Duality and Geodesics for Probabilistic Frames
aa r X i v : . [ m a t h . F A ] M a y Duality and Geodesics for Probabilistic Frames
Clare Wickman
Johns Hopkins University Applied Physics Laboratory
Kasso Okoudjou
Department of Mathematics, University of Maryland, College Park
Keywords: frames, probabilistic frames, optimal transport, Wasserstein metric, duality
Abstract
Probabilistic frames are a generalization of finite frames into the Wasserstein space of proba-bility measures with finite second moment. We introduce new probabilistic definitions of duality,analysis, and synthesis and investigate their properties. In particular, we formulate a theory oftransport duals for probabilistic frames and prove certain properties of this class. We also inves-tigate paths of probabilistic frames, identifying conditions under which geodesic paths betweentwo such measures are themselves probabilistic frames. In the discrete case this is related toranks of convex combinations of matrices, while in the continuous case this is related to thecontinuity of the optimal transport plan.
Frames are redundant spanning sets of vectors or functions that can be used to represent signalsin a faithful but nonunique way and that provide an intuitive framework for describing and solvingproblems in coding theory and sparse representation. We refer to [5, 4, 19] for more details onframes and their applications. To set the notations for this paper, we recall that a set of columnvectors Φ “ t ϕ i u Ni “ Ă R d is a frame if and only if there exist 0 ă A ď B ă 8 such that @ x P R d , A k x k ď N ÿ i “ x x , ϕ i y ď B k x k . Throughout this paper we abuse notation by also using Φ to denote r ϕ . . . ϕ N s J , the analysisoperator of the frame. The (optimal) bounds in the above inequality are the smallest and largesteigenvalues of the frame operator S Φ “ Φ J Φ . X be a metrizable, locally compact space and ν bepositive, inner regular Borel measure for X supported on all of X . Let H be a Hilbert space. Thena set of vectors η ix , i P t , ¨ ¨ ¨ , n u , x P X ( Ă H is a rank- n (continuous) frame if, for each x P X ,the vectors η ix , i P t , ¨ ¨ ¨ , n u ( are linearly independent, and if there exist 0 ă A ď B ă 8 suchthat @ f P H , A k f k ď n ÿ i “ ż X |x η ix , f y| dν p x q ď B k f k . In this paper, we are concerned with a different generalization of frames called probabilisticframes. Developed in a series of papers [8, 10, 9], probabilistic frames are an intuitive way to gener-alize finite frames to the space of probability measures with finite second moment. The probabilisticsetting is particularly compelling, given recent interest in probabilistic approaches to optimal coding,such as [15, 20]. In the new setting, the defining characteristics of a frame amount to a restrictionon the mean and covariance matrix of the probability measure. Because of this characterization, anatural space to explore probabilistic frames is the Wasserstein space of probability measures withfinite second moment, a metric space with distance defined by an optimal transport problem.Before we give the definitions and the concepts needed to state our results we first observe thatin the simplest example, each finite frame can be associated with a probabilistic frame. Indeed,let Φ “ t ϕ i u Ni “ be a frame and let t α i u Ni “ Ă p , q be such that ř Ni “ α i “
1. Then the canonical α -weighted probabilistic frame associated with Φ is the probability measure µ Φ ,α given by dµ Φ ,α p x q “ N ÿ i “ α i δ ϕ i p x q . More generally, a probabilistic frame µ for R d is a probability measure on R d for which there exist0 ă A ď B ă 8 such that for all x P R d , A k x k ď ż R d x x , y y dµ p y q ď B k x k . Tight (finite, continuous, or probabilistic) frames are those for which the frame bounds are equal.While the work of this paper is limited to probabilistic frames on R d , of interest is also the possibleextension of these ideas to probabilistic frames on infinite dimensional spaces, as outlined in [17].Probabilistic frames form a subclass of the continuous frames defined above. Indeed, definingthe support of a probability measure µ on R d as the set:supp p µ q : “ ! x P R d s.t. for all open sets U x containing x, µ p U x q ą ) , it is not difficult to prove that the support of any probabilistic frame is canonically associatedwith a rank-one continuous frame. And conversely, certain continuous frames can be rewritten asprobabilistic frames. However, despite this equivalence, there is a strong difference in the tools2vailable in the different settings.We shall investigate probabilistic frames in the setting of the Wassertein metric defined on P p R d q , the set of probability measures µ on R d with finite second moment: M p µ q : “ ż R d k x k dµ p x q ă 8 . By [10, Theorem 5], µ is a probabilistic frame if and only if it has finite second moment and thelinear span of its support is R d . This characterization can be restated in terms of the probabilisticframe operator for µ , S µ , which for all y P R d satisfies: S µ y “ ż R d x x , y y x dµ p x q . Equating S µ with its matrix representation ş R d xx J dµ p x q , the requirement that the support of µ span R d is equivalent to this matrix being positive definite.The (2-)Wasserstein distance, W between two probability measures µ and ν in P p R d q is: W p µ, ν q : “ inf γ $’&’% ij R d ˆ R d k x ´ y k dγ p x, y q : γ P Γ p µ, ν q ,/./- , where Γ p µ, ν q is the set of all joint probability measures γ on R d ˆ R d such that for all A, B Ă B p R d q , γ p A ˆ R d q “ µ p A q and γ p R d ˆ B q “ ν p B q . The Monge-Kantorovich optimal transport problem is thesearch for the set of joint measures which induce the infimum; any such joint distribution is calledan optimal transport plan. A special subclass of transport plans are those given by deterministictransport maps (or deterministic couplings), where ν can be written as the pushforward of µ by amap T, denoted ν “ T µ. That is, for all ν -integrable functions φ , ż R d φ p y q dν p y q “ ż R d φ p T p x qq dµ p x q . When µ is absolutely continuous with respect to Lebesgue measure [2, p. 150], then W p µ, ν q : “ inf T "ż R d k x ´ T p x q k dµ p x q : T µ “ ν * . Equipped with the 2-Wasserstein distance, P p R d , W q is a complete, separable metric space. Con-vergence in P p R d q is the usual weak convergence of probability measures, combined with conver-gence of the second moments.A few structural statements can be