On a Parametric Spectral Estimation Problem

On a P arametric Sp ectral Estimation Problem ⋆ B. Zh u ∗ ∗ Dep artment of Information Engine ering, Un iversity of Padova, V ia Gr adenigo 6/B, 35131 Padova, Italy (e-mail: zhubin@dei.unip d.it) Abstract: W e consider an open question pos ed in Zh u and Baggio (2017) on the uniqueness of the so lution to a pa r ametric sp ectral estimation pro blem. Keywor ds: Sp ectral estimation, g eneralized moment proble m, global inv erse function theor em, sp ectral facto rization. 1. INTR ODUCTION In this pap er, we consider a spectr al estimation pr oblem sub jected to a gener alized moment constraint , a frame- work pioneere d by Byrnes, Georgio u, and Lindquist in Byrnes et al. (2000 ); Georg iou and Lindquist (2003). The formulation of the problem can be seen a s a gener alization of earlie r work on r ational c ovarianc e extension , c f. e.g., Kalman (1982); Georgiou (1983); Byrnes et al. (1995, 1998, 2001b), and Nevanlinna–Pick interp olation , cf. Geor giou (1987); Byrnes et al. (200 1a) and r eferences ther ein. A standar d setup of the problem is as follows. Suppo se we hav e a zer o-mean wide-sense stationary v ector sig na l y ( t ) with an unknown sp ectral density matrix Φ( z ). In order to estimate the sp ectrum, we p erfo r m the following steps. Step 1 . F eed the signal y ( t ) into a filter bank with a transfer function G ( z ) = ( z I − A ) − 1 B (1) to g et an output x ( t ). The cor resp onding time domain representation is just x ( t + 1) = Ax ( t ) + B y ( t ) . (2) W e hav e so me extr a sp ecifications on the sys tem matrices, w hich include • A ∈ C n × n is Sch ur s table, i.e., all its eig env al- ues hav e mo duli less than 1; • B ∈ C n × m is of full column r ank with n ≥ m ; • The pair ( A, B ) is r e achable . Step 2 . Co mpute an estimate of the steady-state cov ar i- ance matrix Σ := E x ( t ) x ( t ) ∗ of the s tate vec- tor x ( t ); cf. e.g., F er rante et al. (2012) for suc h structured cov ariance estimation pro blem. Hence we hav e Z G Φ G ∗ = Σ , (3) where the function is integrated on the unit circle T ag ainst the normalized Leb esg ue measure , i.e., Z F := Z π − π F ( e iθ ) dθ 2 π . ⋆ This work w as supported by the China Scho larshi p Council (CSC) under Fi le No. 201506230140. This simplified notation will be adopted through- out. Step 3 . Given the estimated Σ > 0, find a sp ectral density Φ s uch that the g eneralized moment constra int (3) is satisfied. W e must p oint out tha t existence of a bo unded and co ercive Φ satis fying (3 ) is not triv ial in general. Such feasibility pro blem w as addre ssed in Georgiou (2002), see also e.g ., F erran te et al. (2 010, 2012). In this pa per , we shall a lw ays as s ume the feasibility in the sense explained next. Let C ( T ; H m ) denote the space of m × m Hermitia n matrix-v alued contin uous functions o n the unit cir cle and let H n be the vector s pace o f n × n Her mitian matrices. Define the linear ope r ator Γ : C ( T ; H m ) → H n Φ 7→ Z G Φ G ∗ . (4) W e shall denote the image/r a nge of this map by im Γ for short. Then we as s ume that the cov ar iance matrix Σ ∈ im Γ. According to (F erra n te et al., 20 12, Pro po sition 3.1), im Γ is a linear spa ce with real dimension m (2 n − m ). Given a p os itiv e definite Σ ∈ im Γ, there ar e in g eneral infinitely many