Optimal properties of the canonical tight probabilistic frame
OOPTIMAL PROPERTIES OF THE CANONICAL TIGHTPROBABILISTIC FRAME
DESAI CHENG AND KASSO A. OKOUDJOU
Abstract.
A probabilistic frame is a Borel probability measure withfinite second moment whose support spans R d . A Parseval probabilisticframe is one for which the associated matrix of second moment is theidentity matrix in R d . Each probabilistic frame is canonically associ-ated to a Parseval probabilistic frame. In this paper, we show that thiscanonical Parseval probabilistic frame is the closest Parseval probabilis-tic frame to a given probabilistic frame in the 2 − Wasserstein distance.Our proof is based on two main ingredients. On the one hand, we showthat a probabilistic frame can be approximated in the 2 − Wassersteinmetric with (compactly supported) finite frames whose bounds can becontrolled. On the other hand we establish some fine continuity prop-erties of the function that maps a probabilistic frame to its canonicalParseval probabilistic frame. Our results generalize similar ones for fi-nite frames and their associated Parseval frames. Introduction
The notion of probabilistic frames was first introduced in [8] in the settingof probability measures on the unit sphere, and was later generalized toprobability measures on R d in [10]. In essence, this theory is a generalizationof the theory of finite frames which has seen a wealth of activities in recentyear, [6, 7, 11, 12, 14].1.1. Review of finite frame theory.
Before we give the definition andsome elementary properties of probabilistic frames, we recall that a set Φ = { ϕ i } Ni =1 ⊂ R d is a frame for R d if and only if there exist 0 < A ≤ B < ∞ such that A (cid:107) x (cid:107) ≤ N (cid:88) i =1 (cid:104) x, ϕ i (cid:105) ≤ B (cid:107) x (cid:107) ∀ x ∈ R . The frame Φ is a tight frame if we can choose A = B . Furthermore, if A = B = 1, Φ is called a Parseval frame. In the sequel the set of framesfor R d with N vectors will be denoted by F ( N, d ), and simply F when thecontext is clear. The subset of frames with frame bounds 0 < A ≤ B < ∞ will be denoted F A,B ( N, d ) , or simply F A,B . We equip the set F ( N, d ) with
Date : November 11, 2018.1991
Mathematics Subject Classification. a r X i v : . [ m a t h . C A ] M a y D. CHENG AND K. A. OKOUDJOU the metric(1) d (Φ , Ψ) = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) i =1 (cid:107) ϕ i − ψ i (cid:107) = (cid:118)(cid:117)(cid:117)(cid:116) d (cid:88) i =1 (cid:107) R i − P i (cid:107) where Φ = { ϕ i } Ni =1 , Ψ = { ψ i } Ni =1 ) ∈ F ( M, d ) , { R i } di =1 , { P i } di =1 ⊂ R N denotethe rows of Φ, and those of Ψ, respectively.Let Φ = { ϕ i } Ni =1 be a frame for R d . Throughout the paper we shall abusenotation and denote the synthesis matrix of the frame by Φ, the d × N whose i th column is ϕ i . The matrix S := S Φ = ΦΦ T = N (cid:88) i =1 (cid:104)· , ϕ i (cid:105) ϕ i is the frame matrix . It is known that Φ = { ϕ i } Ni =1 is a frame for R d if andonly if S is a positive definite matrix. Moreover, the smallest eigenvalueof S is the optimal lower frame bound, while its largest eigenvalue is theoptimal upper frame bound. Φ is a tight frame if and only if S is a multipleof the d × d identity matrix. In particular, Φ is a Parseval frame if and onlyif S = I .If Φ is a frame, then S is positive definite and thus invertible. Conse-quently, Φ † = { ϕ † i } Ni =1 = { S − / ϕ i } Ni =1 is a Parseval frame, leading to following reconstruction formula: x = N (cid:88) i =1 (cid:68) x, ϕ † i (cid:69) ϕ i = N (cid:88) i =1 (cid:104) x, ϕ i (cid:105) ϕ † i ∀ x ∈ R d . In addition, Φ † is the unique Parseval frame which solves the followingproblem [5, Theorem 3.1]:(2) min { d (Φ , Ψ) = N (cid:88) i =1 (cid:107) ϕ i − ψ i (cid:107) : Ψ = { ψ i } Ni =1 ⊂ R d , P arseval f rame } . To be specific,
Theorem 1.1. [5, Theorem 3.1] If Φ = { ϕ i } Ni =1 is a frame for R d , then Φ † = { ϕ † i } Ni =1 = { S − / ϕ i } Ni =1 isthe unique solution to (2) . In Section 2, and for the sake of completeness, we give a new and simpleproof of this result and we refer to [2, 3, 4] for related results.
PTIMAL PROPERTIES OF THE CANONICAL TIGHT PROBABILISTIC FRAME 3
Probabilistic frames.
The main goal of this paper is to characterizethe minimizers of an optimal problem analog of (2) for probabilistic frames.To motivate the definition of a probabilistic frame, we note that given aframe Φ = { ϕ i } Ni =1 ⊂ R d , then the discrete probability measure µ Φ = N N (cid:88) k =1 δ ϕ k has the property that its support ( { ϕ k } Nk =1 ) spans R d and that it has finitesecond moment, i.e., (cid:90) R d (cid:107) x (cid:107) dµ Φ ( x ) = N N (cid:88) k =1 (cid:107) ϕ k (cid:107) < ∞ . The probability measure µ Φ is an example of a probabilistic frame that wasintroduced in [8, 10].More specifically, a Borel probability measure µ is a probabilistic frame ifthere exist 0 < A ≤ B < ∞ such that for all x ∈ R d we have(3) A (cid:107) x (cid:107) ≤ (cid:90) R d |(cid:104) x, y (cid:105)| dµ ( y ) ≤ B (cid:107) x (cid:107) . The constants A and B are called lower and upper probabilistic frame bounds ,respectively. When A = B, µ is called a tight probabilistic frame . In partic-ular, when A = B = 1, µ is called a Parseval probabilistic frame .A special class of probabilistic frames that will be considered in the sequelconsists of discrete measures µ Φ ,w = (cid:80) Ni =1 w i δ ϕ i where Φ = { ϕ i } Ni =1 ⊂ R d ,and w = { w i } Ni =1 ⊂ [0 , ∞ ) is a set of weights such that (cid:80) Ni =1 w i = 1. Aprobability measure such as µ Φ ,w will be termed finite probabilistic frame ,if and only if it is a probabilistic frame for R d . When the context is clearwe will simply write µ for µ Φ ,w . We shall also identify a finite probabilisticframe µ Φ ,w with the frame Φ w = {√ w i ϕ i } Ni =1 , as both have the same framebounds. We refer to the surveys [9, 15] for an overview of the theory ofprobabilistic frames.We shall prove an analog of Theorem 1.1 by endowing the set of proba-bilistic frames with the Wasserstein metric. Let P := P ( B , R d ) denote thecollection of probability measures on R d with respect to the Borel σ -algebra B . Let P := P ( R d ) = (cid:26) µ ∈ P : M ( µ ) := (cid:90) R d (cid:107) x (cid:107) dµ ( x ) < ∞ (cid:27) be the set of all probability measures with finite second moments. For µ, ν ∈ P , let Γ( µ, ν ) be the set of all Borel probability measures γ on R d × R d whose marginals are µ and ν , respectively, i.e., γ ( A × R d ) = µ ( A )and γ ( R d × B ) = ν ( B ) for all Borel subset A, B in R d . The space P is D. CHENG AND K. A. OKOUDJOU equipped with the 2-
Wasserstein metric given by(4) W ( µ, ν ) := min (cid:26) (cid:90) R d × R d (cid:107) x − y (cid:107) dγ ( x, y ) , γ ∈ Γ( µ, ν ) (cid:27) . The minimum defined by (4) is achieved at a measure γ ∈ Γ( µ, ν ), that is: W ( µ, ν ) = (cid:90) R d × R d (cid:107) x − y (cid:107) dγ ( x, y ) . We refer to [1, Chapter 7], and [16, Chapter 6] for more details on theWasserstein spaces.1.3.
