What is a sheaf? Michael Robinson. SIMPLEX Program

Transcription

1 What is a sheaf? SIMPLEX Program 215 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4. International License.

2 Session objectives What is heterogeneous data fusion? What is a sheaf? What happens if I don't have a sheaf? 3

3 Axiom 1: There is a set of data sources A list of sensors A collection of tables Grad school High school Undergrad Postdoc 4

4 Undergrad GR GPA Stipend Salary UG GPA High school Grad school Emp ID HS GPA GR GPA Possible signal spaces UG GPA UG I D HS GPA Possible columns or keys HS ID GR I D Different sensors will read out in attributes UG GPA Axiom 2: There is a list of possible attributes Postdoc 5

5 Stipend Salary Undergrad GR GPA ID UG GPA Grad school Emp High school GR GPA UG GPA UG GPA HS GPA UG I D HS GPA Topology comes from shared attributes HS ID GR I D Axioms 3 and 4: The data sources are topologized Postdoc 6

6 Stipend GR GPA UG GPA GR GPA GR I D Salary GR GPA UG GPA Undergrad Emp ID UG GPA UG G PA High school Grad school UG G PA HS GPA UG I D HS GPA Transformations are permitted HS ID UG GPA Essentially by forming subtables HS G PA UG G PA Axiom 5: Data sources can be compared via restriction Postdoc 7

7 Stipend GR GPA GR I D UG GPA Salary GR GPA Grad school UG GPA Undergrad GR GPA UG GPA $2k Emp ID High school 3.5 UG G PA UG G PA HS GPA UG I D HS GPA Transformations are permitted HS ID UG GPA Essentially by forming subtables HS G PA UG G PA Axiom 5: Data sources can be compared via restriction $45k Postdoc 3.5 8

8 Stipend Salary GR GPA GR GPA Salary UG GPA Undergrad HS GPA High school UG GPA Emp ID 91 Grad school UG G PA $2k UG GPA HS ID HS GPA GR GPA Problems can arise if this isn't satistifed! HS G PA GR I D Table joins: can find rows of a bigger table UG I D HS GPA UG GPA Axioms 6 and 7: Data sources can be combined in a unique way $45k Postdoc $45k 9

9 Stipend Salary GR GPA GR GPA Salary UG GPA Undergrad HS GPA High school UG GPA Emp ID 91 Grad school UG G PA $2k UG GPA UG I D HS GPA HS GPA GR GPA Might result in empty tables (no rows) HS ID UG GPA Table joins: HS G PA GR I D Axioms 6 and 7: Data sources can be combined in a unique way $45k Postdoc $45k 1

10 Stipend Salary GR GPA GR GPA Salary Emp ID Undergrad UG GPA Grad school UG G PA UG GPA High school X HS GPA $2k UG GPA HS GPA GR GPA Might result in empty tables (no rows) HS ID UG GPA Table joins: UG I D HS GPA GR I D Axioms 6 and 7: Data sources can be combined in a unique way $45k Postdoc $45k 11

11 Stipend GR GPA UG GPA GR GPA GR I D Salary GR GPA UG GPA Emp ID Undergrad UG GPA UG G PA High school Grad school UG G PA HS GPA UG GPA Specifies transformations UG I D HS GPA HS G PA Specifies the attribute spaces HS ID UG G PA Mathematical representation Postdoc 12

12 Key/table duality Two obvious viewpoints: Table-centric Tables on vertices (leads to sheaf) Bottom up Restrictions go from low dimensional simplices to higher dimensional simplices Key-centric Keys on vertices (leads to cosheaf) Top down Extensions go from high dimensional simplices to lower dimensional simplices 13

13 Building a cosheaf model Keys are vertices, and tables are simplices (so the base space is different) Stipend Grad school High school Undergrad GR GPA HS GPA UG GPA Postdoc Salary Idea credit: Jose Perea (Michigan) (Note: ID fields ignored for simplicity) 14

14 What happens when the axioms fail? Axiom 1 or 2: Data isn't in a set Axiom 3 or 4: Sources not topologized No basis for combining data sources Axiom 5: No transformations to identify commonalities between observations You won't be able to do much computationally Although information about a given entity might be available through different sources, they can't be joined Axiom 6 or 7: Cannot uniquely fuse Even if two observations are comparable, it is impossible to infer anything else about nearby observations This often happens in databases two rows with overlapping keys and matching values doesn't mean they're the same! 15

15 Now, abstractly... A sheaf of on a (data type) (topological space)

16 Simplicial complexes... higher dimensional simplices (tuples of vertices) v2 {v1,v2} {v2,v3} v1 {v1,v2,v3} v4 {v3,v4} {v1,v3} v3 17

