A universal deviation inequality for random polytopes

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A universal deviation inequality for random polytopes

Victor-Emmanuel Brunel

To cite this version:

Victor-Emmanuel Brunel. A universal deviation inequality for random polytopes. 2013. �hal-

00903687v2�

A universal deviation inequality for random polytopes

Victor-Emmanuel Brunel

CREST, Paris (France) - University of Haifa, Haifa (Israel) [email protected]

Abstract

We consider the convex hull of a finite sample of i.i.d. points uni- formly distributed in a convex body inR^d,d≥2. We prove an expo- nential deviation inequality, which leads to rate optimal upper bounds on all the moments of the missing volume of the convex hull, uniformly over all convex bodies ofR^d, with no restriction on their volume, loca- tion in the space and smoothness of the boundary. As a consequence of this inequality, we use an extension of an identity due to Efron, to prove uniform upper bounds on the moments of the number of vertices of this convex hull, and we show that the rates of convergence of these bounds are tight.

Keywords.

convex, convex hull, deviation inequality, metric entropy, ran- dom polytope, vertices

1 Introduction

Probabilistic properties of random polytopes have been studied extensively

in the literature in the last fifty years. Consider a convex body K in

, and

a set of n i.i.d. random points uniformly distributed in K. The convex hull

of these random points is a random polytope. Its number of vertices and

its missing volume, i.e., the volume of its complement in K, have been first

analyzed in the seminal work of R´enyi and Sulanke [13, 14]. They derived

the asymptotics of the expected missing volume in the case d = 2, when K is

supposed to be either a polygon with a given number of vertices, or a convex

set with smooth boundary. More recently, considerable efforts were devoted

to understanding the behavior of the expected missing volume. Thus, several

particular examples of K were studied, including a d-dimensional simple

polytope

[1], a d-dimensional polytope [2] and a d-dimensional Euclidean ball [5]. In [7], it is shown that the expected missing volume is maximal when K is an ellipsoid, see also the references therein. B´ar´ any and Larman [3]

showed that if K has volume one, then the expected missing volume has the same asymptotic behavior as the volume of the (1/n)-wet part of K , defined as the union of all caps of K (a cap being the intersection of K with a half space) of volume at most 1/n. This reduces the initial probabilistic problem to computation of such a deterministic volume, which is a specific analytic problem that was extensively studied. When K has a smooth boundary, a key point was the introduction of the affine surface area, see [16, 18], which leads to the result that the expected missing volume is of the order n

. When K is a polytope, it is of the order (ln n)

/n [3]. In addition, [3]

proves that the expected missing volume, in dimension d, is minimal for simple polytopes, and maximal for ellipsoids. As a conclusion, the properties of the expected missing volume are now very well-understood. Much less is known about its higher moments and deviation probabilities. In particular, using a jackknife inequality for symmetric functions of n random variables, Reitzner [12] proved that if K is a d-dimensional smooth convex body, the variance of the missing volume is bounded from above by n

, and he conjectured that this is the actual order of magnitude for the variance. In addition, he proved that the second moment of the missing volume is exactly of the order n

, with explicit constants in terms of the affine surface area of K. Vu [19] obtained deviation inequalities for general convex bodies of volume one, involving quantities such as the volume of the wet part, and derived precise deviation inequalities in the cases when K is a polytope, and when it has a smooth boundary. These inequalities involve constants which depend on K in an unknown way. The main tools are martingale inequalities and, as a consequence, upper bounds on the moments of the missing volume are proved, again with implicit constants depending on K.

Let V

stand for the missing volume of the convex hull of n i.i.d. points uniformly distributed in a convex body K. If K has volume one, there exist positive constants α, c and ǫ

such that:

P h

| V

−

[V

] | ≥ √ λv

≤ 2e

+ e

,

∀ ǫ ∈ α(ln n)/n, ǫ

, λ ∈ 0, n | K(ǫ) | ,

where v = 36ng(ǫ)

| K(ǫ) | , g(ǫ) = sup {| F | : F star-shaped ⊆ K(ǫ) } , and K(ǫ) is the ǫ-wet part of K defined in [3]. If K has a smooth boundary and

1Ad-dimensional simple polytope is a convex polytope such that each of its vertices is adjacent to exactlydedges.

is of volume one, Vu [19] showed the existence of positive constants c and α, which depend on K, such that for any λ ∈

0, (α/4)n

, the following holds:

P

"

| V

−

[V

] | ≥

αλn

#

≤ 2 exp( − λ/4) + exp

− cn

. This last inequality allows one to derive upper bounds on the variance and on the q-th moment of the missing volume, respectively of orders n

and n

, for q > 0, for a smooth convex body K of volume one, up to constant factors depending on K in an unknown way. Note that these two inequalities proved by Vu remain true for K of any positive volume, if

| V

−

[V

] | is replaced by | V

−

[V

] | / | K | . The aim of the present paper is to prove a universal deviation inequality and upper bounds on the moments of the missing volume, i.e., with no restriction on the volume and boundary structure of K, and with constants which do not depend on K. The only assumptions on K are compactness and convexity.