made about probabilistic frames as a subset of P p R d q . Forbrevity, the probabilistic frames for R d are denoted by PF p R d q , and PF p A, B, R d q denotes the setof probabilistic frames in PF p R d q with optimal upper frame bound less than or equal to B andoptimal lower frame bound greater than or equal to A, with 0 ă A, B ă 8 . Then PF p A, B, R d q is3 nonempty, convex, closed subset of P p R d q . The nonemptiness and convexity are trivial to show.With respect to closedness, let t µ n u be a sequence in PF p A, B, R d q converging to µ P P p R d q .Let y “ argmin y P S d ´ ż R d x x , y y dµ p x q . Because x x , y y ď k x k k y k ď k y k p ` k x k q , it follows by definition of weak convergence in P p R d q that ż R d x x , y y dµ n p x q Ñ ż R d x x , y y dµ p x q . Since for all n , the values of ş R d x x , y y dµ n p x q are bounded above and below by B and A , re-spectively, µ is an element of PF p A, B, R d q . Taking A “ B , this also shows the closedness of P F p A, R d q “ P F p A, A, R d q , the set of tight probabilistic frames with frame bound A. However, theset of probabilistic frames itself is not closed, since one can construct a sequence of probabilisticframes whose lower frame bounds converge to zero: for example, a sequence of zero-mean, Gaussianmeasures with covariances n I, n P N . The goal of this paper is to investigate two main topics on probabilistic frames in the setting of theWasserstein space. The first topic is the notion of duality. For a finite frame, Φ “ t ϕ i u Ni “ Ă R d , aset Ψ “ t ψ i u Ni “ Ă R d is said to be a dual frame to it if for every x P R d ,x “ N ÿ i “ x x , ϕ i y ψ i . It is known that the redundancy of frames implies among other things the existence of manydual frames. While much attention has been paid to the so-called canonical dual frame, certainrecent investigations have focused on alternate duals. For example, Sobolev duals were consideredin [3, 14] in relation to Σ ´ ∆ quantization. Another example is the construction of dual framesfor reconstruction of signals in the presence of erasures [16]. In this paper, we introduce two othertype of dualities, one dictated by the optimal transport problem, and the other grounded in theprobabilistic setting we are working in. These two approaches will be developed in Section 2.The second goal of the paper is to investigate paths of probabilistic frames. Indeed, looking atthe geodesic between any two probabilistic frames, it is natural to ask if the all probability measuresalong this path are probabilistic frames. This will be developed in Section 3.4 Duality, Analysis, and Synthesis in the Set of Probabilistic Frames
Duality, analysis, and synthesis are well-studied objects in finite frame theory. Sobolev duals havebeen proposed for use in reducing error in Σ∆ quantization [3], and the authors of [15] have foundoptimal dual frames for random erasures. Through the lens of optimal transport, extra nuances canbe found in the probabilistic setting.Given a frame Φ “ t ϕ i u Ni “ as above, any possible dual frame to Φ can be written as: t ψ i u Ni “ “ t S ´ ϕ i ` β i ´ N ÿ k “ x S ´ ϕ i , ϕ k y β k u Ni “ (1)where t β i u Ni “ Ă R d and S Φ is the frame operator for Φ [5, Theorem 5.6.5]. When β i “ i, wehave the canonical dual to Φ , which consists of the columns of the Moore-Penrose pseudoinverse ofits analysis operator. Inspired by the definition of duality above and this enumeration of the set ofall possible duals to finite frames, we introduce a new notion of duality in the probabilistic contextin this section. Definition 1.
Let µ be a probabilistic frame on R d . We say that a probability measure ν P P p R d q is a transport dual to µ if there exists γ P Γ p µ, ν q such that ij R d ˆ R d xy J dγ p x, y q “ I. We denote the set of transport duals to µ by D µ : “ $’&’% ν P P p R d q ˇˇˇ D γ P Γ p µ, ν q with ij R d ˆ R d xy J dγ p x, y q “ I ,/./- . We let Γ D µ Ă Γ p µ, ν q be the set of joint distributions on R d ˆ R d with first marginal µ ( π γ “ µ )for which ť R d ˆ R d xy J dγ p x, y q “ I. This is the set of couplings (joint distributions) which inducethe duality.We recall that the canonical dual to a probabilistic frame µ defined in [8, 10, 9], was given by˜ µ : “ p S ´ µ q µ, yielding the reconstruction formula x “ ş R d x x , y y S µ yd ˜ µ p y q . It is easily seen that thecanonical dual ˜ µ is an example of transport dual to µ . Indeed, it is clear that γ “ p ι ˆ S ´ µ q µ P Γ p µ, ˜ µ q , where ι signifies the identity, and ij R d ˆ R d xy J dγ p x, y q “ ż R d x p S ´ µ x q J dµ p x q “ S µ S ´ µ “ I. Therefore, for a given probabilistic frame µ , ˜ µ P D µ .
5n fact, for a given probabilistic frame µ, there are other transport duals corresponding to similardeterministic couplings. Generalizing the set of duals for discrete frames outlined in (1) leads tothe following construction: Theorem 2.
Let µ be a probabilistic frame for R d , and let h : R d Ñ R d be any function in L p R d , µ q : “ t f : R d Ñ R d | ş k f p x q k dµ p x q ă 8u . Then ψ h µ P D µ , where ψ h : R d Ñ R d is definedby ψ h p x q “ S ´ µ x ` h p x q ´ ş R d x S ´ µ x , y y h p y q dµ p y q . Proof.
Consider µ, ψ h µ as above. Define γ : “ p ι, ψ h q µ P Γ p µ, ψ h µ q . Then ij R d ˆ R d xy J dγ p x, y q “ ż R d x „ S ´ µ x ` h p x q ´ ż R d x S ´ µ x , z y h p z q dµ p z q J dµ p x q“ I ` ż R d xh p x q J dµ p x q ´ ij R d ˆ R d x p S ´ µ x q J zh p z q J dµ p x q dµ p z q “ I The restriction of the set of transport duals D µ to lie inside P p R d q is necessary, unlike in thefinite frame case. One might consider the following simple example. Let t e i u di “ Ă R d denote thestandard orthonormal basis. Let t ϕ i u d ` i “ be given by ϕ i “ ? i i e i , i P t , ¨ ¨ ¨ , d u , and let ϕ d ` “ . Take the weights α i “ i , i P N . Define µ “ ´ d δ ` d ÿ i “ α i δ ϕ i . Let t ψ i u i “ be given by ψ i “ b i i e `rp i ´ q mod d s , i P N . Let µ “ ř i “ α i δ ψ i . Then µ P P p R d q , but M p µ q “ ÿ i “ i k ψ i k “ ÿ i “ i i i “ 8 . Hence, µ R P p R d q . However, letting γ P P p R d ˆ R d q be given by γ “ d ÿ i “ α i δ p ϕ i ,ψ i q ` ÿ i “ d ` α i δ p ,ψ i q , it is clear that γ P Γ p µ , µ q , and ij R d ˆ R d xy J dγ p x, y q “ d ÿ i “ i ? i i c i i e i e J i “ I. This example shows that the Bessel-like restriction in the definition of transport duals, requiring6hem to lie in P p R d q , is necessary. Given this restriction, we can assert the following theorem: Theorem 3.