Our contributions.
The investigation of probabilistic frames is stillat its initial stage. For example, in [17] the authors introduced the notion oftransport duals and used the setting of the Wasserstein metric to investigatethe properties of such probabilistic frames. In particular, this setting offersthe flexibility to find (non-discrete) probabilistic frames which are duals to agiven probabilistic frame. Transport duals are the probabilistic analogues ofalternate duals in frame theory [7, 13]. The main contribution of this paper(Theorem 2.12) is to investigate the properties of the canonical Parsevalprobabilistic frame associated to a given probabilistic frame, see Section 2for definitions. To prove this result we approximate a given probabilisticframe with one that is compactly supported and whose frame bounds arecontrolled in a precise way (Theorem 2.7). In the process of proving ourmain result, we prove a number of results that are of interest on their ownright. For example, in Section 2 we establish a number of new results aboutthe canonical Parseval frame Φ † associated to a frame Φ.2. Optimal Parseval probabilistic frames
Before proving our main result in Section 2.3, we revisit the canonical Par-seval frame Φ † associated to a given frame Φ = { ϕ k } Nk =1 ⊂ R d . In particular,Section 2.1 considers the continuity properties of the map F (Φ) = Φ † . InSection 2.2 we show how a probabilistic frame can be approximated in the2-Wasserstein metric by a sequence of finite frames whose bounds are con-trolled by those of the initial probabilistic frame. While such approximationfor probability measures in the 2-Wasserstein metric is well known [16, The-orem 6.18], our key contribution here is the control of the frame bounds ofthe approximating sequence.2.1. Continuity properties of the canonical Parseval frame.
In thissection we revisited the canonical Parseval frame Φ † associated to a givenframe Φ = { ϕ k } Nk =1 ⊂ R d . First, we give a new and elementary proof ofTheorem 1.1. Proof.
Proof of Theorem 1.1 We first note that a frame Ψ ⊂ R d is Parseval ifthe rows of its synthesis matrix are orthonormal. Furthermore, Ψ ⊂ R d is aParseval frame if and only if U Ψ is a Parseval frame for any d × d orthogonalmatrix U . PTIMAL PROPERTIES OF THE CANONICAL TIGHT PROBABILISTIC FRAME 5
Now let Φ = { ϕ i } Ni =1 be a frame for R d . Write S = ΦΦ T = U DU T forsome orthogonal matrix U . Observe that U T Φ is the matrix of Φ writtenwith respect to the orthonormal basis given by the rows of U T . In addition,the rows of U T Φ are pairwise orthogonal. Let Ψ = { ψ i } Ni =1 ⊂ R d be anyParseval frame, then d (Φ , Ψ) = d ( U T Φ , U T Ψ) = d (cid:88) i =1 (cid:107) R i − P i (cid:107) , where { R i } di =1 ⊂ R N and { P i } di =1 ⊂ R N denote respectively the rows of U T Φ and U T Ψ. Consequently, findingmin { d (Φ , Ψ) = N (cid:88) i =1 (cid:107) ϕ i − ψ i (cid:107) : Ψ = { ψ i } Ni =1 ⊂ R d , P arseval f rame } is equivalent to findingmin { d (cid:88) i =1 (cid:107) R i − P i (cid:107) : { P i } Ni =1 ⊂ R N , o rthonormal set } where { R i } di =1 form an orthogonal set of vectors in R N .But Φ † = { S − / ϕ i } Ni =1 is a Parseval frame, so its rows form an orthonor-mal set in R N . Consequently, Φ † is a solution to (2). The uniqueness followsby observing that the (unique) closest orthonormal set to a given orthogonalvectors { u i } di =1 ⊂ R N is { u i (cid:107) u i (cid:107) } di =1 . Consequently,min { d (Φ , Ψ) = N (cid:88) i =1 (cid:107) ϕ i − ψ i (cid:107) : Ψ = { ψ i } Ni =1 ⊂ R d , P arseval f rame } = d (cid:88) k =1 (1 − λ − / k ) where { λ k } dk =1 ⊂ (0 , ∞ ) are the eigenvalues of S = ΦΦ T . (cid:3) In the remaining part of section we study the continuity properties of thefunctions that maps a given frame to its canonical Parseval frame. This map F : F ( N, d ) → F ( N, d )given by(5) F (Φ) = F ( { ϕ i } Ni =1 ) = S − / ( { ϕ i } Ni =1 ) = { S − / ϕ i } Ni =1 . In fact, our results show that for 0 < A ≤ B , F is uniformly continuouson F A,B , the set of frames with frame bounds between A and B . Morespecifically, Theorem 2.1.
Let < A ≤ B < ∞ , and δ > be given. Then there exists (cid:15) > such that given any frame Φ = { ϕ i } Ni =1 , with frame bounds between A and B , and N := N Φ ≥ , for any frame Ψ = { ψ i } Ni =1 such that d (Φ , Ψ) < (cid:15) we have d ( F (Φ) , F (Ψ)) < δ. D. CHENG AND K. A. OKOUDJOU
Before proving this theorem, we establish a number of preliminary resultsand make the following remark that will be used in the sequel.
Remark . Let Φ = { ϕ i } Ni =1 ∈ F ( N, d ) be a frame. Then, S = ΦΦ T = ODO T where O is a d × d orthogonal matrix and D is a positive definitediagonal matrix. Fix the orthonormal basis of R d whose columns form thematrix O and write each frame vector ϕ i in this basis. The synthesis matrixof the frame Φ in the basis O is[Φ] O = O T Φ . Let { R i } di =1 be the rows of [Φ] O . We shall refer to { R i } di =1 as simply therows of Φ. Lemma 2.3.
Let
Φ = { ϕ i } Ni =1 ∈ F ( N, d ) . Denote by { R i } di =1 the rows of Φ as described by Remark 2.2. Let (cid:15) > and Ψ = { ψ i } Ni =1 ∈ F ( N, d ) besuch that d (Φ , Ψ) < (cid:15) . Denote by { P i } di =1 the rows of Ψ when written in theorthonormal basis O . Then (a) (cid:12)(cid:12) (cid:107) R i (cid:107) − (cid:107) P i (cid:107) (cid:12)(cid:12) < (cid:15). Furthermore, √ A − (cid:15) < (cid:107) P i (cid:107) < √ B + (cid:15) for each i = 1 , , . . . , d. (b) d (Φ , F (Φ)) ≥ (cid:118)(cid:117)(cid:117)(cid:116) d (cid:88) i =1 (cid:107) R i − R i (cid:107) R i (cid:107) (cid:107) . (c) For each i ∈ { , . . . , d } we have (cid:13)(cid:13)(cid:13)(cid:13) P i (cid:107) P i (cid:107) − R i (cid:107) R i (cid:107) (cid:13)(cid:13)(cid:13)(cid:13) < (cid:15) √ A . (d)
For each i ∈ { , . . . , d } we have ≤ (cid:107) P i − R i (cid:107) R i (cid:107) (cid:107) − (cid:107) P i − P i (cid:107) P i (cid:107) (cid:107) ≤ (cid:15) √ A c + (cid:15) A , where c = max (1 − √ A + (cid:15), √ B + (cid:15) − .Proof. (a) This is trivial so we omit it.(b) This follows immediately from the fact that the rows of a Parse-val frame are an orthonormal set when written with respect to anyorthonormal basis and R i (cid:107) R i (cid:107) is the closest unit norm vector to R i .(c) Since, d (Φ , Ψ) < (cid:15), we know that (cid:12)(cid:12) (cid:107) P i (cid:107) − (cid:107) R i (cid:107) (cid:12)(cid:12) < (cid:15) . Hence (cid:13)(cid:13)(cid:13)(cid:13) P i (cid:107) P i (cid:107) ·(cid:107) R i (cid:107)− R i (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) P i (cid:107) P i (cid:107) ·(cid:107) R i (cid:107)− P i (cid:13)(cid:13)(cid:13)(cid:13) + (cid:107) P i − R i (cid:107) = (cid:12)(cid:12) (cid:107) P i (cid:107)−(cid:107) R i (cid:107) (cid:12)(cid:12) + (cid:107) P i − R i (cid:107) < (cid:15). The result follows by recalling that (cid:107) R i (cid:107) ≥ √ A .(d) It is clear that (cid:107) P i − P i (cid:107) P i (cid:107) (cid:107) = |(cid:107) P i (cid:107) − | ≤ max (1 − √ A + (cid:15), √ B + (cid:15) −
1) = c . By part (c) we know that (cid:107) P i (cid:107) P i (cid:107) − R i (cid:107) R i (cid:107) (cid:107) < (cid:15) √ A . Using PTIMAL PROPERTIES OF THE CANONICAL TIGHT PROBABILISTIC FRAME 7 the fact hat P i (cid:107) P i (cid:107) is the closest unit norm vector to P i , we see that (cid:107) P i − P i (cid:107) P i (cid:107) (cid:107) ≤ (cid:107) P i − R i (cid:107) R i (cid:107) (cid:107) ≤ (cid:107) P i − P i (cid:107) P i (cid:107) (cid:107) + 2 (cid:15) √ A .
