Divergence for log concave functions
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1 Divergence or log concave unctions Umut Caglar The Euler International Mathematical Institute June 22nd, 2013 Joint work with C. Schütt and E. Werner
2 Outline 1 Introduction 2 Main Theorem 3 -divergence 4 -divergences or convex bodies 5 -divergences or s-concave unctions
3 Introduction K is a convex body in R n
4 Introduction K is a convex body in R n Aine surace area as 1 (K = K κ(x 1 n+1 dµ(x
5 Introduction K is a convex body in R n Aine surace area as 1 (K = K κ(x 1 n+1 dµ(x L p aine surace area as p (K = K κ n+p p n(p 1 x,n(x n+p (Blaschke, Leichtweiss, Lutwak, Meyer-Werner, Schütt-Werner... dµ
6 Introduction : R n [0, is log concave i = e ψ, ψ : R n R convex.
7 Introduction : R n [0, is log concave i = e ψ, ψ : R n R convex. Theorem 1 (Artstein-Klartag-Schütt-Werner, 12 Let : R n [0, be a log concave unction such that dx = 1. supp( ( ln det (Hess ( ln dx 2 [ Ent( Ent(g ] where g(x = (2π n 2 e x 2 2. Note that Ent( = ln dx and Ent(g = ln(2πe n 2.
8 Introduction Let : R n [0, be a log concave unction. The polar (Artstein-Klartag-Milman o is (x = in y e x,y (y
9 Introduction Let : R n [0, be a log concave unction. The polar (Artstein-Klartag-Milman o is (x = in y e x,y (y Functional Blaschke Santaló inequality (Ball; Artstein-Klartag-Milman ( ( dx dx (2π n
10 Main Theorem Theorem 2 (C.-Schütt-Werner, 13 Let : (0, R be a convex unction. Let : R n [0, be a log concave unction. (,x e 2 ( det (Hess ( ln dx dx ( dx dx
11 Main Theorem Theorem 2 (C.-Schütt-Werner, 13 Let : (0, R be a convex unction. Let : R n [0, be a log concave unction. (,x e 2 ( det (Hess ( ln dx dx ( dx dx I is concave, the inequality is reversed. I is linear, there is equality.
12 Main Theorem Theorem 2 (C.-Schütt-Werner, 13 Let : (0, R be a convex unction. Let : R n [0, be a log concave unction. (,x e 2 ( det (Hess ( ln dx dx ( dx dx I is concave, the inequality is reversed. I is linear, there is equality. I is strictly convex or concave, there is equality or (x = Ce Ax,x, where C > 0 and A is an n n positive-deinite matrix.
13 Main Theorem Theorem 2 = Theorem 1 (,x ( e det (Hess ( ln dx 2 dx ( dx dx
14 Main Theorem Theorem 2 = Theorem 1 (,x ( e det (Hess ( ln dx 2 dx ( dx dx Put (t = ln t in Theorem 2: [ ], x 2 ln + ln (det (Hess ( ln
15 Main Theorem Theorem 2 = Theorem 1 (,x ( e det (Hess ( ln dx 2 dx ( dx dx Put (t = ln t in Theorem 2: [ ], x 2 ln + ln (det (Hess ( ln = n dx 2 Ent( + ln (det (Hess ( ln
16 Main Theorem Theorem 2 = Theorem 1 (,x ( e det (Hess ( ln dx 2 dx ( dx dx Put (t = ln t in Theorem 2: [ ], x 2 ln + ln (det (Hess ( ln = n dx 2 Ent( + ln (det (Hess ( ln ( ln dx
17 Main Theorem Theorem 2 = Theorem 1 (,x ( e det (Hess ( ln dx 2 dx ( dx dx Put (t = ln t in Theorem 2: [ ], x 2 ln + ln (det (Hess ( ln = n dx 2 Ent( + ln (det (Hess ( ln ( ln dx ln (det (Hess ( ln 2 Ent( + ln (e n dx
18 Main Theorem Theorem 2 = Theorem 1 (,x ( e det (Hess ( ln dx 2 dx ( dx dx Put (t = ln t in Theorem 2: [ ], x 2 ln + ln (det (Hess ( ln = n dx 2 Ent( + ln (det (Hess ( ln ( ln dx ln (det (Hess ( ln 2 Ent( + ln (e n dx 2 Ent( + ln (2πe n
19 -divergence The expressions o Theorem 2
20 -divergence The expressions o Theorem 2 Deinition 3 (Csiszar, Morimoto, Ali-Silvery (X, µ is a measure space. P = pµ and Q = qµ are measures on X that are absolutely continuous with respect to the measure µ. : (0, R is a convex or a concave unction.
