Original: Indeed, every time we differentiate a Ψ t term we get a X t term.
Revised: Indeed, every time we differentiate a Ψ t term we get a X t term.

Original: By the inductionhypothesis X ( l (cid:48) ) t =0{+=
+} = 0 for l (cid:48) < l and therefore R = 0 .
Revised: By the induction hypothesis X ( l (cid:48) ) t {+=
+} = 0 for l (cid:48) < l and therefore R = 0 .

Original: We conclude that X ( l ) satisﬁes (110) justas X = X (0) .
Revised: We conclude that X ( l ) satisﬁes (111) just as X = X (0) .

Original: Differentiating Div X = 0 with respect to t we also conclude Div X ( l ) = 

Revised: Differentiating Div X = 0 with respect to t we also conclude Div X ( l ) =

Original: and asbefore we conclude that X ( l ) ( s )
Revised: and as before we conclude that X ( l ) ( s )

Original: = {+
+}
Revised: = {+
+}

Original: for s / ∈ ( ε (cid:48) , 1 − ε (cid:48) )
Revised: for s / ∈ ( ε (cid:48) , 1 − ε (cid:48) )

Original: d .
Revised: d .

Original: As above this implies X ( l ) t =0 ( s )
Revised: As above this implies X ( l ) t =0 ( s )

Original: = 0 .
Revised: = 0 .

Original: Thisends the proof if M = R d .
Revised: This ends the proof if M = R d .

Original: We now discuss the necessary extensions for the general case.
Revised: We now discuss the necessary extensions for the general case.

Original: So we now assume that f t , Φ t : R d →
Revised: So we now assume that f t , Φ t : R d →

Original: M for some Riemannian manifold ( M, g ) .
Revised: M for some Riemannian manifold ( M, g ) .

Original: The main strategy is to establish that (108) holds wherethe deﬁnitions of the involved quantities are adapted suitably.
Revised: The main strategy is to establish that (109) holds where the deﬁnitions of the involved quantities are adapted suitably.

Original: Then we can conclude the proof asabove.
Revised: Then we can conclude the proof as above.

Original: Note that if we could ﬁnd orthonormal coordinates locally the same proof applies.
Revised: Note that if we could ﬁnd orthonormal coordinates locally the same proof applies.

Original: However,this is general not possible so we need to argue more carefully.
Revised: However, this is general not possible so we need to argue more carefully.

Original: We deﬁne Ψ t : R d → R d and
Revised: We deﬁne Ψ t : R d → R d and

Original: X t as above, i.e., Ψ t = (Φ t ) − 1 ◦ f t and ∂ t Ψ t =
Revised: X t as above, i.e., Ψ t = (Φ t ) − 1 ◦ f t and ∂ t Ψ t =

Original: X t (Ψ t ) .
Revised: X t (Ψ t ) .

Original: Then we deﬁne the matrix functionsΛ t , Ω t ∈ R d × d by
Revised: Then we deﬁne the matrix functions Λ t , Ω t ∈ R d × d by

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: By assumption Λ t and Ω t are diagonal.
Revised: By assumption Λ t and Ω t are diagonal.

Original: We consider a (inverse) chart η : U → M where U ⊂ R d .
Revised: We consider a (inverse) chart η : U → M where U ⊂ R d .

Original: We now argue locally on η ( U ) ⊂
Revised: We now argue locally on η ( U ) ⊂

Original: M
Revised: M

Original: but we do not denote domain restriction of the function toimprove readability.
Revised: but we do not denote domain restriction of the function to improve readability.

Original: We deﬁne
Revised: We deﬁne

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Maps ˜Φ t and ˜ f t map R d to itself and we can consider their usual derivatives.
Revised: Maps ˜Φ t and ˜ f t map R d to itself and we can consider their usual derivatives.

Original: Moreover, we haveΨ t 
Revised: Moreover, we have Ψ t =

Original: = (˜Φ t ) − 1 ◦ ˜ f t .
Revised: (˜Φ t ) − 1 ◦ ˜ f t .

Original: Observe that (using η ˜Φ = Φ ) we can rewrite Λ t using the induced metric throughη on R d
Revised: Observe that (using η ˜Φ = Φ ) we can rewrite Λ t using the induced metric through η on R d

Original: [Equation]
Revised: [Equation]

Original: where we deﬁned G : U → R d × d by G ( z )
Revised: where we deﬁned G : U → R d × d by G ( z )

Original: = ( dη ) ∗ ( z )( dη )( z ) which captures the pullback metricon R d .
Revised: = ( dη ) ∗ ( z )( dη )( z ) which captures the pullback metric on R d .

