Summary: Graphs are structures commonly used in computer science that model the interactions among entities. I will start from introducing the basic formulations of graph based machine learning, which has been a popular topic of research in the past decade and led to a powerful set of techniques. Particularly, I will show examples on how it acts as a generic data mining and predictive analytic tool. In the second part, I am going to discuss applications of such learning techniques in media analytics: (1) image analysis, where visually coherent objects are isolated from images; (2) social analysis of videos, where actors' social properties are predicted from videos. Materials in this part are based on our recent publications in highly selective venues (papers on https://sites.google.com/site/leiding2010/ ). Bio: Lei Ding is a researcher making sense of large amounts of data in all media types. He currently works in Intent Media as a scientist, focusing on data analytics and applied machine learning in online advertising. Previously, he has worked in several research institutions including Columbia University, UIUC and IBM Research on digital / social media analysis and understanding. He received a Ph.D. degree in Computer Science and Engineering from The Ohio State University, where he was a Distinguished University Fellow.

1. Graph based machine learning
with applications to media analytics
Lei Ding, PhD
9-1-2011
with collaborators at

2. Outline
• Graph based machine learning
– Basic structures
– Algorithms
– Examples
• Applications in media analytics
– Social analysis of videos
– Content analysis of images

3. Outline
• Graph based machine learning
– Basic structures
– Algorithms
– Examples
• Applications in media analytics
– Social analysis of videos
– Content analysis of images

4. What is a graph
Not the graph we are going to talk about

5. What is a graph
• A graph is composed of
– Vertices (nodes): pixels, actors in videos, genes, ads, etc.
– Edges: their relations
– In machine learning, we are interested in predicting some quantity
(a class label, or a continuous value) at each unlabeled vertex

6. What is a graph
• A graph is composed of
– Vertices (nodes): pixels, actors in videos, genes, ads, etc.
– Edges: their relations
– In machine learning, we are interested in predicting some quantity
(a class label, or a continuous value) at each unlabeled vertex
• Broadly speaking, there are two kinds of graphs
undirected directed

7. Graph based machine learning for
media analytics
• Oftentimes, media content can be represented using graphs
• Therefore, challenging inference problems with media content
can be answered by learning on graphs

8. Social content model
Content network
encodes content
similarity (videos,
audios, etc.)
Content generation
process
Social network
encodes peoples’
social connections
Can be used for media genre classification, media recommendation, etc.

9. Graph based machine learning
• On undirected graphs
– Optimization based approaches (e.g. energy minimization)
– Probabilistic models (e.g. random fields)
• On directed graphs
– Optimization based approaches (e.g. directed energy minimization)
– Probabilistic models (e.g. latent Dirichlet allocation, Bayesian networks)

10. Relations
• How are they related to traditional stats learning (e.g. logistic regression)
(Sutton McCallum, 2007)

11. Graph based machine learning
• On undirected graphs
– Optimization based approaches (e.g. energy minimization)
– Probabilistic models (e.g. random fields)
• On directed graphs
– Optimization based approaches (e.g. directed energy minimization)
– Probabilistic models (e.g. latent Dirichlet allocation, Bayesian networks)

12. Learning on undirected graphs
• Classification methods
– We have some labeled data, and
want to predict labels for others
– e.g. manifold regularization
• Clustering methods
– We would like to partition data
into clusters
– e.g. spectral clustering

13. Constructing data graphs
• How to transform a dataset ({xi}, i=1..m) into a graph

14. Affinity matrix
• A graph is usually represented using an affinity matrix W,
where the corresponding entry is 1 if two vertices are
connected, and 0 otherwise

15. Graph Laplacians
• L=D-W, where W is an affinity matrix, D is a diagonal matrix of
row sums
• Discretization of Laplace-Beltrami operator on manifolds, which
is the sum of second order derivatives on tangent space (more
details later)

16. Function on graph
• A vector can be used to represent a function over the graph
– We can encode what we already know or what we want to predict in a
label function
– For example in this graph, a vertex can represent a person, and the
function can represent if he is a likely customer
0 1
1
1
0
0
f = [ 1, 1, 0, 0, 1, 0 ] T

17. Eigenvectors reviewed

18. Properties of graph Laplacians
• Symmetric and positive semi-definite
• Graph Laplacian induces a smoothness term
– Transposed label function f * Laplacian matrix L * label function f (always
non-negative)
– Smoothness term (fTLf) measures how much the function f varies with
respect to the underlying graph
– We have labels on some vertices, and want to predict labels on other
vertices. A smooth function (small fTLf) typically predicts well
• Laplacian eigenvectors with small eigenvalues can be used for
data clustering / classification, data set parametrization, image
segmentation, etc.

