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PatternTheory - Pattern Theory the Mathematics of...

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Pattern Theory: the Mathematics of Perception Prof. David Mumford Division of Applied Mathematics Brown University International Congress of Mathematics Beijing, 2002
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Outline of talk I. Background: history, motivation, basic definitions II. A basic example – Hidden Markov Models and speech; and extensions III. The “natural degree of generality” – Markov Random Fields; and vision applications IV. Continuous models: image processing via PDE’s, self-similarity of images and random diffeomorphisms URL: www.dam.brown.edu/people/mumford/Papers /ICM02powerpoint.pdf or /ICM02proceedings.pdf
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Some History Is there a mathematical theory underlying intelligence? 40’s – Control theory (Wiener-Pontrjagin), the output side: driving a motor with noisy feedback in a noisy world to achieve a given state 70’s – ARPA speech recognition program 60’s-80’s – AI, esp. medical expert systems, modal, temporal, default and fuzzy logics and finally statistics 80’s-90’s – Computer vision, autonomous land vehicle
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Statistics vs. Logic Gauss – Gaussian distributions, least squares relocating lost Ceres from noisy incomplete data Control theory – the Kalman-Wiener-Bucy filter AI – Enhanced logics < Bayesian belief networks Vision – Boolean combinations of features < Markov random fields Plato : “If Theodorus, or any other geometer, were prepared to rely on plausibility when he was doing geometry, he'd be worth absolutely nothing.” Graunt – counting corpses in medieval London
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What you perceive is not what you hear : ACTUAL SOUND 1. The ?eel is on the shoe 2. The ?eel is on the car 3. The ?eel is on the table 4. The ?eel is on the orange PERCEIVED WORDS 1. The heel is on the shoe 2. The wheel is on the car 3. The meal is on the table 4. The peel is on the orange (Warren & Warren, 1970) Statistical inference is being used!
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Why is this old man recognizable from a cursory glance? His outline is lost in clutter, shadows and wrinkles; except for one ear, his face is invisible. No known algorithm will find him.
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Pr( | , ).Pr( | ) or Pr( , ) o h h h x x x x θ θ θ MODEL: observable variables, hidden (not directly observable) variables, parameters in model o h x x θ = = = VARIABLES: ( ) ( ) ˆ arg max Pr( | ) or ˆ arg max Pr( , | ) h o h x x x x α θ α α θ α θ θ ML PARAMETER ESTIMATION: The Bayesian Setup, I
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BAYES’S RULE: ˆ Pr( | , ).Pr( | ) ˆ Pr( | , ) ˆ Pr( | ) ˆ Pr( | , ).Pr( | ) o h h h o o o h h x x x x x x x x x θ θ θ θ θ θ = μ This is called the “ posterior ” distribution on x h Sampling Pr( x o ,x h | θ ), “synthesis” is the acid test of the model The central problem of Statistical learning theory: The complexity of the model and the Bias-Variance dilemma * Minimum Description Length—MDL, * Vapnik’s VC dimension The Bayesian Setup, II
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A basic example: HMM’s and speech recognition I. Setup s k S k+1 S k-1 x k x k+1 x k-1 s k = sound in window around time k t are the observables x k = part of phoneme being spoken at time k t are the hidden vars.
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