Original: We can express this concisely and get similarly for Ω t
Revised: We can express this concisely and get similarly for Ω t

Original: [Equation]

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: We now consider as above ∂ t Ω t and get using the last display
Revised: We now consider as above ∂ t Ω t and get using the last display

Original: [Equation]
Revised: [Equation]

Original: Next we calculate
Revised: Next we calculate

Original: [Equation]
Revised: [Equation]

Original: t [Equation] [Equation] Using the relations (105), (106) (for ˜ f ) and (122) in (121) we obtain
Revised: Using the relations (106), (107) (for ˜ f ) and (123) in (122) we obtain

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: Plugging the relations in (123) we obtain
Revised: Plugging the relations in (124) we obtain

Original: [Equation]
Revised: [Equation]

Original:  Thus we established that the relation (107) also holds in the manifold case.
Revised: k Λ t [Equation] Thus we established that the relation (108) also holds in the manifold case.

Original: The rest of the proof isthe same.
Revised: The rest of the proof is the same.

Original: The proof of Corollary 2 is trivial.
Revised: The proof of Corollary 2 is trivial.

Original: Proof of Corollary 2.
Revised: Proof of Corollary 2.

Original: Apply Theorem 4 with f t = f 0 constant.
Revised: Apply Theorem 4 with f t = f 0 constant.

Original: The assumption that Φ t is analyticin t can be dropped as explained in the proof of Theorem 4.
Revised: The assumption that Φ t is analytic in t can be dropped as explained in the proof of Theorem 4.

Original: For reference, we now state the uniqueness result for second order hyperbolic partial differentialequations.
Revised: For reference, we now state the uniqueness result for second order hyperbolic partial differential equations.

Original: Let U ⊂ R n open, bounded and let U T =
Revised: Let U ⊂ R n open, bounded and let U T =

Original: U × (0 , T ) .
Revised: U × (0 , T ) .

Original: Consider the boundary problem
Revised: Consider the boundary problem

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation] [Equation] where f : U T → R and g, h : U → R are given functions which we assume to be C 1 and g = 0 on∂U .
Revised: where f : U T → R and g, h : U → R are given functions which we assume to be C 1 and g = 0 on ∂U .

Original: The function u : U T → R d is the unknown.
Revised: The function u : U T → R d is the unknown.

Original: The operator is assumed to be an elliptic operatorgiven by
Revised: The operator is assumed to be an elliptic operator given by

Original: [Equation]
Revised: [Equation]

Original: 129) where we assume a ij ∈ C 1 ( ¯ U T ) , a ij = a ji , and that there is θ > 0{+
+} such that[Equation]
Revised: where we assume a ij ∈ C 1 ( ¯ U T ) , a ij = a ji , and that there is θ > {+
+} such that

Original: [Equation]
Revised: [Equation]

Original: for all ( x, t ) ∈ U T and ξ ∈ R n .
Revised: for all ( x, t ) ∈ U T and ξ ∈ R n .

Original: Then the following result holds.
Revised: Then the following result holds.

Original: Theorem 9 (Theorem 4 in Section 7.2 in [13]) .
Revised: Theorem 9 (Theorem 4 in Section 7.2 in [13]) .

Original: Under the assumptions above there is a unique weaksolution u of the system (126) with boundary values as in(127) and (128)  .
Revised: Under the assumptions above there is a unique weak solution u of the system (127) with boundary values as in (128) and (129) .

Original: For our purposes it is not necessary to deﬁne weak solution let us just emphasize that any classicalsolution is a weak solution so this implies uniqueness of classical solutions.
Revised: For our purposes it is not necessary to deﬁne weak solution let us just emphasize that any classical solution is a weak solution so this implies uniqueness of classical solutions.

Original: The key obstacle to improve upon this result and to remove the compact support condition on X isthat the resulting PDE in equation (112) is well posed for the Cauchy initial value problem but itis not well posed for the Dirichlet problem or for mixed Dirichlet and Neumann boundary data.
Revised: The key obstacle to improve upon this result and to remove the compact support condition on X is that the resulting PDE in equation (113) is well posed for the Cauchy initial value problem but it is not well posed for the Dirichlet problem or for mixed Dirichlet and Neumann boundary data.

Original: Inparticular, solutions are, in general, not unique.
Revised: In particular, solutions are, in general, not unique.

