An introduction to chaining, and applications to sublinear algorithms

Transcription

1 An ntroducton to channg, and applcatons to sublnear algorthms Jelan Nelson Harvard August 28, 2015

2 What s ths talk about?

3 What s ths talk about? Gven a collecton of random varables X 1, X 2,...,, we would lke to say that max X s small wth hgh probablty. (Happens all over computer scence, e.g. Chernon (Chernoff+Unon) bound)

4 What s ths talk about? Gven a collecton of random varables X 1, X 2,...,, we would lke to say that max X s small wth hgh probablty. (Happens all over computer scence, e.g. Chernon (Chernoff+Unon) bound) Today s topc: Beatng the Unon Bound

5 What s ths talk about? Gven a collecton of random varables X 1, X 2,...,, we would lke to say that max X s small wth hgh probablty. (Happens all over computer scence, e.g. Chernon (Chernoff+Unon) bound) Today s topc: Beatng the Unon Bound Dsclamer: Ths s an educatonal talk, about deas whch aren t mne.

6 T B l n 2 A frst example

7 A frst example T B l n 2 Random varables (Z x ) Z x = g, x for a vector g wth..d. N (0, 1) entres

8 A frst example T B l n 2 Random varables (Z x ) Z x = g, x for a vector g wth..d. N (0, 1) entres Defne gaussan mean wdth g(t ) = E g sup Z x

9 A frst example T B l n 2 Random varables (Z x ) Z x = g, x for a vector g wth..d. N (0, 1) entres Defne gaussan mean wdth g(t ) = E g sup Z x How can we bound g(t )?

10 A frst example T B l n 2 Random varables (Z x ) Z x = g, x for a vector g wth..d. N (0, 1) entres Defne gaussan mean wdth g(t ) = E g sup Z x How can we bound g(t )? Ths talk: four progressvely tghter ways to bound g(t ), then applcatons of technques to some TCS problems

11 Gaussan mean wdth bound 1: unon bound g(t ) = E sup Z x = E sup g, x

12 Gaussan mean wdth bound 1: unon bound g(t ) = E sup Z x = E sup g, x Z x s a gaussan wth varance one

13 Gaussan mean wdth bound 1: unon bound g(t ) = E sup Z x = E sup g, x Z x s a gaussan wth varance one E sup Z x = = 0 u 0 P(sup Z x > u)du P(sup Z x > u) } {{ } 1 u + T e u2 /2 du + u log T (set u = 2 log T ) P(sup Z x > u) }{{} T e u2 /2 (unon bound) du

14 Gaussan mean wdth bound 1: unon bound g(t ) = E sup Z x = E sup g, x Z x s a gaussan wth varance one E sup Z x = = 0 u 0 P(sup Z x > u)du P(sup Z x > u) } {{ } 1 u + T e u2 /2 du + u log T (set u = 2 log T ) P(sup Z x > u) }{{} T e u2 /2 (unon bound) du

15 Gaussan mean wdth bound 1: unon bound g(t ) = E sup Z x = E sup g, x Z x s a gaussan wth varance one E sup Z x = = 0 u 0 P(sup Z x > u)du P(sup Z x > u) } {{ } 1 u + T e u2 /2 du + u log T (set u = 2 log T ) P(sup Z x > u) }{{} T e u2 /2 (unon bound) du

16 Gaussan mean wdth bound 2: ε-net g(t ) = E sup g, x Let S ε be ε-net of (T, l 2 )

17 Gaussan mean wdth bound 2: ε-net g(t ) = E sup g, x Let S ε be ε-net of (T, l 2 ) g, x = g, x + g, x x (x = argmn y T x y 2 ) g(t ) g(s ε ) + E g sup g, x x }{{} ε g 2

18 Gaussan mean wdth bound 2: ε-net g(t ) = E sup g, x Let S ε be ε-net of (T, l 2 ) g, x = g, x + g, x x (x = argmn y T x y 2 ) g(t ) g(s ε ) + E g sup g, x x }{{} ε g 2 log S ε + ε(e g g 2 2 )1/2 log 1/2 N (T, l 2, ε) +ε n }{{} smallest ε net sze

19 Gaussan mean wdth bound 2: ε-net g(t ) = E sup g, x Let S ε be ε-net of (T, l 2 ) g, x = g, x + g, x x (x = argmn y T x y 2 ) g(t ) g(s ε ) + E g sup g, x x }{{} ε g 2 log S ε + ε(e g g 2 2 )1/2 log 1/2 N (T, l 2, ε) }{{} smallest ε net sze +ε n Choose ε to optmze bound; can never be worse than last slde (whch amounts to choosng ε = 0)

