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Learning from ShadowsLearning from Shadows

Dimensionality Reduction and its Dimensionality Reduction and its Application in Artificial Intelligence, Application in Artificial Intelligence,

Signal Processing and RoboticsSignal Processing and Robotics

Ali GhodsiDepartment of Statistics and Actuarial

Science

University of WaterlooOctober 2006

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Dimensionality ReductionDimensionality Reduction

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Dimensionality ReductionDimensionality Reduction

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Manifold and Hidden VariablesManifold and Hidden Variables

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Data RepresentationData Representation

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Data RepresentationData Representation

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11 11 11 11 11

11 00 11 00 11

11 11 11 11 11

11 0.50.5 0.50.5 0.50.5 11

11 11 11 11 11

Data RepresentationData Representation

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644 by 103

644 by 2

2 by 103

23 by 28 23 by 28

-2.19

-0.02

-3.19

1.02

2 by 12 by 1

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16Hastie, Tibshirani, Friedman 2001

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The Big PictureThe Big Picture

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Uses of Dimensionality Uses of Dimensionality ReductionReduction

(Manifold Learning)(Manifold Learning)

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DenoisingDenoising

Mika et. al. 1999

Zhu and Ghodsi 2005

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Tenenbaum, V de Silva, Langford 2001

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Roweis and. Saul 2000

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Arranging words: Each word was initially represented by a high-dimensional vector that counted the number of times it appeared in different encyclopedia articles. Words with similar contexts are collocated

Roweis and Saul 2000

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Hinton and Roweis 2002

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Embedding of Sparse Music Embedding of Sparse Music Similarity GraphSimilarity Graph

Platt, 2004

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Pattern Recognition Pattern Recognition

Ghodsi, Huang, Schuurmans 2004

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Pattern RecognitionPattern Recognition

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ClusteringClustering

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Glasses vs. No GlassesGlasses vs. No Glasses

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Beard vs. No BeardBeard vs. No Beard

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Beard DistinctionBeard Distinction

Ghodsi , Wilkinson, Southey 2006

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Glasses DistinctionGlasses Distinction

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Multiple-Attribute MetricMultiple-Attribute Metric

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Reinforcement LearningReinforcement Learning

Mahadevan and Maggioini, 2005

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Semi-supervised LearningSemi-supervised Learning

Use graph-based discretization of manifold to infer missing labels.

Build classifiers from bottom eigenvectors of graph Laplacian.

Belkin & Niyogi, 2004; Zien et al, Eds., 2005

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Learning CorrespondencesLearning Correspondences

How can we learn manifold structure that is shared across multiple data sets?

Ham et al, 2003, 2005

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Mapping and Robot LocalizationMapping and Robot Localization

Bowling, Ghodsi, Wilkinson 2005

Ham, Lin, D.D. 2005

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Action Respecting Action Respecting EmbeddingEmbedding

Joint Work with

Michael Bowlingand

Dana Wilkinson

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Modelling Temporal Data and Modelling Temporal Data and ActionsActions

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OutlineOutline

• Background– PCA– Kernel PCA

• Action Respecting Embedding (ARE)– Prediction and Planning– Probabilistic Actions

• Future Work

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Principal Component Analysis Principal Component Analysis (PCA)(PCA)

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Principal Component Analysis Principal Component Analysis (PCA)(PCA)

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Kernel MethodsKernel Methods

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Kernel TrickKernel Trick

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Observed, Feature and Embedded Observed, Feature and Embedded SpacesSpaces

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Kernel PCA Kernel PCA

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ProblemProblem

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IdeaIdea

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Action Respecting Embedding Action Respecting Embedding (ARE)(ARE)

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Action Respecting ConstraintAction Respecting Constraint

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Preserve distances between each point and its k nearest neighbors.

Local Distances ConstraintLocal Distances Constraint

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Preserve local distances

Local Distances ConstraintLocal Distances Constraint

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Semidefinite ProgrammingSemidefinite Programming

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ExperimentExperiment

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Experiment 1Experiment 1

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Experiment 2Experiment 2

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Experiment 3Experiment 3

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Experiment 4Experiment 4

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Experiment 5Experiment 5

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PlanningPlanning

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PlanningPlanning

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PlanningPlanning

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ExperimentExperiment

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Probabilistic Actions

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Future workFuture work

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