In addition, we derive, from our deviation inequality, moment inequali- ties for the number of vertices of the random convex hull. To our knowledge, these inequalities are new, and the corresponding rates turn to be tight, uni- formly on all convex bodies.

2 Statement of the problem and notation

Let d ≥ 2 be an integer. We denote by | · | the Lebesgue measure in

, ρ the Euclidean distance in

, B

the unit Euclidean ball with center 0, and β

its volume.

If G ⊆

and ǫ > 0, we denote by G

= { x ∈

: ρ(x, G) ≤ ǫ } the closed ǫ-neighborhood of G. Here, ρ(x, G) = inf

y∈G

ρ(x, y).

When G

and G

are two subsets of

, we denote by G

△ G

their symmetric difference, and the Hausdorff distance between G

and G

is defined as:

d

(G

, G

) = inf { ǫ > 0 : G

⊆ G

, G

⊆ G

} .

For brevity, we call a convex body a compact and convex subset of

with

positive Lebesgue measure. We denote by K

the class of all convex bodies

in

, and by K

the set of all convex bodies that are included in B

. For a given K ∈ K

, consider a sample of n i.i.d. random points X

, . . . , X

, uniformly distributed in K. We denote by ˆ K

the convex hull of X

, . . . , X

. This is a random polytope whose missing volume is denoted by V

, i.e., V

= | K \ K ˆ

| . We denote respectively by

and

the joint probability measure of (X

, . . . , X

) and the corresponding expectation operator. We are interested in deviation inequalities for V

, i.e., in bounding from above the probability

P_K

[V

> ǫ | K | ],

for ǫ > 0. This yields, as a consequence, upper bounds for the moments

[V

], q > 0. In order to obtain a deviation inequality, we use the metric entropy of the class K

. We first prove that it is sufficient to obtain a devia- tion inequality for K ∈ K

, by a scaling argument. The deviation inequality that we prove is uniform on the class K

, hence it is of much interest in a statistical framework. If one aims to recover K from the observation of the sample points X

, . . . , X

, using ˆ K

as an estimator, the risk, measured in terms of the Nikodym distance (defined as the Lebesgue measure of the symmetric difference), can be bounded from above uniformly on K

, with no assumption on the volume, boundary structure and location in

of K.

3 Deviation inequality for random polytopes

Theorem 1.

There exist two positive constants C

and C

, which depend on d only, such that:

sup

K∈Kd

P_K

"

n | K \ K ˆ

|

| K | − C

n

!

> x

#

≤ C

e

, ∀ x > 0. (1) Theorem 1 involves constants which depend at least exponentially on the dimension d. This seems to be the price for getting a uniform deviation inequality on K

. Note that the missing volume is normalized here by the volume of K . Theorem 1 may be refined by normalizing the missing volume by another functional of K, which could be expressed in terms of the affine surface area of K, as in [17] where only the first moment of V

is considered.

Theorem 1 allows one to derive upper bounds for all the moments of the

missing volume. Indeed, applying Fubini’s theorem leads to the following

corollary.

Corollary 1.

For every positive number q, there exists some positive con- stant A

, which depends on d and q only, such that

E_K h

| K \ K ˆ

|

≤ A

| K |

n

, ∀ K ∈ K

. (2) Note that no restriction is made on K except for its compactness and convexity. In particular, its boundary may not be smooth, and K may be located anywhere in the space, not necessarily in some given compact set.

In this sense, the exponential deviation inequality (1) and the inequality on the moments (2) are universal. Combining this corollary with the lower bound for the minimax risk given in [11] in a statistical framework yields the following result.

Corollary 2.

For every positive number q, there exist some positive con- stants a

and A

, which depend on d and q only, such that

a

n

≤ sup

K∈Kd

E_K

"

| K \ K ˆ

|

| K |

!q#

≤ A

n

. In order to prove Theorem 1, we first state two lemmas.

Lemma 1.

Let K ∈ K

. There exists an ellipsoid E in

such that K ⊆ E and | E | ≤ d

| K | .