sp ectral densities that would so lve (3). The mainstream appr oach to day to remedy such ill-p osednes s is to first intro duce a pr io r matrix spec tral density Ψ, which repre sent s our guess of the “ true” densit y Φ. Then one tries to define an en tropy-like dista nce index d (Φ , Ψ) betw een tw o sp ectral densities , and to find the “b est” Φ by so lving the constrained o ptimiza tion problem minimize Φ ∈S m d (Φ , Ψ) sub ject to (3) , where S m is the family of m × m bo unded and co ercive sp ectral densities. Due to the page limit, we refer the readers to e.g., Zhu and B aggio (2017) for a brief r eview of the literature in this direction. In this work how ever, we attempt to attack the problem in a dire c tio n different from optimization, a s a contin uation of the work in F erra n te et al. (201 0), where a para metric family of sp ectra l densities was in tro duced, and a ce rtain map from the parameter s pace to the space o f g eneralized moments was s tudied. The question whether a solution to the pa rametric spec tral estima tio n pr oblem in fact exists was essentially left op en in F errante et al. (2010) until recently , suc h exis tence r esult has b een w or ked out in Zhu and Ba ggio (2017). In this pap er, we try to appr oach the question of uniqueness of the solution and even well- po sedness of the problem. The main to o l here is the global inv erse function theor em of Hada mard that is rep or ted e.g., in Gordon (19 72). Howev er, we do not claim to have answered such q uestions to a sa tisfactory level. Instead, we only provide a p oss ible wa y to the answer. The outline of this pap er is as follows. In Sectio n 2, we review the problem formulation and characteriz e the solution in a sp ecial cas e . A sp ectral facto r ization problem is dis c ussed in Section 3, whose re s ult will b e us eful for the developmen t in Section 4, where we pr esent our main results. 2. A P ARAMETRIC FORMULA TION AND THE SOLUTION IN A SP ECIAL CASE Let us first define the set L + := { Λ ∈ H n : G ∗ ( z )Λ G ( z ) > 0 , ∀ z ∈ T } , (5) which obviously contains a ll the Hermitian p ositive def- inite matr ic es, since G ( z ) is of full co lumn ra nk for a ny z ∈ T which r eadily fo llows from the problem setup. By the contin uous dependenc e of eigenv alues on the matrix ent ries , one can verify that L + is an ope n subset of H n . F or Λ ∈ L + , ta ke W Λ as the unique sta ble a nd minimum phase (right ) spe c tral factor of G ∗ Λ G (F err ante et al., 2010, Lemma 11.4.1 ), i.e., G ∗ Λ G = W ∗ Λ W Λ . (6) The s pectr al factor W Λ can b e written as W Λ ( z ) = L −∗ B ∗ P A ( z I − A ) − 1 B + L, (7) where P is the unique stabilizing solution of the Disc r ete- time Algebr aic Ricca ti Equation (D ARE) Π = A ∗ Π A − A ∗ Π B ( B ∗ Π B ) − 1 B ∗ Π A + Λ , (8) and L is the right Cholesky factor of the p os itiv e matrix B ∗ P B , i.e., B ∗ P B = L ∗ L (9) with L being low er triangular having real a nd po sitive diagonal entries. It is worth p ointing out that the D ARE (8) a bove is not a standard o ne, a s Λ ∈ L + can b e indefinite. A for mal pro of for the existence of a stabilizing solution ca n b e found in the app endix of (Avven ti, 2 011, Paper A). T o av oid any r e dundancy in the parameter ization, we have to define the s et L Γ + := L + ∩ im Γ. This is due to a s imple geometric