The result follows by squaring the last inequality. (cid:3)
Finally, we have the following technical lemma, that contains the keyargument in the proof of Theorem 2.1.
Lemma 2.4.
Given < A ≤ B < , fix Φ = { ϕ i } Ni =1 ∈ F A,B . Let (cid:15), δ > be such that δ √ d − (cid:15) √ A > and √ A − (cid:15) > . Let Ψ = { ψ i } Ni =1 be such that d (Φ , Ψ) < (cid:15) , and d ( S − / Φ , S − / Ψ) = d (Φ † , Ψ † ) > δ . Then, d (cid:88) i =1 ( (cid:107) P i − R (cid:48) i (cid:107) − (cid:107) P i − P i (cid:107) P i (cid:107) (cid:107) ) ≥ min ( Cd (cid:48) , C ) , where d (cid:48) = δ √ d − (cid:15) √ A , C = min( √ A − (cid:15), , and { R (cid:48) i } di =1 ⊂ R d is the set ofthe rows of S − / Ψ .Proof. We first show that there exists k then (cid:107) P k − R (cid:48) k (cid:107) − (cid:107) P k − P k (cid:107) P k (cid:107) (cid:107) ≥ min ( (cid:107) R (cid:48) k − P k (cid:107) P k (cid:107) (cid:107) · min ( (cid:107) P k (cid:107) , , (cid:107) P k (cid:107) ) . Since d ( S − / Φ , S − / Ψ) ≥ δ , then (cid:107) R k (cid:107) R k (cid:107) − R (cid:48) k (cid:107) ≥ δ √ d for some k . ByLemma 2.3 we know that (cid:107) P k (cid:107) P k (cid:107) − R k (cid:107) R k (cid:107) (cid:107) < (cid:15) √ A . It follows from the triangleinequality that (cid:107) P k (cid:107) P k (cid:107) − R (cid:48) k (cid:107) ≥ δ √ d − (cid:15) √ A = d (cid:48) . Suppose that C = min( √ A − (cid:15),
1) = 1, or equivalently, √ A − (cid:15) ≥ (cid:107) P i (cid:107) ≥ i .Since the angle (cid:92) R (cid:48) k P k (cid:107) P k (cid:107) P i > π/
2, it follows that (cid:107) P k − R (cid:48) k (cid:107) > (cid:107) P k − P k (cid:107) P k (cid:107) (cid:107) + (cid:107) P k (cid:107) P k (cid:107) − R (cid:48) k (cid:107) . But since, (cid:107) P k (cid:107) P k (cid:107) − R (cid:48) k (cid:107) ≥ δ √ d − (cid:15) √ A , we conclude that (cid:107) P k − R (cid:48) k (cid:107) − (cid:107) P k − P k (cid:107) P k (cid:107) (cid:107) > (cid:107) P k (cid:107) P k (cid:107) − R (cid:48) k (cid:107) ≥ d (cid:48) = Cd (cid:48) and we are done. D. CHENG AND K. A. OKOUDJOU
Assume now C = √ A − (cid:15) < (cid:107) P k (cid:107) + η ≤
1, where η is defined inFigure 1. Figure 1. Q is the orthogonal projection of R (cid:48) k onto P k ,and η = (cid:107) Q − P k (cid:107) P k (cid:107) (cid:107) .Then, (cid:13)(cid:13)(cid:13)(cid:13) P k − R (cid:48) k (cid:107) − (cid:107) P k − P k (cid:107) P k (cid:107) (cid:13)(cid:13)(cid:13)(cid:13) = (1 − ( (cid:107) P k (cid:107) + η )) + 2 η − η − (1 − (cid:107) P k (cid:107) ) = 2 η (cid:107) P k (cid:107) = (cid:13)(cid:13)(cid:13)(cid:13) P k (cid:107) P k (cid:107) − R (cid:48) k (cid:13)(cid:13)(cid:13)(cid:13) (cid:107) P k (cid:107) . The the conclusion follows from (cid:107) P k (cid:107) P k (cid:107) − R (cid:48) k (cid:107) ≥ d (cid:48) .Now assume (cid:107) P k (cid:107) + η > η ≤
1, where η is defined in Figure 2. (cid:13)(cid:13)(cid:13)(cid:13) P k − R (cid:48) k (cid:107) −(cid:107) P k − P k (cid:107) P k (cid:107) (cid:13)(cid:13)(cid:13)(cid:13) = (( (cid:107) P k (cid:107) + η ) − +2 η − η − (1 −(cid:107) P k (cid:107) ) = 2 η (cid:107) P k (cid:107) and the rest of the proof is similar to the one given above.If η > η is defined in Figure 3, then the angle ∠ P k R (cid:48) k > π hence (cid:107) P k − R (cid:48) k (cid:107) > (cid:107) P k (cid:107) + 1. We know (cid:107) P k − P k (cid:107) P k (cid:107) (cid:107) ≤ (cid:107) P k − R (cid:48) k (cid:107) − (cid:107) P k − P k (cid:107) P k (cid:107) (cid:107) > (cid:107) P k (cid:107) ≥ C Figure 2. Q is the orthogonal projection of R (cid:48) k onto P k , and η = (cid:107) Q − P k (cid:107) P k (cid:107) (cid:107) . (cid:3) We are now ready to prove Theorem 2.1.
Proof of Theorem 2.1.
Assume by way of contradiction that there exists δ > (cid:15) > (cid:15) = { ϕ i,(cid:15) } N (cid:15) i =1 , ∈ F A,B . and Ψ (cid:15) = { ψ i,(cid:15) } N (cid:15) i =1 such that d (Φ (cid:15) , Ψ (cid:15) ) < (cid:15) and d ( S − / (cid:15) Φ (cid:15) , S − / (cid:15) Ψ (cid:15) ) > δ. Furthermore, choose (cid:15) small enough so that δ √ d − (cid:15) √ A > √ A − (cid:15) > d (cid:88) i =1 ( (cid:107) P i − R i (cid:107) R i (cid:107) (cid:107) − (cid:107) P i − P i (cid:107) P i (cid:107) (cid:107) ) < min ( Cd (cid:48) , C )where C and d (cid:48) are as in Lemma 2.4(such (cid:15) exists by Lemma 2.3).Hence d (cid:88) i =1 ( (cid:107) P i − R i (cid:107) R i (cid:107) (cid:107) − (cid:107) P i − P i (cid:107) P i (cid:107) (cid:107) ) < d (cid:88) i =1 ( (cid:107) P i − R (cid:48) i (cid:107) − (cid:107) P i − P i (cid:107) P i (cid:107) (cid:107) ) Figure 3. Q is the orthogonal projection of R (cid:48) k onto P k , and η = (cid:107) Q − P k (cid:107) P k (cid:107) (cid:107) .Consequently, (cid:80) di =1 (cid:107) P i − R i (cid:107) R i (cid:107) (cid:107) < (cid:80) di =1 (cid:107) P i − R (cid:48) i (cid:107) contradicting that R (cid:48) i are the rows of the closest Parseval frame to Ψ (cid:15) = { ψ i,(cid:15) } N (cid:15) i =1 . (cid:3) Approximation of probabilistic frames in the − Wassersteinmetric.