20 Gaussan mean wdth bound 3: ε-net sequence S k s a ( 1 /2 k )-net of T, k 0 π k x s closest pont n S k to x T, k x = π k x π k 1 x

21 Gaussan mean wdth bound 3: ε-net sequence S k s a ( 1 /2 k )-net of T, k 0 π k x s closest pont n S k to x T, k x = π k x π k 1 x wlog T < (else apply ths slde to ε-net of T for ε small) g, x = g, π 0 x + k=1 g, kx

22 Gaussan mean wdth bound 3: ε-net sequence S k s a ( 1 /2 k )-net of T, k 0 π k x s closest pont n S k to x T, k x = π k x π k 1 x wlog T < (else apply ths slde to ε-net of T for ε small) g, x = g, π 0 x + k=1 g, kx g(t ) E sup g, π 0 x g } {{ } 0 + k=1 E g sup g, k x

23 Gaussan mean wdth bound 3: ε-net sequence S k s a ( 1 /2 k )-net of T, k 0 π k x s closest pont n S k to x T, k x = π k x π k 1 x wlog T < (else apply ths slde to ε-net of T for ε small) g, x = g, π 0 x + k=1 g, kx g(t ) E sup g, π 0 x g + k=1 E g sup g, k x } {{ } 0 { k x : x T } N (T, l 2, 1 /2 k ) N (T, l 2, 1 /2 k 1 ) (N (T, l 2, 1 /2 k )) 2

24 Gaussan mean wdth bound 3: ε-net sequence S k s a ( 1 /2 k )-net of T, k 0 π k x s closest pont n S k to x T, k x = π k x π k 1 x wlog T < (else apply ths slde to ε-net of T for ε small) g, x = g, π 0 x + k=1 g, kx g(t ) E sup g, π 0 x g + k=1 E g sup g, k x } {{ } 0 { k x : x T } N (T, l 2, 1 /2 k ) N (T, l 2, 1 /2 k 1 ) (N (T, l 2, 1 /2 k )) 2 g(t ) k=1 (1 /2 k ) log 1/2 N (T, l 2, 1 /2 k ) 0 log 1/2 N (T, l 2, u)du (Dudley s theorem)

25 Gaussan mean wdth bound 4: generc channg Agan, wlog T <. Defne T 0 T 1 T k = T T 0 = 1, T k 2 2k (call such a sequence admssble )

26 Gaussan mean wdth bound 4: generc channg Agan, wlog T <. Defne T 0 T 1 T k = T T 0 = 1, T k 2 2k (call such a sequence admssble ) Exercse: show Dudley s theorem s equvalent to g(t ) nf {Tk } admssble k=1 2k/2 sup d l2 (x, T k ) (should pck T k to be the best ε = ε(k) net of sze 2 2k )

27 Gaussan mean wdth bound 4: generc channg Agan, wlog T <. Defne T 0 T 1 T k = T T 0 = 1, T k 2 2k (call such a sequence admssble ) Exercse: show Dudley s theorem s equvalent to g(t ) nf {Tk } admssble k=1 2k/2 sup d l2 (x, T k ) (should pck T k to be the best ε = ε(k) net of sze 2 2k ) Fernque 76 : can pull the sup x outsde the sum g(t ) nf {Tk } sup k=1 2k/2 d l2 (x, T k ) def = γ 2 (T, l 2 )

28 Gaussan mean wdth bound 4: generc channg Agan, wlog T <. Defne T 0 T 1 T k = T T 0 = 1, T k 2 2k (call such a sequence admssble ) Exercse: show Dudley s theorem s equvalent to g(t ) nf {Tk } admssble k=1 2k/2 sup d l2 (x, T k ) (should pck T k to be the best ε = ε(k) net of sze 2 2k ) Fernque 76 : can pull the sup x outsde the sum g(t ) nf {Tk } sup k=1 2k/2 d l2 (x, T k ) def = γ 2 (T, l 2 ) equvalent upper bound proven by Fernque (who mnmzed some ntegral over all measures over T ), but reformulated n terms of admssble sequences by Talgarand