Let µ be a probabilistic frame. Then:(i) Each ν P D µ is also a probabilistic frame.(ii) D µ is a compact subset of P p R d q with respect to the weak topology. In particular, D µ is aclosed subset of PF p R d q with respect to the weak topology on P p R d q . Proof. (i) Suppose ν P D µ Ă P p R d q . Since D µ Ă P p R d q by definition, it is sufficient to showthat supp p ν q spans R d . Let us assume, on the contrary, that there exists z P R d , z ‰
0, such that z K w for all w P span p supp p ν qq . Pick γ P Γ p µ, ν q such that ť xy J dγ p x, y q “ I. Because for all x P supp p ν q ,z J x “ , k z k “ ij x z , x yx z , y y dγ p x, y q “ ij x z , x yx z , y y r supp p ν qˆ R d s p x, y q dγ p x, y q “ D µ : “ t γ P Γ p µ, ν q s.t. ť xy J dγ p x, y q “ I u . It canbe shown by Prokhorov’s Theorem that Γ D µ is precompact [22, Chapter 4]. That is, given t γ n u Ă Γ D µ , there exists a subsequence t γ n k u converging weakly to a limit γ P P p R d ˆ R d q . With this in mind, if t ν n u is a sequence in D µ , we can choose the corresponding t γ n u P Γ D µ , and let t ν n k u be the second marginals of a subsequence t γ n k u . For all ϕ P C p R d ˆ R d q satisfyingfor some C ą | ϕ p x, y q| ď C p ` k x k ` k y k q , ij ϕ p x, y q dγ n k p x, y q ÝÑ ij ϕ p x, y q dγ p x, y q . In particular, for all such ϕ “ ϕ p x q , ij ϕ p x q dγ n k p x, y q “ ż ϕ p x q dν n k p x q ÝÑ ij ϕ p x q dγ p x, y q “ ż ϕ p x q d p π γ qp x q . Thus ν n k converges weakly in P p R d q to π γ “ : ν, so that t ν n u contains a weakly convergentsubsequence. Therefore D µ is precompact.Now let t ν n u be any convergent sequence in D µ which has a limit ν and which forms thesecond marginals of t γ n u Ă Γ D µ . Take again a convergent subsequence t γ n k u with limit γ necessarily in Γ p µ, ν q . Since | x i y j | ď p k x k ` k y k q , it follows that ij x i y j dγ n k p x, y q ÝÑ ij x i y j dγ p x, y q . Then, since for each n k , ť x i y j dγ n k p x, y q ” δ i,j , it follows that ť x i y j dγ p x, y q “ δ i,j , and7herefore ν P D µ . This shows that D µ is also closed, and is therefore compact. The closednessin P F p R d q then follows naturally.From the definition of transport duals, it is clear that their construction depends on the creationof a probability distribution on the product space which has a predetermined second-momentsmatrix and first and second marginals. This is, in general, a very difficult problem, which becomesa bit more tractable for probabilistic frames supported on finite, discrete sets by appealing to toolsfrom linear algebra.Suppose we have two frames Φ “ t ϕ i u Ni “ and Ψ “ t ψ j u Mj “ , and two sets of positive weights, t α i u Ni “ and t β j u Mj “ , summing to unity. Let µ Φ ,α : “ ř Ni “ α i δ ϕ i , and let µ Ψ ,β : “ ř Mj “ β j δ ψ j . In thiscase, any joint distribution γ for µ Φ ,α and µ Ψ ,β satisfies dγ p x, y q “ N ÿ i “ M ÿ j “ A i,j δ ϕ i p x q δ ψ j p y q where A P R N ˆ M with N ÿ i “ A i,j “ β j , M ÿ j “ A i,j “ α i , A i,j ě @ i, j, and N ÿ i “ N ÿ j “ A i,j “ . That is, there is a one-to-one correspondence between Γ p µ Φ ,α , µ Ψ ,β q and this set of “doubly stochas-tic” matrices, which we denote by DS p α, β q . Thus, to show that µ Φ ,α P D µ Ψ ,β , one must constructa matrix A P DS p α, β q solving Φ J A Ψ “ I. Regarding this question, we have the following result:
Theorem 4.
Given frames t ϕ i u Ni “ and t ψ j u Mj “ for R d with analysis operators Φ and Ψ , there exists A P DS p α, β q with Φ J A Ψ “ I if and only if there is no triplet p B, u, v q with B P R d ˆ d , u P R M ,v P R N such that ϕ J i Bψ j ` u i ` v j ě trace p B q ` u J α ` v J β ă Proof.
Recall that we must solve the systemΦ J A Ψ “ I, A i,j ě M ÿ j “ A i,j “ α i , N ÿ i “ A i,j “ β j (2)Defining, for a matrix B , vec p B q to be the vector formed by stacking the columns of B , we mayrewrite the problem in terms of the Kronecker product. Using the following variables, K “ Ψ J b Φ J , “ vec p A q , z N “ r . . . s J P R N , z M “ r . . . s J P R M , and t “ vec p I q , I P R d ˆ d , we have: $’’’’’&’’’’’% Ka “ t p z J N b I p M ˆ M q q a “ β p I p N ˆ N q b z J M q a “ αa i ě @ i P t , . . . , M N u We can combine the equations above, letting K “ »—– K p z J N b I p M ˆ M q qp I p N ˆ N q b z J M q fiffifl and t “ »—– tβα fiffifl Then the problem simplifies to solving K a “ t such that a i ě @ i P t , . . . , M N u . By Farkas’Lemma [12, Lemma 1], either this system has a solution or there exists y P R d ` M ` N such that y J K ě y J t ă y as y “ »—– buv fiffifl , with b P R d , u P R M , and v P R N , and let b “ vec p B q with B P R d ˆ d . Then Equations (3) and (4) hold if and only if b J K ` u J p z J N b I p M ˆ M q q ` v J p I p N ˆ N q b z J M q ě b J t ` u J β ` v J α ă vec p Φ B Ψ J q ` vec p z N u J I p M ˆ M q q J ` vec p I p N ˆ N q vz J M q J ě b J t ` u J β ` v J α ă ϕ J i Bψ j ` u i ` v j ě @ i, jtrace p B q ` u J β ` v J α ă Corollary 5.