In this section we prove some of the technical results needed toestablish our main result. The key idea is that a probabilistic frame µ withframe bounds A, B can be approximated in the Wasserstein metric by afinite probabilistic frame whose bounds are arbitrarily close to
A, B . Weprove this statement in Proposition 2.7 and point out that it is a refinementof a well-known result, e.g., [16, Theorem 6.18]. But first, we prove a fewnew results about finite probabilistic frames that are of interest in their ownright. In particular, Lemma 2.5 will be a very useful technical tool that weshall often use. It shows that given a finite frame we may replace any framevector by a finite number of new vectors so as to leave unchanged the frameoperator. More specifically,
Lemma 2.5.
Given a frame
Φ = { ϕ i } Ni =1 with frame operator S Φ . Fix i ∈ { , , . . . , N } and consider the new set of vectors Φ i = { ϕ k } Nk =1 ,k (cid:54) = i ∪ { a j ϕ i } pj =1 = { ϕ (cid:48) k } N + p − k =1PTIMAL PROPERTIES OF THE CANONICAL TIGHT PROBABILISTIC FRAME 11 where (cid:80) pj =1 a j = 1 . Then, Φ i ∈ F ( N + p − , d ) , that is, Φ i is a frame for R d and its frame operator is S Φ . Furthermore, N (cid:88) k =1 (cid:107) ϕ k − ϕ † k (cid:107) = N + p − (cid:88) k =1 (cid:107) ϕ (cid:48) k − ϕ (cid:48) † k (cid:107) where ϕ † k = S − / ϕ k and ϕ (cid:48) † k = S − / ϕ (cid:48) k Proof.
It is easy to see that for each x ∈ R d we have N (cid:88) k =1 | (cid:104) x, ϕ k (cid:105) | = i − (cid:88) k =1 | (cid:104) x, ϕ k (cid:105) | + p (cid:88) j =1 a j | (cid:104) x, ϕ i (cid:105) | + N (cid:88) k = i +1 | (cid:104) x, ϕ k (cid:105) | . (cid:3) We now use Lemma 2.5 and Theorem 1.1 to find the closest Parsevalframe to a finite probabilistic frame in the 2-Wasserstein metric.
Proposition 2.6.
Let µ Φ ,w be a finite probabilistic frame with bounds A and B , where Φ = { ϕ i } Ni =1 ⊂ R d and w = { w i } Ni =1 ⊂ [0 , ∞ ) . Then the closestfinite Parseval probabilistic frame to Φ is Φ † = { S − / ϕ i } Ni =1 and it satisfies W ( µ Φ ,w , µ Φ † ,w ) = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) i =1 w i (cid:107) ϕ i − ˜ ϕ i (cid:107) ≤ (cid:113) d max(( √ A − , ( √ B − ) where ˜ ϕ i = S − / ϕ i .Proof. We first prove that W ( µ Φ ,w , µ Φ † ,w ) ≤ d max(( √ A − , ( √ B − ) . Let Φ w = {√ w i ϕ i } Ni =1 . Let S = Φ w Φ Tw = ODO T be the frame operatorof Φ w . Consider the columns of O as an orthonormal basis for R d . Writingthe vectors √ w k ϕ k with respect to this basis leads to Φ (cid:48) w = O T Φ w whereΦ w = | ... |√ w ϕ ... √ w n ϕ m | ... | Let { P k,w } dk =1 and { R k,w } dk =1 respectively denote the rows of Φ (cid:48) w and Φ w .Notice that √ A ≤ (cid:107) P k,w (cid:107) ≤ √ B, ∀ k = 1 , , . . . , d. It is easily seen thatmin u ∈ R d , (cid:107) u (cid:107) =1 (cid:107) P k,w − u (cid:107) = (cid:107) P k,w − P k,w (cid:107) P k,w (cid:107) (cid:107) = |(cid:107) P k,w (cid:107)− | ≤ max(( √ A − , ( √ B − ) . But by construction, (cid:104) P k,w , P (cid:96),w (cid:105) = 0 for k (cid:54) = (cid:96) , and P k,w (cid:107) P k,w (cid:107) = λ − / k P k,w where λ k is the k th eigenvalue of S . Consequently, { λ − / k P k,w } dk =1 represents the rows of the canonical tight frame S − / Φ w written in the orthonormalbasis O . Therefore, d (Φ w , S − / Φ w ) = d (cid:88) k =1 (cid:107) P k,w − λ − / k P k,w (cid:107) ≤ d max(( √ A − , ( √ B − ) . Clearly, W ( µ Φ ,w , µ S − / Φ ,w ) ≤ N (cid:88) i =1 w i (cid:107) ϕ i − S − / ϕ i (cid:107) = d (Φ w , S − / Φ w ) ≤ d max(( √ A − , ( √ B − ) . Suppose there exists a finite probabilistic Parseval frame µ Ψ ,v where Ψ = { ψ i } Mi =1 ⊂ R d , v = { v i } Mi =1 ⊂ [0 , ∞ ) such that W ( µ Φ ,w , µ Ψ ,v ) < N (cid:88) i =1 w i (cid:107) ϕ i − S − / ϕ i (cid:107) . Let γ ∈ Γ( µ Φ ,w , µ Ψ ,v ) be such that W ( µ Φ ,w , µ Ψ ,v ) = (cid:90) (cid:90) R d (cid:107) x − y (cid:107) dγ ( x, y ) . Note that γ is a discrete measure with γ ( x, y ) = (cid:80) i,j w (cid:48) i,j δ ϕ i ( x ) δ ψ i ( y ) with (cid:80) j w (cid:48) i,j = w i and (cid:80) i w (cid:48) i,j = v j .Furthermore, by assumption W ( µ Φ ,w , µ Ψ ,v ) = (cid:88) i,j w (cid:48) i,j (cid:107) ϕ i − ψ j (cid:107) < N (cid:88) i =1 w i (cid:107) ϕ i − S − / ϕ i (cid:107) . Notice since (cid:80) i w (cid:48) i,j = v j the frame Ψ (cid:48) = { (cid:113) w (cid:48) i,j ψ j } i,j is a Parseval frame.Since (cid:80) j w (cid:48) i,j = w i , it easy to see that (cid:80) j w (cid:48) i,j w i = 1. We now use Lemma 2.5.For each i, replace √ w i ϕ i with { (cid:113) w (cid:48) i,j ϕ i } j . This results in a frame Φ (cid:48) = { (cid:113) w (cid:48) i,j ϕ i } i,j . Consequently, d (Φ (cid:48) , Ψ (cid:48) ) = d (Φ w , Ψ v ) < d (Φ w , Φ † w ) where Ψ v is a Parseval frame. This is a contradiction. (cid:3) The next result is one of our key technical results. It allows us to approx-imate a probabilistic frame in the 2-Wasserstein metric with a compactlysupported finite probabilistic frame whose bounds are controlled by those ofthe original probabilistic frame.
Theorem 2.7.
Let µ be a probabilistic frame with frame bounds A and B ,and (cid:15) > . Then, there exists a finite probabilistic µ Φ with frame bounds A (cid:48) , B (cid:48) such that A (cid:48) ≥ A − (cid:15) , B (cid:48) ≤ B + (cid:15) and (cid:107) µ − µ Φ (cid:107) W < (cid:15). To establish this result we first prove the following two Lemmas.
PTIMAL PROPERTIES OF THE CANONICAL TIGHT PROBABILISTIC FRAME 13
Lemma 2.8.