29 Gaussan mean wdth bound 4: generc channg Proof of Fernque s bound g(t ) E sup g, π 0 x g } {{ } 0 + E g sup g, k x }{{} Y k k=1 (from before) t, P(Y k > t2 k/2 k x 2 ) e t2 2 k /2 (gaussan decay)

30 Gaussan mean wdth bound 4: generc channg Proof of Fernque s bound g(t ) E sup g, π 0 x g } {{ } 0 + E g sup g, k x }{{} Y k k=1 (from before) t, P(Y k > t2 k/2 k x 2 ) e t2 2 k /2 (gaussan decay) P( x, k Y k > t2 k/2 k x 2 ) k (22k ) 2 e t2 2 k /2

31 Gaussan mean wdth bound 4: generc channg Proof of Fernque s bound g(t ) E sup g, π 0 x g } {{ } 0 + E g sup g, k x }{{} Y k k=1 (from before) t, P(Y k > t2 k/2 k x 2 ) e t2 2 k /2 (gaussan decay) P( x, k Y k > t2 k/2 k x 2 ) k (22k ) 2 e t2 2 k /2 E g sup Y k = k 0 P(sup Y k > u)du k

32 Gaussan mean wdth bound 4: generc channg E g sup Y k = k = γ 2 (T, l 2 ) 0 0 P(sup P(sup Y k > u)du k Y k > t sup 2 k/2 k x 2 )dt k (change of varables: u = t sup 2 k/2 k x 2 tγ 2 (T, l 2 )) k = γ 2 (T, l 2 ) [t + (2 2k ) 2 e t2 2 k /2 dt] γ 2 (T, l 2 ) t k=1 k Concluson: g(t ) γ 2 (T, l 2 ) Talagrand: g(t ) γ 2 (T, l 2 ) (won t show today) ( Majorzng measures theorem )

33 Gaussan mean wdth bound 4: generc channg E g sup Y k = k = γ 2 (T, l 2 ) 0 0 P(sup P(sup Y k > u)du k Y k > t sup 2 k/2 k x 2 )dt k (change of varables: u = t sup 2 k/2 k x 2 tγ 2 (T, l 2 )) k = γ 2 (T, l 2 ) [t + (2 2k ) 2 e t2 2 k /2 dt] γ 2 (T, l 2 ) t k=1 k

34 Gaussan mean wdth bound 4: generc channg E g sup Y k = k = γ 2 (T, l 2 ) 0 0 P(sup P(sup Y k > u)du k Y k > t sup 2 k/2 k x 2 )dt k (change of varables: u = t sup 2 k/2 k x 2 tγ 2 (T, l 2 )) γ 2 (T, l 2 ) [2 + γ 2 (T, l 2 ) 2 k ( ) (2 2k ) 2 e t2 2 k /2 dt] k=1 k

35 Gaussan mean wdth bound 4: generc channg E g sup Y k = k = γ 2 (T, l 2 ) 0 0 P(sup P(sup Y k > u)du k Y k > t sup 2 k/2 k x 2 )dt k (change of varables: u = t sup 2 k/2 k x 2 tγ 2 (T, l 2 )) γ 2 (T, l 2 ) [2 + γ 2 (T, l 2 ) 2 k ( ) (2 2k ) 2 e t2 2 k /2 dt] k=1 k Concluson: g(t ) γ 2 (T, l 2 )

36 Gaussan mean wdth bound 4: generc channg E g sup Y k = k = γ 2 (T, l 2 ) 0 0 P(sup P(sup Y k > u)du k Y k > t sup 2 k/2 k x 2 )dt k (change of varables: u = t sup 2 k/2 k x 2 tγ 2 (T, l 2 )) γ 2 (T, l 2 ) [2 + γ 2 (T, l 2 ) 2 k ( ) (2 2k ) 2 e t2 2 k /2 dt] k=1 k Concluson: g(t ) γ 2 (T, l 2 ) Talagrand: g(t ) γ 2 (T, l 2 ) (won t show today) ( Majorzng measures theorem )

37 Are these bounds really dfferent? γ 2 (T, l 2 ): nf {Tk } sup k=1 2k/2 d l2 (x, T k ) Dudley: nf {Tk } k=1 2k/2 sup d l2 (x, T k ) 0 log 1/2 N (T, l 2, u)du

38 Are these bounds really dfferent? γ 2 (T, l 2 ): nf {Tk } sup k=1 2k/2 d l2 (x, T k ) Dudley: nf {Tk } k=1 2k/2 sup d l2 (x, T k ) 0 log 1/2 N (T, l 2, u)du Dudley not optmal: T = B l n 1