17 Simplicial complexes The attachment diagram shows how simplices fit together v2 {v1,v2} {v2,v3} v1 {v1,v2,v3} v4 {v3,v4} {v1,v3} v3 18

18 A sheaf is... A set assigned to each simplex and... 2 Each such set is called the stalk over its simplex This is a sheaf of vector spaces on a simplicial complex 3 19

19 A sheaf is... a function assigned to each simplex inclusion () 1 2 (1 ) () 1 Each such function is called a restriction 2 ( 1) (1 ) ( )

20 A sheaf is... so the diagram commutes. () 1 2 (1 ) () 1 (1 ) ( 1) ( ) (1 ) = 2 (1 ) ( ) ( )

21 A global section is... An assignment of values from each of the stalks that is consistent with the restrictions 1 1 () 1 () 1 () (1 ) ( 1) 1 (1 1) () (2 1-1) (1 )

22 Some sections are only local They might not be defined on all simplices or disagree with restrictions 1 2 () 1 () 2 (1 ) 1 11 X ( 1) X 1 (1 1) () (2 1-1) N/A (1 )

23 Flabbiness If all local sections defined on vertices extend to global sections, the sheaf is called flabby (or flasque) These sheaves don't have interesting invariants They are good for decomposing other sheaves 3 3 Example: Vertex- or edge-weighted graphs with no further constraints Flabby sheaves mean there is a lack of constraints imposed by the model 24

24 Queue model example

25 A queue as a sheaf Contents of the shift register at each timestep N = 3 shown

26 A single timestep Contents of the shift register at each timestep N = 3 shown (9,2) (1,9,2) (1,9) (1,1,9) (1,1) (5,1,1) (5,1) (2,5 1 1

27 Wireless network example

28 Wireless activation sheaf Each node in the wireless complex has a unique ID The stalk over a face consists of the set of nodes adjacent to that face, and the special symbol 4 6 {1, 2, }

29 Wireless activation sheaf Each node in the wireless complex has a unique ID The stalk over a face consists of the set of nodes adjacent to that face, and the special symbol 4 6 {1, 2, }

30 Wireless activation sheaf Each node in the wireless complex has a unique ID The stalk over a face consists of the set of nodes adjacent to that face, and the special symbol {1, 2, 3, 4, }

31 Wireless activation sheaf Each node in the wireless complex has a unique ID The stalk over a face consists of the set of nodes adjacent to that face, and the special symbol Note: adjacent means has a higher-dimensional face in common {2, 3, 4, }

32 Wireless activation sheaf Restrictions map node IDs via identity whenever possible, and otherwise send to restriction {2, 3, 4, } 4 6 {1, 2, 3, 4, }

33 Wireless activation sheaf Sheaves model node activation and traffic-passing protocols, by encoding local consistency constraints Node activation patterns when interference is possible. ( means no activity) Some possible sections Link complex {,1,2} {,2,3} {,1,2} {,1,2,3} {,2,3} Activation sheaf

34 Shared situational awareness

35 Forming mosaics Multiple, overlapping images can be assembled into a mosaic by stitching together similar regions Many algorithms exist Most are robust to perspective (or other) changes Overlapping maps (Image courtesy of NASA/JPL) 36

36 Shared visual situational awareness (In collaboration with UCLA)... Time Potential features: Embrace multi-modality: categorical and image data together could yield improved object recognition and tracking Objects in scene correspond to categorical-valued partial sections 37

37 Heterogeneous fusion among homogeneous sensors A sensor transforms a physical region into data Camera 1 Images Camera 2 Physical sensor footprints Images Sensor data space 38

38 Heterogeneous fusion among homogeneous sensors Camera 1 Images Camera 2 Images Summarization Detections Physical sensor footprints Sensor data space 39

39 Heterogeneous fusion among homogeneous sensors This construction the data together with the transformations is a sheaf Camera 1 Images Camera 2 Globalization Fused targets Images Detections Physical sensor footprints Sensor data space 4

40 Data Structures as Sheaves SIMPLEX Program 215 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4. International License.