Let µ be a probabilistic frame with frame bound A and B .Given (cid:15) > , there exists a probabilistic frames ν with compact support andframe bounds A (cid:48) , B (cid:48) such that (a) W ( µ, ν ) < (cid:15) , (b) A (cid:48) ≥ A − (cid:15) , and B (cid:48) = B .Proof. (a) Let µ be a probabilistic frame with frame bound A and B .Given (cid:15) >
0, there exists R > (cid:90) R d \ B (0 ,R ) (cid:107) x (cid:107) dµ ( x ) < (cid:15). Let ν be the measure defined for each Borel set A ⊂ R d by ν ( A ) = µ ( A (cid:92) B (0 , R ) + µ ( R d \ B (0 , R )) δ . Clearly, ν is a probabilistic measure with compact support.We consider the marginal γ of µ and ν defined for each Borel sets A, B ⊂ R d by γ ( A × B ) = (cid:26) µ ( A (cid:84) B (0 , R ) (cid:84) B ) + µ ( A (cid:84) B c (0 , R ) i f ∈ Bµ ( A (cid:84) B (0 , R ) (cid:84) B ) i f (cid:54)∈ B Since ν is supported in B (0 , R ) (cid:90) (cid:90) R d (cid:107) x − y (cid:107) dγ ( x, y ) = (cid:90) (cid:90) R d × B (0 ,R ) (cid:107) x − y (cid:107) dγ ( x, y )= (cid:90) (cid:90) B (0 ,R ) × B (0 ,R ) (cid:107) x − y (cid:107) dγ ( x, y )+ (cid:90) (cid:90) B c (0 ,R ) × B (0 ,R ) (cid:107) x − y (cid:107) dγ ( x, y ) . However, we know (cid:90) B (0 ,R ) × B (0 ,R ) (cid:107) x − y (cid:107) dγ ( x, y ) = 0since, when restricted to B (0 , R ) × B (0 , R ), γ is supported only onthe diagonal where (cid:107) x − y (cid:107) = 0 . Moreover, (cid:90) B c (0 ,R ) × B (0 ,R ) (cid:107) x − y (cid:107) dγ ( x, y ) = (cid:90) (cid:90) B c (0 ,R ) × B (0 ,R ) \{ } (cid:107) x − y (cid:107) dγ ( x, y )+ (cid:90) (cid:90) B c (0 ,R ) ×{ } (cid:107) x − y (cid:107) dγ ( x, y )= 0 + (cid:90) (cid:90) B c (0 ,R ) ×{ } (cid:107) x − y (cid:107) dγ ( x, y ) < (cid:15). Therefore, W ( µ, ν ) < (cid:15). (b) The upper bound B is obtained trivially as ν is µ restricted to B (0 , R ).For x ∈ R d we have (cid:82) | (cid:104) x, y (cid:105) | dν ( y ) = (cid:82) B (0 ,R ) | (cid:104) x, y (cid:105) | dµ ( y ) . From the fact that (cid:82) R d \ B (0 ,R ) (cid:107) x (cid:107) dµ ( x ) ≤ (cid:15) it follows that (cid:90) R d \ B (0 ,R ) | (cid:104) x, y (cid:105) | dµ ( y ) ≤ (cid:107) x (cid:107) (cid:15). (cid:3) Suppose that µ is a probabilistic frame supported in a ball B (0 , R ). Let r > Q = [0 , r ) d . Choose points { c k } Mk =1 ⊂ R d with c = 0such that B (0 , R ) = ∪ Mk =0 Q k where Q k = c k + Q . Observe that Q k ∩ Q (cid:96) = ∅ whenever k (cid:54) = (cid:96) . Let µ ,Q = (cid:80) Mk =1 µ ( Q k ) δ c k .Next partition each cube Q k uniformly into cube of size r/ µ ,Q as above. Iterate this process to construct asequence of probability measures µ n,Q . Lemma 2.9.
Let µ be a probabilistic frame with bounds A and B , whichsupported in a ball B (0 , R ) . For r > let { µ n,Q } ∞ n =1 be a sequence ofprobability measures as constructed above. Then, lim n →∞ W ( µ, µ n,Q ) = 0 . Furthermore, there exists N such that for all n ≥ N , µ n,Q is a finite proba-bilistic frame whose bounds are arbitrarily close to those of µ .Proof. Let d = max x ∈ Q k (cid:107) x − c k (cid:107) . Given, x ∈ Q k , x = c k + a k , where (cid:107) a k (cid:107) ≤ d . PTIMAL PROPERTIES OF THE CANONICAL TIGHT PROBABILISTIC FRAME 15
For any x ∈ R d , (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) B (0 ,R ) (cid:104) x, y (cid:105) dµ ( y ) − M (cid:88) k =1 (cid:104) x, c k (cid:105) µ ( Q k ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) k =1 (cid:90) Q k (cid:104) x, y (cid:105) dµ ( y ) − M (cid:88) k =1 (cid:104) x, c k (cid:105) µ ( Q k ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) k =1 (cid:90) Q k ( (cid:104) x, y (cid:105) − (cid:104) x, c k (cid:105) ) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ M (cid:88) k =1 (cid:90) Q k (cid:12)(cid:12) (cid:104) x, y (cid:105) − (cid:104) x, c k (cid:105) (cid:12)(cid:12) dµ ( y )= M (cid:88) k =1 (cid:90) Q k | (cid:104) x, c k + a k (cid:105) − (cid:104) x, c k (cid:105) | dµ ( y )= M (cid:88) k =1 (cid:90) Q k | (cid:104) x, a k (cid:105) + 2 (cid:104) x, c k (cid:105) (cid:104) x, a k (cid:105) | dµ ( y ) ≤ (cid:107) x (cid:107) M (cid:88) k =1 µ ( Q k )( (cid:107) a k (cid:107) + 2 (cid:107) c k (cid:107)(cid:107) a k (cid:107) ) ≤ ( d + 2 d ( R + d )) (cid:107) x (cid:107) . Note that by the iterative construction of µ n,Q we get that for each x ∈ R d (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R d (cid:104) x, y (cid:105) dµ ( y ) − (cid:90) R d (cid:104) x, y (cid:105) dµ n,Q ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( d n + 2 d n ( R + d n )) (cid:107) x (cid:107) where lim n →∞ d n = 0. It follows that given (cid:15) >
0, we can find
N > n ≥ N , (cid:90) R d (cid:104) x, y (cid:105) dµ n,Q ( y ) > (cid:90) R d (cid:104) x, y (cid:105) dµ ( y ) − (cid:15) (cid:107) x (cid:107) > (cid:107) x (cid:107) ( A − (cid:15) )which concludes that µ n,Q is a a finite probabilistic frame whose lower boundis at least A − (cid:15) . Furthermore, (cid:90) R d (cid:104) x, y (cid:105) dµ n,Q ( y ) < (cid:90) R d (cid:104) x, y (cid:105) dµ ( y ) + (cid:15) (cid:107) x (cid:107) ≤ (cid:107) x (cid:107) ( B + (cid:15) )which implies that the upper frame bound µ n,Q is at most B + (cid:15) .Next, fix n ≥ N and let γ n ( x, y ) be the measure on R d × R d be definedfor any Borel sets A, B ⊂ R d by: γ n ( A × B ) = (cid:88) k : c k ∈ B µ ( A ∩ Q k ) = M (cid:88) k =1 µ | Qk × δ c k ( A × B ) where A, B c k denoting the centers of the cubes Q k . It is easy to see that γ n ∈ Γ( µ, µ n,Q ) and so W ( µ, µ n,Q ) ≤ (cid:90) (cid:90) (cid:107) x − y (cid:107) dγ n ( x, y )= M (cid:88) k =1 (cid:90) (cid:90) (cid:107) x − y (cid:107) d ( µ | Qk × δ c k )( x, y )= M (cid:88) k =1 (cid:90) Q k (cid:107) x − c k (cid:107) dµ ( x ) ≤ M (cid:88) k =1 µ ( Q k ) (cid:90) Q k d n dµ ( x ) ≤ d n and the result follows from the fact that lim n →∞ d n = 0. (cid:3) We can now give a proof of Theorem 2.7.
Proof of Theorem 2.7.
Let µ be a probabilistic frame with frame bounds A and B , and (cid:15) >
0. By Lemma 2.8 let ν be a compactly supportedprobabilistic frame with frame bounds between A − (cid:15)/ B and suchthat W ( µ, ν ) < (cid:15)/ µ Φ ,w whoseframe bounds are within (cid:15)/ ν and such that W ( ν, µ Φ ,w ) < (cid:15)/ W ( µ, µ Φ ,w ) < (cid:15) which concludes the proof. (cid:3) Corollary 2.10.