39 Are these bounds really dfferent? γ 2 (T, l 2 ): nf {Tk } sup k=1 2k/2 d l2 (x, T k ) Dudley: nf {Tk } k=1 2k/2 sup d l2 (x, T k ) 0 log 1/2 N (T, l 2, u)du Dudley not optmal: T = B l n 1 sup x Bl n 1 g, x = g, so g(t ) log n Exercse: Come up wth admssble {T k } yeldng γ 2 log n (must exst by majorzng measures)

40 Are these bounds really dfferent? γ 2 (T, l 2 ): nf {Tk } sup k=1 2k/2 d l2 (x, T k ) Dudley: nf {Tk } k=1 2k/2 sup d l2 (x, T k ) 0 log 1/2 N (T, l 2, u)du Dudley not optmal: T = B l n 1 sup x Bl n 1 g, x = g, so g(t ) log n Exercse: Come up wth admssble {T k } yeldng γ 2 log n (must exst by majorzng measures) Dudley: log N (B l n 1, l 2, u) ( 1 /u 2 ) log n for u not too small (consder just coverng ( 1 /u 2 )-sparse vectors wth u 2 n each coordnate). Dudley can only gve g(b l n 1 ) log 3/2 n.

41 Are these bounds really dfferent? γ 2 (T, l 2 ): nf {Tk } sup k=1 2k/2 d l2 (x, T k ) Dudley: nf {Tk } k=1 2k/2 sup d l2 (x, T k ) 0 log 1/2 N (T, l 2, u)du Dudley not optmal: T = B l n 1 sup x Bl n 1 g, x = g, so g(t ) log n Exercse: Come up wth admssble {T k } yeldng γ 2 log n (must exst by majorzng measures) Dudley: log N (B l n 1, l 2, u) ( 1 /u 2 ) log n for u not too small (consder just coverng ( 1 /u 2 )-sparse vectors wth u 2 n each coordnate). Dudley can only gve g(b l n 1 ) log 3/2 n. Smple vanlla ε-net argument gves g(b l n 1 ) poly(n).

42 Hgh probablty So far just talked about g(t ) = E g sup Z x But what f we want to know sup Z x s small whp, not just n expectaton?

43 Hgh probablty So far just talked about g(t ) = E g sup Z x But what f we want to know sup Z x s small whp, not just n expectaton? Usual approach: bound E g sup Zx p for large p and do Markov ( moment method ) Can bound moments usng channg too; see (Drksen 13)

44 Applcatons n computer scence Fast RIP matrces (Candès, Tao 06), (Rudelson, Vershynn 06), (Cheragch, Guruswam, Velngker 13), (N., Prce, Wootters 14), (Bourgan 14), (Havv, Regev 15) Fast JL (Alon, Lberty 11), (Krahmer, Ward 11), (Bourgan, Drksen, N. 15), (Oymak, Recht, Soltanolkotab 15) Instance-wse JL bounds (Gordon 88), (Klartag, Mendelson 05), (Mendelson, Pajor, Tomczak-Jaegermann 07), (Drksen 14) Approxmate nearest neghbor (Indyk, Naor 07) Determnstc algorthm to estmate graph cover tme (Dng, Lee, Peres 11) Lst-decodablty of random codes (Wootters 13), (Rudra, Wootters 14)...

45 A channg result for quadratc forms Theorem [Krahmer, Mendelson, Rauhut 14] Let A R n n be a famly of matrces, and let σ 1,..., σ n be ndependent subgaussans. Then E sup Aσ 2 2 E Aσ 2 A A σ 2 γ 2 2(A, l2 l 2 ) + γ 2 (A, l2 l 2 ) F (A) + l2 l 2 (A) F (A) ( X s dameter under X -norm)

46 A channg result for quadratc forms Theorem [Krahmer, Mendelson, Rauhut 14] Let A R n n be a famly of matrces, and let σ 1,..., σ n be ndependent subgaussans. Then E sup Aσ 2 2 E Aσ 2 A A σ 2 γ 2 2(A, l2 l 2 ) + γ 2 (A, l2 l 2 ) F (A) + l2 l 2 (A) F (A) ( X s dameter under X -norm) Won t show proof today, but t s smlar to boundng g(t ) (wth some extra trcks). See edu/ mnlek/madalgo2015/, Lecture 3.

