Tracking as Dimensionality Reduction

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Tracking as Dimensionality Reduction

• Ali Rahimi (Intel Research Seattle)
• Ben Recht (Caltech)
• Brian Ferris (University of Washington)
• Dieter Fox (University of Washington)

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Tracking as Dimensionality Reduction
11 and smooth
Want these
Given only this
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Ferris, Fox, Lawrence, IJCI 06
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Taylor, Rahimi, Bachrach, Shrobe, IPSN 06
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Rahimi, Recht, NIPS 06
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Rahimi, Recht, CVPR 05
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WiFi localization
RFID tags
Sensor networks
RF tracking devices
Video
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Lessons

• Utilize time information
• Avoid local minima
• Use plausible models

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Lesson 1 Use Time
Low-dimensional latent variables
Sensor
High-dimensional Observations
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Lesson 1 Use Time

• For smooth time series data
• ST-Isomap (Jenkins) gt Isomap
• DGPLVM (WangFleetHertzman) gt GPLVM
• RahimiRecht gt KPCA

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The Sensetable
Signal strength measurements from tag
RFID tag
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Sensetable Manifold Learning
LLE
KPCA
Ground Truth
Isomap
Ground Truth
ST-Isomap
RahimiRecht 06
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Lesson 2 Avoid Local Minima

• What good is a local max of the posterior?

(It may be arbitrarily improbable)
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Lesson 2 Avoid Local Minima
Wi-Fi localization ground truth
Recovered by DGPLVM with PPCA initializer
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Lesson 2 Avoid Local Minima
Wi-Fi localization ground truth
Recovered by DGPLVM with Isomap initializer
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Lesson 2 Avoid Local Minima
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Lesson 3 Manifold Learning is a Red Herring for
Tracking

• Pros No local mins, fast.
• Cons
• Inapplicable assumptions Isometry, LLE
• Only asymptotic guarantees (not MAP estimate, for
  example)

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An Algorithm

• Dynamical model
• MAP estimate of f
• No local mins
• Approximation guarantees

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Why Its Hard
Hairy hairy integral
This integral is hard in general
EM cant even get us a local max of this!
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Steal Some Good Ideas
Manifold Learning f is 11
Nonlinear System ID (EM) prior on X
Y is obtained via change of variable on X
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Two assumptions to simplify marginalization
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Final cost function
Outputs should be likely
Mapping cant collapse to zero
Represent g with kernels on observations
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APPROXIMATE OPTIMUM 1 Using dual relaxation
Not convex in C. Convex in ZCC. Solve by
SDP. Extract solution by SVD. Gives bounds on
solution.
APPROXIMATE OPTIMUM 2 Using spectral relaxation
Find g that gives most likely X while matching
moments of prior. Results in dynamic-augmented
KPCA. Exact in the linear case.
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The Sensetable
Signal strength measurements from tag
RFID tag
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Sensetable Manifold Learning
LLE
KPCA
Ground Truth
Isomap
Ground Truth
ST-Isomap
RahimiRecht 06
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Applying g
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Tracking with KPCA
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Unsupervised Learning in Sensor Networks
Unknown sensor locations Unknown smooth
measurement model Unknown smooth path
Output as a funtcion of distance for various
sensors.
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target position
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True trajectory
Recovered Up to scale and 90 rotation
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Unknown measurement process
Measurements
g
Unknown sensor locations Unknown smooth
measurement model Unknown smooth path
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Conclusion

• Lessons for learning trackers
• Utilize time
• Avoid local mins
• Plausible models
• Requests
• Quantify errors
• Online updates

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Request Online Updates

• f changes over time.

RF Signal strengths After a chair moves
voltage
time
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Request Quantify errors

• Needed for
• Algorithm development
• Parameter tuning

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Visualization vs. Tracking
From the Isomap page
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Request Quantify errors

• Dont penalize errors in scale, translation,
  rotation.
• Do penalize incorrect choice of subspace,
  folding, changes in topology, nonlinear
  stretching, etc
• We use affine registration error

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LEARNING TO TRACK in UNCALIBRATED SENSOR
NETWORKS Mapping scalar sensor outputs to
location
Trajectory through sensor net
Measurements from all nodes
Recovered trajectory
Test trajectory
Applying g
Applying g from KPCA
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LEARNING TO TRACK in VIDEOS Mapping appearance
to pose
True rotation
Recovered rotation
Unsup w/ dynamics
Observed image sequence