41 Session objectives How do sheaves extend well-known data structures? How do I translate between sheaf-based data structures? What can I do once I have a sheaf? 3

42 What is a sheaf? A A' A A' a a 4

43 What is a sheaf? A A A A' A A' a a 5

44 What isn't a sheaf? A x If labels on the graph are not systematically related to one another... x Consistency checks become ad hoc A x Cross-modality inference is no longer possible 6 x? A' x a

45 Vertex- or (hyper)edge-weighted (hyper)graphs Vertex weighted sheaves Hyperedge-weighted cosheaves Vertex has nontrivial stalk Toplex has nontrivial stalk All restrictions are zero maps All extensions are zero maps The resulting sheaf is flabby The resulting cosheaf is coflabby* * This is such a fun term but I don't see it much in use 7

46 Flow sheaves Start with a collection of paths along which material flows Label each track segment with amount of material on that segment Time

47 Flow sheaves Conservation law enforced at each vertex Depending on precisely how we count material (in ℕ or, for instance), we might write the conservation law as a+b = c+d or a+b-c-d = a c b d e d+e = f f 9

48 Flow sheaf Each degree n vertex is assigned a free sr-module* of rank (n 1) for material measured in a semiring sr Restriction maps are projections sr (3,2,1,4) 4 sr U sr sr sr sr 3 Flow sheaf sr sr sr V sr sr sr (4,3,7) 7 sr * or just a vector space if it seems easier 1

49 Local consistency A flow sheaves encodes a notion of consistency between adjacent faces 3 1 Cannot label this edge in a way consistent with the data It's an indication that a flow was incorrectly measured somewhere in this vicinity 8 11

50 Inferential ambiguity Depending on how we make measurements, we might not get the full story of the flow Many possible inferences 3 1? 2 3? Inconsistent: no inference possible 3 2? 1 2?? Exactly one inference 12

51 Bayesian networks

52 Probability spaces Start with a set of random variables X,X1, Xn Consider the set P(X,X1,, Xn) of all joint probability distributions over these random variables These are the nonnegative measures (generalized functions) f = f(x,x1,, Xn) with unit integral This is not a vector space adding probability distributions doesn't yield another distribution 14

53 Marginalization cosheaf There is a natural map P(X,X1,, Xn) P(X,X1,, Xn-1) via marginalization, namely f(x,x1,, Xn-1) = f(x,x1,, Xn) dxn There similar maps for marginalizing out the other random variables, too This yields a cosheaf on the complete n-simplex! 15

54 Bayes' rule Conditional probabilities produce maps going the other way For instance, P(X,X1,, Xn-1) P(X,X1,, Xn) is parameterized by functions C F(X,X1,, Xn) = C(X,X1,, Xn) f(x,x1,, Xn-1) usually, one writes the arguments to C like C = C(Xn X,X1,, Xn-1) So conditional probabilities yield a sheaf on part of the n-simplex 16

55 Small Bayes net example Consider two binary random variables X and Y with a given conditional C(Y X) X P(X) P(X) Y ( ) p( ) p(1 ) p( 1) p(1 1) P(X,Y) ( P(X,Y) ) P(Y) Marginalization cosheaf P(Y) Conditional sheaf = C(Y X) 17

56 Linear translation-invariant filters

57 How does a sheaf model a signal? C((-2,),) -1 C((-1,),) Simplicial complex for C((-1,1),) The sheaf of continuous functions C((,1),) C((,2),) 1 19

58 How does a sheaf model a signal? Computing global section C((-2,),) Set of real-valued continuous functions on (-2,) C((-1,),) Restriction map chops off portion of a function C((-1,1),) C((,1),) Space of global sections is the set of continuous functions on C((,2),)

59 A sheaf morphism... takes data in the stalks of two sheaves

60 A sheaf morphism... and relates them through linear maps ½ -½ -1 ( )

61 A sheaf morphism... so the diagram commutes! ½ -½ ( )

62 A sampling morphism is... Computing global section C((-2,),) Evaluate at -1 C((-1,),) C((-1,1),) Evaluate at C((,1),) C((,2),) Evaluate at 1

63 An ambiguity sheaf is... The collection of kernels of these sampling maps The set of continuous functions on (-2,) that vanish at -1-1 Z-1(-2,) C((-2,),) C((-1,),) C((-1,),) Z(-1,1) C((-1,1),) Evaluate at C((,1),) C((,1),) Z1(,2) C((,2),) 25 Evaluate at -1 Evaluate at 1

64 An ambiguity sheaf is... C((-1,),) Computing global section Sections of the ambiguity sheaf are functions that appear to be all zero under the sampling Z-1(-2,) Z(-1,1) C((,1),) Z1(,2)

65 The general sampling theorem Given a sheaf S and a sampling of it, construct the ambiguity sheaf A Theorem: Perfect reconstruction of global sections of S is possible if and only if the only global section of A is the zero function No ambiguities means it's possible to reconstruct 27