Original: Furthermore, there are no general uniqueness resultsfor ﬁrst order systems as in (109).
Revised: Furthermore, there are no general uniqueness results for ﬁrst order systems as in (110).

Original: Note that the existence of a non-trivial divergence free solution X 0of (109) does not imply that a non-constant ﬂow Φ t exists because this is not sufﬁcient to deﬁne theﬂow for positive times.
Revised: Note that the existence of a non-trivial divergence free solution Xof (110) does not imply that a non-constant ﬂow Φ t exists because this is not sufﬁcient to deﬁne the ﬂow for positive times.

Original: We now prove Theorem 5. 
Revised: We illustrate the inﬂuence of the boundary condition further below, when we prove Theorem 5. F.3 Proof of Theorem In this section we show that a family of simple mixing functions is locally identiﬁable for most parameter values even when the mixing is not known close to the boundary. Note that actually we can construct a set of parameter values for which this holds giving a slightly stronger result that we state now.

Original: Proof of  Theorem 5.
Revised: Theorem 5 will be simple consequence of this result. Theorem 10. Let f : C d → R d be given by f ( x ) = RDx , where R ∈ O

Revised: ( d ) and D =

Revised: Diag( µ 1 , . . . , µ d ) with µ i > 0 and µ − 2 i are linearly independent over the rational numbers Q .

Revised: Suppose that Φ t is a smooth invariant deformation in F OCT such that Φ 0 = f , (Φ t ) ∗ ν = f ∗ ν , and Φ t is analytic in t .

Revised: Then Φ t = f on C d , i.e., Φ t is constant in time.

Original:  The initial part of the proof proceeds as in the proof of Theorem 4 and we keepusing the same notation.
Revised: Proof. The initial part of the proof proceeds as in the proof of Theorem 4 and we keep using the same notation.

Original: In particular Ψ t = (Φ t ) − 1 ◦ f and X t is given by ∂ t Ψ t =
Revised: In particular Ψ t = (Φ t ) − 1 ◦ f and X t is given by ∂ t Ψ t =

Original: X t ◦ Ψ t .
Revised: X t ◦ Ψ t .

Original: Note, that now f t is constant in t and therefore Ω t =
Revised: Note, that now f t is constant in t and therefore Ω t =

Original: Df (cid:62) t Df t is constant in t .
Revised: Df (cid:62) t Df t is constant in t .

Original: We now investigatethe boundary conditions for equation 112.
Revised: We now investigate the boundary conditions for equation 113.

Original: Note that
Revised: Note that

Original: [Equation]
Revised: [Equation]

Original: So Ψ t preserves ν
Revised: So Ψ t preserves ν

Original: and we conclude that Ψ t ((0 , 1) d )
Revised: and we conclude that Ψ t ((0 , 1) d )

Original: = (0 , 1) d .
Revised: = (0 , 1) d .

Original: Let us denote by
Revised: Let us denote by

Original: [Equation]
Revised: [Equation]

Original: 132) the boundary hyperplanes and write D = ∂C d = ∂ (0 , 1) d
Revised: the boundary hyperplanes and write D = ∂C d = ∂ [Equation]

Original:  = (cid:83) i D i .
Revised: 0 , 1) d = (cid:83) i D i .

Original: As Ψ t maps (0 , 1) d bijectivelyto itself we conclude that
Revised: As Ψ t maps (0 , 1) d bijectively to itself we conclude that

Original: [Equation]
Revised: [Equation]

Original: We now focus on t = 0 and use the shorthand X = X 0 .
Revised: We now focus on t = 0 and use the shorthand X = X 0 .

Original: Then the differential equation (110) impliesthat
Revised: Then the differential equation (111) implies that

Original: [Equation]
Revised: [Equation]

Original: We conclude that the function X i solves the following mixed Dirichlet and Neumann type boundaryproblem
Revised: We conclude that the function X i solves the following mixed Dirichlet and Neumann type boundary problem

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: Recall here that a j = Λ i / Λ j .
Revised: Recall here that a j = Λ i / Λ j .

Original: So far we have not used any speciﬁc assumption except that Φ t is acontinuous deformation and Φ 0 ∈ F OCT .
Revised: So far we have not used any speciﬁc assumption except that Φ t is a continuous deformation and Φ 0 ∈ F OCT .

Original: So the existence of non-trivial continuous deformationsimplies that a certain hyperbolic PDE has a non-trivial solution.
Revised: So the existence of non-trivial continuous deformations implies that a certain hyperbolic PDE has a non-trivial solution.