If there does not exist B P R d ˆ d such that α J Φ B Ψ J β ´ trace p B q ą then by Farkas’ Lemma the system of (2) (and its equivalents) is not solvable and the desired matrix A P DS p α, β q exists. A true converse has proven elusive. However, we can identify a few related conditions under whichno transport duals whatsoever can be constructed. In particular, in the case that the frames areuniformly weighted, we have the following zero-centroid condition.
Theorem 6.
Again, take z N : “ r . . . s J P R N . Suppose that Ψ “ t ψ i u Ni “ Ă R d is a frame suchthat ř Ni “ ψ i “ , then µ Ψ , N z N has no equal-weight transport dual supported on a set of of cardinality d. Proof.
Given Ψ as above, let t v j u dj “ Ă R N denote the columns of the analysis operator Ψ , andlet t u i u di “ Ă R N denote the rows of some A P DS p d z d , N z N q . Ψ will have a transport dual ofcardinality d if and only if for some A, A Ψ “ rrx u i , v j yss is invertible. (Here, Q “ rr q i,j ss denotesthe entrywise definition of Q. ) Each u i “ s ` λ i , where s “ r Nd ¨ ¨ ¨ Nd s P R d , and N ÿ k “ λ ik “ i P t , ..., d u d ÿ i “ λ ik “ k P t , ..., N u so that t λ i u di “ has zero centroid as well and is therefore linearly dependent. Let Λ “ r λ . . . λ d s J . Then det p A Φ q “ d ź i “ x u i , v i y “ d ź i “ x s ` λ i , v i y “ d ź i “ x λ i , v i y “ det p ΛΨ q “ v i K s for all i P t , . . . , d u and since rank p Λ q ď d ´ . As a consequence, Theorem 6 implies that no equiangular tight frame in R has a transportdual of cardinality 2. Remark . One interesting aspect of the transport duals in the context of finite discrete probabilisticframes, i.e., finite frames, is the existence of pairs of dual frames with different cardinalities. Forexample, one can consider the probabilistic frame given by dµ “ δ ϕ ` δ ϕ ` δ ϕ with ϕ “r s J , ϕ “ r ?
32 12 s J , and ϕ “ r s J . Then the probabilistic frame ν given by dν “ δ ψ ` δ ψ with ψ “ r ´? ¨p `? ´? s J and ψ “ r ´ ´? ´? s J is a transport dual for µ with support ofdifferent cardinality. The role of transport duals in problems such as reconstruction in the presenceof erasure will be the object of future investigations.10 .2 Analysis and Synthesis in the Probabilistic Context In [8, 10, 9], the analysis and synthesis operators for probabilistic frames were defined analogouslyto those of continuous frames. Given a probabilistic frame µ , the analysis operator was defined [10,2.2] as A µ : R d Ñ L p R d , µ q given by x ÞÑ x x , ¨ y . Its synthesis operator was A ˚ µ : L p R d , µ q Ñ R d given by f ÞÑ ż R d xf p x q dµ p x q . The foregoing construction of transport duals, on the other hand, begs a more probability-theoreticdefinition of analysis and synthesis. As defined above, the analysis operator A µ is independent ofthe measure µ . Indeed, it is not clear from this definition how one could do “analysis” with oneprobabilistic frame followed by “synthesis” with another. However, finite frame theory itself givesus a clue about how to think about analysis and synthesis in the probabilistic context. Example 1.
Consider two frames for R d , t ϕ i u Ni “ and t ψ i u Ni “ . Let t e i u Ni “ Ă R N be an orthonormalbasis for R N . Then the analysis operator for Φ , A Φ : R d Ñ R N is given by A Φ p x q “ Φ x “ N ÿ i “ x x , ϕ i y e i for x P R d . The synthesis operator for Ψ , A ˚ Ψ : R N Ñ R d , is given by A ˚ Ψ p y q “ Ψ J y “ N ÿ i “ x y , e i y ψ i for y P R N . Then we can compose the operators simply by writing A ˚ Ψ A Φ p x q “ N ř i “ x x , ϕ i y ψ i . If, however, wechoose some σ and π in Π N , the set of permutations on N-element sets, and instead choose to doanalysis and synthesis with the two frames as A ˚ Ψ A Φ p x q “ N ÿ i “ x x , ϕ σ p i q y ψ π p i q , then it will be as if we had chosen two different finite frames to work with. This is because theordering of the frame vectors is implicitly tied to the ordering of the reference basis t e i u Ni “ . Order matters! From the example, it is clear that even given the fixed reference basis, wecannot truly speak of a single analysis operator for the set t ϕ i u Ni “ , without imposing an order onit relating it to the fixed reference basis. Similarly, for a probabilistic frame µ, there must be areference measure η playing the role of the reference basis, and this will still lead to a family ofanalysis operators, each corresponding to a joint distribution γ P Γ p µ, η q . The orthogonality of the11eference basis in the above example turns out not to be necessary; its function is to match upframe coefficients with the appropriate vectors. What is key is that transport plans exist betweenthe probabilistic frame and the reference measure and that the support of the reference measure issufficient to “glue” together arbitrary probabilistic frames through analysis and synthesis.To make this idea of coefficient-matching rigorous, some technicalities about conditional prob-abilities are necessary. Conditional probabilities can be defined via the Rokhlin DisintegrationTheorem [2, Theorem 5.3.1]. If µ P P p R M ˆ R N q and ν “ µ “ π µ , then one can find a Borelfamily of probability measures t µ x u x P R M Ă P p R N q which is µ -a.e. uniquely determined such that µ “ ş R M µ x dµ p x q . In the language of conditional probability, for any f P C b p R M ˆ R N q , it is thenmeaningful to write ij R M ˆ R N f p x, y q d µ p x, y q “ ż R N ż R M f p x, y q d µ p y | x q dµ p x q , with the understanding that µ p¨| x q is defined µ -a.e. Gluings can then be constructed, which allowus to use conditional probabilities with respect to a common reference measure to construct a jointdistribution between previously unrelated measures. Lemma 8.
