Generic uniqueness of a structured matrix factorization and applications in blind source separation

0.75 MB·

by  Ignat Domanov

1
Generic uniqueness of a structured matrix
factorization and applications in blind source
separation
Ignat Domanov and Lieven De Lathauwer, Fellow, IEEE
Abstract—Algebraic geometry, although little explored in sig- solution of (1) becomes unique. Commonly used constraints
6
1 nal processing, provides tools that are very convenient for include sparsity [1], constant modulus [2] and Vandermonde
investigating generic properties in a wide range of applications.
0 structure [3].
Genericpropertiesarepropertiesthathold“almosteverywhere”.
2 Sufficient conditions for uniqueness can be deterministic or
We present a set of conditions that are sufficient for demon-
n strating the generic uniqueness of a certain structured matrix generic. Deterministic conditions concern particular matrices
a factorization.Thissetofconditionsmaybeusedasachecklistfor M and S. Generic conditions concern the situation that can
J genericuniquenessindifferentsettings.Wediscusstwoparticular be expected in general; a generic property is a property that

6 applications in detail. We provide a relaxed generic uniqueness holds everywhere except for a set of measure 0. (A formal
condition for joint matrix diagonalization that is relevant for
definition will be given in Subsection I-C below.)
] independent component analysis in the underdetermined case.
G Wepresentgenericuniquenessconditionsforarecentlyproposed To illustrate the meaning of deterministic and generic
A class of deterministic blind source separation methods that rely uniqueness let us consider decomposition (1) in which X ∈
on mild source models. For the interested reader we provide CK×N, M ∈ CK×R and the columns of S ∈ CN×R are
.
h someintuitiononhowtheresultsareconnectedtotheiralgebraic obtainedbysamplingtheexponentialsignalszt−1,...,zt−1at
at geometric roots. t=1,...,N. Then (S)nr =(zrn−1), i.e. S is1a VandermRonde
m IndexTerms—structuredmatrixfactorization,structuredrank matrix. A deterministic condition under which decomposition
decomposition, blind source separation, direction of arrival,
[ (1) is unique (up to trivial indeterminacies) is [3]: (i) the
uniqueness, algebraic geometry Vandermonde matrix S has strictly more rows than columns
1
and its generators z are distinct and (ii) the matrix M has
v j
3 I. INTRODUCTION full column rank. (In this paper we say that an K×R matrix
7 A. Blind source separation and uniqueness has full column rank if its column rank is R, which implies
1 K ≥R.)Thisdeterministicconditioncaneasilybeverifiedfor
1 The matrix factorization X = MST is well known in the any particular M and S. A generic variant is: (i) the Vander-
0 blind source separation (BSS) context: the rows of ST and X monde matrix S has N >R and (ii) the (unstructured) matrix
1. represent unknown source signals and their observed linear MhasK ≥R.Indeed,underthesedimensionalityconditions
0 mixtures, respectively. The task of the BSS problem is to
the deterministic conditions are satisfied everywhere, except
6 estimate the source matrix S and the mixing matrix M from in a set of measure 0 (which contains the particular cases of
1
: X. If no prior information is available on the matrices M or coinciding generators zr and the cases in which the columns
v S,thentheycannotbeuniquelyidentifiedfromX.Indeed,for of M are not linearly independent despite the fact that M is
i any nonsingular matrix T,
X square or even tall). Note that generic properties do not allow
r X=MST =(MT)(ST−T)T =MST. (1) onetomakestatementsaboutspecificmatrices;theyonlyshow
a the general picture.
Applications may involve particular constraints on M and/or As mentioned before, BSS has many variants, which dif-
S, so that in the resulting class of structured matrices the fer in the types of constraints that are imposed. Different
constraints usually mean different deterministic uniqueness
This work was supported by Research Council KU Leuven: C1 project conditions, and the derivation of these is work that is difficult
c16/15/059-nD, CoE PFV/10/002 (OPTEC) and PDM postdoc grant, by
to automate. In this paper we focus on generic uniqueness
F.W.O.: project G.0830.14N, G.0881.14N, by the Belgian Federal Science
Policy Office: IUAP P7 (DYSCO II, Dynamical systems, control and op- conditions. We propose a framework with which generic
timization, 2012-2017), by EU: The research leading to these results has uniqueness can be investigated in a broad range of cases.
received funding from the European Research Council under the European
Indeed, it will become clear that if we limit ourselves to
Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Advanced
Grant: BIOTENSORS (no. 339804). This paper reflects only the authors’ generic uniqueness, the derivation of conditions can to some
views and the Union is not liable for any use that may be made of the extent be automated. We discuss two concrete applications
containedinformation.
which may serve as examples.
The authors are with Group Science, Engineering and Technol-
ogy, KU Leuven-Kulak, E. Sabbelaan 53, 8500 Kortrijk, Belgium. Our approach builds on results in algebraic geometry. Al-
Lieven De Lathauwer is also with Dept. of Electrical Engineering gebraic geometry has so far been used in system theory in
ESAT/STADIUS KU Leuven, Kasteelpark Arenberg 10, bus 2446, B-3001
[4],[5]anditalsohasdirectapplicationsintensor-basedBSS
Leuven-Heverlee, Belgium (e-mail: [email protected];
[email protected]). via the generic uniqueness of tensor decompositions [6]–[8].
2
Ourpapermakesacontributioninfurtherconnectingalgebraic the other hand, a row of Brat is modeled as values taken by
geometry with applications in signal processing. one particular function pn(·) at R points ζ ,...,ζ .
qn(·) 1 R
The structure that we consider in our study for the N ×R
B. Notation matrix B(z) is more general than the rational structure of
ThroughoutthepaperFdenotesthefieldofrealorcomplex Brat(z) in the sense that we additionally allow (possibly
nonlinear)transformationsofζ ,...,ζ .Formally,weassume
numbers; bold lowercase letters denote vectors, while bold 1 R
that the columns of B(z) are sampled values of known vector
uppercase letters represent matrices; a column of a matrix
functions of the form
A and an entry of a vector b are denoted by a and b ,
j j
respectively; the superscripts ·∗, ·T and ·H are used for the (cid:20)p (f(ζ)) p (f(ζ))(cid:21)T
b(ζ)= 1 ... N , ζ ∈Fl, (3)
conjugate, transpose, and Hermitian transpose, respectively; q (f(ζ)) q (f(ζ))
1 N
“⊗” denotes the Kronecker product.
at points ζ ,...,ζ ∈Fl, such that
1 R
C. Statement of the problem and organization of the paper p1(f(ζ1)) ... p1(f(ζR))
q1(f(ζ1)) q1(f(ζR))
A structured matrix factorization. In this paper we consider B(z)=[b(ζ1)...b(ζR)]= ... ... ... ,
the following structured factorization of a K×N matrix Y,  
pN(f(ζ1)) ... pN(f(ζR))
Y =A(z)B(z)T, z∈Ω (2) qN(f(ζ1)) qN(f(ζR))
where
where Ω is a subset of Fn and A(z) and B(z) are known
f(ζ)=(f (ζ),...,f (ζ))∈Fl,
matrix-valued functions defined on Ω. 1 l
f ,...,f are scalar functions of l variables.