result. More precisely , the adjoint map of Γ in (4) is given b y (cf. F er rante et al. (201 0)) Γ ∗ : H n → C ( T ; H m ) X 7→ G ∗ X G, (10) and w e have the relation (im Γ) ⊥ = ker Γ ∗ = { X ∈ H n : G ∗ ( z ) X G ( z ) = 0 , ∀ z ∈ T } . (11) Hence fo r any Λ ∈ L + , we hav e the ortho g onal decomp o- sition Λ = Λ Γ + Λ ⊥ with Λ Γ ∈ im Γ a nd Λ ⊥ in the orthog onal complement. In view of (11), the part Λ ⊥ do es not contribute to the function v a lue o f G ∗ Λ G o n the unit circ le , and we simply hav e L Γ + = Π im Γ L + , where Π im Γ denotes the o rthogona l pro jection op er ator onto the linear spa ce im Γ. F rom this p oint on, we sha ll take the prior Ψ ∈ S m to b e contin uous o n T , whic h would facilitate reasoning. W e can now define a parametric family of sp ectra l densities S := { Φ Λ = W − 1 Λ Ψ W −∗ Λ : Λ ∈ L Γ + } . (12) W e have the map Λ 7→ W Λ 7→ W − 1 Λ Ψ W −∗ Λ from the parameter Λ ∈ L Γ + to the density function Φ Λ . R emark 1. In the sca lar case, the form of sp ectra l dens ities in the family (12 ) r educes to Φ Λ = Ψ G ∗ Λ G , which is pr ecisely the solutio n (4.3) in Geo rgiou and Lindquist (20 03) o f a co nstrained optimizatio n pr oblem in terms of the L agrang e m ultiplier Λ. An alterna tive matricial par ametrization has b een pr op osed and studied in Geo rgiou (200 6). Our pr oblem is formu lated as follows. Pr oblem 2. Given the filter bank G ( z ) in (1), the prior Ψ ∈ S m contin uous, a nd a p ositive de finite matr ix Σ ∈ im Γ, find a sp ectral dens ity in the par ametric family S defined in (12 ) such that Z G Φ Λ G ∗ = Σ . (13) The a bove problem has an equiv alent formulation. Define im + Γ := im Γ ∩ H + ,n where H + ,n is the o pe n set of n × n Hermitian p ositive definite matrices. Consider the map ω : L Γ + → im + Γ Λ 7→ Z G Φ Λ G ∗ . (14) Then Pro blem 2 is asking: what is the pr e ima ge of Σ ∈ im + Γ under the map ω ? As shown in Zh u and B a ggio (2017), this is a contin uous su rje ctive map betw een o pe n subsets of the linear s pace im Γ, and thus a solution to Problem 2 always exists. T he question now is whether the solution is unique. W e show next that uniqueness is indeed true if the pr io r Ψ has a sp ecial structure. 2.1 Wel l-p ose dness given a sc alar prior In the cas e of a scalar prior, in which we take Ψ( z ) = ψ ( z ) I m , where the sc a lar-v alued function ψ ( z ) ∈ S 1 is contin uous, the ma p ω would r educe to ˜ ω : L Γ + → im + Γ Λ 7→ Z ψ G ( G ∗ Λ G ) − 1 G ∗ , (15) and the family of sp ectral densities b ecomes ˜ S := { Φ Λ = ψ ( G ∗ Λ G ) − 1 : Λ ∈ L Γ + } . (16) According to F err ante et al. (201 0), solution to Problem 2 under a scalar prior exists and is unique. W e shall next show that given a cont inuous prior ψ , the map ˜ ω is a C 1 diffe omorphism 1 betw een L Γ + and im + Γ, which in particu- lar, means that the solution Λ dep ends contin uously on the cov aria nce da ta Σ, and thus the problem is well-posed in the sense of Hadamard. The pro of is an a pplication of the global inv erse function theor em of Hadamard that app ear s e.g., in Gordon (1972). The or em 3. (Hadamar d). Let M 1 and M 2 be connected, oriented, b oundary-le s s n - dimensional manifolds of cla ss C 1 , and supp ose that M 2 is simply connected. Then a C 1 map f : M 1 → M 2 is a diffeomorphism if and only if f is prop er and the J acobian