Gluing Lemma [2, Lemma 5.3.2]
Let γ P P p R K ˆ R M q , γ P P p R K ˆ R N q suchthat π γ “ π γ “ µ . Then there exists µ P P p R K ˆ R M ˆ R N q such that π , µ “ γ and π , µ “ γ . Moreover, if γ “ ş R K γ x dµ , γ “ ş R K γ x dµ , and µ “ ş R K µ x dµ arethe disintegrations of γ , γ , and µ with respect to µ , then the first statement is equivalent to µ x P Γ p γ x , γ x q Ă P p R M ˆ R N q for µ -a.e. x P R K . Now let us consider a probabilistic frame µ and another probability measure η and take γ P Γ p µ, η q . From Lemma 8, there is a set of conditional probability measures t γ p¨| w qu w P R d that areuniquely defined η -a.e. To proceed with the construction of analysis and synthesis in the probabilisticcontext, we will first establish a useful fact. Recall that L p R N ˆ R M , R K , γ q : “ t f : R N ˆ R M Ñ R K | ij k f p x, y q k dγ p x, y q ă 8u . Then, by condition Jensen’s inequality, if f f P L p R d ˆ R d , R d , γ q , it follows that g p w q : “ ş R d f p y, w q dγ p y | w q is in L p R d , R d , η q .Finally, since h p z, w q : “ k z k P L p R d ˆ R d , R , γ q for any γ P Γ p µ, η q provided that µ P P p R d q , itfollows that the vector-valued function ş zdγ p z | w q lies in L p R d , R d , η q .To define analysis and synthesis operators which are more closely tied to their probabilisticframes, a reference measure must be chosen; take an absolutely continuous η P P p R d q whosesupport is R d . Given µ P PF p R d q , we define families of analysis and synthesis operators for µ withrespect to η . Definition 9. t A γµ u γ P Γ p µ,η q is the family of analysis operators, and for each γ P Γ p µ, η q we have:12 γµ : R d Ñ L p R d , R d , η q , is given by A γµ p x qp w q “ ż R d x x , y y dγ p y | w q . Similarly, the family of synthesis operators, t Z γµ u γ P Γ p µ,η q is defined for each γ P Γ p µ, η q by Z γµ : L p R d , R , η q Ñ R d , given by Z γµ p f q “ ij R d ˆ R d zf p w q dγ p z | w q dη p w q The class of reference measure η was chosen such that, for any probabilistic frame µ, the probabilisticanalysis and synthesis operators can be constructed using deterministic couplings between η and µ .There are several interesting ways to pair disparate types of probabilistic frames with one an-other. A useful technique is the transport of an absolutely continuous measure to a discrete measureusing power (Voronoi) cells. Following [18], we define maps which can be used for these pairings.It is an interesting fact due to Brenier that the Voronoi mapping we will describe, T wP , is in fact anoptimal map between the two measures it couples, µ and T wP | µ, when µ is absolutely continuouswith respect to Lebesgue measure [18, Theorem 1]. Definition 10.
Given a probability measure µ on R d , a finite set P of points in R d and w : P Ñ R ` a weight vector, the power diagram or weighted Voronoi diagram of p P, w q is a decomposition of R d into cells corresponding to each member of P . Given p P P , a point x P R d belongs to Vor wP p p q ifand only if for every q P P , k x ´ p k ´ w p p q ď k x ´ q k ´ w p q q . Let T wP be the map that assigns to each x in a power cell Vor wP p p q to p , the “center” of thatpower cell. We call T wP the weighted Voronoi mapping. T wP | µ “ ÿ p P P µ p Vor wP p p qq δ p . Let η be an absolutely continuous measure in P p R d q , and let ν “ ř p P P λ p δ p be a discretemeasure in P p R d q supported on a finite set of points P with weights t λ p u summing to unity. Thenwe say that a vector weight w : P Ñ R ` is adapted to p η, ν q if for all p P P , λ p “ η p Vor wP p p qq “ ş Vor wP p p q dη p x q . Example 2.
Now given discrete frames Φ “ t ϕ i u Mi “ and Ψ “ t ψ j u Nj “ for R d , and η a referencemeasure in Definition 9, choose γ “ p ι, T w Φ q η and γ “ p ι, T w Ψ q η , where the weights w and w are adapted to p µ Φ , η q and p µ Ψ , η q , respectively. Then Z γ µ Ψ p A γ µ Φ p x qq “ ż x x , T w Φ p y q y T w Ψ p y q dη p y q . xample 3. Recovering the old definitions of analysis and synthesis
In the special case M “ N , we could choose P “ t p i u Ni “ Ă R d and w adapted to p µ P , η q .Then let f Ψ : P Ñ Ψ be given by f Ψ p p i q “ ψ i , and let f Φ : P Ñ Φ be similarly defined. Then if γ “ p ι, f Φ ˝ T w P q η and γ “ p ι, f Ψ ˝ T w P q η , it follows that Z γ µ Ψ p A γ µ Φ p x qq “ ż x x , f Φ ˝ T w P p y q y f Ψ ˝ T w P p y q dη p y q “ N ÿ i “ x x , ϕ i y ψ i . Hence, we have recovered the analysis and synthesis operation of finite frames.
Example 4.
Discrete dual to absolutely continuous probabilistic frame
Finally, choose a frame contained in the support of η , say t ψ i u Ni “ . Let T w Ψ be the transportmap between η and µ Ψ , as constructed above. Choose t ϕ i u Ni “ to be any dual to t ψ i u Ni “ , and let f : Ψ Ñ Φ be given by f p ψ i q “ ϕ i . Then γ “ p ι, f ˝ T w Ψ q η P P p R d ˆ R d q is a joint transport planin Γ p η, µ Ψ q such that ť xy J dγ p x, y q “ ş xT w Ψ p x q dη p x q “ I , so that η and µ Ψ are dual to one anotherin PF p R d q . A number of important questions in finite frame theory involve determining distances betweenframes and constructing new frames. In this section we consider geodesics in P p R d q and investi-gate conditions under which probability measures on these paths are probabilistic frames. As weshall prove, in the case of discrete probabilistic frames, this question is equivalent to one of ranksof convex combinations of matrices. Furthermore, for probabilistic frames with density, a sufficientcondition for geodesic measures to be probabilistic frames is the continuity of the optimal deter-ministic coupling. This question has ramifications for constructions of paths of frames in general,for frame optimization problems, and for our understanding of the geometry of P F p R d q . In constructing paths of probabilistic frames, minimal paths between frames in P p R d q are a naturalplace to start since PF p R d q is not closed. We follow the construction of geodesics in the Wassersteinspace given in [13]. To this end, given t P r , s define Π t : R d ˆ R d Ñ R d as Π t p x, y q “ p x, p ´ t q x ` ty q . For µ , µ P P p R d q , take γ P Γ p µ , µ q to be an optimal transport plan for µ and µ with respect to the 2-Wasserstein distance. Then let the interpolating joint probability measure be γ t on R d ˆ R d , given by: ij R d ˆ R d F p x, y q dγ t p x, y q “ ij R d ˆ R d F p Π t p x, y qq dγ p x, y q F P C b p R d ˆ R d q . In particular, for F P C b p R d q , ij R d ˆ R d F p x q dγ t p x, y q “ ij R d ˆ R d F p x q dγ p x, y q “ ż R d F p x q dµ p x q . Given t P r , s let µ t be the probability measure such that for all G P C b p R d q : ż R d G p y q dµ t p y q “ ij R d ˆ R d G p y q dγ t p x, y q “ ij R d ˆ R d G pp ´ t q x ` ty q dγ p x, y q , (8)we call µ t a geodesic measure with respect to µ and µ . Indeed, the mapping t Ñ µ t is truly ageodesic of the 2-Wasserstein distance in the sense that W p µ , µ t q ` W p µ t , µ q “ W p µ , µ q . Recall that a probability measure µ on R d is a probabilistic frame if it is an element of P p R d q and if S µ is positive definite. It is easy to show that µ t , as constructed by the method above, alwaysmeets the first requirement. Lemma 11.
