1. Semi-Supervised Learning in Gigantic Image Collections
Rob Fergus (New York University)
Yair Weiss (Hebrew University)
Antonio Torralba (MIT)
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2. Gigantic Image Collections
What does the world look like?
Gigantic Image Collections
Object Recognition for large-scale image search
High level image statistics
Our goal is to develop techniques for image search that can be applied to the billions of images on the Internet
We’re interested in developing object recognition techniques that can scale to the billions of images on the Internet.

3. Spectrum of Label Information
Human annotations
Noisy labels
Unlabeled
One property of images on the internet is that they have a wide range of label information. A tiny fraction have been labeled by humans so have reliable labels, but a much larger fraction have a some kind of noisy label. In other words, there is some kind of text associated with the image, from the image name or surrounding HTML text that gives some cue as to what’s in the image but is not that accurate. And of course, we also have a large amount of data with no labels at all. So we’d like a framework that can make use of all these types of labels.

4. Semi-Supervised Learning
Classification function should be smooth with respect to data density
Data
Supervised
Semi-Supervised
And one such technique is semi-supervised learning. Consider a toy dataset with just two labels

5. Semi-Supervised Learning using Graph Laplacian
[Zhu03,Zhou04]
is n x n affinity matrix (n = # of points)
We consider approaches of this type that are based on the Graph Laplacian. Here each image is a vertex in a graph and the weight of the edge between the vertices is given by an affinity defined as follows. So for n points we have an n by n affinity matrix W, from which we can compute the normalized graph Laplacian L, like so.
Graph Laplacian:

6. SSL using Graph Laplacian
Want to find label function f that minimizes:
y = labels
If labeled, , otherwise
Smoothness
Agreement with labels
In SSL, we solve for a label function f over the data points. The graph Laplacian measures the smoothness of the label function f while the second term constrains f to agree with the labels, using a weighting lambda according the reliability of the label. We can rewrite these lambda in matrix form like so and then the optimal f is given by the solution to a this n by n linear system.
Solution:
n x n system
(n = # points)

7. Eigenvectors of Laplacian
Smooth vectors will be linear combinations of eigenvectors U with small eigenvalues:
[Belkin & Niyogi 06,
Schoelkopf & Smola 02,
Zhu et al 03, 08]
Instead of directly solving the n by n linear system, we can instead model f as a linear combination of the smallest few eigenvectors of the Laplacian. So U are the eigenvectors and \alpha are the coefficients. These smallest eigenvectors are smooth with respect to the data density. The smallest is just a DC term, but the 2nd smallest splits the data horizontally and the 3rd splits it vertically.

8. Rewrite System Let U = smallest k eigenvectors of L = coeffs.
k is user parameter (typically ~100)
Optimal is now solution to k x k system:
So if we use the k smallest eigenvectors as a basis, k being a value we select, typically being 100 or so, then we can rewrite the linear system we had previously as follows. This now means that instead of solving an n by n system, we just need to solve a k by k system for the coefficients alpha, from which we can compute the label function f.

9. Computational Bottleneck
Consider a dataset of 80 million images
Inverting L
Inverting 80 million x 80 million matrix
Finding eigenvectors of L
Diagonalizing 80 million x 80 million matrix
But if we want to scale to really large datasets, then we have a problem. Consider a dataset of 80 million images. If we directly solve the original linear system, then we need to invert an 80 million by 80 million matrix. If we use the eigenvector basis, then finding the eigenvectors themselves is going to require diagonalizing this 80 million by 80 million matrix.

