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Geometry in (Very) High Dimensions

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Geometry in (Very) High Dimensions Geometry in (Very) High Dimensions Renato Feres Washington University in St. Louis Department of Mathematics 1/20 The RGB color cube 2/20 The space of musical chords (Dmitry Tymoczko) 3-dimensional space of three-note chords, with major and minor triads found near the center. (An orbifold, much like a 3-dim Moebius band.) Tymoczko is composer and professor of music at Princeton University. 3/20 Primary faces The 6 basic emotions Based on research by Paul Ekman Other facial expressions of emotions are combinations of these 6, just as colors are combinations of R, B, G (a 6-dimensional space.) Illustration by Scott McCloud From Making Comics by McCloud (2006) 4/20 Combinations of primary faces 5/20 Thermodynamic space To specify the kinetic state of one mole of gas we need: 23 3 velocity components for each of 6.022 10 molecules. × So we need more than 18 1023 dimensions. × 6/20 Distance in Cartesian n-dimensional space Rn n Space R consists of all n-tuples of real numbers: x x1,...,xn . = ( ) Distance between x x1,...,xn and y y1,...,yn : = ( ) = ( ) 2 2 d x, y » y1 x1 yn xn ( ) = ( − ) +⋯+( − ) Euclid (c. 300 BC) Descartes (17th century) 7/20 Cubes and Balls in Cartesian n-space The n-cube C r of side length 2r: set of x such that r xi r. ( ) − ≤ ≤ It is also the Cartesian product: Cn r r, r r, r (n times.) ( ) = − × ⋯ × − [ ] [ ] 8/20 n-dimensional balls The pythagorean formula in dimension n is 2 2 x x1 xn distance from x to origin 0,..., 0 . = ¼ + ⋯ + = The n-ballS S of radius r is the set of x at distance r from( the) origin: ≤ Bn r x Rn x r . = ∈ ∶ ≤ ( ) { S S } 9/20 Testing our geometric intuition ... 10/20 Similar arrangement in dimension 3 11/20 What happens to the red ball as n goes to infinity? Now, do the same in arbitrary dimensions: Begin with a cube of dimension n and side length 1. n 1 Pack 2 balls inside, each of radius 4 . Add a (red) ball at the center that touches all the other balls. Let rn be the radius of the red ball. What happens to rn when n goes to infinity? 12/20 rn →∞ In dimension 2, 1 2 1 2 1 1 r2 2 1 . ¾ 4 4 4 4 = + − = √ − In dimension n, 1 2 1 2 1 1 rn n 1 = ¾ 4 + ⋯ + 4 − 4 = 4 − √ 13/20 Volumes The fundamental volume is that of the cube. We define it to be n n Vol C a 2 Vol(n-cube of side length a) a = = By filling the ball( ( with~ )) small cubes and taking a limit, n n n Vol B r Vol B 1 r . = Using multivariate calculus( and( )) induction( ( in)) the dimension, n n 2 Bn π ~ 1 2πe Vol 1 n . = Γ 2 1 nπ ¾ n ⎛ ⎞ ( ( )) + ∼ √ This is very small for big n. ⎝ ⎠ 14/20 Example: dimension 100 Volume of cube 1 = 40 Volume of inscribed ball 2.37 10− = × 15/20 How are volumes distributed? (1. Shells) Fraction of volume of n-ball contained in the outer shell of thickness a: n Vol Bn r Vol Bn r a rn r a n a − n − − n− 1 1 . Vol B r = r = − − r ( ( )) ( ( )) ( ) ( ( )) Therefore, for arbitrarily small a, as n → , ∞ Vol shell of thickness a 1. Vol Bn r → ( ) The volume inside a ball (for big( n())) is mostly near the surface. 16/20 How are volumes distributed? (2. Central slabs) Consider the n-dimensional ball of fixed volume equal to 1. It can be shown that: n radius rn ∼ ½2πe about 96% of the volume lies in the (relatively very thin) slab n 1 1 x ∈ B rn ≤ x1 ≤ . ∶ −2 2 ( ) This is true for the slab centered on any n 1 -dimensional subspace! − 17/20 ( ) Hyper-area of slices Associated to the n-dimensional cube of side 1, define: 1 e n 1,..., 1 , a vector of unit length; = √ Vn 1 t( hyper-area) of slice perpendicular to e above t. − = ( ) Hyper-area of slice above t (big n) a 1.2 0.8 0.4 0 t -1.5 -1 -0.5 0 0.5 1 1.5 Theorem (CLT) x1 ⋅⋅⋅+ xn 6 6t2 Vn 1 t Voln 1 x + t ∼ e− − = − ∶ n = ¾π ( ) W √ W ¡ 18/20 If we identify Probability ⇔ Volume ... Theorem (Central Limit Theorem) − 1 1 If X1,X2,... are random numbers (uniformly distributed) on 2 , 2 , x X1 + ⋯ + Xn 6 6t2 lim Prob x e− dt n n ≤ S x ¾π →∞ = − W √ W 600 Probability experiment: 500 Compute 5000 sample values of the random number 400 300 then plot the distribution of values of Y (histogram) 200 100 0 -1.5 -1 -0.5 0 0.5 1 1.5 19/20 Morals to draw from the story Concepts in dimension 2, 3 often generalize to n 3; > But peculiar things can happen for big n. Insight is regained by thinking probabilistically. Probability theory can be “interpreted” geometrically. Algebra, analysis, combinatorics, modern physical theories, music, and many other subjects use geometric ideas to represent concepts that may not seem geometric at first. This is reflected in the many flavors of modern geometry: Riemannian and pseudo-Riemannian geometries; algebraic geometry; symplectic and non-commutative geometry; information geometry, etc., etc., etc. 20/20.
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