For any measure µ t , t P r , s , on the geodesic between two probabilistic frames µ and µ , M p µ t q ă 8 . Showing that S µ t is positive definite, or, equivalently, that the support of µ t spans R d dependson the characteristics of the support of the measures at the endpoints. For this reason, it is naturalto divide the analysis into two parts: the discrete case and the absolutely continuous case. In both,a monotonicity property that characterizes optimal transport plans will play a key role. For the canonical discrete probabilistic frames with uniform weights, we have:
Lemma 12. [2, Theorem 6.0.1]
Given µ “ µ Φ and µ “ µ Ψ , discrete probabilistic frames withsupports of equal cardinality N, uniformly weighted, the Monge-Kantorovich problem simplifies, anddenoting by Γ p N q the set of matrices with row and column sums identically N : W p µ , µ q “ min A P Γ p N q N ÿ i “ N ÿ j “ a i,j k ϕ i ´ ψ j k and, by the Birkhoff-von Neumann Theorem, the optimal transport matrix A is a permutation matrixcorresponding to some σ P Π N , i.e.: W p µ , µ q “ min σ P Π N N N ÿ i “ k ϕ i ´ ψ σ p i q k
15n this case, for some optimal σ P Π N , S µ t : “ N N ÿ i “ rp ´ t q ϕ i ` tψ σ p i q srp ´ t q ϕ i ` tψ σ p i q s J . (9)The optimality of σ implies that σ maximizes N ř i “ x ϕ i , ψ σ p i q y among all elements of Π N , and thiscrucial fact motivates the definition of a monotonicity condition. Definition 13.
A set S Ă R d ˆ R d is said to be cyclically monotone if, given any finite subset tp x , y q , ..., p x N , y N qu Ă S, for every σ P S N holds the inequality: N ÿ i “ x x i , y i y ě N ÿ i “ x x i , y σ p i q y . With this definition in hand, the main result of this section can be stated:
Theorem 14.
Let t ϕ i u Ni “ and t ψ i u Ni “ be frames for R d . If Ψ : Φ has no negative eigenvaluesand tp ϕ i , ψ i qu Ni “ is cyclically monotone, then every measure on the geodesic between the canonicalprobabilistic frames µ Φ and µ Ψ is a probabilistic frame. The proof of this theorem will follow from Lemma 11 and Proposition 16, proven below. Toprove Proposition 16, the following lemma from matrix theory is necessary:
Lemma 15. [21, Theorem 2] Let A and B be m ˆ n complex matrices, m ě n . Let rank p A q “ rank p B q “ n . If B : A has no nonnegative eigenvalues, then every matrix in h p A, B q : “ tp ´ t q A ` tB, t P r , su has rank n . Similarly, if A and B are n ˆ n complex matrices with rank n , we can define in r p A, B q : “ tp I ´ T q A ` T B u , where T is a real diagonal matrix with diagonal entries in r , s . Then, if B ´ A is such that all itsprincipal minors are positive, then every matrix in r p A, B q will have rank n . Combining the cyclical monotonicity condition with Lemma 15, we can state the following resultwhich gives sufficient conditions for a geodesic between discrete probability measures in P p R d q tobe a path of frames. We note that little can be claimed about the spectra of the frame operatorsalong the path (i.e., the frame bounds of the probabilistic frames along the geodesic) in general,other than their boundedness away from zero. Proposition 16.
Let t ϕ i u Ni “ and t ψ i u Ni “ be frames for R d with analysis operators Φ and Ψ .Denoting by P si : the Moore-Penrose pseudoinverse of Ψ , if Ψ : Φ has no negative eigenvalues, andif tp ϕ i , ψ i qu Ni “ is a cyclically monotone set, then every measure µ t on the geodesic between µ Φ and µ Ψ has support which spans R d . roof. Each measure on the geodesic µ t will be supported on a new set of vectors, namely tp ´ t q ϕ i ` tψ σ p i q u Ni “ , and will be a probabilistic frame provided this set of vectors spans R d .Equivalently, µ t will be a probabilistic frame if the probabilistic frame operator S µ t is positivedefinite. Let P σ be the N ˆ N permutation matrix corresponding to σ P Π N , where now σ is theoptimal permutation for the Wasserstein distance. Let Ψ σ “ P σ Ψ. A quick calculation shows: S µ t “ N ` p ´ t q Φ J ` t Ψ J σ ˘ pp ´ t q Φ ` t Ψ σ q . Ψ and Ψ σ have rank d , and to show that S µ t is positive definite, it remains to prove that everymatrix in the set h p Φ , Ψ σ q : “ tp ´ t q Φ ` t Ψ σ u t Pr , s has rank d . By Lemma 15, a sufficient conditionfor this to be true is that Ψ : σ Φ be positive semi-definite. Finally, we note that if tp ϕ i , ψ i qu Ni “ is acyclically monotone set, then P σ “ I , the identity, is an optimal permutation, and then Ψ : σ Φ “ Ψ : Φis positive definite by assumption.
Proof.
Proof of Theorem 14
With Lemma 11 showing that measures on the geodesic have finite second moment and Propo-sition 16 showing that the support of these measures spans R d , Theorem 14 is now proved.Certain dual frame pairs immediately satisfy the conditions laid out in Theorem 14.