10. Large Scale SSL - Related work
Nystrom method: pick small set of landmark points
Compute exact eigenvectors on these
Interpolate solution to rest
Other approaches include:
[see Zhu ‘08 survey]
Data
Landmarks
A number of techniques exist for doing large scale semi-supervised learning. Many of these are similar to the Nystrom method which computes the exact eigenvectors on a small set of landmarks and then interpolates the remaining points to give an approximate solution. A variety of other approaches exist that adaptively group the datapoints and compute eigenvectors on these groupings.
Mixture models (Zhu and Lafferty ‘05), Sparse Grids (Garcke and Griebel ‘05), Sparse Graphs (Tsang and Kwok ‘06)

11. Our Approach
So our approach takes a different route.

12. Overview of Our Approach
Compute approximate eigenvectors
Density
Data
Landmarks
Ours
Nystrom
In Nystrom we reduce the number of data points down to set a set of landmarks. By contrast in our approach we consider the limit as the number of points goes to infinity and we have a continuous density. A key points is that our approach is linear in the # of examples.
Limit as n  ∞
Reduce n
Linear in number of data-points
Polynomial in number of landmarks

13. Consider Limit as n  ∞
Consider x to be drawn from 2D distribution p(x)
Let Lp(F) be a smoothness operator on p(x), for a function F(x)
Smoothness operator penalizes functions that vary in areas of high density
Analyze eigenfunctions of Lp(F) 2
where
Let’s consider a toy 2D dataset. As n goes to infinity, then we have 2D density. We define an operator that measures smoothness of a continuous label function capital F. Notice that is a continuous analogue of the graph Laplacian. Nearby locations x1 and x2 which have high affinity will similar values in F, if F is smooth. We now analyze the eigenfunctions of this operator. Lp of F.

14. Eigenvectors & Eigenfunctions
To get an intuition as to what these eigenfunction looks like we show them in the bottom row. Notice that they are continuous functions that capture the same structure as the discrete eigenvectors.

15. Key Assumption: Separability of Input data
p(x1)
Claim:
If p is separable, then:
Eigenfunctions of marginals are also eigenfunctions of the joint density, with same eigenvalue
A key assumption we make in our method is that the input distribution is separable. So for our toy 2D data, we assume that the joint density is modeled as a product of the two marginal distributions px1 and px2. And you can show that eigenfunctions of these marginals are eigenfunctions of the joint density.
p(x2)
[Nadler et al. 06,Weiss et al. 08]
p(x1,x2)

16. Numerical Approximations to Eigenfunctions in 1D
300,000 points drawn from distribution p(x)
Consider p(x1)
p(x1)
So let’s look at how we compute the eigenfunctions for one of these marginal distributions. Given a large set of observed data, which we assume is drawn from the density, we can form a histogram hx1 which will be an approximation to the true marginal.
Histogram h(x1)
p(x)
Data

17. Numerical Approximations to Eigenfunctions in 1D
Solve for values of eigenfunction at set of discrete locations (histogram bin centers)
and associated eigenvalues
B x B system (B = # histogram bins, e.g. 50)
Then we can solve for the eigenfunctions of the 1D distribution using the histogram. We solve for the values of the eigenfunction g and their associated eigenvalues at the locations of the bin centers using the following equation. The size of this system is given by the # of histogram bins which is small, e.g. 50 or so.

18. 1D Approximate Eigenfunctions
So if we solve this system, then what do the eigenfunctions look like? Well, here are the smallest 3 eigenfunctions of the marginal of the 1st dimension. You can see that the first one is fairly constant in regions of high density and then changes rapidly in the middle. The second has an extra kink and the 3rd has two kinks.
1st Eigenfunction
of h(x1)
2nd Eigenfunction
of h(x1)
3rd Eigenfunction
of h(x1)

19. Separability over Dimension
Build histogram over dimension 2: h(x2)
Now solve for eigenfunctions of h(x2)
And we do the same thing for the 2nd dimension. Here the marginal is a similar to a gaussian and these are the eigenfunctions.
1st Eigenfunction
of h(x2)
2nd Eigenfunction
of h(x2)
3rd Eigenfunction
of h(x2)

