From Steiner formulas for cones to concentration of intrinsic volumes

STEINER FORMULAS FOR CONES AND INTRINSIC VOLUMES 7Indeed, the raw second moment of a chi-square random variable X k with k degrees of freedom equals k 2 + 2k.Combine these two displays to reachVar[V C ] = E [ ‖Π C (g )‖ 4 ] − δ(C) 2 − 2δ(C) = E [ ‖Π C (g )‖ 4 ] − ( E [ ‖Π C (g )‖ 2 ]) 2 − 2δ(C)where the second identity follows from Proposition 4.3. Identify the variance of ‖Π C (g )‖ 2 to complete theproof of (4.3). To establish (4.4), note that Var[V C ] = Var[d − V C ] = Var[V C ◦], and then apply (4.3) to therandom variable V C ◦.□4.4. A bound for the variance of the intrinsic volumes. Proposition 4.4 also allows us to produce a generalbound on the variance of the intrinsic volumes of a cone.Theorem 4.5 (Variance bound for intrinsic volumes). Let C ∈ C d be a closed convex cone. ThenThe operator ∧ returns the minimum of two numbers.Var[V C ] ≤ 2 ( δ(C) ∧ δ(C ◦ ) ) .The example in Section 6.3 demonstrates that the constant two in (4.5) cannot be reduced.Proof. To bound the variance of V C , we plan to invoke the Gaussian Poincaré inequality [Bog98, Thm. 1.6.4]to control the variance of ‖Π C (g )‖ 2 . This inequality states thatVar[H(g )] ≤ E [ ‖∇H(g )‖ 2 ] .for any function H : R d → R whose gradient is square-integrable with respect to the standard normal measure.We apply this result to the functionH(x) = ‖Π C (x)‖ 2 with ‖∇H(x)‖ 2 = 4‖Π C (x)‖ 2 .The gradient calculation is justified by (7.5). We determine thatVar [ ‖Π C (g )‖ 2 ] ≤ 4E [ ‖Π C (g )‖ 2 ] = 4δ(C)where the second identity follows from Proposition 4.3. Introduce this inequality into (4.3) to see thatVar[V C ] ≤ 2δ(C). We can apply the same argument to see thatVar [ ‖Π C ◦(g )‖ 2 ] ≤ 4δ(C ◦ ).Substitute this bound into (4.4) to conclude that Var[V C ] ≤ 2δ(C ◦ ).In principle, a random variable taking values in {0,1,2,...,d} can have variance larger than d 2 /3—considerthe uniform random variable. In contrast, Theorem 4.5 tells us that the variance of the intrinsic volumerandom variable V C cannot exceed d for any cone C. This observation has consequences for the tail behaviorof V C . Indeed, Chebyshev’s inequality implies that{P |V C − δ(C)| > λ √ }δ(C)≤ Var[V C ]λ 2 δ(C) ≤ 2 λ 2 .That is, most of the mass of V C is located near the statistical dimension.4.5. Exponential moments of the intrinsic volumes. In the previous section, we discovered that theintrinsic volume random variable V C is often close to its mean value. This observation suggests that V Cmight exhibit stronger concentration. A standard method for proving concentration inequalities for a randomvariable is to calculate its exponential moments. The master Steiner formula allows us to accomplish this task.Proposition 4.6 (Exponential moments of the intrinsic volumes). Let C ∈ C d be a closed convex cone. For eachparameter ζ ∈ R,Ee ζV C= Ee ξ‖Π C (g )‖ 2 where ξ = 1 2(1 − e−2ζ ) .Proof. Fix a number ξ < 1 2 . With the choice f (a,b) = eξa , Corollary 3.2 shows thatd∑Ee ξ‖Π C (g )‖ 2 [ ] d∑d∑= EeξX k · vk (C) = (1 − 2ξ) −k/2 v k (C) = e ζk v k (C) = Ee ζV C.k=0k=0We have used the familiar formula for the exponential moments of a chi-square random variable X k with kdegrees of freedom. The penultimate identity follows from the change of variables ζ = − 1 2log(1 − 2ξ), whichestablishes a bijection between ξ < 1 2and ζ ∈ R.□k=0□