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Statistical perturbation theory for spectral clustering Harrachov, 2007 A. Spence and Z. Stoyanov

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Plan of the Talk A. Clustering (Brief overview). B. Deterministic Perturbation Theory. C. Statistical Perturbation Theory.

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Graph Clustering 3 4 1 2 6 7 5

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3 4 1 2 6 7 5

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Graph Clustering + Perturbation 3 4 1 2 6 7 5 ?

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Gene Expression Data Clustering An Application There are over 10 000 genes expressed in any one tissue; DNA arrays typically produce very noisy data. 1.Genes in same cluster behave similarly? 2. Genes in different clusters behave differently? 1.Genes in same cluster behave similarly? 2. Genes in different clusters behave differently? Issues:

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Bi-partite Graphs 1 2 3 4 1 2 3

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Matrix Form

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A Real Data Matrix (Leukemia)

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Spectral Clustering: General Idea Discrete Optimisation Problem (NP - Hard) Discrete Optimisation Problem (NP - Hard) Real Optimisation Problem (Tractable) Real Optimisation Problem (Tractable) Approximation Exact - Impractical Heuristic - Practical

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Discrete Optimisation SVD Active Inactive Active Solution: Singular Value Decomposition of W scaled

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Clustering Algorithm: Summary ACTIVE INACTIVE

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Literature

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Types of Graph Matrices

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How we Cluster

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Leukemia Data

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Clustered Leukemia Data

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Inaccuracies in the Data (Perturbation Theory)

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Perturbation Theory (Deterministic Noise)

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Deterministic Perturbation (Symmetric Matrix)

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Linear Solve

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Taylor Expansions

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Rectangular Case Symmetric

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Random Perturbations (plan) The Model Issues with the Theory A Possible Solution via Simulations? Experiments

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The Model 3 4 1 2 6 7 5

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Difficulties with Random Matrix Theory (RMT)

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Deterministic Perturbation Stochastic Perturbation (simple eigenvector)

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Deterministic Perturbation Stochastic Perturbation (simple eigenvalues)

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PP Plot -Test for Normality (Largest eigenvalue of a Symmetric Matrix)

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Simulated Random Perturbation (Largest eigenvalue of a Symmetric Matrix)

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Deterministic Perturbation Stochastic Perturbation (simple eigenvectors)

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Results for Laplacian Matrices

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Functional of the Eigenvector

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Results for h T v 2

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PP Plot of h T v’(0) - Test for Normality (h = e j )

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Histogram of h T v’(0) - Simulations (h = e j )

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PP Plot of Simulated v [j] ( ) (Distribution close to Normal)

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Histogram of Simulated v [j] ( ) (Distribution close to Normal)

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Extension to the Rectangular Case

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Probability of “Wrong Clustering”

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Issues with Numerics

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Efficient Simulations

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Solution via Simulations?

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Solution via Simulations? (Algorithm)

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Comparing: Direct Calculation Vs. Repeated Linear Solve

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