Original: Unfortunately, this type of boundaryvalue problem for hyperbolic equations is not well posed and has not always a unique solution.
Revised: Unfortunately, this type of boundary value problem for hyperbolic equations is not well posed and has not always a unique solution.

Original: Wenow show that in the speciﬁc setting of Theorem 5 uniqueness holds.
Revised: We now show that in the speciﬁc setting of the theorem uniqueness holds.

Original: In this case f is linear and
Revised: In this case f is linear and

Original: [Equation]
Revised: [Equation]

Original: so
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: This implies
Revised: This implies

Original: [Equation]
Revised: [Equation]

Original: in particular a j is constant.
Revised: in particular a j is constant.

Original: So the equation (135) becomes a constant coefﬁcient hyperbolic equationwhich can be solved explicitly.
Revised: So the equation (136) becomes a constant coefﬁcient hyperbolic equation which can be solved explicitly.

Original: We can now use Theorem 1 from [11] (and a simple scaling argument) we conclude that the system(135) has a unique solution which is X i = 

Revised: We can now use Theorem 1 from [11] (and a simple scaling argument) we conclude that the system (136) has a unique solution which is X i =

Original: (actually this result is for X i = 0 on ∂D
Revised: (actually this result is for X i = 0 on ∂D

Original: but the proof isstill valid).
Revised: but the proof is still valid).

Original: To give an intuition, we note that separation of variable is possible in this setting and allsolutions to the boundary value problem(135) and (136)  (i.e., without the boundary condition (137)for D i ) can be expressed as a linear combination of the form
Revised: To give an intuition, we note that separation of variable is possible in this setting and all solutions to the boundary value problem (136) and (137) (i.e., without the boundary condition (138) for D i ) can be expressed as a linear combination of the form

Original: [Equation]
Revised: [Equation]

Original: where ϕ j are eigenfunctions of the problem ϕ
Revised: where ϕ j are eigenfunctions of the problem ϕ

Original: (cid:48)(cid:48)
Revised: (cid:48)(cid:48)

Original: j = λ j ϕ j on (0 , 1) and ϕ (cid:48)
Revised: j = λ j ϕ j on (0 , 1) and ϕ (cid:48)

Original: j (0) =
Revised: j (0) =

Original: ϕ (cid:48)
Revised: ϕ (cid:48)

Original: j (1) = 0 .
Revised: j (1) = 0 .

Original: It is easyto see that those are given by cos( πm j t ) where m j ∈ N 

Revised: It is easy to see that those are given by cos( πm j t ) where m j ∈ N

Original: and then ϕ (cid:48)(cid:48)
Revised: and then ϕ (cid:48)(cid:48)

Original: j ( s j ) =
Revised: j ( s j ) =

Original: π 2 m 2 j ϕ j .
Revised: π 2 m 2 j ϕ j .

Original: Solving forf we ﬁnd from (135) that f satisﬁes the ode
Revised: Solving for f we ﬁnd from (136) that f satisﬁes the ode

Original: [Equation]
Revised: [Equation]

Original: Using now that X i (0)
Revised: Using now that X i (0)

Original: = 

Revised: =

Original: (by (137))
Revised: (by (138))

Original: we conclude f (0) = 0 and therefore 
Revised: we conclude f (0) = 0 and therefore f

Original: [Equation]
Revised: [Equation]

Original: [Equation] where α = (cid:113)(cid:80) j (cid:54) =
Revised: ( παs i ) (144) where α = (cid:113)(cid:80) j (cid:54) =

Original: i m 2 j µ 

Revised: i m 2 j µ

Original: i /µ 2 j , or equivalently0 =
Revised: i /µ 2 j , or equivalently

Original: [Equation]
Revised: [Equation]

Original: [Equation] − [Equation]
Revised: =

Original: [Equation]
Revised: [Equation]

Original: Now the condition X i ( s ) = 

Revised: Now the condition X i ( s ) =

Original: for all s ∈ [0 , 1] d with s i = 1 is satisﬁed if and only if f (1) = 0 whichholds iff α ∈ N 0 .
Revised: for all s ∈ [0 , 1] d with s i = 1 is satisﬁed if and only if f (1) = 0 which holds iff α ∈ N 0 .

Original: Note that this argument also implies to solutions that are sums of functions as in(141) by linear independence.
Revised: Note that this argument also implies to solutions that are sums of functions as in (142) by linear independence.