47 Instance-wse bounds for JL Corollary (Gordon 88, Klartag-Mendelson 05, Mendelson, Pajor, Tomczak-Jaegermann 07, Drksen 14) For T S n 1 and 0 < ε < 1/2, let Π R m n have ndependent subgaussan ndependent entres wth mean zero and varance 1/m for m (g2 (T )+1)/ε 2. Then E sup Πx < ε Π

48 Proof of Gordon s theorem Instance-wse bounds for JL For x T let A x denote the m mn matrx: x 1 x n 0 0 A x = x 1 x n 0 0 m x 1 x n

49 Proof of Gordon s theorem Instance-wse bounds for JL For x T let A x denote the m mn matrx: x 1 x n 0 0 A x = x 1 x n 0 0 m x 1 x n Then Πx 2 2 = A xσ 2 2, where σ s formed by concatenatng rows of Π (multpled by m).

50 Proof of Gordon s theorem Instance-wse bounds for JL For x T let A x denote the m mn matrx: x 1 x n 0 0 A x = x 1 x n 0 0 m x 1 x n Then Πx 2 2 = A xσ 2 2, where σ s formed by concatenatng rows of Π (multpled by m). A x A y = A x y = ( 1 / m) x y 2 γ 2 (A T, l2 l 2 ) = γ 2 (T, l 2 ) g(t )

51 Proof of Gordon s theorem Instance-wse bounds for JL For x T let A x denote the m mn matrx: x 1 x n 0 0 A x = x 1 x n 0 0 m x 1 x n Then Πx 2 2 = A xσ 2 2, where σ s formed by concatenatng rows of Π (multpled by m). A x A y = A x y = ( 1 / m) x y 2 γ 2 (A T, l2 l 2 ) = γ 2 (T, l 2 ) g(t ) F (A T ) = 1, l2 l 2 (A T ) = 1/ m

52 Proof of Gordon s theorem Instance-wse bounds for JL For x T let A x denote the m mn matrx: x 1 x n 0 0 A x = x 1 x n 0 0 m x 1 x n Then Πx 2 2 = A xσ 2 2, where σ s formed by concatenatng rows of Π (multpled by m). A x A y = A x y = ( 1 / m) x y 2 γ 2 (A T, l2 l 2 ) = γ 2 (T, l 2 ) g(t ) F (A T ) = 1, l2 l 2 (A T ) = 1/ m Thus E Π sup Πx g2 (T )/m + g(t ) / m + 1 / m

53 Proof of Gordon s theorem Instance-wse bounds for JL For x T let A x denote the m mn matrx: x 1 x n 0 0 A x = x 1 x n 0 0 m x 1 x n Then Πx 2 2 = A xσ 2 2, where σ s formed by concatenatng rows of Π (multpled by m). A x A y = A x y = ( 1 / m) x y 2 γ 2 (A T, l2 l 2 ) = γ 2 (T, l 2 ) g(t ) F (A T ) = 1, l2 l 2 (A T ) = 1/ m Thus E Π sup Πx g2 (T )/m + g(t ) / m + 1 / m Set m (g2 (T )+1)/ε 2

54 Consequences of Gordon s theorem m (g2 (T )+1)/ε 2 T < : g 2 (T ) log T (JL) T a d-dm subspace: g 2 (T ) d (subspace embeddngs) T all k-sparse vectors: g 2 (T ) k log(n/k) (RIP)

55 Consequences of Gordon s theorem m (g2 (T )+1)/ε 2 T < : g 2 (T ) log T (JL) T a d-dm subspace: g 2 (T ) d (subspace embeddngs) T all k-sparse vectors: g 2 (T ) k log(n/k) (RIP) more applcatons to constraned least squares, manfold learnng, model-based compressed sensng,... (see (Drksen 14) and (Bourgan, Drksen, N. 15))

56 Channg sn t just for gaussans

57 Channg wthout gaussans: RIP (Rudelson, Vershynn 06) Restrcted sometry property useful n compressed sensng. T = {x : x 0 k, x 2 = 1}. Theorem (Candès-Tao 06, Donoho 06, Candés 08) If Π satsfes (ε, k)-rip for ε < 2 1 then there s a lnear program whch, gven Πx and Π as nput, recovers x n polynomal tme such that x x 2 O( 1 / k) mn y 0 k x y 1.