W.l.o.g. we can assume that the parameter vector z = 1 l
[z1 ... zn]T is ordered such that B(z) depends on the last The functions f1,...,fl are subject to an analyticity assump-
s≤n entries, while A(z) depends on m≥0 entries that are tion that will be specified in Theorem 1 further. Although
not necessarily the first or the last. That is, our general result in Theorem 1 will be formulated in terms
of functions f ,...,f in l variables, in the applications in
A(z)=A(z ,...,z ), B(z)=B(z ,...,z ) 1 l
i1 im n−s+1 n SectionsIII–IVwewillonlyneedentry-wisetransformations:
for some 1≤i <i <···<i ≤n. In general, the entries
1 2 m
used to parameterize A and B are allowed to overlap so that f(ζ)=f(ζ ,...,ζ )=(f (ζ ),...,f (ζ )) (4)
1 l 1 1 l l
m+s ≥ n. The case where A and B depend on separated
with f ,...,f analytic functions in one variable.
parameter sets corresponds to m + s = n; in this case A 1 l
As an example of how the model for B(z) can be used,
depends strictly on the first m of the entries of z.
consider R vectors that are obtained by sampling the expo-
OurstudyislimitedtoK×RmatricesA(z)thatgenerically
have full column rank. We do not make any other assump- nential signals eiζ1(t−1),...,eiζR(t−1) (with ζ1,...,ζR ∈ R)
at t=1,...,N. In this case B(z) is an N×R Vandermonde
tions on the form of A(z). In particular, we do not impose
matrix with unit norm generators; its rth column is b(ζ ) =
restrictions on how the entries depend on z. We are however r
more explicit about the form of the N × R matrix B(z). [1eiζr ... eiζr(N−1)]T.Wehaveeiζr(n−1) = pqnn((ff((ζζrr)))),where
We assume that each of its columns br(z) is generated by f(ζ) = eiζ, pn(x) = xn−1, and qn(x) = 1 for ζ ∈ R and
l parameters that are independent of the parameters used to x∈C.
generatetheothercolumns,i.e.,B(z)=[b (ζ ) ... b (ζ )] Generic uniqueness of the decomposition. We interpret
1 1 R R
withζ ,...,ζ ∈Fl.Notethattheindependenceimpliesthat factorization (2) as a decomposition into a sum of structured
1 R
s=Rl and that [ζT ... ζT]T and [z ... z ]T are the rank-1 matrices
1 R n−s+1 n
same up to index permutation. R
(cid:88)
For the sake of exposition, let us first consider a class of Y =A(z)B(z)T = ar(z)b(ζr)T, z∈Ω, (5)
matricesB(z)thatissmallerthantheclassthatwewillbeable r=1
to handle in our derivation of generic uniqueness conditions. where a (z) denotes the rth column of A(z). It is clear that
r
Namely, let us first consider matrices Brat(z), of which the in (5) the rank-1 terms can be arbitrarily permuted. We say
n-th row is obtained by evaluating a known rational function that decomposition (5) is unique when it is only subject to
pqnn((··)) at some points ζ1,...,ζR ∈Fl, 1≤n≤N: this trivial indeterminacy. We say that decomposition (5) is
p1(ζ1) ... p1(ζR) genericallyuniqueifitisuniqueforagenericchoiceofz∈Ω,
q1(ζ1) q1(ζR) that is
Brat(z)= ... ... ... , µn{z∈Ω: decomposition (5) is not unique}=0, (6)
pN(ζ1) ... pN(ζR)
qN(ζ1) qN(ζR) whereµn isameasurethatisabsolutelycontinuous(a.c.)with
where respect to the Lebesgue measure on Fn.
In this paper we present conditions on the polynomials
p ,...,p , q ,...,q are polynomials in l variables.
1 N 1 N p ,...,p , q ,...,q , the function f and the set Ω which
1 N 1 N
NotethatwemodelacolumnofBrat throughthevaluestaken guarantee that decomposition (5) is generically unique. As
by N functions p1(·),...,pN(·) at one particular point ζ . On a technical assumption, since in the case where µ (Ω) = 0
q1(·) qN(·) r n
3
condition (6) cannot be used to infer generic uniqueness from Further,
a subset of Ω, we assume that µ (Ω)>0.
n Range(r)={r(x): q (x)···q (x)(cid:54)=0, x∈Cl}⊂CN
Organizationandresults.InSectionIIwestatethemainre- 1 N
sultofthispaperingeneralterms,namely,Theorem1presents denotes the set of all values of r(x) for x∈Cl. We say that
conditionsthatguaranteethatthestructureddecomposition(5) the set Range(r) is invariant under scaling if
is generically unique. The proof of Theorem 1 is given in
Range(r)⊇λ·Range(r) for all λ∈C.
Appendix A. Besides the technical derivation, Appendix A
provides some intuition behind the high-level reasoning and Theorem 1. LetΩbeasubsetofFn andµ (Ω)>0.Assume
n
makes the connection with the trisecant lemma in algebraic that
geometry, for readers who are interested. In Sections III–IV
1) thematrixA(z)hasfullcolumnrankforagenericchoice
we use Theorem 1 to obtain new uniqueness results in the
of z∈Ω, that is,
context of two different applications. This is done by first
expressing the specific BSS problem as a decomposition of µ {z∈Ω: rankA(z)<R}=0;
n
the form (5), for which the list of conditions in Theorem 1
2) the coordinate functions f ,...,f of f can be repre-
ischecked. SectionIIIconcernsan applicationinindependent 1 l
sented as
component analysis. More precisely, it concerns joint matrix
f (ζ) f (ζ)
diagonalization in the underdetermined case (more sources f (ζ)= 1,num ,...,f (ζ)= l,num ,
than observations) and presents a new, relaxed bound on the 1 f1,den(ζ) l fl,den(ζ)
number of sources under which the solution of this basic
where the functions
subproblem is generically unique. This bound is a simple
expression in the number of matrices and their dimension. f1,num(ζ),f1,den(ζ),...,fl,num(ζ),fl,den(ζ)
Section IV presents generic uniqueness results for a recently are analytic on Cl;
introduced class of deterministic blind source separation al- 3) there exists ζ0 ∈Cl such that detJ(f,ζ0)(cid:54)=0;
gorithms that may be seen as a variant of sparse component
4) the dimension of the subspace spanned by the vectors of
analysiswhichmakesuseofanon-discretedictionaryofbasis
form (7) is at least N(cid:98),
functions. Appendix B contains the short proof of a technical
lemma in Section IV. The paper is concluded in Section V. dimspan{r(x): q1(x)···qN(x)(cid:54)=0, x∈Cl}≥N(cid:98);
5) rankJ(r,x)≤(cid:98)l for a generic choice of x∈Cl;
II. MAINRESULT 6) R≤N(cid:98)−(cid:98)l or R≤N(cid:98)−(cid:98)l−1, depending on whether the
set Range(r) is invariant under scaling or not.
Thefollowingtheoremisourmainresultongenericunique-
Then decomposition (5) is generically unique.
ness of decomposition (5). It states that, generically, the R
structuredrank-1termsoftheK×N matrixYcanbeuniquely Proof. See Appendix A.
recovered if K ≥ R and R ≤ N(cid:98) −(cid:98)l. Here, N(cid:98) ≤ N is
Assumptions 1–6 can be used as a checklist for demon-
a lower bound on the dimension of the linear vector space
span{r(x) : q (x)···q (x) (cid:54)= 0, x ∈ Fl} generated by strating the generic uniqueness of decompositions that can be
1 N
put in the form (2). We will discuss two application examples
vectors of the form
in Sections III–IV. We comment on the following aspects of
r(x)=(cid:20)p1(x) ... pN(x)(cid:21)T . (7) assumptions 2–6.
q (x) q (x) •InthispaperwewilluseTheorem1inthecasewheref(ζ)
1 N
is of the form (4). For such f the matrix J(f,ζ) is diagonal,
(Note that the definition of r(x) does not involve a nonlinear
yielding that detJ(f,ζ) = f(cid:48)(ζ )···f(cid:48)(ζ ). Moreover, in
transformation f, even when such a nonlinear transformation 1 1 l l
this paper f ,...,f are non-constant, so detJ(f,ζ) is not
1 l
is used for modelling b(ζ).) On the other hand, the value
identically zero. Thus, assumption 3 in Theorem 1 will hold
(cid:98)l ≤ l is an upper bound on the number of “free parameters” automatically.