Let µ be a probabilistic Parseval frame and (cid:15) > . Then,there exists a finite Parseval probabilistic frame µ Φ ,w with W ( µ, µ Φ ,w ) < (cid:15). Proof.
This follows from Proposition 2.6 and Theorem 2.7. (cid:3)
Remark . Since the set of finite Parseval frames is dense in the set of allParseval frames in the Wasserstein metric, by Proposition 2.6 since there isno finite Parseval frame closer to Φ than Φ † = { S − / ϕ i } Ni =1 , there are noParseval frame closer to Φ than Φ † .2.3. The closest Parseval frame in the − Wasserstein distance.
Inthis section we prove and state of our main result, Theorem 2.12. We recallthat if µ is a probabilistic frame for R d , then its probabilistic frame operator(equivalently, the matrix of second moments associated to µ ) S µ : R d → R d , S µ ( x ) = (cid:90) R d (cid:104) x, y (cid:105) ydµ ( y ) PTIMAL PROPERTIES OF THE CANONICAL TIGHT PROBABILISTIC FRAME 17 is positive definite, and thus S − / µ exists. We define the push-forward of µ through S − / µ by µ † ( B ) = µ ( S / B )for each Borel set in R d . Alternatively, if f is a continuous bounded functionon R d , (cid:90) R d f ( y ) dµ † ( y ) = (cid:90) R d f ( S − / µ y ) dµ ( y ) . It then follows that x = S − / µ S µ S − / µ ( x ) = (cid:90) R d (cid:68) S − / µ x, y (cid:69) S − / µ y dµ ( y ) = (cid:90) R d (cid:104) x, y (cid:105) y dµ † ( y )implying that µ † is a Parseval probabilistic frame [9, 15]. In particular, S µ † = I where I is the identity matrix on R d . As was the case with thecanonical Parseval frame Φ † of a given frame Φ, µ † is the (unique) closestParseval probabilistic frame to µ . Theorem 2.12.
Let µ be a probabilistic frame on R d with probabilistic frameoperator S µ . Then µ † is the (unique) closest probabilistic Parseval frame to µ in the − Wasserstein metric, that is (6) µ † = arg min W ( µ, ν ) where ν ranges over all Parseval probabilistic frames. Before proving this theorem, we need to establish a few preliminary re-sults. We start by extending Theorem 2.1 to finite probabilistic frames inthe Wasserstein metric. In particular, this extension allows use to deal withfinite probabilistic frames of different cardinalities.
Theorem 2.13.
Let < A ≤ B < ∞ , and δ > be given. Then thereexists (cid:15) > such that given any finite probabilistic frame µ Φ ,w = (cid:80) Ni =1 w i δ ϕ i with frame bounds between A and B , N := N Φ ≥ , Φ = { ϕ i } Ni =1 ⊂ R d ,and weights w = { w i } Ni =1 ⊂ [0 , ∞ ) , for any finite probabilistic frame µ Ψ ,η = (cid:80) Mi =1 η i δ ψ i , M := M Ψ ≥ , where Ψ = { ψ i } Mi =1 ⊂ R d , and weights η = { η i } Ni =1 ⊂ [0 , ∞ ) if W ( µ Φ ,w , µ Ψ ,η ) < (cid:15), then we have W ( F ( µ Φ ,w ) , F ( µ Ψ ,η )) < δ. Proof.
Fix δ >
0. By Theorem 2.1 we know that there exists (cid:15) such thatgiven a frame X = { x i } Mi =1 ( M ≥ A and B , and Y = { y i } Mi =1 is a frame such that d ( X, Y ) = (cid:118)(cid:117)(cid:117)(cid:116) M (cid:88) i =1 (cid:107) x i − y i (cid:107) < (cid:15) then d ( F ( X ) , F ( Y )) = d ( S − / X X, S − / Y Y ) < δ. Let µ Φ ,w = (cid:80) Ni =1 w i δ ϕ i be a finite probabilistic frame with frame boundsbetween A and B , N ≥
2, Φ = { ϕ i } Ni =1 ⊂ R d , and weights w = { w i } Ni =1 ⊂ [0 , ∞ ). Then by Theorem 2.6, µ Φ † ,w where Φ † = { S − / ϕ i } Ni =1 is the closestParseval frame to µ Φ ,w .Let µ Ψ ,v where Ψ = { ψ i } Mi =1 , M ≥ W ( µ Φ ,w , µ Ψ ,η ) < (cid:15) .Choose γ ∈ Γ( µ Φ ,w , µ Ψ ,v ) such that W ( µ Φ ,w , µ Ψ ,η ) = (cid:90) (cid:90) R d × R d (cid:107) x − y (cid:107) dγ ( x, y ) < (cid:15) . Identify γ with { w i,j } N,Mi,j =1 . Then, W ( µ Φ ,w , µ Ψ ,η ) = (cid:90) (cid:90) R d × R d (cid:107) x − y (cid:107) dγ ( x, y ) = M (cid:88) i =1 N (cid:88) j =1 w i,j (cid:107) ϕ i − ψ j (cid:107) < (cid:15) . Observe that Φ (cid:48) = {√ w i,j ϕ i } M,Ni,j =1 is a frame whose frame bounds are thesame as those for µ Φ ,w . Similarly, Ψ (cid:48) = {√ w i,j ψ j } M,Ni,j =1 is a frame whoseframe bounds are the same as those for µ Ψ ,η . Furthermore, d (Φ (cid:48) , Ψ (cid:48) ) = W ( µ Φ ,w , µ Ψ ,η ) < (cid:15) which implies that d ( F (Φ (cid:48) ) , F (Ψ (cid:48) )) = M,N (cid:88) i,j =1 (cid:107) S − / ( √ w i,j ϕ i ) − S − / ( √ w i,j ψ j ) (cid:107) < δ . However, (cid:88) i,j (cid:107) S − / ( √ w i,j ϕ i ) − S − / ( √ w i,j ψ j ) (cid:107) = (cid:88) i,j w i,j (cid:107) S − / ϕ i − S − / ψ j (cid:107) But since w i,j = γ ( { ϕ i } , { ψ j } ) we have (cid:80) j w i,j = w i and (cid:80) i w i,j = v j wesee that W ( F ( µ Φ ,w ) , F ( µ Ψ ,η )) = W ( µ Φ † ,w , µ Ψ † ,v ) ≤ (cid:88) i,j w i,j (cid:107) S − / ϕ i − S − / ψ j (cid:107) . (cid:3) Let
DP F ( A, B ) denote the set of all discrete (finite) probabilistic framesin R d whose lower frame bounds are less than or equal to A and whose upperbounds are greater or equals to B . It follows from the above result that F isuniformly continuous from DP F ( A, B ) into itself when equipped with theWasserstein metric. Consequently, we can prove the following result.
Proposition 2.14.
Let µ be a probabilistic frame with frame bounds A and B . Let µ k := µ Φ k ,w k , where Φ k := Φ k,w k = { ϕ k } N k k =1 and ν k := µ Ψ k ,v k , where Ψ k := Ψ k,v k = { ψ k } M k k =1 be two sequences of finite probabilistic frames in PTIMAL PROPERTIES OF THE CANONICAL TIGHT PROBABILISTIC FRAME 19 R d such that lim k →∞ W ( µ, µ Φ k ) = lim k →∞ W ( µ, µ Ψ k ) = 0 . Furthermore,suppose that the frame bounds of µ Φ k are between A/ and B + A/ . Then lim k →∞ F ( µ Φ k ) = lim k →∞ F ( µ Ψ k ) . Proof.