deter mina n t o f f never v anishes. Conditions on the domain and codoma in of ˜ ω can b e verified easily . In fact, the set L Γ + = L + ∩ im Γ is easily seen to b e op en a nd path-connected since b oth L + and im Γ a re such. The simple connectedness of im + Γ has b een rep orted in (Zhu and Baggio , 2017, Prop ositio n 1 ). The fact that ˜ ω is of cla s s C 1 can b e s een alo ng the pr o of of (Zhu a nd Bagg io, 2017, Lemma 1). Mor eov er, pr op erness of the mo r e general map ω has b een proven in (F erra nte et a l., 2010, The o rem 11.4.1 ). Therefore, it is only left to chec k the Jacobian of ˜ ω . The next r esult can b e viewed as an interpretation o f (F er r ante et al., 20 1 0, Theo rem 11.4.2). Here and in the s equel, we shall intro duce the notation Φ( z ; Λ) to denote a sp e ctral density function that depe nds o n the para meter Λ, and us e it interc hang e a bly with Φ Λ ( z ). Pr op osition 4. The Jacobian determina nt of ˜ ω never v a n- ishes in L Γ + , a nd hence the map ˜ ω is a diffeomorphism. Pro of. F rom (Zhu and Bagg io, 20 17, Lemma 1), the differential o f ˜ ω at Λ ∈ L Γ + is δ ˜ ω (Λ; δ Λ) = − Z ψ G ( G ∗ Λ G ) − 1 ( G ∗ δ Λ G )( G ∗ Λ G ) − 1 G ∗ (17) such that δ Λ ∈ im Γ. Our target is to show that δ ˜ ω (Λ; δ Λ) = 0 = ⇒ δ Λ = 0 . T o this e nd, first notice that the middle part of the int egr and in (17) is just the differen tial of the sp ectr a l density Φ Λ = ψ ( G ∗ Λ G ) − 1 w.r.t. Λ : δ Φ( z ; Λ; δ Λ) := − ψ ( G ∗ Λ G ) − 1 ( G ∗ δ Λ G )( G ∗ Λ G ) − 1 . Then the condition δ ˜ ω (Λ; δ Λ) = 0 means that δ Φ( z ; Λ; δ Λ) ∈ ker Γ = (im Γ ∗ ) ⊥ , which in view of (10), reads h G ∗ X G, δ Φ ( z ; Λ; δ Λ) i = tr Z G ∗ X G δ Φ ( z ; Λ; δ Λ) = 0 , ∀ X ∈ H n . In particular , following (F err a nt e et al., 2010, Eqns. 11.4 4– 11.45), cho o sing X = δ Λ would lead to G ∗ δ Λ G ≡ 0 , ∀ z ∈ T , which by (11), implies that δ Λ ∈ (im Γ) ⊥ . Since at the same time δ Λ ∈ im Γ, it is neces sary that δ Λ = 0 . The r e st is just an application of Theorem 3. R emark 5. The unique solution in ˜ S to the sp ectral estimation pr oblem has an interesting characterization in terms o f an optimization problem; cf. (Avven ti, 2011, Paper A) for details. 1 The word “diffeomorphism” in the sequel should alwa ys be under- stoo d i n the C 1 sense. Hence the attributive C 1 will b e omitted. A difficulty aris es when one tries to extend the analysis in the previous pro po sition to the more gener al map ω , as it w ould en tail the differentiation of the sp ectral factor W Λ in (6) w.r.t. the pa rameter Λ . Such a difficulty can b e bypassed by introducing a sp ectral factor ization a s will b e discussed next. 3. A DIFFEOMO RPHIC SPE CTRAL F A CTORIZA TIO N F o llowing the lines of Avven ti (20 11), g iven the stabilizing solution P o f the D ARE (8), let us intro duce a change of v a riables by setting C := L −∗ B ∗ P. (18) Then it is not difficult to r ecov er the relation L = C B fo r the Cholesky factor in (9). In this wa y , the sp ectral factor (7) ca n b e rewritten as W Λ ( z ) = C A ( z I − A ) − 1 B + C B = z C G, (19) where the seco nd equality holds b ecause of the identit y A ( z I − A ) − 1 + I = z ( z I − A ) − 1 . In v iew of this, the factorization (6) can then b e rewritten as G ∗ Λ G = G ∗ C ∗ C G, ∀ z ∈ T . (20) This relatio n has also been expresse d in (F er