Manifold learning with application to object recognition
Transcript of Manifold learning with application to object recognition
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Advanced Perception David R. Thompson
manifold learning with applications to object recognition
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1. why learn manifolds?
2. Isomap
3. LLE
4. applications
agenda
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types of manifolds
exhaust manifold
low-D surface embedded in high-D space
Sir Walter Synnot Manifold
1849-1928
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Find a low-D basis for describing high-D data. X → X' S.T. dim(X') << dim(X)
uncovers the intrinsic dimensionality (invertible)
manifold learning
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plenoptic function / motion / occlusion
manifolds in vision
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appearance variation
manifolds in vision
images from hormel corp.
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deformation
manifolds in vision
images from www.golfswingphotos.com
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1. data compression
2. “curse of dimensionality”
3. de-noising
4. visualization
5. reasonable distance metrics *
why do manifold learning?
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reasonable distance metrics
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reasonable distance metrics
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reasonable distance metrics
?
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reasonable distance metrics
?
linear interpolation
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reasonable distance metrics
?
manifold interpolation
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1. why learn manifolds?
2. Isomap
3. LLE
4. applications
agenda
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Isomap For n data points, and a distance matrix D,
Dij =
...we can construct a m-dimensional space to preserve inter-point distances by using the top eigenvectors of D scaled by their eigenvalues.
yi= [ λ
1v
1i , λ
2v
2i , ... , λ
mv
mi ]
j
i
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Infer a distance matrix using distances along the manifold.
Isomap
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Isomap 1. Build a sparse graph with K-nearest neighbors
Dg =
(distance matrix issparse)
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Isomap 2. Infer other interpoint distances by finding shortest paths on the graph (Dijkstra's algorithm).
Dg =
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Isomap 3. Build a low-D embedded space to best preserve the complete distance matrix.
Error function:
Solution – set points Y to top eigenvectors of Dg
L2 norm
inner product distances in new coordinate system
inner product distances in graph
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Isomap shortest-distance on a graph is easy to compute
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Isomap results: hands
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- preserves global structure
- few free parameters
- sensitive to noise, noise edges
- computationally expensive (dense matrix eigen-reduction)
Isomap: pro and con
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Find a mapping to preserve local linear relationships between neighbors
Locally Linear Embedding
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Locally Linear Embedding
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1. Find weight matrix W of linear coefficients:
Enforce sum-to-one constraint with the Lagrange Multiplier:
LLE: Two key steps
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2. Find projected vectors Y to minimize reconstruction error
must solve for whole dataset simultaneously
LLE: Two key steps
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We add constraints to prevent multiple / degenerate solutions:
LLE: Two key steps
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cost function becomes:
the optimal embedded coordinates are given by bottom m+1 eigenvectors of the matrix M
LLE: Two key steps
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LLE: Result preserves local topology
PCA
LLE
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- no local minima, one free parameter - incremental & fast
- simple linear algebra operations
- can distort global structure
LLE: pro and con
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● Laplacian Eigenmaps (Belkin 2001)● spectral method similar to LLE● better preserves clusters in data
● Kernel PCA
● Kohonen Self-Organizing Map (Kohonen, 1990)
● iterative algorithm fits a network of pre-defined connectivity
● simple, fast for on-line learning● local minima● lacking theoretical justification
Others you may encounter
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the “curvier” your manifold, the denser your data must be
No Free Lunch
bad OK!
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Manifold learning is a key tool in your object recognition toolbox
A formal framework for many different ad-hoc object recognition techniques
conclusions