Theorem 2.7 ensures the existence of the finite probabilistic frames µ Φ k .Let δ > (cid:15) > ν and any k ≥ W ( µ Φ k , ν ) < (cid:15) = ⇒ W ( F ( ν ) , F ( µ Φ k )) < δ. Choose N (cid:15) > k > N (cid:15) , W ( µ, µ Φ k ) < (cid:15) and W ( µ, µ Ψ k ) < (cid:15) . Thus, for k ≥ N (cid:15) , W ( µ Φ k , µ Ψ k ) < (cid:15) , which implies that for all k ≥ N (cid:15) , W ( F ( µ Φ k ) , F ( µ Ψ k )) < δ . It easily follows that lim k →∞ F ( µ Φ k ) =lim k →∞ F ( µ Ψ k ). (cid:3) We can now use this proposition to extend the definition of the map F to all probabilistic frames. Let µ be a probabilistic frame with bounds0 < A ≤ B < ∞ . Let { µ Φ k } ∞ k =1 be a sequence of finite probabilistic frameswith bounds between A/ B + A/ lim k →∞ W ( µ Φ k , µ ) = 0.Then, F ( µ ) = lim k →∞ F ( µ Φ k )is well-defined. Before proving Theorem 2.12 we first identify the minimizerof (6) with F ( µ ). Theorem 2.15.
Let µ be a probabilistic frame on R d with probabilistic frameoperator S µ . Then F ( µ ) is the unique closest probabilistic Parseval frame to µ in the − Wasserstein metric, that is F ( µ ) is the unique solution to (6) .Proof. Set Q = min W ( µ, ν ) where ν ranges over all Parseval probabilisticframes.Let δ >
0, and µ be a probabilistic frame wth frame bounds A and B . ByTheorem 2.7, there exists a sequence of finite probabilistic frame µ Φ k withframe bounds between A and B + A where Φ k := Φ k,w ( k ) = { ϕ k } N k k =1 ⊂ R d , w ( k ) = { w n } N k n =1 ⊂ (0 , ∞ ), and N k ≥ k →∞ W ( µ, µ Φ k ) = 0.Observe that for all k ≥ W ( µ, F ( µ Φ k )) ≤ W ( µ, F ( µ )) + W ( F ( µ ) , F ( µ Φ k )) . Choose (cid:15) > K ≥ W ( µ, µ Φ K ) < (cid:15). Thus, W ( F ( µ ) , F ( µ Φ K )) < δ. Consequently, W ( µ, F ( µ Φ K )) ≤ W ( µ, F ( µ )) + W ( F ( µ ) , F ( µ Φ K )) < W ( µ, F ( µ )) + δ. Since F ( µ Φ K ) is a Parseval frame we conclude that F ( µ ) minimizes (6).We now prove that F ( µ ) is the unique minimizer of (6) by consideringthree cases. Case 1. If µ is a finite frame Φ = { ϕ i } Ni =1 ⊂ R d , it is known that S − / Φ isthe (unique) closest Parseval frame to Φ, see Theorem 1.1, and [5, Theorem3.1].
Case 2. If µ = µ Φ ,w , where Φ = { ϕ i } Ni =1 ⊂ R d , and w = { w i } Ni =1 ⊂ [0 , ∞ ).Then, µ Φ † ,w where Φ † = S − / Φ is the unique closest Parseval probabilisticframe to Φ. Indeed, we already know that µ Φ † ,w achieves the minimumdistance Proposition 2.6. We now prove that it is unique. We argue bycontradiction and assume that there exists another Parseval probabilisticframe ν that achieves this distance.First, we assume that ν = µ κ,v where κ = { κ (cid:48) i } Mi =1 ⊂ R d with weights v = { v i } Mi =1 ⊂ [0 , ∞ ). Let γ ∈ Γ( µ, ν ) such that W ( µ, ν ) = (cid:90) (cid:90) (cid:107) x − y (cid:107) dγ ( x, y ) . For all i , j let w i,j = γ ( ϕ i , κ (cid:48) j ). Let Q = (cid:80) Ni =1 w i (cid:107) ϕ i − ϕ † i (cid:107) , where ϕ † i = S − / ϕ i . Since κ also achieved this distance we clearly have Q = (cid:80) i,j w i,j (cid:107) ϕ i − κ (cid:48) j (cid:107) .We now use Lemma 2.5. For each i , we replace the vector ϕ i and itsweight w i by M copies of itself (i.e., ϕ i ) each weighted by w i,j . Apply thesame procedure to Φ † , and to κ , except that for the latter we break eachvector κ (cid:48) j into N copies of itself with weights w i,j . Denote by F , F , and F the three resulting frames. We note that the vectors in each of these framescan be considered to have weight 1.It follows from Theorem 1.1 that the finite frame F = {√ w i,j κ (cid:48) j } i,j is the(unique) closest Parseval frame to F = {√ w i,j ϕ i } i,j , which we also know is F = {√ w i,j ϕ † i } i,j . Therefore, µ κ,v = µ Φ † ,w .Next, we assume that ν is not discrete. Choose a sequence of finite Par-seval frames { ν n } ∞ n such that lim n →∞ W ( ν n , ν ) = 0. Hence, Q = W ( µ, F ( µ )) = W ( µ Φ ,w , ν ) = lim n →∞ W ( µ Φ ,w , ν n ) . We now prove that lim n →∞ W ( ν n , µ Φ † ,w ) = 0 . Let δ > N ≥ n > NW ( ν n , µ Φ ,w ) < Q + δ. Suppose by contradiction that lim n →∞ W ( ν n , µ Φ † ,w ) >
0. Thus, there is (cid:15) > k ≥
1, there exists n > max( k, N ) such that W ( ν n , µ Φ † ,w ) > (cid:15). For n given above, let γ n ∈ Γ( ν n , µ Φ ,w ) be such that W ( ν n , µ Φ ,w ) = (cid:90) (cid:90) R d (cid:107) x − y (cid:107) dγ n ( x, y ) . PTIMAL PROPERTIES OF THE CANONICAL TIGHT PROBABILISTIC FRAME 21
Since ν n is a finite probabilistic frame we may assume further that ν n = µ u n ,v where u n = { ψ i } Mi =1 ⊂ R d and v = { v i } Mi =1 ⊂ [0 , ∞ ). For the sake ofsimplicity in notations, we omit the dependence of both ψ i and v i on n . Let w n,j,k = γ n ( ϕ j , ψ k ).Now consider the finite frames { u (cid:48) j } j = {√ w n,j,k ψ k } j,k and Φ (cid:48) = {√ w n,j,k ϕ j } j,k .Note that W ( µ Φ (cid:48) , µ Φ (cid:48)† ) = Q . Now we consider the rows of these frameswritten with respect to the eigenbasis of the frame operator S := S Φ (cid:48) of Φ (cid:48) .Because, W ( ν n , µ Φ † ,w ) > (cid:15), then (cid:80) j,k w n,j,k (cid:107) ψ k − S − / ϕ j (cid:107) > (cid:15) .Using this and Lemma 2.4 we have the following estimates: W ( µ Φ ,w , ν n ) ≥ W ( µ Φ ,w , ν ) + min( (cid:15) d · M, M )where A is the lower frame bound of Φ and M = min(1 , √ A ).Consequently, W ( µ Φ ,w , ν n ) − Q ≥ min( (cid:15) d · M, M ) > . But, this contradicts the fact that Q = W ( µ Φ ,w , ν ) = lim n →∞ W ( ν Φ ,w , ν n ) . Hence, lim n →∞ W ( ν n , µ S − / Φ ,w ) = 0, and ν = µ Φ † ,w . Case 3:
