On the Topology of Finite Metric Spaces

Transcription

1 On the Topology of Finite Metric Spaces Meeting in Honor of Tom Goodwillie Dubrovnik Gunnar Carlsson, Stanford University June 27, 2014

2 Data has shape The shape matters

3 Shape of Data Regression

4 Shape of Data Clusters

5 Shape of Data Predator-Prey model

6 Shape of Data

7 Shape of Data Flares

8 Shape of Data Normally defined in terms of a metric Euclidean distance, Hamming, correlation distance, etc. Encodes similarity

9 Topology Two tasks: Represent shape - triangulation, cell structures, etc. - can be viewed as a form of compression. Measure shape - algebraic topology.

10 Representing Shape Cluster Analysis

11 Representing Shape Scatterplots

12 Representing Shape Can we extend methods for compressing shape (triangulation etc.) to the point cloud world?

13 Topological Mapping 1. Obvious idea: try nerves of coverings 2. Works for coverings of sets, w/o topology 3. How to create coverings?

14 Topological Mapping 1. Use a reference map f : X R, where R is a reference space with systematic families of coverings 2. Pull back standard coverings from R 3. Typical choices for R - R, R n, S 1, trees. All have systematic families of coverings 4. Wrinkle - break up each set in the covering into its connected components, get a higher resolution cover 5. For point clouds, need to replace connected components by outcome of clustering scheme

15 Mapping

16 Mapping Typical one dimensional filters: Density estimators Measures of data depth, e.g. x X d(x, x ) 2 Eigenfunctions of graph Laplacian for Vietoris-Rips graph PCA or MDS coordinates User defined, data dependent filter functions

17 Mapping

18 Mapping Cell Cycle Microarray Data Joint with M. Nicolau, Nagarajan, G. Singh

19 Mapping RNA hairpin folding data Joint with G. Bowman, X. Huang, Y. Yao, J. Sun, L. Guibas, V. Pande, J. Chem. Physics, 2009

20 Mapping Diagram of gene expression profiles for breast cancer M. Nicolau, A. Levine, and G. Carlsson, Proc. Natl. Acad. Sci. 2011

21 Mapping Comparison with hierarchical clustering

22 Different platforms - importance of coordinate free approach

23 Different platforms - importance of coordinate free approach

24 Fragile X splits into distinct disorders

25 Mapping Serendipity - copy number variation reveals parent child relations

26 Measuring Shape of Data Need to extend homology to more general setting including point clouds Method called persistent homology Developed by Edelsbrunner, Letscher, and Zomorodian and Zomorodian-Carlsson

27 Measuring Shape of Data How to define homology to point clouds sensibly? Finite sets are discrete Statisticians suggest an approach

28 Measuring Shape of Data Dendrogram

29 Measuring the Shape of Data 1. Can be constructed from Vietoris-Rips complexes VR(X, r). 2. Vertex set is X. 3. {x 0, x 1,..., x k } spans a simplex if d(x i, x j ) r for all i, j. 4. Creates increasing family of simplicial complexes, increasing with r. 5. Construct the diagram of sets π 0 (VR(X, r)), parametrized by r. This is the dendrogram.

30 Measuring Shape of Data 1. The diagram π 0 VR(X, ) represents the components of X. 2. Suggests that the diagrams H i (VR(X, ) ) represent the higher homology of X. 3. Can we classify the resulting diagrams as we can for the dendrogram?

31 Measuring Shape of Data Definition; A persistence vector space is a functor from the non-negative real numbers (regarded as an ordered set) to vector spaces over a field k. Denote the category of such by Pers(k) With certain finiteness hypotheses, they can be classified up to isomorphism.

32 Measuring Shape of Data If we restrict the persistence parameter to integer values, have an equivalence of categories Pers Modgr(k[t]), where k is the ground field, given by {V n } n n V n The k-vector space structure is that of the infinite sum, action of t given by t (v 0, v 1,...) = (0, σ(v 0 ), σ(v 1 ),...) where σ is the shift in the persistence vector space.

33 Measuring Shape of Data 1. Means we can classify N-persistence vector spaces up to isomorphism, since k[t] is a graded PID. 2. Classification of finite objects in terms of barcodes, i.e. finite collections of intervals, possibly infinite to the right. 3. Finite intervals correspond to torsion summands, infinite intervals to free summands. 4. Similar construction using finitely presented modules over k[r + ] (a coherent ring) gives an equivalence with the category of R-persistence vector spaces.

