1. From Signal to Symbols
Approximate Category Theory and Conceptual Regularization
Anthony Bak
SF Data Mining, November 2015

2. Disclaimer
Everything in this presentation is stolen (with permission). No ideas, images or
formulas are my own excepting errors which are surely mine.
M. Ovsjanikov, M. Ben-Chen, J. Solomon, A. Butscher and L. Guibas. Functional Maps: A Flexible
Representation of Maps Between Shapes. ACM Transactions on Graphics. 31(4), 2012 – Siggraph 2012
F. Wang, Q. Huang, and L. Guibas. Image Co-Segmentation via Consistent Functional Maps. The 14th
International Conference on Computer Vision (ICCV). Sydney, Australia, December 2013.
Q. Huang, F. Wang, L. Guibas, Functional map networks for analyzing and exploring large shape
collections, ACM Transactions on Graphics (TOG), Volume 33, Issue 4, July 2014
R. Rustamov, D. Romano, A. Reiss, and L. Guibas. Compact and Informative Representation of Functional
Connectivity for Predictive Modeling. Medical Image Computing and Computer Assisted Intervention
Conference (MICCAI), 2014
Syntactic and Functional Variability of a Million Code Submissions in a Machine Learning MOOC, J.
Huang, C. Piech, A. Nguyen, L. Guibas. In the 16th International Conference on Artificial Intelligence in
Education (AIED 2013) Workshop on Massive Open Online Courses (MOOCshop) Memphis, TN, USA,
July 2013.
Special thanks to Leo Guibas who kindly sent me his images and presentation on
this material. http://geometry.stanford.edu/

3. Disclaimer
Everything in this presentation is stolen (with permission). No ideas, images or
formulas are my own excepting errors which are surely mine.
M. Ovsjanikov, M. Ben-Chen, J. Solomon, A. Butscher and L. Guibas. Functional Maps: A Flexible
Representation of Maps Between Shapes. ACM Transactions on Graphics. 31(4), 2012 – Siggraph 2012
F. Wang, Q. Huang, and L. Guibas. Image Co-Segmentation via Consistent Functional Maps. The
14th International Conference on Computer Vision (ICCV). Sydney, Australia, December 2013.
Q. Huang, F. Wang, L. Guibas, Functional map networks for analyzing and exploring large shape
collections, ACM Transactions on Graphics (TOG), Volume 33, Issue 4, July 2014
R. Rustamov, D. Romano, A. Reiss, and L. Guibas. Compact and Informative Representation of Functional
Connectivity for Predictive Modeling. Medical Image Computing and Computer Assisted Intervention
Conference (MICCAI), 2014
Syntactic and Functional Variability of a Million Code Submissions in a Machine Learning MOOC, J.
Huang, C. Piech, A. Nguyen, L. Guibas. In the 16th International Conference on Artificial Intelligence in
Education (AIED 2013) Workshop on Massive Open Online Courses (MOOCshop) Memphis, TN, USA,
July 2013.
Special thanks to Leo Guibas who kindly sent me his images and presentation on
this material. http://geometry.stanford.edu/

4. Big Picture
We want to bridge the gap between Human and Computer understanding of
sensor data

5. Big Picture
We want to bridge the gap between Human and Computer understanding of
sensor data
Human understanding and situational reasoning comes from
Having models of the world consisting of symbols (Cars, words etc.)

6. Big Picture
We want to bridge the gap between Human and Computer understanding of
sensor data
Human understanding and situational reasoning comes from
Having models of the world consisting of symbols (Cars, words etc.)
Relating sensor input to past experience and the model

7. Big Picture
We want to bridge the gap between Human and Computer understanding of
sensor data
Human understanding and situational reasoning comes from
Having models of the world consisting of symbols (Cars, words etc.)
Relating sensor input to past experience and the model
Here we present a way to build the symbols from signals by looking for invariants
of the collection. Formally

8. Big Picture
We want to bridge the gap between Human and Computer understanding of
sensor data
Human understanding and situational reasoning comes from
Having models of the world consisting of symbols (Cars, words etc.)
Relating sensor input to past experience and the model
Here we present a way to build the symbols from signals by looking for invariants
of the collection. Formally
We build networks relating signals to each other

