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Three Algorithms for Nonlinear Dimensionality Reduction

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Outline

- Problem
- PCA (Principal Component Analysis)
- MDS (Multidimentional Scaling)
- Isomap (isometric mapping)
- A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science, 292(22), 2319-2323, 2000.
- LLE (locally linear embedding)
- Nonlinear Dimensionality Reduction by Locally Linear Embedding. Science, 292(22), 2323-2326, 2000.
- Eigenmap
- Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering. NIPS01.

Problem

- Given a set x1, …, xk of k points in Rl, find a set of

points y1, …, yk in Rm(m << l) such that yi “represents” xi as accurately as possible.

- If the data xi is placed in a super plane in high dimensional space, the traditional algorithms, such as PCA and MDS, work well.
- However, when the data xi is placed in a nonlinear manifold in high dimensional space, then the linear algebra technique can not work any more.
- A nonlinear manifold can be roughly understood as a distorted super plane, which may be twisted, folded, or curved.

PCA (Principal Component Analysis)

- Reduce dimensionality of data by transformingcorrelated variables (bands) into a smaller number of uncorrelated components
- Reveals meaningful latent information
- Best preserves the variance as measured in the high-dimensional input space.
- Nonlinear structure is invisible to PCA

Rotate axes to create two orthogonal (uncorrelated) components

PC1

Band 2

PC2

“Reflected”

X- and y-axes

Band 1

PCA: algorithm description

- Step 1: Calculate the average x of xi .
- Step 2: Estimate the Covariance Matrix by
- Step 3: Letλp be the p-th eigenvalue (in decreasing order) of the matrixM, and vpi be the i-th component of the p-th eignvector. Then set the p-th componet of the d-dimentional coordinate vector yiequal to

MDS

- Step 1: Given the distance d(i, j) between i and j.
- Step 2: From d(i, j), get the covariance matrix M by

- Step3: The same as PCA

An example of embedding of a two dimentional manifold into a three dimentional space

Not the true distance

The true distance

Isomap: basic idea

- Learn the global distance by the local distance.
- The local distance calculated by the Euclidean distance is relatively accurate because a patch in the nonlinear manifold looks like a plane when it is small, and therefore the direct Euclidean distance approximates the true distance in this small patch.
- The global distance calculated by the Euclidean distance is not accurate because the manifold is curved.
- Best preserve the estimated distance in the embedded space in the same way as MDS.

Isomap: algorithm description

Step 1: Construct neighborhood graph

Define the graph over all data points by connecting points i and j if they are closer than ε (ε-Isomap), or if i is one of the n nearest neighbors of j (k-Isomap). Set edge lengths equal to dX(i,j).

Step 2: Compute shortest paths

Initialize dG(i,j)= dX(i,j) if i and j are linked by an edge; dG(i,j)= ∞

otherwise. Then compute the shortest path distances dG(i,j) between all

pairs of points in weighted graph G. LetDG=( dG(i,j) ).

Step 3: Construct d-dimensional embedding

Letλp be the p-th eigenvalue (in decreasing order) of the matrixτ(DG), and vpi be the i-th component of the p-th eignvector. Then set the p-th componet of the d-dimentional coordinate vector yiequal to .

An example: each picture, a 4096 (64*64)-dimensional point, can be mapped into 2-dinesional plane

Another example: the 3-dimentional points are maped into 2-dimentional plane

LLE: basic idea

- Learn the local linear relation by the local data
- The local data is relatively linear because a patch in the nonlinear manifold looks like a plane when it is small.
- Globally the data is not linear because the manifold is curved.
- Best preserve the local linear relation in the embedded space in the similar way as PCA.

LLE: algorithm description

Step 1: Discovering the Adjacency Information

For each xi find its n nearest neighbors, .

Step 2: Constrcting the Approximation Matrix

Choose Wij by minimizing

Under the condition that

Step 3: Compute the Embedding

The embedding vectors yi can be found by minimizing

An example: 4096-dimentional face pictures are embedded into a 2-dimentional plane

Eigenmap: Basic Idea

- Use the local information to decide the embedded data.
- Motivated by the way that heat transmits from one point to another point.

Eigenmap

Step 1: Construct neighborhood graph

The same as Isomap.

Step 2: Compute the weights of the graph

If node i and node j are connected, put

Step 3: Construct d-dimensional embedding

Compute the eigenvalues and eigenvectors for the generalized eigenvector problem: , where D is a diagonal matrix, and

Cont.

Let f0,…,fk-1 be the solutions of the above equation,

ordered increasingly according to their eignvalues,

Lf0=λ0Df0

Lf1=λ1Df1

…

Lfk-1=λk-1Dfk-1

Then yi is determined by the ith component of the d

eigenvectors f1,…,fd .

An example: 256-dimentional speech data is represented in a 2-dimentional plane

Conclusion

- Isomap, LLE and Eigenmap can find the meaningful low-dimensional structure hidden in the high-dimensional observation.
- These three algorithms work well especially in the nonlinear manifold. In such a case, the linear methods such as PCA and MDS can not work.

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