actually needed to parameterize a generic vector of the form
• For the reader who wishes to apply Theorem 1 in cases
(7). (Indeed, although r(x) is generated by l independent
where f is not of the form (4), we recall the definition of an
parameters, it may be possible to do it with less in particular
analytic (or holomorphic) function of several variables used
cases. For instance, let N = 3, q1(x) = q2(x) = q3(x) = 1 in assumption 2. A function f : Cl → C of l complex
andp (x)=x +x ,p (x)=x −x ,p (x)=x +x ,sothat
1 1 3 2 (cid:20) 2 3 (cid:21)3 1 2 variablesisanalytic[9,page4]ifitisanalyticineachvariable
1 0 1
r(x)=Wx with W = 0 1 −1 . Since rank(W)=2, separately, that is, if for each j = 1,...,l and accordingly
1 1 0 fixed ζ ,···ζ ,ζ ,...,ζ the function
r(x) can be parameterized by 2<3 independent parameters.) 1 j−1 j+1 l
In the theorem and throughout the paper we use J(r,x)∈ z (cid:55)→f(ζ ,···ζ ,z,ζ ,...,ζ )
1 j−1 j+1 l
FN×l and J(f,ζ) ∈ Fl×l to denote the Jacobian matrices of
is analytic on C in the classical one-variable sense. Examples
r and f, respectively,
of analytic functions of several variables can be obtained by
∂pi ∂f taking compositions of multivariate polynomials and analytic
(J(r,x)) = qi , (J(f,ζ)) = i.
ij ∂x ij ∂ζ functionsinonevariables,e.g.f(ζ1,ζ2)=sin(cos(ζ1ζ2))+ζ1.
j j
4
• To check assumption 4 in Theorem 1 it is sufficient to “H” in (8) is replaced by the superscript “T” (for quantities
present (or prove the existence of) N(cid:98) linearly independent x, M are s that can be either real valued or complex valued);
vectors {r(xi)}Ni(cid:98)=1. It is clear that larger N(cid:98) yield a better 2) in [11], [17] (where x, M are s are complex valued). In
bound on R in assumption 6. In all cases considered in this [8], [11], [17] the matrix equations in (8) were interpreted as
paper N(cid:98) = N. The situation N(cid:98) < N may appear when the a so-called canonical polyadic decomposition of a (partially
N ×1 vector-function b(ζ) models a periodic, (locally) odd symmetric) tensor. In the following theorems we interpret the
or even function, etc. equations in (8) as matrix factorization problem (2). The new
• The goal of assumption 5 is to check whether generic interpretation only relies on elementary linear algebra; it does
signals of the form (7) can be re-parameterized with fewer not make use of advanced results on tensor decompositions
(i.e. (cid:98)l < l) parameters. In this case, the Jacobian J(r,x) has while it does lead to more relaxed bounds on R than in [11],
indeed rank strictly less than l. It is clear that assumption 5 in [17] for I ≥5. We consider the variants τ (cid:54)=0, 1≤p≤P,
p
Theorem 1 holds trivially for (cid:98)l =l and that smaller (cid:98)l yield a and τ1 =0 in Theorems 2 and 3, respectively.
betterboundonRinassumption6.Inthispaperweseteither
Theorem 2. Assume that τ (cid:54)=0 and
(cid:98)l=l(namelyintheproofofTheorem5)or,inthecasewhere 1
itisclearthatJ(r,x)doesnothavefullcolumnrank(namely R≤min(2P,(I−1)2). (9)
in the proof of Theorems 2 and 6), (cid:98)l=l−1.
• Although the Theorem holds both for F=C and F=R, Then (8) has a unique solution for generic matrices M and
we formulated assumptions 3, 4 and 5 in Theorem 1 for ζ0 ∈ D1,...,DP, i.e.,
Cl and x∈Cl. In these assumptions Cl can also be replaced
by Rl. We presented the complex variants, even for the case µk{(vec(D),vec(M)): solution of (8) is not unique}=0,
F = R, since they may be easier to verify than their real (10)
where D denotes the P × R matrix with entries d , k =
counterparts,asζ0 andxareallowedtotakevaluesinalarger pr
set. On the other hand, the analyticity on Cl in assumption 2 IR+PR, and µk is a measure that is a.c. with respect to the
is a stronger assumption than analyticity on Rl and is needed Lebesgue measure on Ck .
in the form it is given. Proof. (i) First we rewrite the equations in (8) as matrix
decomposition (5)1. In step (ii) we will apply Theorem 1
III. ANAPPLICATIONININDEPENDENTCOMPONENT to (5).
ANALYSIS R
Since CH = (cid:80) d∗ m mH, the pth equation in (8) is
We consider data described by the model x = Ms, where p pr r r
r=1
x is the I-dimensional vector of observations, s is the R- equivalent to the following pair of equations
dimensional unknown source vector and M is the I-by-R
unknown mixing matrix. We assume that the sources are ReC = Cp+CHp =(cid:88)R Red m mH,
mutually uncorrelated but individually correlated in time. It is p 2 pr r r
r=1
knownthatthespatialcovariancematricesoftheobservations
satisfy [10] ImC = Cp−CHp =(cid:88)R Imd m mH.
p 2i pr r r
R r=1
(cid:88)
C =E(x xH )=MD MH = d m mH,
1 t t+τ1 1 1r r r Since vec(mmH)=m∗⊗m, we further obtain that
r=1
... (8) vec(ReCp)T =
(cid:88)R [Redp1...RedpR][m∗1⊗m1...m∗R⊗mR]T,
CP =E(xtxHt+τP)=MDPMH = dPrmrmHr , vec(ImCp)T =
r=1 [Imd ...Imd ][m∗⊗m ...m∗ ⊗m ]T.
in which D = E(s sH ) is the R-by-R diagonal matrix p1 pR 1 1 R R
p t t+τp
with the elements of the vector (d ,...,d ) on the main Hence, the P equations in (8) can be rewritten as Y =ABT,
p1 pR
diagonal.TheestimationofMfromtheset{C }isknownas where
p
Second-Order Blind Identification (SOBI) [10] or as Second-
Y =
Order Blind Identification of Underdetermined Mixtures (SO-
BIUM) [11] depending on whether the matrix M has full [vec(ReC )...vec(ReC ) vec(ImC )...vec(ImC )]T,
1 P 1 P
column rank or not. Variants of this problem are discussed (cid:20)D+D∗(cid:21)
in, e.g., [12], [13], [14], [15, Chapter 7]. It is clear that if the A= D−2D∗ ∈RK×R, K =2P, and
matrices M and D ,...,D satisfy (8), then the matrices 2i
M = MΛP and D11 = PTPD1P,...,DP = PTDPP also B=[m∗1⊗m1 ... m∗R⊗mR]∈RN×R, N =I2.
satisfy (8) for any permutation matrix P and diagonal unitary
Now we choose l, ζ, p , q , and f such that the columns of
n n
matrix Λ. We say that (8) has a unique solution when it is
B are of the form (3). Note that the trivial parameterization
only subject to this trivial indeterminacy.
Genericuniquenessofsolutionsof(8)hasbeenstudied1)in
1Our derivation of a matrix version of (8) is similar to the derivation in
[16]and[8,Subsection1.4.2]inthecasewherethesuperscript [17,Subsection5.2].