20. From Eigenfunctions to Approximate Eigenvectors
Take each data point
Do 1-D interpolation in each eigenfunction
Very fast operation
Eigenfunction value
Having obtained the eigenfunctions, how do we compute the approximate eigenvectors? For each data point we just do a 1D interpolation in the eigenfunctions to find the eigenvector value. This is a very quick operation. 1 50
Histogram bin

21. Preprocessing Need to make data separable Rotate using PCA PCA
One important pre-processing operation is that we rotate the data to make it more separable, as assumed by our algorithm. Currently we do this we PCA, although other options are possible.
Not separable
Separable

22. Overall Algorithm
Rotate data to maximize separability (currently use PCA)
For each of the d input dimensions:
Construct 1D histogram
Solve numerically for eigenfunctions/values
Order eigenfunctions from all dimensions by increasing eigenvalue & take first k
Interpolate data into k eigenfunctions
Yields approximate eigenvectors of Laplacian
Solve k x k least squares system to give label function

23. Experiments on Toy Data

24. Nystrom Comparison
With Nystrom, too few landmark points result in highly unstable eigenvectors
Here we compare Nystrom to our eigenfunction approach. When only a few landmarks are used, Nystrom fails to capture the structure of the data. But our eigenfunction approach uses all the datapoints and we get the correct solution.

25. Nystrom Comparison
Eigenfunctions fail when data has significant dependencies between dimensions
But if the input data has significant dependencies between dimensions, like in this example, then our eigenfunction approach fails. Here Nystrom, despite the small set of landmarks, gets the correct solution.

26. Experiments on Real Data

27. Experiments
Images from 126 classes downloaded from Internet search engines, total 63,000 images
Dump truck Emu
Labels (correct/incorrect) provided by Alex Krizhevsky, Vinod Nair & Geoff Hinton, (CIFAR & U. Toronto)
In our first set of experiments we use 63,000 images from a set of 126 classes downloaded from image search engines. Human labels for these images have been gathered by Alex Krizhevsky and Geoff Hinton.

28. Input Image Representation
Pixels not a convenient representation
Use Gist descriptor (Oliva & Torralba, 2001)
L2 distance btw. Gist vectors rough substitute for human perceptual distance
We represent each image with a single global descriptor. We use the Gist descriptor of Oliva & Torralba which captures the energy of gabor filters at different scales & orientations over the image. We use PCA to project it down into 64 d.
Apply oriented Gabor filters
over different scales
Average filter energy
in each bin

29. Are Dimensions Independent?
Joint histogram for pairs of dimensions from raw 384-dimensional Gist
Joint histogram for pairs of dimensions after PCA to 64 dimensions
PCA
MI is mutual information score. 0 = Independent

30. Real 1-D Eigenfunctions of PCA’d Gist descriptors
Input Dimension

31. Protocol Task is to re-rank images of each class (class/non-class)
Use eigenfunctions computed on all 63,000 images
Vary number of labeled examples
Measure precision @ 15% recall

32. Total number of images
4800
5000
8000
6000

33. Total number of images
4800
5000
8000
6000

34. Total number of images
4800
5000
8000
6000

35. Total number of images
4800
5000
8000
6000

36. 80 Million Images

37. Running on 80 million images
PCA to 32 dims, k=48 eigenfunctions
For each class, labels propagating through 80 million images
Precompute approximate eigenvectors (~20Gb)
Label propagation is fast <0.1secs/keyword

38. Japanese Spaniel
3 positive
3 negative
Labels from CIFAR set

39. Airbus, Ostrich, Auto

40. Summary
Semi-supervised scheme that can scale to really large problems – linear in # points
Rather than sub-sampling the data, we take the limit of infinite unlabeled data
Assumes input data distribution is separable
Can propagate labels in graph with 80 million nodes in fractions of second
Related paper in this NIPS by Nadler, Srebro & Zhou
See spotlights on Wednesday

42. Future Work Can potentially use 2D or 3D histograms instead of 1D
Requires more data
Consider diagonal eigenfunctions
Sharing of labels between classes