rante et al., 2010, Equa tion 11.29 ). In the sequel, we shall also call the m × n matrix C a “ sp ectral factor”. As rep orted in (Avvent i, 201 1, Section A.5.5), it is p ossible to build a home omorphi c factoriza tion by c a refully choos- ing the set where the factor C lives. More pr ecisely , le t the set C + ⊂ C m × n contain tho se ma trices C that sa tisfy the following tw o c onditions • C B is lower tria ng ular with real a nd p ositive diagonal ent ries , • A − B ( C B ) − 1 C A has eig env alues strictly inside the unit cir cle. Define the map h : L Γ + → C + Λ 7→ C via (1 8) . (21) Then acc ording to (Avven ti, 2 011, Theo rem A.5.5), the map h o f sp ectra l factor ization is a homeomorphism. W e shall next str e ngthen this r esult by showing that the map h is in fact a diffeomorphism using Theorem 3. 3.1 Char acterization of diffe omorphism W e are g oing to apply Theorem 3 to the inv erse of h h − 1 : C + → L Γ + C 7→ Λ := Π im Γ ( C ∗ C ) . (22) Those tec hnical re q uirements on the do main and co doma in of h − 1 can b e verified without difficult y . The set C + is an op en subset of the linear space C :=  C ∈ C m × n : C B is lower triang ular with r eal diag o nal entries } , whose real dimensio n coincides with im Γ (cf. Avventi (2011)). The fact that C + is also path-co nnected is a consequence o f h b eing a homeomorphism. F urther more, the pro o f of L Γ + being simply connected can be adapted easily fr om (Zhu and B aggio, 20 17, Pro po sition 1). The map h − 1 is actually smo oth (hence of co urse C 1 ) bec ause it is a comp osition o f the quadr atic map C 7→ C ∗ C and the pro jection Π im Γ , b oth of which are smo o th. The fact that h − 1 is prop er has also b een rep orted in Avvent i (2011). Ther efore, it remains to in vestigate the Jacobian of h − 1 . In order to carr y out explic it computation, it is necessary to choo se bas e s for the tw o linea r spaces C and im Γ. Let M := m (2 n − m ), and let { Λ 1 , Λ 2 , . . . , Λ M } and { C 1 , . . . , C M } b e orthono rmal bases o f im Γ a nd C , resp ec- tively . Then one ca n para meterize Λ ∈ L Γ + and C ∈ C + as Λ( x ) = x 1 Λ 1 + x 2 Λ 2 + · · · + x M Λ M , C ( y ) = y 1 C 1 + y 2 C 2 + · · · + y M C M , (23) for so me x j , y j ∈ R , j = 1 , . . . , M . The map h − 1 can then be expres sed co or dinate-wisely as x j = h Λ j , C ( y ) ∗ C ( y ) i . (24) Then the partial deriv ativ es ca n b e c omputed a s ∂ x j ∂ y k = h Λ j , C ∗ k C ( y ) + C ∗ ( y ) C k i , (25) which is the ( j, k ) element o f the Jaco bian matr ix denoted as J h − 1 ( y ). W e need some ancillar y res ults in or der to show that h − 1 has everywhere nonv anishing Jacobian. Pr op osition 6. If v ∈ C n is such that v ∗ G ( z ) = 0 for a ll z ∈ T , then v = 0. Pro of. The condition that v ∗ G ( z ) = 0 fo r all z ∈ T implies tha t v ∗ Z GG ∗ v = 0 . Under our problem setting stated in Section 1 , we have R GG ∗ > 0 and thus the a ssertion of the prop osition follows. T o see the fact of p ositive definiteness, note fir st that the following expansio n ho lds G ( z ) = ( z I − A ) − 1 B = z − 1 ∞ X k =0 z − k A k B , for | z | ≥ 1 , (26) since A is stable. Then b y the Parsev al identit y , we hav e Z GG ∗ = ∞ X k =0 A k B B ∗ ( A ∗ ) k = R R ∗ , where R = [ B , AB , . . . , A k B , . . . ]. The ab ove is the unique solution o f the discrete-time