5
b(ζ)=ζ∗⊗ζ with ζ ∈CI is not of the form (3) because of TABLEI
the conjugation. However, since for m=Rem+iImm, UPPERBOUNDSONTHENUMBEROFSOURCESINSOBI
m∗⊗m=(Rem−iImm)⊗(Rem+iImm), I 3 4 5 6 7 8 9
Theorem2 F=C 4 9 16 25 36 49 64
the parameterization [11,Eq.(15)] F=C 4 9 14 21 30 40 51
[8,Proposition1.11] F=R* 3 6 10 15 21 28 36
b(ζ)=([ζ ... ζ ]T −i[ζ ... ζ ]T⊗
1 I i+1 2I *orF=Cifthesuperscript“H”in(8)isreplacedbythesuperscript“T”
([ζ ... ζ ]T +i[ζ ... ζ ]T), ζ ∈Rl
1 I i+1 2I
with l = 2I, is of the form (3). As a matter of fact, each
Assuming that R ≤ P, we check up to which value of
component of b(ζ) is a polynomial p in ζ ,...,ζ , 1≤n≤
n 1 l R condition (9) in Theorem 2 and conditions R(R −1) ≤
N, so we can set f(ζ)=ζ, and q (ζ)=···=q (ζ)=1.
1 N I2(I −1)2/2 in [11] and R ≤ (I2 −I)/2 in [8] hold. The
It is clear that the matrix A can be parameterized indepen-
results are shown in Table I. Note that under the condition in
dently of B by m = 2PR real parameters, namely, by the
[11] the mixing matrix M can be found from an eigenvalue
entries of the P ×R matrices D+D∗ and D−D∗. Thus, the
2 2i decomposition in the exact case. Hence, it is not surprising
equations in (8) can be rewritten as decomposition (5) with
that this condition is more restrictive. The condition in [8]
z∈Ω=Rn, where n=m+s=2PR+lR=2PR+2IR.
is more restrictive since, if D is complex, the unsymmetric
p
Moreover, one can easily verify that (8) has a unique solution
matrix MD MH has more distinct entries than the complex
p
if and only if decomposition (5) is unique. In turn, since,
symmetric matrix MD MT.
p
obviously, (10) is equivalent to
µ {(vec((D+D∗)/2),vec((D−D∗)/2i),
n
IV. ANAPPLICATIONINDETERMINISTICSIGNAL
Rem ,Imm ,...,Rem ,Imm ):
1 1 R R
SEPARATIONUSINGMILDSOURCEMODELS
solution of (8) is not unique}=0,
A. Context and contribution
it follows that (10) can be rewritten as (6).
(ii) To prove (6) we check assumptions 1–6 in Theorem We have recently proposed tensor-based algorithms for the
1. Assumption 1: it is clear that if D is generic, then, by deterministic blind separation of signals that can be modeled
the assumption 2P ≥ R, the matrix A has full column rank. as exponential polynomials (i.e., sums and/or products of
Assumptions 2–3 are trivial since f is the identity mapping. exponentials,sinusoidsand/orpolynomials)[18]orasrational
Assumption 4: since the rank-1 matrices of the form mmH functions [19]. These signal models are meant to be little
spanthewholespaceofI×I matricesandb(Rem,Imm)= restrictive; on the other hand, they enable a unique source
vec(mmH) it follows that assumption 4 holds for N(cid:98) = I2. separationundercertainconditions.Theapproachissomewhat
Assumption 5: an elementary computation shows that for a related to sparse modelling [1]. In sparse modelling, matrix
genericζ,J(r,x)[xI+1 ... x2I −x1 ... −xI]=0,implying M in (1) is known but has typically more columns than rows
that rank(J(r,x)) ≤ l−1, so we set (cid:98)l = l−1. Assumption while most of the entries of S are zero. That is, the nonzero
6: since N(cid:98)−(cid:98)l=I2−√2I+1, ass√umption 6 holds by (9) since entries of S make sparse combinations of the columns of M
λr(ζ)=λb(ζ)=b( λζ)=r( λζ). (called the “dictionary”) to model X. The uniqueness of the
model depends on the degree of linear independence of the
Nowweconsiderthecaseτ1 =0.Theonlydifferencewith columns of M and the degree of sparsity of the rows of S
the case τ1 (cid:54)=0 is that the diagonal matrix D1 =E(stsHt+τ1) [1]. In [18], [19] on the other hand, the basis vectors are
is real, yielding that (8) can be parameterized by R real and estimated as well, by optimization over continuous variables.
IR+(P −1)R complex parameters, or equivalently, by n= By way of example, in the case of sparse modelling of a
R+2IR+2(P −1)R real parameters. sine wave, the columns of M could be chosen as sampled
versions of sin((ω +k∆ω)t) for a number of values k (say
Theorem 3. Assume that τ =0 and R≤min(2P −1,(I− 0
1
1)2). Then (8) has a unique solution for generic real matrix k =−K,...,−1,0,1,...,K sothatR=2K+1),andω0and
∆ω are fixed. On the other hand, in [18] one optimizes over
D and generic complex matrices M and D ,...,D , i.e.,
1 2 P
a continuous variable ω to determine the best representation
(cid:110) (cid:16)
µ d ,...,d , vec((D+D∗)/2),vec((D−D∗)/2i), sin(ωt); in this way the accuracy is not bounded by ∆ω.
n 11 1R
In [18], [19] deterministic uniqueness conditions are given
Rem ,Imm ,...,Rem ,Imm ):
1 1 R R for exponential polynomial and rational source models. Here,
solution of (8) is not unique}=0, weproposegenericuniquenessconditionsforthecasethatthe
mixing matrix has full column rank.
where D∈C(P−1)×R denotes a matrix with entries d (p>
pr We actually consider a more general family of models,
1),n=(2I+2P−1)R,andµ isameasurethatisa.c.with
n namely we assume that the source signals s (t),...,s (t)
respect to the Lebesgue measure on Rn . 1 R
can be modeled as the composition of a known multivariate
Proof. The proof is essentially the same as that of Theorem rational function and functions of the type t, cos(ωt + φ),
2. sin(ωt+φ), and at. We assume that the discrete-time signals
6
are obtained by sampling at the points t = 1,...,N. The where
observed data are a mixture of the sources: p (x) xn x +n(cid:18) 1−x2
  n = 1 + 2 Q (x ) 5−
s1(1) ... s1(N) q (x) n x +n n 4 1+x2
X=M ... ... ... =MST. (11) n 3 R (x )5 2x5 (cid:19)+P (x ),
s (1) ... s (N) n 4 1+x n 6
R R 5
ζ ζ
B. An example f(ζ1,ζ2,ζ3,ζ4,ζ5,ζ6)=[ζ1 ζ2 ζ3 tan 24 tan 25 cosζ6]T.
To simplify the presentation we will consider the concrete
Thus, b(ζ) = [s(1) ... s(N)]T is of the form (3) and l =
case where the source signals can be modelled as
6. Now we check assumptions 1–6 in Theorem 1: 1) holds
s (t)= atr +br+tcos(α t+φ )+cos(β t), t∈R (12) by our assumption (iii); 2) and 3) are trivial; 4) holds for
r t cr+t r r r N(cid:98) =N sincethevectorsb(ζ1,ζ2,...,ζ6)−b(0,ζ2,...,ζ6)=
foraprioriunknownparametersar,br,cr,αr,φr andβr.That [ζ11 ... ζN1N]T spantheentirespaceFN;5)holdsfor(cid:98)l=l=6;
is, sr(t) is the composition of the known rational function 6) holds by assumption (i).
x x +x
R(x ,...,x )= 1 + 3 2x +x
1 6 x2 x4+x2 5 6 C. Separation of exponential polynomials and separation of
and the functions x (t)=at, x (t)=t, x (t)=b , x (t)= rational functions
1 r 2 3 r 4
c ,x (t)=cos(α t+φ ),andx =cosβ t.Thegeneralcase
r 5 r r 6 r The cases where the sources in (11) can be expressed as
can be studied similarly.
sampled exponential polynomials
In the remaining part of this subsection we show that if (i)
Rgen≤erNic,−an6d, ((iiii)i)ththeepmariaxminegtemrsaatrri,xbMr, chra,sαfru,llφcro,launmdnβrranarke, s(n)=(cid:88)F (p0f +p1fn+···+pdffndf)anf =
then the mixing matrix and the sources s1(t),...,sR(t) can f=1 (17)
be uniquely recovered. (cid:88)F
P (n)an, n=1,...,N
Werewrite(11)asmatrixdecomposition(5).WesetY =X f f
and A(z) = M. It is clear that the signals in (12) can be f=1
parameterized as and sampled rational functions
s(t)= ζt1t + ζζ23++ttcos(ζ4t+ζ5)+cos(ζ6t), t∈R, (13) s(n)= ab00++ab11nn++······++baqpnnqp, n=1,...,N (18)
where ζ =[ζ ... ζ ]T =[a b c α φ β]T, so we set b(ζ)=
1 6 were studied in [18] and [19], respectively.