9. Big Picture
We want to bridge the gap between Human and Computer understanding of
sensor data
Human understanding and situational reasoning comes from
Having models of the world consisting of symbols (Cars, words etc.)
Relating sensor input to past experience and the model
Here we present a way to build the symbols from signals by looking for invariants
of the collection. Formally
We build networks relating signals to each other
We transport information through the network

10. Big Picture
We want to bridge the gap between Human and Computer understanding of
sensor data
Human understanding and situational reasoning comes from
Having models of the world consisting of symbols (Cars, words etc.)
Relating sensor input to past experience and the model
Here we present a way to build the symbols from signals by looking for invariants
of the collection. Formally
We build networks relating signals to each other
We transport information through the network
Concepts emerge as fixed points in the network

11. Objects and Their Functions
In the network nodes will represent objects we are trying to study. Information (or
annotations) of our objects will be encoded as functions. For example
Segmentation or "part indicators"

12. Objects and Their Functions
In the network nodes will represent objects we are trying to study. Information (or
annotations) of our objects will be encoded as functions. For example
Segmentation or "part indicators"
Geometric properties eg. Eigenfunctions of Laplace-Beltrami operator,
curvature

13. Objects and Their Functions
In the network nodes will represent objects we are trying to study. Information (or
annotations) of our objects will be encoded as functions. For example
Segmentation or "part indicators"
Geometric properties eg. Eigenfunctions of Laplace-Beltrami operator,
curvature
Descriptors (eg. SIFT)

14. Objects and Their Functions
In the network nodes will represent objects we are trying to study. Information (or
annotations) of our objects will be encoded as functions. For example
Segmentation or "part indicators"
Geometric properties eg. Eigenfunctions of Laplace-Beltrami operator,
curvature
Descriptors (eg. SIFT)
etc.

15. Information Transport
We will assume real valued functions for the rest of the discussion and write C(P)
for the space of real valued functions on P. Given two objects, L, C and a map
φ : L → C.

16. Information Transport
We will assume real valued functions for the rest of the discussion and write C(P)
for the space of real valued functions on P. Given two objects, L, C and a map
φ : L → C.
We get a map Tφ : C(C) → C(L) by composition.
f ∈ C(C) → f ◦ φ ∈ C(L)

17. Information Transport
We will assume real valued functions for the rest of the discussion and write C(P)
for the space of real valued functions on P. Given two objects, L, C and a map
φ : L → C.
We get a map Tφ : C(C) → C(L) by composition.
f ∈ C(C) → f ◦ φ ∈ C(L)
Tφ is a linear operator.

18. Information Transport
Information is transported between objects by applying a linear operator
TCL : C(C) → C(L).
We relax the condition that the linear maps are induced from maps on the
underlying objects.

19. Network Regularization
We want to use the network of relationships between objects to constrain the
space of possible solutions. To that end we require

20. Network Regularization
We want to use the network of relationships between objects to constrain the
space of possible solutions. To that end we require
The transport of information from C to L does not depend on the path taken.
C(C) C(B)
C(A) C(L)
TCB
TCA TBL
TAL

21. General Procedure
To apply this to an example
Construct a network consisting of similar objects

22. General Procedure
To apply this to an example
Construct a network consisting of similar objects
Use transport of similarity measures to fit our linear transformations

23. General Procedure
To apply this to an example
Construct a network consisting of similar objects
Use transport of similarity measures to fit our linear transformations
Use these transformations to transport information through the network to
solve some problem

24. Example
Image Co-Segmentation

25. Problem
Task: Jointly segment a set of related images

26. Problem
Task: Jointly segment a set of related images
Same object with different viewpoints and scales

27. Problem
Task: Jointly segment a set of related images
Same object with different viewpoints and scales
Similar objects of the same class

28. Problem
Task: Jointly segment a set of related images
Same object with different viewpoints and scales
Similar objects of the same class
Images provide weak supervision of each other.

29. Images to Image Network
We create a (sparse) similarity graph using a gaussian kernel function on GIST
image descriptors for each image. To each edge we assign the weight
wij = exp(
−||gi − gj||2
2σ
) σ = median(||gi − gj||)
We connect each image to its k = 30 most similar neighbors.