Proposition 17. If t ϕ i u Ni “ is the canonical dual frame to t ψ i u Ni “ , then tp ϕ i , ψ i qu Ni “ is cyclicallymonotone.Proof. Let S “ Ψ J Ψ. Then suppose that Φ J “ S ´ Ψ J . For any permutation σ P Π N , let P σ denote the matrix such that for @ x “ r x . . . x N s J P R N , P σ x “ r x σ p q . . . x σ p N q s J . Then, N ÿ i “ x ϕ i , ψ i ´ ψ σ p i q y “ N ÿ i “ x S ´ ψ i , ψ i ´ ψ σ p i q y“ N ÿ i “ p ψ i ´ ψ σ p i q q J S ´ ψ i “ Tr pp Ψ ´ P σ Ψ q S ´ Ψ J q“ Tr pp I N ´ P σ q Ψ S ´ Ψ J qě S ´ Ψ J “ I dN , the N ˆ N diagonal matrix with d leading ones on thediagonal and zeros else, because S ´ Ψ J is the Moore-Penrose pseudoinverse of Ψ. Therefore, theidentity is an optimal permutation, i.e., the set tp ϕ i , ψ i qu Ni “ is cyclically monotone.17 roposition 18. Let t β i u Ni “ Ă R d be such that tp β i , ψ i qu Ni “ d ` is cyclically monotone. Then use t β i u Ni “ to define t ϕ i u Ni “ , one of the dual frames to t ψ i u Ni “ as given in (1) . Then tp ϕ i , ψ i qu Ni “ iscyclically monotone.Proof. Take t ϕ i u Ni “ to be a dual of the form given in Equation (1). Let W be the matrix whoserows are the t β i u Ni “ . Then, noting that Φ J “ p S ´ Ψ J ` W J p I N ´ Ψ S ´ Ψ J qq , N ÿ i “ x ψ i ´ ψ σ p i q , ϕ i y “ Tr pp I N ´ P σ q ΨΦ J q“ Tr pp I N ´ P σ q Ψ p S ´ Ψ J ` W J p I N ´ Ψ S ´ Ψ J qqq“ Tr pp I N ´ P σ q I dN ` p I N ´ P σ q Ψ W J p I N ´ I dN qq“ Tr pp I N ´ P σ q I dN q ` N ÿ i “ d ` x ψ i ´ ψ σ p i q , β i yě tp ϕ i , ψ i qu Ni “ is cyclically monotone. Proposition 19. If t ϕ i u Ni “ is the canonical dual frame to t ψ i u Ni “ , or if t ϕ i u Ni “ is a dual frameto t ψ i u Ni “ of the form given in (1) , with the t h i u Ni “ ordered so that tp h i , ψ i qu Ni “ d ` is cyclicallymonotone, then Ψ : σ Φ is positive definite, where σ is the optimal permutation for the Wassersteindistance. Consequently, any path on the geodesic joining t ψ i u Ni “ and t ϕ i u Ni “ is a probabilistic frame.Proof. By definition, Ψ : σ “ p P σ Ψ q : “ p Ψ J P J σ P σ Ψ q ´ Ψ J P J σ “ p Ψ J Ψ q ´ Ψ J P J σ This is a permutation of the matrix whose columns are canonically dual to the rows of Ψ σ . If t ϕ i u Ni “ is any dual of t ψ i u Ni “ , then Ψ J Φ “ I d . Therefore, if σ is the identity, then Ψ : σ Φ “ p Ψ J Ψ q ´ Ψ J Φ “p Ψ J Ψ q ´ , which is positive definite. It remains to show that the optimal permutation is the iden-tity. But this is clear: Proposition 17 shows that if t ϕ i u Ni “ is the canonical dual to t ψ i u Ni “ , then tp ϕ i , ψ i qu Ni “ is cyclically monotone, and Proposition 18 shows that if t ϕ i u Ni “ is any dual to t ψ i u Ni “ which meets the above condition, then tp ϕ i , ψ i qu Ni “ is cyclically monotone.There are other frame and dual-frame pairs which can easily be shown to meet the aboveconditions. Consider the finite sequences t ϕ i u Ni “ Ă R d and t ψ i u Ni “ Ă R d with respective analysisoperators Φ and Ψ. Then the finite sequences are disjoint if Φ p R d q Ş Ψ p R d q “ t u . Proposition 20. If t ϕ i u Ni “ and t ψ i u Ni “ are disjoint frames for R d , associated canonically withthe probabilistic frames µ Φ and µ Ψ , then every measure on the geodesic between µ Φ and µ Ψ is aprobabilistic frame. roof. Given v P R d , consider: N ÿ i “ x v , p ´ t q ϕ i ` tψ i y “ N ÿ i “ x v , p ´ t q Φ J e i ` t Ψ J e i y “ N ÿ i “ xp ´ t q Φ v ` t Ψ v , e i y “ k p ´ t q Φ v ` t Ψ v k R N ě C rp ´ t q k Φ v k ` t k Ψ v k s for some C ą , since the frames are disjoint. Since the two sequences in question are finite frames,choosing the minimum of the two lower frame bounds, say A , the last quantity can be boundedbelow by p ´ t ` t q C ¨ A k v k , yielding the result.Finally, in the following result control of the distance between the elements of a one frame and thoseof the canonical dual of the other by a coherence-like quantity guarantees the frame properties forthe frames on the geodesic. Proposition 21.
Let t ψ i u Ni “ be a dual frame to a frame t ϕ i u Ni “ Ă S d ´ . Let S Φ denote the frameoperator. For each i , let z i “ ψ i ´ S ´ ϕ i , and let a : “ min i ‰ j x ϕ i , S ´ p ϕ i ´ ϕ j q y . If max j k z j k ď aN ,then the optimal σ for the mass transport problem is the identity.Proof. First, we note that a ě . If a “ , then our hypothesis guarantees that k z i k “ k ψ i ´ S ´ ϕ i k “ i , so that Ψ is the canonical dual to Φ , and in this case our result holds by Proposition 17.Therefore, it only remains to consider the case when a ą . For all u, v P R d , N ř i “ x u , z i yx v , ϕ i y “
0. Then given σ P S N , let n σ be the number of elementsnot fixed by σ . Then if σ is the identity, n σ “ N ÿ i “ x ϕ i , S ´ x σ p i q ` z σ p i q y “ T r p Ψ J P σ Φ q “ d If σ is not the identity, then N ÿ i “ x ϕ i , S ´ ϕ σ p i q ` z σ p i q y “ ÿ i ‰ σ p i q x ϕ i , S ´ p ϕ σ p i q ´ ϕ i q ` z σ p i q ´ z i y ` d ď d ´ an σ ` ÿ i ‰ σ p i q x ϕ i , z σ p i q ´ z i yď d ´ p ´ N q an σ Since, given the hypothesis, for all i, j , x ϕ i , z j y ď k ϕ i kk z j k “ k z j k ď aN . Thus Tr p Ψ J P σ Φ q ď d ´ p ´ N q an σ ď d “ T r p Ψ J Φ q for all σ, and it follows that the identity is the optimal transportmap for the Wasserstein metric. 19 .3 Absolutely Continuous Probabilistic Frames The question of the nature of the optimal transport plan for the 2-Wasserstein distance is simplerfor absolutely continuous measures. From [2, Theorem 6.2.10 and Proposition 6.2.13], which gathertogether a long list of characteristics, two key facts about this plan can be extracted, which arecollected in the following lemma.