[s(1) ... s(N)]T.First,webringb(ζ)intotheform(3).Then
The following two theorems complement results on generic
we will check assumptions 1–6 in Theorem 1.
uniqueness from [18] and [19]. In contrast to papers [18] and
The following identities are well-known:
[19]wedonotexploitspecificpropertiesofHankelorLo¨wner
1−tan2 ζ 2tanζ matrices in our derivation. We only use the source models
cosζ = 2, sinζ = 2 . (14)
1+tan2 ζ 1+tan2 ζ (17)–(18) for verifying the assumptions in Theorem 1.
2 2
We will need the following generalization of (14). Theorem5. AssumethatthemixingmatrixMhasfullcolumn
rank and that
Lemma4. ThereexistapolynomialP andrationalfunctions
n
Q and R such that R≤N −(d +...d +2F), (19)
n n 1 F
(cid:18) (cid:19)
ζ
cosζn=P (cosζ)=Q tan , (15) then M and R generic sources of form (17) can be uniquely
n n 2 recovered from the observed data X=MST.
(cid:18) (cid:19)
ζ
sinζn=R tan . (16) Proof. We set
n 2
Proof. See Appendix B. ζ =[a1 p01 ... pd11 ... aF p0F ... pdFF]T ∈Fl,
l=(2+d )+···+(2+d )=d +···+d +2F
From (13) and Lemma 4 it follows that 1 F 1 F
s(n)= ζ1n + ζ2+ncos(ζ n+ζ )+cos(ζ n)= andchecktheassumptionsinTheorem1forY =X,A(z)=
n ζ +n 4 5 6 M and b(ζ) = [s(1) ... s(N)]T: 1)–3) are trivial; 4) since
3
ζn ζ +n the vectors b(ζ,1,0,...,0) = [ζ ... ζN]T span the entire
1 + 2 (cosζ ncosζ −sinζ nsinζ )+cos(ζ n)=
n ζ +n 4 5 4 5 6 space FN, we set N(cid:98) = N; 5) we set (cid:98)l = l; 6) holds by (19)
3
ζn ζ +n(cid:32) (cid:18) ζ (cid:19)1−tan2 ζ5 since Range(r) is invariant under scaling.
1 + 2 Q tan 4 2 −
n ζ3+n n 2 1+tan2 ζ5 Theorem6. AssumethatthemixingmatrixMhasfullcolumn
2
(cid:18) ζ (cid:19) 2tanζ5 (cid:33) p (f(ζ)) rank, q ≥1, and that
R tan 4 2 +P (cosζ )= n ,
n 2 1+tan2 ζ5 n 6 qn(f(ζ)) R≤N −(p+q+1), (20)
2
7
then M and R generic sources of form (18) can be uniquely in S = {b(ζ) : q (f(ζ))···q (f(ζ)) (cid:54)= 0}, where b(ζ) is
1 N
recovered from the observed data X=MST. definedin(3).ThelatterproofisgiveninSubsectionA-C.The
step from the deterministic formulation in Subsection A-A to
Proof. We set
the generic result in Subsection A-C is taken in Subsection
ζ =[a ... a b ... b ]T ∈Fl, l=p+q+2 A-B.
0 p 0 q
andchecktheassumptionsinTheorem1forY =X,A(z)= A. A deterministic uniqueness result
M and r(ζ) = b(ζ) = [s(1) ... s(N)]T: 1)–3) are trivial; Theorem 7. Assume that
4) since an N ×N matrix with (k+1)th column (for k = 1) the matrix A has full column rank;
0,...,N −1) given by 2) the columns b ,...,b of the matrix B satisfy the
1 R
following condition:
b(1,0,...,0,k,1,0,...,0)=[(k+1)−1 ... (k+N)−1]T,
(cid:124) (cid:123)(cid:122) (cid:125) if at least two of the values λ ,...,λ ∈F
p+1 1 R (22)
are nonzero, then λ b +···+λ b (cid:54)∈S.
1 1 R R
is nonsingular [20, p. 38], we set N(cid:98) = N; 5) an elementary
computation shows that for a generic x, J(r,x)x = 0, Then decomposition (21) is unique.
implying that rank(J(r,x)) ≤ l−1, so we set (cid:98)l = l−1; 6) Proof. We need to show that if there exist A and B such that
holds by (20) since Range(r) is invariant under scaling.
R
We assume that the matrix M is generic and compare the Y =ABT =(cid:88)a bT, b ∈S (23)
r r r
bounds in Theorem 5 and Theorem 6 with the generic bounds
r=1
in [18] and [19], respectively. Since M is generic, it has
then decompositions (21) and (23) coincide up to permutation
full column rank if and only if R ≤ K. Thus, we compare
of the rank-1 terms.
the bound R ≤ min(N −(d +...d +2F),K) with the
1 F First we show that assumption 2 implies that B has full
bound R(d +...d +F) ≤ (cid:98)N+1(cid:99), 2 ≤ K in [18], and
1 F 2 column rank. Assume that there exist λ ,...,λ for which
the bound R ≤ min(N − (p + q + 1),K) with the bound 1 R
λ b +···+λ b =0,suchthatatleastoneofthesevalues
R ≤ 1 (cid:98)N+1(cid:99), 2 ≤ K in [19]. On one hand, the 1 1 R R
max(p,q) 2 being nonzero would imply that B does not have full column
bounds in [18] and [19] can be used in the undetermined case
rank.
(2 ≤ K), while our bounds work only in the overdetermined
case (R ≤ K). On the other hand, roughly speaking, our Thenforanyµ(cid:54)∈{0,−λ1}, λ1µ+µb1+λµ2b2+···+λµRbR =
b ∈ S. Hence, by assumption 2, at most one of the values
bounds are of the form R ≤N −c while the bounds in [18] 1
λ +µ,λ ,...,λ is nonzero. Since µ(cid:54)=−λ , we have that
and [19] are of the form R ≤ N/c, where c is the number 1 2 R 1
λ = ...λ = 0. Since λ b = λ b +···+λ b = 0,
of parameters that describe a generic signal. In this sense our 2 R 1 1 1 1 R R
it follows that λ = 0 or b = 0. One can easily verify that
new uniqueness conditions are significantly more relaxed. 1 1
b = 0 is in contradiction to assumption 2. Hence λ = 0.
1 1
Thus the matrix B has full column rank.
V. CONCLUSION Since the matrices A and B have full column rank, it
Borrowing insights from algebraic geometry, we have pre- follows from the identity
sented a theorem that can be used for investigating generic
Y =ABT =ABT (24)
uniqueness in BSS problems that can be formulated as a par-
ticular structured matrix factorization. We have used this tool thatthematricesAandBalsohavefullcolumnrank.Hence,
for deriving generic uniqueness conditions in (i) SOBIUM-
type independent component analysis and (ii) a class of A†ABT =BT (25)
deterministic BSS approaches that rely on parametric source
where A† = (cid:0)AHA(cid:1)−1AH denotes the left inverse of A.
models. In a companion paper we will use the tool to obtain
By assumption 2, each row of the matrix A†A contains at
genericresultsforstructuredtensorandcoupledmatrix/tensor
most one nonzero entry. Since the matrices B and B have
factorizations.
full column rank, the square matrix A†A is nonsingular.