30. Images to Image Network

31. Image Function Space
Use a super pixel segmentation of the images and build a graph by taking
Nodes are super pixels
Edges are weighted by length of shared boundary
The function space we associate to each image is the space of real valued
functions on this graph.

32. Hierarchical Subspace
On a graph have the graph laplacian L = D − W where D is the diagonal degree
matrix and W is the edge weight matrix.

33. Hierarchical Subspace
On a graph have the graph laplacian L = D − W where D is the diagonal degree
matrix and W is the edge weight matrix.
The eigenvectors (eigenfunctions) of L are ordered by "scale". For example, if
there are k components to the graph then the first k eigenfunctions are
indicator functions on the components. The next smallest eigenvector is used
in many graph cut algorithms.

34. Hierarchical Subspace
On a graph have the graph laplacian L = D − W where D is the diagonal degree
matrix and W is the edge weight matrix.
The eigenvectors (eigenfunctions) of L are ordered by "scale". For example, if
there are k components to the graph then the first k eigenfunctions are
indicator functions on the components. The next smallest eigenvector is used
in many graph cut algorithms.

35. Hierarchical Subspace
On a graph have the graph laplacian L = D − W where D is the diagonal degree
matrix and W is the edge weight matrix.
The eigenvectors (eigenfunctions) of L are ordered by "scale". For example, if
there are k components to the graph then the first k eigenfunctions are
indicator functions on the components. The next smallest eigenvector is used
in many graph cut algorithms.Examples of one-dimensional mappings
u2 u3
u4 u8
Radu Horaud Graph Laplacian Tutorial

36. Hierarchical Subspace
On a graph have the graph laplacian L = D − W where D is the diagonal degree
matrix and W is the edge weight matrix.
The eigenvectors (eigenfunctions) of L are ordered by "scale". For example, if
there are k components to the graph then the first k eigenfunctions are
indicator functions on the components. The next smallest eigenvector is used
in many graph cut algorithms.Examples of one-dimensional mappings
u2 u3
u4 u8
Radu Horaud Graph Laplacian Tutorial
We choose the subspace spanned by the first 30 eigenvectors. This keep the
dimensionality of the problem under control and the hierarchy assures us that this
is reasonable.

37. Eigenfunctions on Images

38. Joint Estimation of Transfer Functions
We do joint estimation of the transfer functions by optimizing over transfer matrices
Tij
Aligning image features associated with each super pixel (e.g.. average RGB
color)
A regularization term that penalizes mapping eigenspaces with very different
eigenvalues to each other
A cycle consistency term

39. Joint Estimation of Transfer Functions
We do joint estimation of the transfer functions by optimizing over transfer matrices
Tij
Aligning image features associated with each super pixel (e.g.. average RGB
color)
A regularization term that penalizes mapping eigenspaces with very different
eigenvalues to each other
A cycle consistency term
This is solvable. So we get our consistent maps Tij

40. Segmentation
The actual segmentation is done by finding
Best cut function
Subject to the consistency between cut functions on other images
.

41. Segmentation
The actual segmentation is done by finding
Best cut function
Subject to the consistency between cut functions on other images
This is a joint optimization problem and also solvable.

42. Experimental Results
Tested on three standard datasets
iCoseg
Very similar or the same object in each class
5-10 images per class
MSRC
Very similar objects in each class
30 images per class
Pascal
Larger scale and variability

43. Experimental Results: iCoseg

44. Experimental Results: iCoseg
Vicente is a supervised method

45. iCoseg: 5 images per class are shown

46. iCoseg: 5 images per class are shown

47. iCoseg: 5 images per class are shown

48. iCoseg: 5 images per class are shown

49. MSRC: 5 images per class are shown

50. MSRC: 5 images per class are shown

51. PASCAL: 10 images per class are shown

52. PASCAL: 10 images per class are shown

53. PASCAL: 10 images per class are shown

54. PASCAL: 10 images per class are shown

55. The Network is the Abstraction
Plato’s cow

56. Summary
Classical view:
A hierarchy moves you from signals, to parts, to symbols,...
"Vertical"
Alternate view:
Symbols emerge from the network of signal relationships as invariants.
"Horizontal"