34 Measuring the Shape of Data - Barcodes One dimensional barcode: 0

35 Measuring the Shape of Data - Barcodes

36 Measuring the Shape of Data - Barcodes 1=3

37 Measuring the Shape of Data - Barcodes

38 Measuring the Shape of Data - Barcodes 1=2

39 Space of high density high contrast image patches in natural images Betti 0 = 1 Betti 0 = 1 Betti 1 = 2 Betti 1 = 2 Betti 2 = 1 Betti 2 = 1

40 Mapping Patches

41 Natural Image Statistics Klein bottle makes sense in quadratic polynomials in two variables, as polynomials which can be written as where f = q(λ(x)) 1. q is single variable quadratic 2. λ is a linear functional 3. D f = 0 4. D f 2 = 1

42 Kleinlet Compression This understanding of density can be applied to develop compression schemes Earlier work, based on primary circle, called Wedgelets, done by Baraniuk, Donoho, et al. Extension to Klein bottle dictionary of patches natural

43 Kleinlet Compression

44 Kleinlet Compression

45 Kleinlet Compression

46 Texture Recognition Texture patches can be sampled for high contrast patches Yields distribution on Klein bottle

47 Texture Recognition Klein bottle has a natural geometry, and supports its own Fourier Analysis Textures provide distributions on the Klein bottle Pdf s can be given Fourier expansions, gives coordinates for texture patches (Jose Perea) Gives methods comparable to state of the art in performance, but in which effect of transformations such as rotation is predictable

48 Texture Recognition Jose Perea - Duke University Klein Bottle and Texture Discrimination

49 Evolution Tree of Life

50 Evolution Phylogenetics studies sets of sequences of various classes of organisms Uses Hamming or weighted versions of Hamming distances as organizing principle Often analyze by finding best approximation to space by trees Is this always justified?

51 Evolution Theorem: Let T be a tree, perhaps with lengths assigned to the edges. Then for any finite subspace of T, the persistent homology vanishes for every i > 0. This means there are no bars in higher degrees.

52 Evolution Barcodes indicating the presence of horizontal evolution

53 Evolution Can study persistence barcodes of metric spaces of trees arising in evolution Presence of large loops can suggests standard model is incomplete Signal of presence of alternate mechanisms, such as horizontal gene transfer Can also estimate various rates from the barcodes, by performing simulations J. Chan, G. C., and R. Rabadan, Proc. Natl. Acad. Sci. 2013

54 Applications of Persistence By using persistence on other quantities (density, centrality,...) can get useful shape invariants Persistence barcodes lie in barcode space, has a metric Persistence gives a map P from M (space of metric spaces with Gromov-Hausdorff metric) to B (barcode space with bottleneck distance) P is distance non-increasing (Chazal, Mémoli, Guibas, Oudot) Can be used to get useful invariants of shapes - X-ray images, for example

55 Applications of Persistence Space of barcodes can be thought of as an infinite algebraic variety Get a ring of algebraic functions which detect barcodes Ring analyzed by Adcock, E. Carlsson, G.C.

56 Coordinatizing barcode space 1. A single bar can be viewed as a pair (x, y), with x y. 2. We let B n denote the set of barcodes with exactly n intervals. 3. B n can be identified as SP n (I), where I is the set of intervals (possibly of length = 0). It is a subset of SP n (R 2 ). 4. Need to assemble them together into a single space B.

57 Coordinatizing barcode space 1. B = n B n/, where the equivalence relation is generated by equivalences of the form {(x 1, y 1 ),..., (x n, y n ), (x n+1, x n+1 )} {(x 1, y 1 ),..., (x n, y n ), } 2. Encodes the fact that we want to ignore bars of length zero. They have no persistence. 3. Want to formulate definition so that we have a good ring of functions.

58 Coordinatizing barcode space 1. Each SP n (I) is a subset of an algebraic variety over R, since SP n (R 2 ) is an algebraic variety. 2. There is a diagram of algebraic varieties so that B embeds inside the corresponding diagram of geometric points. 3. The corresponding diagram of rings has an inverse limit, which can be regarded as the affine coordinate ring of persistence barcodes. 4. The ring is too complicated. How to deal with this situation?

59 Coordinatizing barcode spaces 1. Consider the case of SP (R) 2. The ring is in this case the inverse limit A R of the diagram of rings R[σ 1 ] R[σ 1, σ 2 ] R[σ 1, σ 2, σ 3 ] 3. This ring is manageable, but is still complicated, since it consists of power series in all the σ i s, with certain finiteness conditions on the sums.

60 Coordinatizing barcode space 1. SP n (C) is a complex variety with G m action, and the analogous inverse limit A C to A R above is acted on by G m (C), and therefore by S An element α of A C is said to be K-finite if the span of all its translates by elements of S 1 is finite dimensional 3. The K-finite vectors form a sub algebra, which is a polynomial ring on the elementary symmetric functions σ i. 4. Taking the fixed points of complex conjugation gives the same polynomial ring over R. 5. Analogous construction can be carried out for B

61 Coordinatizing barcode space 1. The corresponding K-finite coordinate ring is isomorphic to R[τ ij ; i 1, j 0] 2. τ ij is equal to the symmetrized sum t (y s x s ) i (y s + x s ) j 3. Can use these to analyze databases of unstructured data. 4. A. Bak and M. Lerner have used them to characterize organic molecules which inhibit certain processes within cancer.

62 Coordinatizing barcode space A. Bak and M. Lerner

63 Coordinatizing barcode spaces 1. Coordinates will also be useful in interpreting multidimensional persistence. 2. Multidimensional persistence classification is like multigrades modules over a multigrades polynomial ring in several variables. 3. Precludes the possibility of constructing barcodes. Classification is actually field dependent, more like a moduli space than a discrete target. 4. Vitally important since one wants to take density into account in building the Vietoris-Rips complex. 5. Can one understand instead the functions on the set of modules?