Thus, each row and each column of A†A contains exactly
APPENDIXA
one nonzero entry. Hence there exist an R×R nonsingular
PROOFOFTHEOREM1
diagonal matrix Λ and an R×R permutation matrix P such
In this appendix we consider the decomposition that A†A=ΛP. From (25) it follows that
R
(cid:88) ΛPBT =A†AB=BT. (26)
Y =ABT = a bT, b ∈S, (21)
r r r
r=1 Substituting (26) into (24) and taking into account that the
where the matrix A has full column rank and S denotes a matrix B has full column rank we obtain
known subset of FN.
APTΛ−1 =A. (27)
In Theorem 7 below, we present two conditions that guar-
antee the uniqueness of decomposition (21). These conditions Equations (26)–(27) imply that decompositions (21) and (23)
will be checked in the proof of Theorem 1 for generic points coincide up to permutation of the rank-1 terms.
8
Theorem 7 has already been proved for the particular cases is not identically zero and hence nonzero almost everywhere,
wheredecomposition(21)representstheCPDofathird-order i.e. the problematic cases occur in a measure-zero situation.
tensor [21, Section IV], the CPD of a partially symmetric of InordertomaketheconnectionwithTheorem1wewillneed
order higher than three [22, Theorem 4.1], the CPD of an the following notations:
unstructured tensor of order higher than three [23, Theorem
Range(b):={b(ζ): q (f(ζ))···q (f(ζ))(cid:54)=0, ζ ∈Fl},
4.2],andthedecompositioninmultilinearrank-(L,L,1)terms 1 N
[18, Theorem 2.4]. Range(r):={r(x): q1(x)···qN(x)(cid:54)=0, x∈Fl}.
Lemma 9. Let assumptions 2–6 in Theorem 1 hold. Then
B. A generic variant of assumption 2 in Theorem 7 assumption2inTheorem7holdsforS =Range(b)andb =
1
Condition (22) means that the subspace span{b ,...,b } b(ζ ),...,b =b(ζ )∈S, where the vectors ζ ,...,ζ ∈
1 R 1 R R 1 R
has dimension R and may intersect the set S only at “trivial” Fl are generic.
points λ b , that is
r r Proof. Since (28)–(29) is equivalent to (22) it is sufficient to
the vectors b1,...,bR are linearly independent and (28) show that µRl(Wb)=µRl(Gb)=0, where
span{b1,...,bR}∩S ⊆{λbr : λ∈F, 1≤r ≤R}. (29) Wb ={[ζ1T ... ζRT]T :b1 =b(ζ1),...,bR =b(ζR)
Property (29) is the key to proving uniqueness of (21). We are linearly dependent},
can easily find span{b1,...,bR} from the matrix Y if it Gb ={[ζ1T ... ζRT]T : (29) does not hold for
can be assumed that the matrix A has full column rank.
b =b(ζ ),...,b =b(ζ )}.
1 1 R R
On the other hand, property (29) means that the only points
in span{b ,...,b } that have the hypothesized structure It is a well-known fact that the zero set of a nonzero analytic
1 R
(encoded in the definition of the set S), are the vectors b , functiononCRl hasmeasurezerobothonCRl andRRl.Thus,
r
1≤r ≤R(uptotrivialindeterminacies).However,conditions toproveµRl(Wb)=µRl(Gb)=0,wewillshowthatthereexist
(22)and(29)aremostoftenhardtocheckforparticularpoints analytic functions w and g of Rl complex variables such that
b ,...,b . The checking may become easier if we focus on
1 R w is not identically zero but vanishes on W , (30)
the generic case, and this is where algebraic geometry comes b
g is not identically zero but vanishes on G . (31)
in. More precisely, if S =V is an algebraic variety, then the b
classical trisecant lemma states that if R is sufficiently small, Weconsiderthefollowingthreecases:1)F=Candf(ζ)=ζ;
then (29) holds for “generic” b1,...,bR ∈S. A set V ⊆CN 2) F=C and f(ζ) is arbitrary; 3) F=R.
isanalgebraicvarietyifitisthesetofsolutionsofasystemof 1) Case F = C and f(ζ) = ζ. In this case b(ζ) = r(ζ),
polynomial equations. It is clear that algebraic varieties form
thus, the sets W and G take the following form:
an interesting class of subsets of CN; however, is not easy to b b
verify whether a given subset of CN is a variety or not. On Wb =Wr ={[ζ1T ... ζRT]T :b1 =r(ζ1),...,bR =r(ζR)
the other hand, it is known that a set obtained by evaluating are linearly dependent},
a known rational vector-function (such as r(x) in (7)) can be G =G ={[ζT ... ζT]T : (29) does not hold for
extended to a variety by taking the closure, i.e., by including b r 1 R
S =Range(r) and b =r(ζ ),...,b =r(ζ )}.
its boundary. This is indeed what we will do in the proof 1 1 R R
of Lemma 9 below. First we give a formal statement of the Here we prove that there exist polynomials d and h
num num
trisecant lemma. in Rl variables such that (30)–(31) hold for w = d and
num
g =h .
Lemma 8. ( [24, Corollary 4.6.15], [25, Theorem 1.4]) Let num
V ⊂ CN be an irreducible algebraic variety and R ≤ First we focus on Gr. Let V denote the Zariski closure of
Range(r) ⊂ CN. Since Range(r) is the image of the open
dimspan{V}−dimV or R ≤ dimspan{V}−dimV −1
(hence irreducible) subset {ζ : q (ζ)···q (ζ)(cid:54)=0, ζ ∈Cl}
depending on whether V is invariant under scaling or not. 1 N
under the rational map
Let G denote a set of points (v ,...,v ) such that
V 1 R
span{v1,...,vR}∩V (cid:54)⊂{λvr : λ∈C, 1≤r ≤R}. r: ζ (cid:55)→(cid:20)p1(ζ),...,pN(ζ)(cid:21)T ,
q (ζ) q (ζ)
1 N
Then the Zariski closure of G is a proper subvariety of
V
itfollowsthatRange(r)isalsoanirreducibleset.HenceV ⊂
V × ··· × V (R times), that is, there exists a polynomial
CN is an irreducible variety and the dimension of V is equal
h(v ,...,v ) in RN variables whose zero set does not
1 R to rankJ(r,ζ) at a generic point ζ ∈Cl [26, p. 186]. Hence,
contain V ×···×V but does contain G .
V
by assumption 5 in Theorem 1,
It is the last sentence in the trisecant lemma that makes it
a powerful tool for proving generic properties. Let us explain dimV ≤(cid:98)l. (32)
in more detail how this works. We can use G to denote a
V Since, by definition, Range(r) consists of all vectors of the
set that poses problems in terms of uniqueness, in the sense
form (7), from assumption 4 in Theorem 1 it follows that
thatspan{v ,...,v }doesnotintersectV onlyinthepoints
1 R
thatcorrespondtothepuresources.Thetrisecantlemmastates dimspanRange(r)=
now that GV belongs to the zero set of a polynomial h that dimspan{r(ζ): q1(ζ)···qN(ζ)(cid:54)=0, ζ ∈Cl}≥N(cid:98).