21 -divergence The expressions o Theorem 2 Deinition 3 (Csiszar, Morimoto, Ali-Silvery (X, µ is a measure space. P = pµ and Q = qµ are measures on X that are absolutely continuous with respect to the measure µ. : (0, R is a convex or a concave unction. The -divergence D (P, Q o P and Q is D (P, Q = X ( p qdµ q
22 -divergence Examples D (P, Q = X ( p qdµ q 1. (t = ln t gives the Kullback-Leibler divergence or relative entropy rom P to Q D KL (P Q = q ln p q dµ. X
23 -divergence Examples D (P, Q = X ( p qdµ q 1. (t = ln t gives the Kullback-Leibler divergence or relative entropy rom P to Q D KL (P Q = q ln p q dµ. 2. The convex or concave unctions (t = t α leads to the Rényi entropy X o order α D α (P Q = 1 ( α 1 ln p α q 1 α dµ X The case α = 1 is the relative entropy D KL (P Q.
24 -divergences or convex bodies (X, µ = ( K, µ K
25 -divergences or convex bodies (X, µ = ( K, µ K Q K = q K µ K, q K = x, N(x P K = p K µ K, p K = κ x,n(x n
26 -divergences or convex bodies (X, µ = ( K, µ K Q K = q K µ K, q K = x, N(x P K = p K µ K, p K = κ x,n(x n Q K and P K are the cone measures o K and K.
27 -divergences or convex bodies (X, µ = ( K, µ K Q K = q K µ K, q K = x, N(x P K = p K µ K, p K = κ x,n(x n Q K and P K are the cone measures o K and K. L p -aine surace areas are α-rényi entropy powers (Werner, 12 as p (K = e n n+p D n+p p (P K Q K with α = p n+p.
28 -divergences or convex bodies The -divergence or a convex body K in R n with respect to the (cone measures P K and Q K was deined by Werner D (P K, Q K = = K K ( ( pk q K q K dµ K κ(x x, N(x n+1 x, N(x dµ K
29 -divergences or s-concave unctions Deinition 4 Let s R. : R n [0, is s-concave i or all λ (0, 1, or all x, y ((1 λx + λy [(1 λ(x s + λ(y s ] 1 s
30 -divergences or s-concave unctions Deinition 4 Let s R. : R n [0, is s-concave i or all λ (0, 1, or all x, y ((1 λx + λy [(1 λ(x s + λ(y s ] 1 s relation to log concave unctions: Let be a log concave unction. Then or all s s = (1 + s ln 1 s + is s-concave and s as s 0
31 -divergences or s-concave unctions Deinition 5 (-divergences or s-concave unctions q (s ( = 1 s [ det p (s =, x Hess ( ln + s ( n+ 1 s,x 1 s 2 ] D (s (s (P, Q (s = det ( 2 1 s,x [ Hess ( ln + s ] 2 n+1+ 1 s ( 1 s, x dx
32 -divergences or s-concave unctions D (s (s (P, Q (s =. [ det ( 2 Hess ( ln + s 1 s,x ] 2 n+1+ 1 s ( 1 s, x dx
33 -divergences or s-concave unctions D (s (s (P, Q (s =. [ det ( 2 Hess ( ln + s 1 s,x ] 2 n+1+ 1 s ( 1 s, x dx s 0
34 -divergences or s-concave unctions D (s (s (P, Q (s =. [ det ( 2 Hess ( ln + s 1 s,x ] 2 n+1+ 1 s ( 1 s, x dx (e,x 2 det [Hess ( ln ] s 0 dx
35 -divergences or s-concave unctions D (s (s (P, Q (s =. [ det ( 2 Hess ( ln + s 1 s,x ] 2 n+1+ 1 s ( 1 s, x dx (e,x 2 det [Hess ( ln ] s 0 dx =: D (P, Q
36 -divergences or s-concave unctions D (s (s (P, Q (s =. [ det ( 2 Hess ( ln + s 1 s,x ] 2 n+1+ 1 s ( 1 s, x dx (e,x 2 det [Hess ( ln ] s 0 dx =: D (P, Q or log concave with the corresponding q = and p = 1 e,x det [Hess ( ln ].
37 -divergences or s-concave unctions Theorem 2 ( D (P, Q dx ( dx dx I is concave, the inequality is reversed. I is linear, there is equality.
38 -divergences or s-concave unctions Theorem 2 ( D (P, Q dx ( dx dx I is concave, the inequality is reversed. I is linear, there is equality. Theorem 1 Let dx = 1. ( D KL (P Q ln dx ln (2πe n
39 -divergences or s-concave unctions relation to convex bodies : R n [0, s-concave. s > 0 such that 1 s N. For such we consider the convex bodies introduced by Artstein-Klartag-Milman K s ( = { (x, y R n R 1 x s : s supp(, y s ( x } s
40 -divergences or s-concave unctions relation to convex bodies : R n [0, s-concave. s > 0 such that 1 s N. For such we consider the convex bodies introduced by Artstein-Klartag-Milman K s ( = { (x, y R n R 1 x s : s supp(, y s ( x } s Proposition 6 D (s (s (P, Q (s = D (P Ks(, Q Ks( s n 2 S 1 s 1
41 -divergences or s-concave unctions Thank you!
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