping the plane Embed onto a sphere and texture-map the sphere Plane Sphere

76. Backup slides

77. Laplacian and related processes As well as embedding objects onto manifolds, we can model many interesting processes on manifolds Example: the way ‘heat’ flows across a manifold can be very informative On a sphere it is

78. Heat flow Heat flow allows us to do interesting things on a manifold Smoothing: Heat flow is a diffusion process (will smooth the data) Characterising the manifold (heat content, heat kernel coefficients...) The Laplacian depends on the geometry of the manifold –We may not know this –It may be hard to calculate explicitly Graph Laplacian

79. Given a set of datapoints on the manifold, describe them by a graph –Vertices are datapoints, edges are adjacency relation Adjacency matrix (for example) Then the graph Laplacian is The graph Laplacian is a discrete approximation of the manifold Laplacian

80. Heat Kernel Using the graph Laplacian, we can easily implement heat- flow methods on the manifold using the heat-kernel Can diffuse a function on the manifold by