9
Since V ⊇Range(r), it follows that Sinceh isapolynomial,thefunctionucanberepresented
num
asaratiou=u /u ,wherethefunctionsu andu
dimspanV ≥dimspanRange(r)≥N(cid:98). (33) num den num den
are defined on the whole space CRl. Since both u and
num
Thus, by assumption 6 in Theorem 1 and (32)–(33), u consistofthecompositionofsomepolynomialsand2Rl
den
R≤N(cid:98) −(cid:98)l≤dimspanV −dimV. functions f1,num(ζ1),f1,den(ζ1),...,fl,num(ζR),fl,den(ζR)
which are analytic on CRL, it follows that u and u
num den
Thus, we have shown that V satisfies the assumptions are analytic on CRL [27, p. 6]. We set g = u . It is clear
num
in Lemma 8. Let now the set G and the polynomial
V that g vanishes on G .
b
h(v ,...,v ) be as in Lemma 8. Since V is the Zariski
1 R Now we prove that g is not identically zero. Since u =
closure of Range(r), it follows that V × ··· × V is the
g/u ,itissufficienttoshowthatuisnotzeroatsomepoint.
den
Zariski closure of Range(r) × ··· × Range(r). Since, by Let ζ0 be a point as in assumption 3 in Theorem 1. Then,
Lemma 8, the zero set of h(v ,...,v ) does not contain
1 R by the inverse function theorem, there exists a neighborhood
V ×···×V, it follows that the zero set of h(v1,...,vR) N(ζ0,ε)⊂Fl ofthepointζ0 suchthatforanyp∈N(ζ0,ε)
does not contain Range(r)×···×Range(r). Hence, there the equation f(ζ) = p has the solution ζ = f−1(p).
exist ζ0,...,ζ0 ∈Cl such that h(r(ζ0),...,r(ζ0))(cid:54)=0. On
1 R 1 R Hence the equation (f(ζ1),...,f(ζR)) = (p1,...,pR)
the other hand, since Range(r) is a subset of V, from the has the solution (ζ ,...,ζ ) = (f−1(p ),...,f−1(p ))
1 R 1 R
definitions of Gr and GV it follows that for all (p ,...,p ) ∈ N(ζ0,ε) × ··· × N(ζ0,ε). Since
1 R
(r(ζ1),...,r(ζR))∈GV for all [ζ1T ... ζRT]T ∈Gr. µRl(N(ζ0,ε)×···×N(ζ0,ε)) = µl(N(ζ0,ε))R > 0 [28,
Theorem B, p.144] and, by step 1), h is not identically
Hence, by Lemma 8, num
zero, there exists a point (p0,...,p0) ∈ N(ζ0,ε)×···×
1 R
h(r(ζ1),...,r(ζR))=0 for all [ζ1T ... ζRT]T ∈Gr. (34) N(ζ0,ε) such that hnum(p01,...,p0R) (cid:54)= 0. Hence, by (36),
u(f−1(p0),...,f−1(p0))=h (p0,...,p0)(cid:54)=0. That is,
Since the function h(r(ζ1),...,r(ζR)) is a composition of 1 R num 1 R
(31) holds for g =u .
the polynomial h in RN variables and RN rational functions num
p1(ζ1),...,pN(ζR), it follows that h(r(ζ ),...,r(ζ )) can 3)CaseF=R.Todistinguishbetweenthecomplexandthe
bqe1(ζw1)ritten aqNs(aζRr)atio of two polynomials1 in the enRtries of realcasewedenoteGb andWb incase3byGb,R andWb,R,
respectively.Similarly,thesetsG andW consideredincase
ζ ,...,ζ , b b
1 R h (ζ ,...,ζ ) 2,i.e.forF=C,aredenotedbyGb,C andWb,C,respectively.
h(r(ζ1),...,r(ζR))= hnum(ζ1,...,ζR) . (35) Let gC = unum and wC = w denote the analytic functions
den 1 R constructed in case 2. Then gC and wC are not identically
By(34)–(35),hnumvanishesonGrandisnotidenticallyzero. zero and gC vanishes on Gb,C and wC vanishes on Wb,C.
That is, (31) holds for g =hnum. SinceWb,R isasubsetofWb,C,itfollowsthatwC vanishes
Now we focus on Wr. By assumption 6 in Theorem on Wb,R. Thus, (30) holds for w =wC.
1, N(cid:98) ≥ R + (cid:98)l, so assumption 4 implies that there exist It has not been proven that set Gb,R is a subset of Gb,C
ζ10,...,ζR0 ∈ Cl such that the vectors r(ζ10),...,r(ζR0) are but in any case Gb,R = (Gb,R∩Gb,C)∪(Gb,R\Gb,C). We
linearly independent. Hence there exists an R×R submatrix show that (31) holds for g = gC · wC. Indeed, by case 2,
Rsub(ζ1,...,ζR) of [r(ζ1) ... r(ζR)] whose determinant gC and hence g = gCwC vanish on Gb,C ⊇ Gb,R ∩ Gb,C.
d(ζ1,...,ζR) is not zero at the point (ζ10,...,ζR0). On the On the other hand, if [ζ1T ... ζRT]T ∈ Gb,R \ Gb,C, then
otherhand,d(ζ1,...,ζR)vanishesonWr bydefinition.Since there exist λ1,...,λR ∈ R, λ ∈ C\R and r ∈ {1,...,R}
a determinant is a multivariate polynomial, and since the en- such that λ b(ζ )+···+λ b(ζ ) = λb(ζ ), yielding that
1 1 R R r
triesofRsub(ζ1,...,ζR)arerationalfunctionsofζ1,...,ζR, [ζ1T ... ζRT]T ∈ Wb,C. Thus, Gb,R \Gb,C ⊆ Wb,C and wC
d(ζ1,...,ζR) can be written as a ratio of two polynomials vanishes on Gb,R\Gb,C as well. That is, (31) holds for g =
dnum and dden in the entries of ζ1,...,ζR. It is clear that gCwC.
d (ζ0,...,ζ0) (cid:54)= 0 and that d vanishes on W . That
num 1 R num r
is, (30) holds for w =d .
num C. Proof of Theorem 1
2) Case F=C and f(ζ) is arbitrary. We restrict ourselves
to the case G . Namely, we use the polynomial h and We show that for a generic z ∈ Ω that satisfies the
b num
the function f to construct an analytic function g = u in conditions in Theorem 1, conditions 1–2 in Theorem 7 are
num
Rl variables that satisfies (31). The function w that satisfies also satisfied for
(30) can be constructed in the same way as g but from the
A=A(z), S ={b(ζ): q (f(ζ))···q (f(ζ))(cid:54)=0}, and
1 N
polynomial d and the function f.
num
b =b(ζ ),...,b =b(ζ ),
First we prove the existence and analyticity of g. From the 1 1 R R
definitionsofGb andGr itfollowsthatif(ζ1,...,ζR)∈Gb, where, by our notational convention from Subsection I-C, the
then (f(ζ1),...,f(ζR)) ∈ Gr. Hence, by case 1 above and vectors ζ1,...,ζR are such that [ζ1T ... ζRT]T coincides with
assumption 2 in Theorem 1, the set Gb is contained in the the last s=Rl entries of z∈Ω. The generic uniqueness that
zero set of the function we want to prove in Theorem 1, then follows from Theorem
u(ζ1,...,ζR)=hnum(f(ζ1),...,f(ζR))= 7. We have the following.
(cid:18)f (ζ ) f (ζ )(cid:19) (36) 1) Condition 1 of Theorem 7 holds for generic z ∈ Ω by
h 1,num 1 ,..., l,num R .
num f (ζ ) f (ζ ) assumption 1 in Theorem 1.
1,den 1 l,den R
10
2)ByLemma9,condition2ofTheorem7holdsforgeneric [14] ——, “Blind source separation via the second characteristic function,”
ζ ,...,ζ ∈Fl, or equivalently, for generic [ζT ... ζT]T ∈ SignalProcessing,vol.80,no.5,pp.897–902,2000.
