Construction of optimal quantizers for Gaussian measures on Banach spaces

Quantization Scheme. Since each N -quantizer α lies necessarily in the finite dimensional subspace span(α) ⊂ E, it is possible to project the random variable X onto an appropriate finite dimensional subspace of E and solve the optimal quantization problem there, which allows to apply the results of the former section. Nevertheless, it is at this point in no way clear how to choose these finite dimensional subspaces and how to “project” X onto them in the general Banach space setting. Therefore, we start by introducing an abstract quantization scheme, which allows to describe the invertible transformation of the quantization problem on E to finite dimensional problems on some l q -spaces, and consequently renders a 3 OPTIMAL QUANTIZATION 24 construction of optimal quantizers by numerical methods possible. To be more precise, this quantization scheme consists of (i) a sequence of finite dimensional random variables in E approximating X, (ii) a sequence of isomorphisms, which map the random variables from (i) a.s. into some finite dimensional lq-space, (iii) a sequence of quantizers in lq. Using these three objects, we will be able to describe every optimal and asymptotically optimal quantizer on some Banach space, with which we will deal in this work, in terms of finite dimensional quantizers on lq. In addition, we refer to the rank of a random variable Y as rkY := dim span(P ). Definition 3.2. For X ∈ L(E), N ∈ N and q ∈ [1,∞], let (i) (Xk)k≥1 be a sequence of finite dimensional random variables such that rkXk ≤ dk and (dk)k≥1 ∈ c00, (ii) Ik : Ek → lk q be linear isomorphisms for subspaces Ek ⊂ E with Pk(Ek) = 1, (iii) βk ⊂ lk q with |βk| ≤ Nk, such that ∏ k≥1Nk ≤ N . Then (Xk, Ik, βk)k≥1 is called Abstract Quantization Scheme for X at level N . Moreover, a sequence of Abstract Quantization Schemes at level N ( X k , I N k , β N k ) k≥1 for N →∞, where the isomorphisms are uniformly bounded by a common constant C > 0, i.e. ‖I k ‖, ‖(I k )−1‖ ≤ C ∀ k,N ∈ N, (3.2) should be denoted Asymptotical Quantization Scheme. W.l.o.g we may assume (dk)k≥1 to be ordered non-increasingly. Moreover we will refer in the case (dk)k≥1 = (d1, 0, . . .) to a Single-Block Design, in the case (dk)k≥1 = (1, . . . , 1, 0, . . .) to a Scalar-Product Design and otherwise only to a Product Design. 3 OPTIMAL QUANTIZATION 25 Note that due to the condition (dk)k≥1 ∈ c00, there are only finite many random variables with rkXk > 0. All the other random variables vanish and therefore we have in fact to deal only with a finite number of random variables in the definition of the Abstract Quantization Scheme. Moreover, the same is true for the product ∏ k≥1Nk ≤ N , where only finite many Nk may be greater than one. In addition, we will not demand an explicit specification of the sequences (dk)k≥1 and (Nk)k≥1, although it is in general a non-trivial task to derive these sequences in an optimal way, and the asymptotically optimal choices for (dk)k≥1 and (Nk)k≥1 exhibit mostly a rather complicated form. Nevertheless, regarding the numerical construction of (asymptotically) optimal quantizers, we get better results for finite N ∈ N by solving numerically a so-called Block-AllocationProblem specially tailored to the available quantizers βk in some l dk q -space (cf. [PP05] or [LPW08]). Moreover, the smallest constant C > 0 which can be achieved by an Asymptotical Quantization Scheme is C = 1. Indeed, we always have 1 = ‖id‖ ≤ ‖I−1 k ‖‖Ik‖ ≤ C , hence the case C = 1 corresponds to the fact that all the Ik’s are isometric isomorphisms. In that special case it is, due to Proposition 3.3, completely equivalent if we consider the quantization problem of Xk on Ek or IkXk on l dk q . Otherwise, we only get a weak equivalence, that is up to a constant. For a given Abstract Quantization Scheme of level N , we may construct in a canonical way an N -quantizer for X by means of the Minkowski sum: Proposition 3.9. For X ∈ L(E) and N ∈ N let (Xk, Ik, βk)k∈N be an Abstract Quantization Scheme of level N . Then