43. Comparison of Approaches

44. Exact
Eigenvectors :
Exact -- Approximate
Eigenvalues
Approximate : : : :
Data
So for this toy dataset on the left, we can compute the exact eigenvectors of the graph Laplacian shown in the center, and also the approximate eigenvectors using our eigenfunction approach. Note the similarity of the eigenvalues of the first 3 eigenvectors, but then the exact ones start mixing across dimension

45. Are Dimensions Independent?
Joint histogram for pairs of dimensions from raw 384-dimensional Gist
Joint histogram for pairs of dimensions after PCA
PCA
MI is mutual information score. 0 = Independent

46. Are Dimensions Independent?
Joint histogram for pairs of dimensions from raw 384-dimensional Gist
Joint histogram for pairs of dimensions after ICA
ICA
MI is mutual information score. 0 = Independent

47. Varying # Eigenfunctions

48. Leveraging Noisy Labels
Images in dataset have noisy labels
Keyword used in from Internet search engine
Can easily be incorporated into SSL scheme
Give weight 1/10th of hand-labeled example

49. Leveraging Noisy Labels

50. Effect of Noisy Labels

51. Complexity Comparison
Key: n = # data points (big, >106) l = # labeled points (small, <100)
m = # landmark points d = # input dims (~100)
k = # eigenvectors (~100) b = # histogram bins (~50)
Nystrom
Select m landmark points
Get smallest k eigenvectors of m x m system
Interpolate n points into k eigenvectors
Solve k x k linear system
Eigenfunction
Rotate n points
Form d 1-D histograms
Solve d linear systems, each b x b
k 1-D interpolations of n points
Solve k x k
linear system
Polynomial in # landmarks
Linear in # data points

52. Semi-Supervised Learning using Graph Laplacian
[Zhu03,Zhou04]
V = data points (n in total)
E = n x n affinity matrix W
We consider approaches of this type that are based on the Graph Laplacian. Here each image is a vertex in a graph and the weight of the edge between the vertices is given by an affinity defined as follows. So for n points we have an n by n affinity matrix W, from which we can compute the normalized graph Laplacian L, like so.
Graph Laplacian:

53. Rewrite System Let U = smallest k eigenvectors of L = coeffs.
k is user parameter (typically ~100)
Optimal is now solution to k x k system:
So if we use the k smallest eigenvectors as a basis, k being a value we select, typically being 100 or so, then we can rewrite the linear system we had previously as follows. This now means that instead of solving an n by n system, we just need to solve a k by k system for the coefficients alpha, from which we can compute the label function f.

54. Consider Limit as n  ∞
Consider x to be drawn from 2D distribution p(x)
Let Lp(F) be a smoothness operator on p(x), for a function F(x):
Analyze
eigenfunctions of Lp(F) 2
where
Let’s consider a toy 2D dataset. As n goes to infinity, then we have 2D density. We define an operator that measures smoothness of a continuous label function capital F. Notice that is a continuous analogue of the graph Laplacian. Nearby locations x1 and x2 which have high affinity will similar values in F, if F is smooth. We now analyze the eigenfunctions of this operator. Lp of F.

55. Numerical Approximations to Eigenfunctions in 1D
Solve for values g of eigenfunction at set of discrete locations (histogram bin centers)
and associated eigenvalues
B x B system (# histogram bins = 50)
P is diag(h(x1))
Then we can solve for the eigenfunctions of the 1D distribution using the histogram. We solve for the values of the eigenfunction g and their associated eigenvalues at the locations of the bin centers using the following equation. The size of this system is given by the # of histogram bins which is small, e.g. 50 or so.
Affinity between discrete locations

56. Real 1-D Eigenfunctions of PCA’d Gist descriptors
Eigenfunction value
Color = Input
dimension
xmin
xmax
Histogram bin 1 50
Input Dimension
Eigenfunction 256