19. Properties of graph Laplacians
• Symmetric and positive semi-definite
• Graph Laplacian induces a smoothness term
– Transposed label function f * Laplacian matrix L * label function f (always
non-negative)
– Smoothness term (fTLf) measures how much the function f varies with
respect to the underlying graph
– We have labels on some vertices, and want to predict labels on other
vertices. A smooth function (small fTLf) typically predicts well
• Laplacian eigenvectors with small eigenvalues can be used for
data clustering / classification, data set parametrization, image
segmentation, etc.
Now we are ready to see the algorithms, but let’s
take a little break to understand things even further

20. Manifolds

21. Manifold perspective of
data modeling

22. Why graphs encode underlying
data geometry
If we consider data as samples from an underlying manifold (which is a fairly weak
assumption), and construct the corresponding adjacency graph, then eigenvectors
of graph Laplacian approximate eigenfunctions of the Laplace-Beltrami operator
of the underlying data manifold
(Belkin Niyogi, 2008)

23. Laplacian eigenvectors
“understand” geometry
(Rustamov, 2007)

24. Spectral clustering
More information in von Luxburg (2007)

25. Spectral clustering explained
• Why the eigenvectors of L with small eigenvalues are used as the new
representation?
• The minimizers fi for the following total smoothness term are eigenvectors
of L with the smallest eigenvalues

26. Results

27. Laplacian eigenmap
• Using Laplacian eigenvectors with the smallest eigenvalues as
the new representation
• Can be seen as a non-linear extension of PCA
(Belkin Niyogi, 2003)

28. Results on real data
• Transform data using Laplacian eigenmap, and use linear
regression on the new representation
(Belkin Niyogi, 2004)

29. Manifold regularization
• A comprehensive regularization framework
• Through applying the representer theorem in functional
analysis, the optimal solution is as follows
(Belkin et al., 2006)

30. Results on real data
(Belkin et al., 2006)

31. Summary
• Learning on graphs provides a set of powerful techniques for
data analysis and predictive analytics that “understand” the
geometry of underlying data
• Spectral clustering – addresses the limitation with traditional
K-means
• Laplacian eigenmap manifold regularization – learn a label
function respecting underlying data geometry, and hence
provide benefits over standard methods like PCA and linear
regression
• Lots of other approaches as well – will talk about label
propagation based on graphs later in this presentation

32. Outline
• Graph based machine learning
– Basic structures
– Algorithms
– Examples
• Applications in media analytics
– Social analysis of videos
– Content analysis of images

33. Applications in media analytics
High-level analysis
Social relational inference
People to communities
Mid-level analysis
Event detection
Visual features to events
Low-level analysis
Segmentation
Pixels to semantic objects

34. Application 1: social analysis of
multimedia data
Friends or foes? Acquaintances or strangers? In same or different teams?

35. Social network learning and analysis

36. Social network learning and analysis

37. Social network learning and analysis
(Ding Yilmaz, 2010; 2011)

38. Application areas
• Social content: given the growing popularity of social media, inferring
relations among people is becoming important
• Visual recognition: social context is shown to help improve
recognition results from images (e.g. Wang et al., ECCV 10)
• Surveillance: social network learning and analysis for surveillance
applications (e.g. Yu et al., CVPR 2009)
• Sociology: necessary step in building intelligent systems for aiding
sociological discovery

39. Basic video processing
• Videos segmented into semantic segments
– Scenes, or visually coherent sets of shots, for movies and TV shows
– Shot detection and merging based on key-frame similarity (Rasheed
Shah, 03)
• Identifying the actors appearing in each segment
– Using scripts and closed captions for movies
– Face detection and recognition for other videos

40. Actor appearance matrix

41. Overall process
Social Relations video-level
A number [-1,+1] for each scene: positive
if actors in a scene are likely in the same Grouping cues
community, negative if otherwise
scene-level
Estimate the likely events in a scene
Event estimates
Dynamic systems represent scenes
Scene models
Feature observations frame-level

42. Key steps

43. Visual features
• Generic optical flow orientation histogram

44. Auditory features

45. Using visual concepts
• Visual concept detection provides useful semantic features for inferring
social relations
• Using Columbia s 374 SVM concept detectors on color/texture/edge
features, a concept score vector is generated for each scene

46. Evidence synthesis by Gaussian processes

47. Learned social affinity
— Learned social network is represented by affinity matrix K

48. Learned social networks

49. RACOM dataset
• Ten example movies: (1) G.I. Joe: The Rise of Cobra (2009); (2) Harry
Potter and the Half-Blood Prince (2009); (3) Public Enemies (2009); (4)
Troy (2004); (5) Braveheart (1995); (6) Year One (2009); (7) Coraline
(2009); (8) True Lies (1994); (9) The Chronicles of Narnia: The Lion, the
Witch and the Wardrobe (2005); (10) The Lord of the Rings: The Return
of the King (2003) .