Ly apunov equation X − AX A ∗ = B B ∗ . (27) Since ( A, B ) is by assumption reachable, R is of full row rank, a nd therefore R GG ∗ > 0 . Pr op osition 7. Given C ∈ C + , the rational matrix equa- tion in the unknown V ∈ C m × n G ∗ ( C ∗ V + V ∗ C ) G = 0 , ∀ z ∈ T (28) has the genera l s olution V = Q C (29) where Q ∈ C m × m is a n arbitrar y consta nt skew-Hermitian matrix. If one further re quires V ∈ C , then (28) ha s only the triv ial solutio n V = 0. Pro of. The equa tion (28) is equiv alent to z ∗ G ∗ ( C ∗ V + V ∗ C ) Gz = 0 , ∀ z ∈ T . (30) Let z C G ( z ) = z C ( z I − A ) − 1 B = P C ( z ) z − n det( z I − A ) , where P C ( z ) := z − n +1 C adj( z I − A ) B a nd adj( · ) denotes the adjugate matrix. Obviously , P C ( z ) is a matrix p oly- nomial in the indeterminate z − 1 , whic h is intended to conform to the engineer ing conv ention. F ro m (26 ), we hav e lim z →∞ z C G = C B = lim z →∞ P C ( z ) , where the second equality ho lds since lim z →∞ z − n det( z I − A ) = 1. More over, the scalar p olyno mia l det P C ( z ) has a ll its r o ots inside D , which can be seen from (19) as z C G is minim um phase, i.e., admits a stable in verse. Define similarly P V ( z ) := z − n +1 V adj( z I − A ) B . Then one can re duce (30) to the matrix p olyno mia l equation P ∗ C ( z ) P V ( z ) + P ∗ V ( z ) P C ( z ) = 0 , ∀ z ∈ T , (31) in which we hav e P ∗ C (0) = h lim z →∞ P C ( z ) i ∗ = ( C B ) ∗ nonsingular b eca us e C ∈ C + . By the identit y theor em for holomorphic functions, if the ab ov e equa tio n ho lds o n T , then it ho lds for any z ∈ C ex c ept for 0 (and ∞ ). Hence the restriction z ∈ T can b e remov ed here. Since P ∗ C is anti-stable and P ∗ C (0) nonsingular , acco rding to (a v ariant of ) (Jeˇ zek, 198 6, Theorem MP1 ), the general solution o f (31) is P V = Q P C , where Q ∈ C m × m is an arbitra r y consta n t skew-Hermitian matrix. This in turn implies that V G ( z ) = QC G ( z ) , ∀ z ∈ T , (32) which in v iew o f Pro po sition 6, further implies that V = QC . T o prov e the remaining part o f the claim, just apply the power series ex pa nsion (26) to (32), and notice tha t all the F o urier co efficients on the tw o sides o f (32) must coincide. This in particular means that V B = Q C B . Since we have C ∈ C + and V ∈ C in addition, both V B and C B a re low er tr iangular and the latter is in vertible. Therefore Q turns out to b e also low er triang ular a nd a t the same time skew-Hermitian, which necessa rily means that Q is equa l to 0 and so is V . The or em 8. The Jac o bian determinant of h − 1 never v an- ishes in C + , and hence the map h in (21) is a diffeomor- phism. Pro of. Supp o se v ∈ R M is such that J h − 1 ( y ) v = 0. W e need to show that v = 0. T o this end, no tice fro m (25 ) that eq uiv a len tly w e have fo r j = 1 , 2 , . . . , M , 0 = M X k =1 v k h Λ j , C ∗ k C ( y ) + C ∗ ( y ) C k i = h Λ j , C ∗ ( v ) C ( y ) + C ∗ ( y ) C ( v ) i , which implies that C ∗ ( v ) C ( y ) + C ∗ ( y ) C ( v ) ⊥ im Γ . In vie w of (11), this in tur n mea ns G ∗ ( z ) [ C ∗ ( v ) C ( y ) + C ∗ ( y ) C ( v )] G ( z ) = 0 , ∀ z ∈ T . By P rop osition 7, the only solution is v = 0. Thus Theor e m 3 is applicable and this completes the pro of. 4. THE GENERAL MAP ω Let us return to the map ω defined in (14 ). W e s hall use the result obtained in the pr evious