Next, we suppose that µ is non discrete probabilistic frame withframe bounds A, and B . Let { µ n } ∞ n =1 = { µ Φ n ,w ( n ) } ∞ n =1 be a sequence offinite probabilistic frames with bounds between A/ B + A/ n →∞ W ( µ n , µ ) = 0. Then F ( µ ) = lim n →∞ F ( µ n ) is such that Q = W ( F ( µ ) , µ ) . Suppose there exists another Parseval frame ν such that Q = W ( ν, µ ). Choose a sequence of finite Parseval { ν n } ∞ n =1 such thatlim n →∞ ν n = ν. Observe that Q = lim n →∞ W ( µ n , F ( µ n )) = lim n →∞ W ( ν n , µ n ). WriteΦ n = { ϕ n,j } Mj =1 and w ( n ) = { w j } Mj =1 , where for simplicity we omit thedependence of M on n . Similarly, { ν n } ∞ n =1 = { ψ n,j } M (cid:48) j = i with weights v ( n ) = { v j } M (cid:48) j =1 .Let γ n ∈ Γ( µ n , ν n ) be such that W ( µ n , ν n ) = (cid:90) (cid:90) (cid:107) x − y (cid:107) dγ n ( x, y ) . Set w j,k = γ n ( ϕ n,j , ψ n,k )We know that W ( µ n , F ( µ n )) = M (cid:88) j =1 w j (cid:107) ϕ n,j − ϕ † n,j (cid:107) = (cid:88) j,k w j,k (cid:107) ϕ n,j − ϕ † n,j (cid:107) = (cid:88) j,k (cid:107)√ w j,k ϕ n,j − √ w j,k ϕ † n,j (cid:107) We also know that W ( µ n , ν n ) = (cid:88) j,k w j,k (cid:107) ϕ n,j − ψ n,k (cid:107) = (cid:88) j,k (cid:107)√ w j,k ϕ n,j − √ w j,k ψ n,k (cid:107)
22 D. CHENG AND K. A. OKOUDJOU
Suppose that lim n →∞ W ( F ( µ n ) , ν n ) >
0. Thus, there exists (cid:15) > n > W ( F ( µ n ) , ν n ) > (cid:15) . Consequently, (cid:15) < (cid:88) j,k w j,k (cid:107) ϕ † n,j − ψ n,k (cid:107) = (cid:88) j,k (cid:107)√ w j,k ϕ † n,j − √ w j,k ψ n,k (cid:107) Hence d (Φ (cid:48)† n , Ψ (cid:48) n ) > (cid:15) where Ψ (cid:48) n = {√ w j,k ψ n,k } .By the same argument as in Lemma 2.4 we conclude that W ( µ n , ν n ) − W ( µ n , F ( µ n )) ≥ min ( M (cid:15) d , M ) . where M = min(1 , (cid:113) A )This contradicts the fact that Since lim n →∞ W ( µ n , ν n ) = Q = lim n →∞ W ( µ n , F ( µ n )).Thus lim n →∞ W ( F ( µ n ) , ν n ) = 0 and so F ( µ ) = ν . (cid:3) By Proposition 2.14 it follows that given a probabilistic frame µ and anysequence Φ k := Φ k,w k = { ϕ k } N k k =1 of finite probabilistic frames in R d suchthat lim k →∞ W ( µ, µ Φ k ) = 0, then F ( µ ) = lim k →∞ F ( µ Φ k ) . Furthermore, itis proved in [18] that if { µ n } n ≥ ⊂ P converges in the Wassertein metric to µ ∈ P , then (cid:107) S µ − S µ n (cid:107) ≤ CW ( µ n , µ ) . All that is needed to prove Theorem 2.12 is to show that F ( µ ) = µ † . Proof of Theorem 2.12.
Let µ be a probabilistic frame with bounds A, B .Let 0 < (cid:15) < A/ ν (cid:15) as in Lemma 2.8. In particular ν (cid:15) is supported on B (0 , R (cid:15) ) with framebounds between A/ B + A/
2, where R (cid:15) > (cid:90) R d \ B (0 ,R (cid:15) ) (cid:107) x (cid:107) dx < (cid:15)/ . Choose a finite probabilistic frame µ (cid:15) with bounds between A and B + A such that W ( µ (cid:15) , ν (cid:15) ) < (cid:15) . By taking a sequence { (cid:15) n } ∞ n =1 ⊂ [0 , ∞ ) withlim n →∞ (cid:15) n = 0, we can pick { µ n } n ≥ := { µ (cid:15) n } n ≥ such that lim n →∞ W ( µ n , µ ) =0. Consequently, lim n →∞ S µ n = S µ , and lim n →∞ S − / µ n = S − / µ in the op-erator norm.We recall that lim n →∞ W ( µ n , µ ) = 0 is equivalent tolim n →∞ (cid:90) f dµ n ( x ) = (cid:90) f dµ ( x )for all continuous function f such that | f ( x ) | ≤ C (1 + (cid:107) x − x (cid:107) ) for some x ∈ R d [16, Theorem 6.9]We know that lim n →∞ F ( µ n ) = lim n →∞ µ † n = F ( µ ) in the Wassersteinmetric. We would like to show that lim n →∞ F ( µ n ) = lim n →∞ µ † n = µ † . PTIMAL PROPERTIES OF THE CANONICAL TIGHT PROBABILISTIC FRAME 23
We show that for all continuous function f such that | f ( x ) | ≤ C (1 + (cid:107) x − x (cid:107) ) for some x ∈ R d lim n →∞ (cid:90) f dµ † n ( x ) = (cid:90) f dµ † ( x ) . | (cid:90) f dµ † n ( x ) − (cid:90) f dµ † ( x ) | = | (cid:90) f ( S − / µ n x ) dµ n ( x ) − (cid:90) f ( S − / µ x ) dµ ( x ) |≤ (cid:90) | f ( S − / µ n x ) − f ( S − / µ x ) | dµ n ( x )+ | (cid:90) f ( S − / µ x ) dµ n ( x ) − (cid:90) f ( S − / µ x ) dµ ( x ) | Let f be continuous with | f ( x ) | ≤ C (1 + (cid:107) x − x (cid:107) ) for some x ∈ R d .Then, f ( S − / µ ) is continuous and satisfies | f ( S − / µ x ) | ≤ C (1+ (cid:107) x − S − / µ x (cid:107) ) ≤ C (1+ (cid:107) S − / µ (cid:107) (cid:107) x − S / µ x (cid:107) ) ≤ C (cid:48) (1+ (cid:107) x − S / µ x (cid:107) )) . Consequently, we can find N such that for all n ≥ N , | (cid:90) f ( S − / µ x ) dµ n ( x ) − (cid:90) f ( S − / µ x ) dµ ( x ) | < (cid:15)/ . Since f is continuous, there exists δ > x, y ∈ B (0 , R (cid:48) ), (cid:107) x − y (cid:107) < δ implies that | f ( x ) − f ( y ) | < (cid:15)/
3, where R (cid:48) > n , and x ∈ B (0 , R ), S − / µ n x, S − / µ x ∈ B (0 , R ).Since, lim n →∞ S − / µ n = S − / µ , there exists N such that for all n ≥ N , (cid:107) S − / µ n x − S − / µ x (cid:107) ≤ (cid:107) S − / µ n − S − / µ (cid:107)(cid:107) x (cid:107) ≤ R (cid:107) S − / µ n − S − / µ (cid:107) < δ. Therefore, for n ≥ N , | f ( S − / µ n x ) − f ( S − / µ x ) | < (cid:15)/ x ∈ B (0 , R ).Consequently, (cid:90) | f ( S − / µ n x ) − f ( S − / µ x ) | dµ n ( x ) = (cid:90) B (0 ,R ) | f ( S − / µ n x ) − f ( S − / µ x ) | dµ n ( x )+ (cid:90) R d \ B (0 ,R ) | f ( S − / µ n x ) − f ( S − / µ x ) | dµ n ( x ) < (cid:15)/ (cid:90) R d \ B (0 ,R ) | f ( S − / µ n x ) − f ( S − / µ x ) | dµ n ( x ) < (cid:15)/ M (cid:90) R d \ B (0 ,R ) (cid:107) x (cid:107) dµ n ( x ) < (cid:15)/ M > f , and µ .It follows that for all n ≥ max( N , N ) , we have | (cid:90) f dµ † n ( x ) − (cid:90) f dµ † ( x ) | < (cid:15) which implies that lim n →∞ (cid:82) f dµ † n ( x ) = (cid:82) f dµ † ( x ) . (cid:3) Acknowledgment
Both authors were partially supported by ARO grant W911NF1610008.K. A. Okoudjou was also partially supported by a grant from the SimonsFoundation
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Department of Mathematics, University of Missouri, Columbia, MO 65211-4100
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