50. Analyzing social networks
• We extend the max-min modularity principle such that it works with the
learned social networks, in order to detect the two communities for each
movie
• We also identify the leaders of each community, which interestingly,
correspond to the hero/villain most of the time

51. Max-min modularity

52. Visual maps

53. Quantitative evaluation

54. Detected social communities

55. Youtube dataset
• 10 videos for soccer games; 10 videos for demonstration;
• The goal here is to predict a grouping cue for each scene.
We evaluate against ground truth labeling

56. Youtube results
• Event categories are considered and labeled in a middle step
– Soccer: (chasing, confronting, hugging, others)
– Demonstration: (marching, confronting, public speaking, others)
• Precision (+) for within-community instances and Precision (-) for across-
community instances are reported separately

57. Application 2: image content analysis
• Interactive whole-object segmentation
– Inputs: an image labeled pixels (seeds) for objects/background
– Outputs: labels for all other pixels
(Ding Yilmaz, 2010)

58. Overview
• To segment whole objects from images given user-supplied seeds
– Different from unsupervised segmentation from a single image,
which typically generates homogeneous regions
– The challenge is to segment objects using a small number of seeds
• In addressing this problem, we have proposed
– Probabilistic hypergraph image model (PHIM)
– Automatic label set augmentation using boundary features
– Multiple view learning synthesizing features

59. Graphs vs. hypergraphs
• Graph based approaches have been popular for interactive segmentation
– Graph cut (Rother et al., 2004)
– Random walk (Grady, 2006)
• Hypergraphs vs. graphs for images
– Higher order relations among pixels that tend to form a segment are
encoded as hyperedges, which are collections of vertices
– Model long-range dependencies among the entities (known and unknown
labels)

60. Our model: PHIM
• We propose to use probabilistic hypergraph image model
(PHIM)
– The relation between a hyperedge and a vertex is probabilistic, based
on probabilities learned from image appearance characteristics
• Vertices: superpixels
• Hyperedges: pair-wise + higher-order (generated by mean-
shift weak segmentation with varying color bandwidths)

61. Our model: PHIM (cont’d)
• Feature vector Fs of a superpixel s contains average LUV color values
• Incidences: kernel density estimator taking superpixel features as the input
• Hyperedge weights: inhomogeneous hyperedges are down-weighted
– Reduces to standard graph based edge weights when the hyperedge is of
size 2

62. Laplacians on PHIM
• Normalized Laplacians on PHIM: induced quadratic form measures the
smoothness of a function with respect to the underlying edge system
– We use probabilistic incidences (hv,e) in defining Laplacians on PHIM
• Notations
– f: vector of function values on vertices (+1 for object; -1 for
background)
– H: probabilistic incidence matrix; W: hyperedge weight matrix
– De: hyperedge degree matrix; Dv: vertex degree matrix

63. How to do segmentation
• Constrained smoothness minimization
– Essentially an interpolation, as we have confidence in user-supplied
segment labels
• This interpolation can also be solved in an iterative manner
using the natural random walk

64. Dataset
• GrabCut dataset of 50 images (Rother et al., 2004)
• Seed pixels are provided in the form of trimaps
• Ground-truth segmentations are supplied

65. Results on segmentation
• Error rates averaged over the GrabCut dataset of 50 images
– PHIM performs better than a standard graph
– Our error rate 5.33% is much better than 7.9% achieved in (Blake et al.,
2006), and is comparable to state-of-the-art results from pixel-level
optimization

66. Comparative results

67. The end
• Thanks!
• References
– Ulrike von Luxburg, A Tutorial on Spectral Clustering, 2007
– Charles Sutton and Andrew McCallum, An Introduction to Conditional Random Fields for
Relational Learning, 2007
– Raif Rustamov, Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation,
2007
– Mikhail Belkin and Partha Niyogi, Laplacian Eigenmaps for Dimensionality Reduction and Data
Representation, 2003
– Mikhail Belkin and Partha Niyogi, Semi-Supervised Learning on Riemannian Manifolds, 2004
– Mikhail Belkin, Partha Niyogi and Vikas Sindwani, Manifold Regularization: A Geometric
Framework for Learning from Labeled and Unlabeled Examples, 2006
– Mikhail Belkin and Partha Niyogi, Convergence of Laplacian Eigenmaps, 2008
– Lei Ding and Alper Yilmaz, Learning Relations Among Movie Characters: A Social Network
Perspective, 2010
– Lei Ding and Alper Yilmaz, Interactive Image Segmentation Using Probabilistic Hypergraphs,
2010
– Lei Ding and Alper Yilmaz, Inferring Social Relations from Visual Concepts, 2011