66 Nyquist-Shannon sampling Encode signals as a sheaf of bandlimited functions BF over, with bandwidth B It's easier to work in the frequency domain: BF = { f C(,ℂ) supp f [-B, B] } Samples are taken at integers, obtained by inverse Fourier transform For instance at n, we sample using the function Mn B 2 n π i x M n (f )= f e dx B 28

67 Nyquist-Shannon sampling The ambiguity sheaf identifies bandlimited functions with zeros at specific locations An = { f BF Mn (f) = } = {Bandlimited functions that are zero at n} A-1 A BF identity identity BF identity BF M-1 ℂ identity identity BF identity BF M A1 BF ℂ 29 identity BF identity M1 ℂ

68 Nyquist-Shannon sampling The ambiguity sheaf identifies bandlimited functions with zeros at specific locations An = { f BF Mn (f) = } = {Bandlimited functions that are zero at n} A-1 BF A A1 BF Global sections of the ambiguity sheaf are bandlimited functions that vanish at every integer There aren't any if B < 1/2 3

69 Filters as sheaf morphisms Theorem: Every discrete-time LTI filter can be encoded as a sequence of two sheaf morphisms S1 S2 S3 Input Internal state Output Sheaf formalism Hardware Shift register Weighted sum 31

70 Input sheaf Sections of this sheaf are timeseries, instead of continuous functions 32

71 Output sheaf The output sheaf is the same 33

72 The internal state Contents of the shift register at each timestep N = 3 shown

73 The internal state Loads a new value with each timestep ( 1) ( 1) ( 1)

74 The internal state Produces average of the shift register at each timestep ( 1) 2 3 ( 1) 2 3 (⅓ ⅓ ⅓) (⅓ ⅓ ⅓) 36 ( 1) (⅓ ⅓ ⅓)

75 Finishing both morphisms Put in a few zero maps!

76 A single timestep Sections of the sheaves linked together give possible input/output combinations of the filter 2 9 (9,2) (1,9,2) (1,9) (1,1,9) (1,1) (5,1,1) (5,1) (2,5 2.7

77 Of course, this extends... Sections of the sheaves linked together give possible input/output combinations of the filter 2 9 (9,2) (1,9,2) (1,9) (1,1,9) (1,1) (5,1,1) (5,1) (2,5 2.7

78 How this formalism helps... Of course, it corresponds nicely to the hardware BUT... It's easy to splice in nonlinear operations It works on nontrivial base spaces: A systematic study of filtering is now possible on cell complexes Sheaves make it easy to invent topological filters that have controlled performance characteristics 4

79 Filtering sinosoids from noise 41

80 Filtering out chirpy signals Loss of signal as frequency changes 42 Phase reversal

81 Filtering out chirpy signals Unstable amplitude 43

82 Filtering out chirpy signals Poor initial estimate of frequency Loss of amplitude filter cannot keep up! 44

83 Circumventing bandwidth limits More averaging in a connected window leads to: More noise cancellation (Good) More distortion to the signal (Bad) Can safely do more averaging by collecting samples at similar places across the entire signal 45

84 Filter block diagram Grouping sheaf construction Stage 1: Topological estimation Averaging filter Output signal Stage 2: Topological filtering Average along rows Time Input signal Point cloud metric construction Neighbors 46

85 Stage 1: Topological estimation Image values Pixel neighborhood Image spatial domain Point cloud in 47 16

86 Stage 2: Grouping sheaf Distance matrix of point cloud Sample number Average neighbors Locations of nearest neighbors Sample number Sample number Sample number Values of the signal at the neighbors neighbors Process execution 48

87 Topological filter results Some low frequency distortion Extremely stable output amplitude 49

88 Compare: standard adaptive filter Unstable amplitude 5

89 Input image 51

90 Fixed frequency image filter 52

91 Topological filter output 53

92 Error contributions Matching Dimension: 2 Error Boxcar filter Topological estimation error Using topological estimator Using true topology Noise 54

93 Error contributions Matching Dimension: 25 Error Boxcar filter Using topological estimator Using true topology Noise 55

94 Error contributions Matching Dimension: 3 Error Boxcar filter Using topological estimator Using true topology Noise 56

95 Further reading... Sanjeevi Krishnan, Flow-cut dualities for sheaves on graphs, Robert Ghrist and Sanjeevi Krishnan, A Topological Max-Flow Min-Cut Theorem, Proceedings of Global Signals. Inf., (213)., Understanding networks and their behaviors using sheaf theory, IEEE Global Conference on Signal and Information Processing (GlobalSIP) 213, Austin, Texas., The Nyquist theorem for cellular sheaves, Sampling Theory and Applications 213, Bremen, Germany. 58