machine learning techniques for computer vision

Part 2: Unsupervised Learning

Machine Learning Techniques for Computer Vision

Microsoft Research Cambridge

ECCV 2004, Prague

Christopher M. Bishop

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Machine Learning Techniques for Computer Vision (ECCV 2004) Christopher M. Bishop

Overview of Part 2• Mixture models• EM• Variational Inference• Bayesian model complexity• Continuous latent variables


The Gaussian Distribution• Multivariate Gaussian

• Maximum likelihood

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Gaussian Mixtures• Linear super-position of Gaussians

• Normalization and positivity require


Example: Mixture of 3 Gaussians

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Maximum Likelihood for the GMM• Log likelihood function

• Sum over components appears inside the log– no closed form ML solution


EM Algorithm – Informal Derivation


EM Algorithm – Informal Derivation• M step equations


EM Algorithm – Informal Derivation• E step equation


EM Algorithm – Informal Derivation• Can interpret the mixing coefficients as prior probabilities

• Corresponding posterior probabilities (responsibilities)


Old Faithful Data Set

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Time betweeneruptions (minutes)


Latent Variable View of EM• To sample from a Gaussian mixture:

– first pick one of the components with probability – then draw a sample from that component– repeat these two steps for each new data point

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Latent Variable View of EM• Goal: given a data set, find • Suppose we knew the colours

– maximum likelihood would involve fitting each component to the corresponding cluster

• Problem: the colours are latent (hidden) variables


Incomplete and Complete Data

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Latent Variable Viewpoint


Latent Variable Viewpoint • Binary latent variables describing which

component generated each data point • Conditional distribution of observed variable

• Prior distribution of latent variables

• Marginalizing over the latent variables we obtain

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Graphical Representation of GMM

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Latent Variable View of EM• Suppose we knew the values for the latent variables

– maximize the complete-data log likelihood

– trivial closed-form solution: fit each component to the corresponding set of data points

• We don’t know the values of the latent variables– however, for given parameter values we can compute

the expected values of the latent variables


Posterior Probabilities (colour coded)

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Over-fitting in Gaussian Mixture Models• Infinities in likelihood function when a component

‘collapses’ onto a data point:

with

• Also, maximum likelihood cannot determine the number K of components


Cross Validation• Can select model complexity using an independent

validation data set• If data is scarce use cross-validation:

– partition data into S subsets– train on S−1 subsets – test on remainder– repeat and average

• Disadvantages– computationally expensive – can only determine one or

two complexity parameters


Bayesian Mixture of Gaussians• Parameters and latent variables appear on equal footing• Conjugate priors

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Data Set Size• Problem 1: learn the function

for from 100 (slightly) noisy examples– data set is computationally small but statistically large

• Problem 2: learn to recognize 1,000 everyday objects from 5,000,000 natural images– data set is computationally large but statistically small

• Bayesian inference – computationally more demanding than ML or MAP

(but see discussion of Gaussian mixtures later)– significant benefit for statistically small data sets


Variational Inference• Exact Bayesian inference intractable• Markov chain Monte Carlo

– computationally expensive– issues of convergence

• Variational Inference– broadly applicable deterministic approximation– let denote all latent variables and parameters– approximate true posterior using a simpler

distribution – minimize Kullback-Leibler divergence


General View of Variational Inference• For arbitrary

where

• Maximizing over would give the true posterior– this is intractable by definition


Variational Lower Bound


Factorized Approximation• Goal: choose a family of q distributions which are:

– sufficiently flexible to give good approximation– sufficiently simple to remain tractable

• Here we consider factorized distributions

• No further assumptions are required!• Optimal solution for one factor, keeping the remainder fixed

– coupled solutions so initialize then cyclically update– message passing view (Winn and Bishop, 2004)


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Lower Bound• Can also be evaluated• Useful for maths/code verification• Also useful for model comparison:


Illustration: Univariate Gaussian• Likelihood function

• Conjugate prior • Factorized variational distribution


Initial Configuration

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After Updating

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After Updating

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Converged Solution

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Variational Mixture of Gaussians• Assume factorized posterior distribution

• No other approximations needed!


Variational Equations for GMM


Lower Bound for GMM


VIBES

Bishop, Spiegelhalter and Winn (2002)


ML Limit• If instead we choose

we recover the maximum likelihood EM algorithm


Bound vs. K for Old Faithful Data


Bayesian Model Complexity


Sparse Bayes for Gaussian Mixture• Corduneanu and Bishop (2001)• Start with large value of K

– treat mixing coefficients as parameters– maximize marginal likelihood– prunes out excess components


Summary: Variational Gaussian Mixtures• Simple modification of maximum likelihood EM code• Small computational overhead compared to EM• No singularities• Automatic model order selection


Continuous Latent Variables• Conventional PCA

– data covariance matrix

– eigenvector decomposition

• Minimizes sum-of-squares projection– not a probabilistic model– how should we choose L ?

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Probabilistic PCA• Tipping and Bishop (1998)• L dimensional continuous latent space

• D dimensional data space

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PCAfactor analysis


Probabilistic PCA• Marginal distribution

• Advantages– exact ML solution– computationally efficient EM algorithm– captures dominant correlations with few parameters– mixtures of PPCA– Bayesian PCA– building block for more complex models

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EM for PCA

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EM for PCA

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EM for PCA

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EM for PCA

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EM for PCA

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EM for PCA

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EM for PCA


Bayesian PCA• Bishop (1998)• Gaussian prior over columns of

• Automatic relevance determination (ARD)

ML PCA Bayesian PCA

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Non-linear Manifolds• Example: images of a rigid object

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Bayesian Mixture of BPCA Models

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Flexible Sprites• Jojic and Frey (2001)• Automatic decomposition of video sequence into

– background model– ordered set of masks (one per object per frame)– foreground model (one per object per frame)


Transformed Component Analysis• Generative model

• Now include transformations (translations)

• Extend to L layers• Inference intractable so

use variational framework

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Bayesian Constellation Model• Li, Fergus and Perona (2003)• Object recognition from small training sets• Variational treatment of fully Bayesian model


Bayesian Constellation Model


Summary of Part 2• Discrete and continuous latent variables

– EM algorithm• Build complex models from simple components

– represented graphically– incorporates prior knowledge

• Variational inference– Bayesian model comparison

machine learning techniques for computer vision

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