58 Channg wthout gaussans: RIP (Rudelson, Vershynn 06) Restrcted sometry property useful n compressed sensng. T = {x : x 0 k, x 2 = 1}. Theorem (Candès-Tao 06, Donoho 06, Candés 08) If Π satsfes (ε, k)-rip for ε < 2 1 then there s a lnear program whch, gven Πx and Π as nput, recovers x n polynomal tme such that x x 2 O( 1 / k) mn y 0 k x y 1. Of nterest to show samplng rows of dscrete Fourer matrx s RIP

59 Channg wthout gaussans: RIP (Rudelson, Vershynn 06) (Unnormalzed) Fourer matrx F, rows: z 1,..., z n δ 1,..., δ n ndependent Bernoull wth expectaton m/n

60 Channg wthout gaussans: RIP (Rudelson, Vershynn 06) (Unnormalzed) Fourer matrx F, rows: z 1,..., z n δ 1,..., δ n ndependent Bernoull wth expectaton m/n Want E δ sup T [n] T k I T 1 m n δ z (T ) =1 (T ) z < ε

61 Channg wthout gaussans: RIP (Rudelson, Vershynn 06) LHS = E δ 1 m E = sup T [n] T k sup δ,δ T π 2 1 m I T {}}{ 1 n E δ δ m z (T ) (T ) z 1 m 2π 1 m E 1 m E δ E g =1 n (δ δ )z (T ) =1 E sup E δ,δ,σ g T sup δ,g T sup x B n,k 2 (T ) z n δ z (T ) =1 (Jensen) n g σ (δ δ )z (T ) =1 n g δ z (T ) =1 (T ) z (T ) z (T ) z (Jensen+trangle neq) n g δ z, x 2 (gaussan mean wdth!) =1

62 Channg wthout gaussans: RIP (Rudelson, Vershynn 06) LHS = E δ sup T [n] T k 1 m E = sup δ,δ T π 2 1 m I T {}}{ 1 n E δ δ m z (T ) (T ) z 1 m 2π 1 m E 1 m E δ E g =1 n (δ δ )z (T ) =1 E sup E δ,δ,σ g T sup δ,g T sup x B n,k 2 (T ) z n δ z (T ) =1 (Jensen) n g σ (δ δ )z (T ) =1 n g δ z (T ) =1 (T ) z (T ) z (T ) z (Jensen+trangle neq) n g δ z, x 2 (gaussan mean wdth!) =1

63 Channg wthout gaussans: RIP (Rudelson, Vershynn 06) LHS = E δ sup T [n] T k 1 m E = sup δ,δ T π 2 1 m I T {}}{ 1 n E δ δ m z (T ) (T ) z 1 m 2π 1 m E 1 m E δ E g =1 n (δ δ )z (T ) =1 E sup E δ,δ,σ g sup δ,g T sup x B n,k 2 T (T ) z n δ z (T ) =1 (Jensen) n g σ (δ δ )z (T ) =1 n g δ z (T ) =1 (T ) z (T ) z (T ) z (Jensen+trangle neq) n g δ z, x 2 (gaussan mean wdth!) =1

64 Channg wthout gaussans: RIP (Rudelson, Vershynn 06) LHS = E δ sup T [n] T k 1 m E = sup δ,δ T π 2 1 m I T {}}{ 1 n E δ δ m z (T ) (T ) z 1 m 2π 1 m E 1 m E δ E g =1 n (δ δ )z (T ) =1 E sup E δ,δ,σ g T sup δ,g T sup x B n,k 2 (T ) z n δ z (T ) =1 (Jensen) n g σ (δ δ )z (T ) =1 n g δ z (T ) =1 (T ) z (T ) z (T ) z (Jensen+trangle neq) n g δ z, x 2 (gaussan mean wdth!) =1

65 Channg wthout gaussans: RIP (Rudelson, Vershynn 06) LHS = E δ sup T [n] T k 1 m E = sup δ,δ T π 2 1 m I T {}}{ 1 n E δ δ m z (T ) (T ) z 1 m 2π 1 m E 1 m E δ E g =1 n (δ δ )z (T ) =1 E sup E δ,δ,σ g T sup δ,g T sup x B n,k 2 (T ) z n δ z (T ) =1 (Jensen) n g σ (δ δ )z (T ) =1 n g δ z (T ) =1 (T ) z (T ) z (T ) z (Jensen+trangle neq) n g δ z, x 2 (gaussan mean wdth!) =1

66 The End

67 June 22 nd +23 rd : workshop on concentraton of measure / channg at Harvard, after STOC 16. Detals+webste forthcomng.