Proof of Lemma 1 can be found in [10] and [8]. The second lemma is based on the Steiner formula for convex bodies. It shows that on K

, the Nikodym distance is bounded from above by the Hausdorff distance, up to some positive constant.

Lemma 2.

There exists some positive constant α

which depends on d only, such that

| G △ G

| ≤ α

d

(G, G

), ∀ G, G

∈ K

.

Proof. Let G ∈ K

. Steiner formula (see Section 4.1 in [15]) states that there exist some positive numbers L

(G), . . . , L

(G), such that

| G

\ G | =

d

X

j=1

L

(G)λ

, λ ≥ 0. (3) Besides the L

(G), j = 1, . . . , d are increasing functions of G. In particular, if G ∈ K

, then L

(G) ≤ L

(B

).

Let G, G

∈ K

, and let λ = d

(G, G

). Since G and G

are included in the

unit ball, λ is not greater than its diameter, so λ ≤ 2. By definition of the Hausdorff distance, G ⊆ G

and G

⊆ G

. Hence,

| G △ G

| = | G \ G

| + | G

\ G | ≤ | G

\ G

| + | G

\ G |

≤ 2

d

X

j=1

L

(B

)λ

≤ λ

d

X

j=1

L

(B

)2

.

The Lemma is proved by setting α

=

j=1

L

(B

)2

.

Note that since δ ≤ 1, Steiner formula (3) implies, for G ∈ K

, that

| G

\ G | ≤ α

δ, (4)

where α

=

j=1

L

(B

).

Proof of Theorem 1

This proof is inspired by Theorem 1 in [9], which derives an upper bound on the risk of a convex hull type estimator of a convex function. Let K ∈ K

. Let E be an ellipsoid which satisfies the properties of Lemma 1, and T an affine transform in

which maps E to the unit ball B

. Note that β

= | det T || E | , so T is invertible. Let us denote K

= T (K) and X

= T (X

), i = 1, . . . , n. Let ˆ K

be the convex hull of X

, . . . , X

. By the definition of T , the following properties hold :

(i) K

∈ K

,

(ii) X

, . . . , X

are i.i.d. uniformly distributed in K

, (iii) T ( ˆ K

) = ˆ K

.

Furthermore, one has the following:

| K \ K ˆ

|

| K | = | K

\ K ˆ

|

| det T || K | = | K

\ K ˆ

| | E | β

| K | ≤ d

β

| K

\ K ˆ

| . (5) Let δ = n

. A δ-net of K

, for the Hausdorff distance, is a collection of subsets of K

such that for each G ∈ K

, there is G

in this collection of sets which satisfies d

(G, G

) ≤ δ. Bronshtein [4] showed that there exists a finite δ-net, of cardinality N

≤ C

δ

for some positive constant C

. Let { G

, . . . , G

} be such a δ-net. Let j

, ˆ j ∈ { 1, . . . , N

} be such that:

d

(K

, G

) ≤ δ and d

( ˆ K

, G

) ≤ δ.

Let ε > 0. By (5) and (ii),

"

| K \ K ˆ

|

| K | > ε

#

≤

| K

\ K ˆ

| > β

d

ε

. (6)

Let us recall that if G, G

and G

are three Borel subsets of

, then the following triangle inequality holds:

| G \ G

| ≤ | G \ G

| + | G

\ G

| . (7) Thus, | K

\ K ˆ

| ≤ | K

\ G

| + | G

\ G

| + | G

\ K ˆ

| and, by the definition of j

and ˆ j, and by Lemma 2 and (6),

P_K

"

| K \ K ˆ

|

| K | > ε

#

≤

| G

\ G

| > β

d

ε − 2α

δ

. (8)

Set ε

=

ε − 2α

δ. (8) implies

"

| K \ K ˆ

|

| K | > ε

#

≤

j=1,...,Nδ:|G_j∗\Gj|>ε^′

P_K′h

ˆ j = j

. (9)

Let j ∈ { 1, . . . , N

} be fixed, such that | G

\ G

| > ε

. Recall that ˆ K

⊆ G

j

, and thus if ˆ j = j, then X

∈ G

, i = 1, . . . , n. So,

P_K′

h

ˆ j = j

≤

h

X

∈ G

≤ 1 − | K

\ G

|

| K

|

!n

≤

1 − 1

β

( | G

\ G

| − | G

\ K

| − | G

\ G

| )

,

using the triangle inequality (7) and the fact that | K

| ≤ β

. Denote by I

= 1 if ε

< β

, and 0 otherwise. Continuing (9), and using (4), one gets:

P_K

"

| K \ K ˆ

|

| K | > ε

#

≤

j=1,...,Nδ:|Gj∗\Gj|>ε^′

1 − ε

β

+ (α

+ α

)δ β

n

≤ N

1 − ε

β

+ (α

+ α

)δ β

n

I

≤ C

exp

δ

− εn

d

+ (3α

+ α

)δn β

≤ C

exp

α

δn − εn d

,

where α

= 1+

d

is a positive constant which depends on d only (recall that δ

= δn). Finally, by choosing ε = α

d

δ + x/n, for any x > 0, and by setting the constant C

= α

d

, one gets (1).