section to a ttack the uniqueness conjecture p osed in Zh u and B aggio (201 7). Given the relation (19 ), the sp ectral density Φ Λ can b e repara meter ized in C as Φ Λ ≡ Φ C := ( C G ) − 1 Ψ( C G ) −∗ . (33) In this wa y , the map ω can b e expre ssed a s a comp ositio n ω = τ ◦ h : ω (Λ) = τ ( h (Λ)) , (34) with h in (21) and τ : C + → im + Γ C 7→ Z G Φ C G ∗ . (35) Since h has b een prov ed to b e a diffeo mo rphism, we can restrict our attent ion to the ma p τ due to the next simple result. Pr op osition 9. Let X , Y , Z be op en s ubs ets of R n . Supp ose we have functions f : X → Y , g : Y → Z and f is a diffeomorphism betw een X and Y . Define the comp osite function h = g ◦ f : X → Z . (36) Then h is a diffeomor phism b etw een X and Z if and o nly if g is a diffeomor phis m betw een Y a nd Z . Pro of. The “if ” part is trivial since a comp osition of t wo diffeomorphisms is a gain a diffeomorphism. T o see the conv erse, for y ∈ Y , let x = f − 1 ( y ) ∈ X and put it into (36) as an argument of h . Then one g e ts g = h ◦ f − 1 , which is aga in a compo sition of t wo diffeomorphisms. Since pr o pe r ness of the map ω has alre a dy be prov en, it remains to show that ω is contin uously differentiable and has everywhere nonv anishing Jacobian. In view of the relation (34) a nd the prev io us pro po sition, it would b e sufficient a nd necessa ry that the map τ p osses ses such tw o prop erties. W e need the next lemma b efore proving the contin uous differentiabilit y . L emma 10. Let a seq ue nc e { Λ k } k ≥ 1 ⊂ L + conv erge to some ¯ Λ ∈ L + . Then ther e exists a r eal n umber µ > 0 s uch that G ∗ ( e iθ )Λ k G ( e iθ ) ≥ µI , ∀ k , θ . Pro of. The claim of the lemma follows fr o m the co n- tin uity of the function G ∗ ( e iθ )Λ G ( e iθ ) in Λ and θ , a nd the uniform conv erg ence of the sequence of functions { G ∗ Λ k G } k ≥ 1 to G ∗ ¯ Λ G . Pr op osition 11. The ma p τ in (35) is of class C 1 . Pro of. W e can pro ceed by mimic king the pr o of of (Zhu and Ba ggio, 2017 , Lemma 1), although the arg umen t here is slightly mor e g eneral. First co mpute the differential o f Φ( z ; C ) w.r.t. C ∈ C + as δ Φ( z ; C ; δ C ) = − ( C G ) − 1 δ C G Φ C − Φ C G ∗ δ C ∗ ( C G ) −∗ , (37) which is easily seen to be c o nt inuous in C and θ ∈ [ − π , π ] for a fixed δ C ∈ C . This means that we ca n take the differential o f the map τ inside the integral in (35 ) δ τ ( C ; δ C ) = Z Gδ Φ( e iθ ; C ; δ C ) G ∗ . (38) Next we s how that the ab ov e different ial is contin uous in C for a fix ed δ C . T o this end, supp ose we have a seq uenc e { C k } k ≥ 1 ⊂ C + that conv erges to some ¯ C ∈ C + as k → ∞ . Due to the r e la tion (20), w e hav e for each k G ∗ Λ k G = G ∗ C ∗ k C k G, ∀ z ∈ T , (39) where Λ k = h − 1 ( C k ) ∈ L Γ + . Since h is a diffeomorphis m by Theor em 8, w e have lim k →∞ Λ k = ¯ Λ := h − 1 ( ¯ C ) . Let λ min ,k ( θ ) be the smallest eigenv alue of G ∗ ( e iθ )Λ k G ( e iθ ), and σ min ,k ( θ ) b e the smallest singular v a lue of C k G ( e iθ ). In vie w of (39), we hav e λ min ,k ( θ ) = σ 2 min ,k ( θ ) By Lemma 10, there exist a rea l nu mber µ > 0 suc h that λ min ,k ( θ ) ≥ µ = ⇒ σ min ,k ( θ ) ≥ √ µ, ∀ k , θ . Then we hav e k δ Φ( e iθ ; C k ; δ C ) k 2 ≤ 2 k ( C k G ) − 1 δ C G Φ C k k 2 ≤ 2 k ( C k G ) − 1 k 3 2 k δ C G k 2 k Ψ k 2 ≤ 2 σ 3 min ,k ( θ ) k δ C