F1s. HencRe, condition 2 of Theorem 7 holds fo1r generiRc z ∈ [15] P. Comon and C. Jutten, Eds., Handbook of Blind Source Separation,
Independent Component Analysis and Applications. Academic Press,
Ω. (Indeed, if Ω(cid:101) denotes a set of points z ∈ Ω such that Oxford,UK,2010.
condition 2 of Theorem 7 does not hold and π denotes the [16] A.Stegeman,J.TenBerge,andL.DeLathauwer,“Sufficientconditions
s
projection onto the last s coordinates of Fn, then, by Lemma for uniqueness in CANDECOMP/PARAFAC and INDSCAL with ran-
domcomponentmatrices,”Psychometrika,vol.71,no.2,pp.219–229,
9,µs{πs(Ω(cid:101))}=0,whichimplies[28,TheoremB,p.144]that June2006.
µn{Ω(cid:101)}=0.) [17] M. Sørensen and L. De Lathauwer, “New Uniqueness Conditions for
theCanonicalPolyadicDecompositionofThird-OrderTensors,”SIAM
J.MatrixAnal.Appl.,vol.36,no.4,pp.1381–1403,2015.
APPENDIXB [18] L. De Lathauwer, “Blind Separation of Exponential Polynomials and
PROOFOFLEMMA4 the Decomposition of a Tensor in Rank-(Lr,Lr,1) Terms,” SIAM J.
MatrixAnal.Appl.,vol.32,no.4,pp.1451–1474,2011.
We use the fact that cosnζ and sinnζ are polynomials in [19] O. Debals, M. Van Barel, and L. De Lathauwer, “Lo¨wner-based blind
sinζ
cosζ [29, p. 642]: signal separation of rational functions with applications,” IEEE Trans.
SignalProcess.(accepted).
(cid:98)n/2(cid:99) [20] R.A.HornandC.R.Johnson,MatrixAnalysis. CambridgeUniversity
(cid:88)
cosnζ = C2k(cos2ζ−1)kcosn−2kζ =P (cosζ), Press,Cambridge,2013.
n n [21] T.JiangandN.D.Sidiropoulos,“Kruskal’spermutationlemmaandthe
k=0 identificationofCandecomp/Parafacandbilinearmodelswithconstant
(cid:98)(n−1)/2(cid:99) modulus constraints,” IEEE Trans. Signal Process., vol. 52, no. 9, pp.
sinnζ (cid:88)
= C2k+1(cos2ζ−1)kcosn−2k−1ζ, 2625–2636,September2004.
sinζ n [22] A. Stegeman, “On uniqueness of the canonical tensor decomposition
k=0 with some form of symmetry,” SIAM J. Matrix Anal. Appl., vol. 32,
where(cid:98)x(cid:99)denotestheintegerpartofx.Substituting(14)into no.2,pp.561–583,2011.
[23] ——,“Onuniquenessofthenthordertensordecompositionintorank-
these equations we obtain that there exist rational functions
1termswithlinearindependenceinonemode,”SIAMJ.MatrixAnal.
Rn and Qn such that (15)–(16) hold. Appl.,vol.31,no.5,pp.2498–2516,2010.
[24] H. Flenner, L. O’Carroll, and W. Vogel, Joins and intersections, ser.
SpringerMonogr.Math. Berlin:Springer,1999.
ACKNOWLEDGMENT [25] L.ChiantiniandC.Ciliberto,“Weaklydefectivevarieties,”Trans.Amer.
TheauthorswishtothankGiorgioOttavianiandEdDewey Math.Soc.,vol.354,pp.151–178,2002.
[26] M.Beltrametti,E.Carletti,D.Gallarati,andG.MontiBragadin,Lectures
for their assistance in algebraic geometry.
on Curves, Surfaces and Projective Varieties. A Classical View of
Algebraic Geometry. EMS Textbooks in Mathematics. European
REFERENCES MathematicalSociety(EMS),Zu¨rich,2009.
[27] R. Gunning and H. Rossi, Analytic Functions of Several Complex
[1] A.M.Bruckstein,D.L.Donoho,andM.Elad,“Fromsparsesolutionsof Variables. Prentice-HallInc.,EnglewoodCliffs,N.J.,1965.
systemsofequationstosparsemodelingofsignalsandimages,”SIAM [28] P. Halmos, Measure Theory. Graduate Texts in Mathematics 18,
Rev.,vol.51,no.1,pp.34–81,2009. Springer,1950.
[2] A.-J. van der Veen and A. Paulraj, “An analytical constant modulus [29] A.P.Prudnikov,Y.A.Brychkov,O.I.Marichev,andN.Queen,Integrals
algorithm,”IEEETrans.SignalProcess.,vol.44,no.5,pp.1136–1155, andseries.Vol.1.Elementaryfunctions. CRC,1998.
May1996.
[3] R.RoyandT.Kailath,“ESPRIT—Estimationofsignalparametersvia
rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal
Process.,vol.37,no.7,pp.984–995,1989.
[4] R. Hermann and C. F. Martin, “Applications of algebraic geometry to
systemstheory–PartI,”IEEETrans.Autom.Control,vol.22,no.1,pp. Ignat Domanov IgnatDomanovreceivedtheMas-
19–25,1977. tersdegreefromDonetskStateUniversity,Ukraine,
[5] C.MartinandR.Hermann,“Applicationsofalgebraicgeometrytosys- the Ph.D degree in Physics and Mathematics from
temstheory,partii:Feedbackandpoleplacementforlinearhamiltonian Institute of Applied Mathematics and Mechanics,
systems,”Proc.IEEE,vol.65,no.6,pp.841–848,June1977. Ukraine,andthePh.D.degreeinEngineeringfrom
[6] A. Cichocki, D. Mandic, C. Caiafa, A.-H. Phan, G. Zhou, Q. Zhao, KU Leuven, Belgium, in 1998, 2004, and 2013,
and L. De Lathauwer, “Tensor decompositions for signal processing respectively.Since2013hehasbeenaPostdoctoral
applications. From two-way to multiway component analysis,” IEEE FellowwiththeKULeuven,Belgium.Hisresearch
SignalProcess.Mag.,vol.32,pp.145–163,March2015. interests include applied linear algebra, tensor de-
[7] P.Comon,“Tensors:Abriefintroduction,”IEEESignalProcess.Mag., compositionsandtensor-basedsignalprocessing.
vol.31,no.3,pp.44–53,2014.
[8] I.DomanovandL.DeLathauwer,“Genericuniquenessconditionsfor
thecanonicalpolyadicdecompositionandINDSCAL,”SIAMJ.Matrix
Anal.Appl.,vol.36,no.4,pp.1567–1589,2015.
[9] S.Krantz,FunctionTheoryofSeveralComplexVariables.Reprintofthe
1992edition. AMSChelseaPublishing,Providence,RI,2001.
[10] A. Belouchrani, K. Abed-Meraim, J.-F. Cardoso, and E. Moulines, “A Lieven De Lathauwer Lieven De Lathauwer re-
blind source separation technique using second-order statistics,” IEEE ceived the Masters degree in electromechanical en-
Trans.SignalProcess.,vol.45,no.2,pp.434–444,Feb1997. gineering and the Ph.D. degree in applied sciences
[11] L. De Lathauwer and J. Castaing, “Blind identification of underdeter- from KU Leuven, Belgium, in 1992 and 1997,
mined mixtures by simultaneous matrix diagonalization,” IEEE Trans. respectively. He is currently Professor with KU
SignalProcess.,vol.56,no.3,pp.1096–1105,March2008. Leuven,Belgium.Dr.DeLathauwerisanAssociate
[12] D.-T. Pham and J.-F. Cardoso, “Blind separation of instantaneous Editor of the SIAM Journal on Matrix Analysis
mixturesofnonstationarysources,”IEEETrans.SignalProcess.,vol.49, and Applications and has been an Associate Editor
no.9,pp.1837–1848,Sep2001. for IEEE Transactions on Signal Processing. His
[13] A. Yeredor, “Non-orthogonal joint diagonalization in the least-squares research concerns the development of tensor tools
sense with application in blind source separation,” IEEE Trans. Signal forengineeringapplications.
Process.,vol.50,no.7,pp.1545–1553,Jul2002.