Lemma 22. [2, Chapter 6.2.3] If µ and µ are absolutely continuous probability measures in P p R d q , then there exists a unique optimal transport plan for the 2-Wasserstein distance which isinduced by a transport map r . This transport map is defined (and injective) µ -a.e. Indeed, thereexists a µ -negligible set N Ă R d such that x r p x q ´ r p x q , x ´ x y ą for all x , x P R d z N . Then we have the following result for absolutely continuous probabilistic frames:
Proposition 23. If µ and µ are absolutely continuous (with respect to Lebesgue measure) prob-abilistic frames for which there exists a linear, positive semi-definite deterministic coupling whichminimizes the Wasserstein distance, then all measures on the geodesic between these frames havesupport which spans R d and will therefore be probabilistic frames.Proof. Given the assumptions, let r p x q denote the linear transformation which induces the coupling µ “ r µ . Defining h t p x q “ p ´ t q x ` tr p x q µ -a.e., the geodesic measure is given by µ t : “ h t µ . (10)Then S µ t “ ş R d h t p x q h t p x q J dµ p x q . If r p x q “ Ax for some A P A d ˆ d , then: S µ t “ ż R d pp ´ t q Ix ` tAx qpp ´ t q Ix ` tAx q J dµ p x q“ pp ´ t q I ` tA q S µ pp ´ t q I ` tA q J Since A must be nonsingular–recall that S µ “ AS µ A J , which is certainly of rank d –by Lemma15, p ´ t q I ` tA will also nonsingular for all t P r , s provided that A has no negative eigenvalues,as we assumed. Example 5.
An example in which the assumptions of the above proposition hold is the case of non-degenerate Gaussian measures on R d . Let µ and µ be zero-mean Gaussians. Let r p x q “ S µ S ´ µ x .According to a result in [7], if X and Y are two zero-mean random vectors with covariances Σ X and Σ Y , respectively, then a lower bound for E p k X ´ Y k q is Tr r Σ X ` Σ Y ´ p Σ X Σ Y q s , and thebound is attained, for nonsingular Σ X , when Y “ Σ ´ X Σ Y X , so that the coupling r is an optimalpositive definite linear deterministic coupling of µ and µ . µ, ν for R d , take r p x q to be the optimaltransport map pushing µ to ν guaranteed by Lemma 22. Define h t p x q “ p ´ t q x ` tr p x q for t P r , s ;then S µ t “ ş h t p x q b h t p x q dµ p x q , with µ t “ p h t q µ . Then we can state the following: Proposition 24.
Given two such probabilistic frames, there exists a set N with µ p N q “ such that h t is injective for all t P r , s on supp p µ qz N .Proof. Given x, y P supp p µ qz N , with N as defined in Lemma 22, suppose h t p x q “ h t p y q for some t P r , s . Then, since: 0 “ x h t p x q ´ h t p y q , x ´ y y“ xp ´ t qp x ´ y q ` t p r p x q ´ r p y qq , x ´ y y“ p ´ t q k x ´ y k ` t x r p x q ´ r p y q , x ´ y y it follows that x r p x q ´ r p y q , x ´ y y “ t ´ t k x ´ y k . This implies that x r p x q ´ r p y q , x ´ y y ď . However, from the proposition above, we also know that x r p x q ´ r p y q , x ´ y y ě
0. Therefore k x ´ y k “
0, and h t is injective on supp p µ qz N .This injectivity claim is crucial for the main result of this section: Theorem 25.
Let µ, ν P P r p R d q , and let r be the unique optimal transport map for the ´ W asserstein distance. Let N be the set of measure zero define in Proposition 24. If r is continuous,and if supp p µ qz N contains an open set, then every geodesic measure µ t is a probabilistic frame.Proof. Since r is continuous and, by Proposition 24, injective outside a set N of measure zero, sois h t for each t . Let x P supp p µ qz N . First, we show that for any ǫ ą h ´ t p B ǫ p h t p x qqq containsan open set containing x .Since h t is continuous at any such x , given ǫ ą
0, there exists δ ą @ x P B δ p x q , k h t p x q´ h t p x q k ă ǫ . Hence for any x P B δ p x q , x P h ´ t p B ǫ p h t p x qqq –i.e., B δ p x q Ă h ´ t p B ǫ p h t p x qqq .Then @ x P supp p µ qz N , consider B k p h t p x qq : µ t p B k p h t p x qqq “ ż „ B k p h t p x qq p h t p y qq dµ p y q“ ż „ h ´ t p B k p h t p x qqq p y q dµ p y q“ µ p h ´ t p B k p h t p x qqqqą x P supp p µ q and, as shown above, h ´ t p B k p h t p x qqqq containsan open set containing x . Thus, we have shown that for any k P N , the open ball of radius k around h t p x q has positive µ t -measure, and therefore h t p x q lies in supp p µ t q . Thus h t p supp p µ qz N q Ă supp p µ t q .Therefore, since h t is injective by Proposition 24 above and continuous on supp p µ qz N and byassumption, there exists open set U Ă supp p µ qz N , by invariance of domain, h t p U q Ă supp p µ t q isopen, we conclude that h t µ has support which spans R d .The question of when r is continuous is the subject of ongoing research. One example is when µ and ν are supported on a bounded convex subset of R d [6]. Acknowledgment
Clare Wickman would like to thank the Norbert Wiener Center for Harmonic Analysis and Appli-cations for its support during this research. Kasso Okoudjou was partially supported by a grantfrom the Simons Foundation (
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