G k F k Ψ k F ≤ K , where the constant K = 2 µ 3 / 2 max θ k δ C G ( e iθ ) k F max θ k Ψ( e iθ ) k F . W e can now b o und the integrand in (38). F or a n y θ ∈ [ − π , π ] and k ≥ 1, w e ha ve     Gδ Φ( e iθ ; C k ; δ C ) G ∗  j ℓ    ≤ k Gδ Φ( e iθ ; C k ; δ C ) G ∗ k F ≤ κ k Gδ Φ( e iθ ; C k ; δ C ) G ∗ k 2 ≤ κ K k G k 2 2 ≤ κ K G max , where G max := max θ ∈ [ − π ,π ] tr  G ( e iθ ) G ∗ ( e iθ )  (40) and κ is a constant for no rm eq uiv a lence. The las t step is a n applica tion of Lebesg ue’s do minated co n vergence theorem to conclude lim k →∞ δ τ ( C k ; δ C ) = δ τ ( ¯ C ; δ C ) , which completes the pro of. W e are now left with the task of investigating whether the Ja cobian of τ v anishes nowhere, which can b e ap- proached via the differential (38). How ever, the tr ick of orthogo nality in the pr o of of P rop osition 4 do es not apply in a stra ightforw ard manner to the ge ne r al map ω . The desired r esult can b e obtained if an additional constraint is imp osed o n the prio r Ψ, and this is rep orted in the next prop osition. Pr op osition 12. If the prior Ψ is such that the eq uality tr Z F ∗ Ψ F = tr Z F Ψ F ∗ (41) holds for any C ∈ C + and any V ∈ C , where the matr ix function F = V G ( C G ) − 1 , then the Jacobia n determinant of τ v anishes nowhere in C + , and hence the map ω is a diffeomorphism. Pro of. Fix C ∈ C + and let δ τ ( C ; V ) = 0 for s o me V ∈ C . In vie w of (38), this would imply that δ Φ( z ; C ; V ) ∈ ker Γ = (im Γ ∗ ) ⊥ , which in view of (10), means h G ∗ X G, δ Φ ( z ; C ; V ) i = tr Z G ∗ X G δ Φ ( z ; C ; V ) = 0 , ∀ X ∈ H n . (42) Cho osing X = C ∗ V + V ∗ C in (42) would le ad to the relation tr Z 2 F Ψ F ∗ + Ψ F ∗ F ∗ + F F Ψ = 0 after some manipulations of the v ariables using (37). The left-hand s ide in the ab ove equa tion is differen t from tr Z ( F + F ∗ ) Ψ ( F + F ∗ ) (43) in only one term. If the equality (41) holds for any C ∈ C + and V ∈ C , then we would hav e the expressio n (43) equal to zero, which, by the same reaso ning as in Pro po sition 4, implies F + F ∗ ≡ 0 , ∀ z ∈ T , which is equiv alent to (28). In view of Pro po sition 7, this in turn implies V = 0 . The ab ov e prop ositio n do es not improve muc h ov er P rop o- sition 4 for the scalar c a se, since the requirement on the prior seems very artificia l and a ma tr ix-v alued Ψ in ge neral do es not satisfy it, as illustrated in the next example. Example 13. Cons ider a static case in whic h n = m , B = I , and the matr ix A void, that is, the tra nsfer function (1) reduces to G = z − 1 I and the o utput of the linear system is identical to the 1-step delay ed input. Let us fix Ψ ≡ diag { 1 , 2 } and C = I , and then (41 ) would reduce to tr V ∗ Ψ V = tr V Ψ V ∗ . The only re quirement on V is b eing low er-tria ng ular with real dia gonal entries. Hence we ca n take, e.g ., V =  1 0 1 2  , and it is stra ightf or ward to chec k that the ab ov e equa lit y do es not hold. Ho wev er, in this overly simplified exa mple, the solutio n to Pr o blem 2 is still unique. Indeed, given Σ ∈ im + Γ and Ψ ∈ S m , one is lo oking for a parameter C ∈ C + such that Z C − 1 Ψ C −∗ = Σ . Clearly , this implies C − 1 L R = L Σ U, where the notation L A denotes the usual Cholesk y fa ctor of A > 0, U is a unitary matr ix, and R := R Ψ > 0. 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