4 Moment inequalities for the number of vertices of random polytopes

Let K ∈ K

. Denote by V

the set of vertices of ˆ K

, and by R

the cardinality of V

, for n ∈

. Efron [6] proved a simple but elegant identity, which connects the normalized expected missing volume

h|K\Kˆn|

|K|

i

and the expected number of vertices

[R

] of the random polytope ˆ K

. Namely, one has

E_K

"

| K \ K ˆ

|

| K |

#

=

[R

]

n + 1 , ∀ n ∈

. (10) This identity is extended to higher moments of | K \ K ˆ

| in the following theorem.

Lemma 3.

Let K ∈ K

, n and q be positive integers,

[R

(R

− 1) . . . (R

− q + 1)]

≤

"

| K \ K ˆ

|

| K |

!q#

(n + q)(n + q − 1) . . . (n + 1). (11) Combining Corollary 1 and Theorem 3 yields the following inequality:

E_K

[R

(R

− 1) . . . (R

− q + 1)] ≤ A

n

, ∀ n ∈

, ∀ q ∈

, where A

is the same constant as in Corollary 1. Since the polynomial x

is a linear combination of the polynomials x(x − 1) . . . (x − k + 1), 0 ≤ k ≤ q, we get the following inequality:

sup

K∈Kd

E_K

[R

] ≤ B

n

, (12)

for some positive constant B

which depends on d and q only.

Let K ∈ K

and n, q be positive integers. By H¨older inequality,

[R

] ≥

[R

]

and, by Efron’s identity (10),

E_K

[R

] ≥ n

"

| K \ K ˆ

|

| K |

#q

.

If the affine surface area of K is positive, which occurs, for instance, when the boundary of K is smooth with positive Gauss curvature, then it is known that

|K\Kˆn−1|

|K|

i

is exactly of the order of n

(see [16]). It is shown in [7] that

h|K\Kˆn−1|

|K|

i

is maximal when K is an ellipsoid, and and the corresponding value is exactly of the order of n

, up to a constant factor which depends on the dimension d only. Therefore, for smooth convex bodies K with positive curvature, the rate of the upper bound in (12) is tight, and one gets the following theorem:

Theorem 2.

Let n and q be positive integers. Then, for some positive constants b

and B

which depend on d and q only,

b

n

≤ sup

K∈Kd

E_K

[R

] ≤ B

n

.

Proof of Lemma 3

Let K ∈ K

and q be a positive integer, and let X

, X

, . . . a sequence of i.i.d. random variables uniformly distributed in K. For precision’s sake, we denote by

K

the n-product of the probability measure

, i.e., the joint probability measure of the random variables X

, . . . , X

, n ∈

. The corresponding expectation operator is denoted by

K

. Denote by

the indicator function, which applies to an event. First, note that the expectation

K

[ | K \ K ˆ

|

] can be rewritten as the integral:

E^⊗n

K

[ | K \ K ˆ

|

] =

K

Z

K^q1

(x

∈ / K ˆ

, . . . , x

∈ / K ˆ

)dx

. . . dx

,

and therefore,

K

[ | K \ K ˆ

|

] = | K |

K

h

X

∈ / K ˆ

, ∀ j = 1, . . . , q

. (13)

Since the event { X

∈ / K ˆ

, ∀ j = 1, . . . , q } contains { X

∈ V

, ∀ j = 1, . . . , q } , (13) becomes

E^⊗n

K

[ | K \ K ˆ

|

] ≥ | K |

K

[X

∈ V

, ∀ j = 1, . . . , q]

= | K |

n+q q

X

1≤i1<...<iq≤n+q

P^⊗n+q

K

X

∈ V

, ∀ j = 1, . . . , q

= | K |

n+q q

E^⊗n+q

K





X

1≤i1<...<iq≤n+q

1

X

∈ V

, ∀ j = 1, . . . , q





= | K |

n+q q

E^⊗n+q

K

R

q

= | K |

K

[R

(R

− 1) . . . (R

− q + 1)]

(n + q)(n + q − 1) . . . (n + 1) , and the theorem is proved.

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