Original: Then the assumption that µ − 2 i are linearly independent over Q impliesthat α = [-
-] (and m j = 0 ) which implies X i = 0 .
Revised: Then the assumption that µ − 2 i are linearly independent over Q implies that α = [-
-]0 (and m j = 0 ) which implies X i = 0 .

Original: Note that this argument only applies at t = 0because it heavily relies on the explicit form of Φ 0 = f .
Revised: Note that this argument only applies at t =because it heavily relies on the explicit form of Φ 0 = f .

Original: However, we can apply the same reasoningto ∂ kt X t inductively (just as in the proof of Theorem 4) and then conclude using the assumption thatΦ t is analytic in t .
Revised: However, we can apply the same reasoning to ∂ kt X t inductively (just as in the proof of Theorem 4) and then conclude using the assumption that Φ t is analytic in t .

Original: The complete argument goes as follows.
Revised: The complete argument goes as follows.

Original: We take the time derivative of equation (108) (recall that∂ t Ω t = 0 as f t = f 0 and get, denoting ˙
Revised: We take the time derivative of equation (109) (recall that ∂ t Ω t = 0 as f t = f 0 and get, denoting ˙

Original: X t = ∂ t X t and ˙Λ t = ∂ t Λ t ,
Revised: X t = ∂ t X t and ˙Λ t = ∂ t Λ t ,

Original: [Equation]
Revised: [Equation]

Original: We have seen that DX 0 = 

Revised: We have seen that DX 0 =

Original: so we infer
Revised: so we infer

Original: [Equation]
Revised: [Equation]

Original: 146) and Div ˙ X t = ∂ t Div X t = 0 .
Revised: [Equation] and Div ˙ X t = ∂ t Div X t = 0 .

Original: The same arguments as before imply ˙ X 0 = 0 on (0 , 1) d .
Revised: The same arguments as before imply ˙ X 0 = 0 on (0 , 1) d .

Original: By inductionall time derivatives of X 0 vanish.
Revised: By induction all time derivatives of X 0 vanish.

Original: This implies that ∂ lt Ψ t =0 ( s ) = 0 for all s and l , i.e., its Taylorexpansion at t = 0 disappears and since we assumed Φ t to be analytic in t so is Ψ t and we concludethat Ψ t ( s )
Revised: This implies that ∂ lt Ψ t =0 ( s ) = 0 for all s and l , i.e., its Taylor expansion at t = 0 disappears and since we assumed Φ t to be analytic in t so is Ψ t and we conclude that Ψ t ( s )

Original: =
Revised: =

Original: Ψ 0 ( s ) =
Revised: Ψ 0 ( s ) =

Original: s and therefore Φ t ( s ) =
Revised: s and therefore Φ t ( s ) =

Original: f ( s ) .
Revised: f ( s ) .

Original: F.3 Proofs for the construction of spurious solutions
Revised: The proof of Theorem 5 is now simple. Proof of Theorem 5. Using Theorem 10 we only need to show that with probability 1 the real numbers µ − 2 i are independent over Q . Note that by assumption µ i has a density.

Revised: Thus, also the distribution of α i = µ − 2 i has a density, i.e., is absolutely continuous with repsect to the Lebesgue measure.

Revised: For a vector of rational numbers ( q 1 , . . .

Revised: , q d ) the set H q = {

Revised: α ∈ R d | (cid:80) q i

Revised: α

Revised: i = 0 } is a codimensionhyperplane and thus has Lebesgue measure 0.

Revised: As Q is countable this implies that N = (cid:83) q ∈ Q d H q is a null set.

Revised: Note that α ∈ N iff α

Revised: i are linearly dependent over Q .

Revised: Since the distribution of α i = µ − 2 i is absolutely continuous with respect to the Lebesgue measure, we conclude that

Revised: [Equation]

Revised: which implies the result.

Revised: F.4 Proofs for the construction of spurious solutions

Original: Finally, we show how ﬂows can be used to construct families of solutions to the ICA problem.
Revised: Finally, we show how ﬂows can be used to construct families of solutions to the ICA problem.

Original: Thissection contains the technical results missing in the overview given in Appendix C.
Revised: This section contains the technical results missing in the overview given in Appendix C.

Original: The ﬁrst construction was described in Lemma 1.
Revised: The ﬁrst construction was described in Lemma 1.

Original: Let us for completeness give a proof (we emphasizeagain that this result is essentially taken from [24]).
Revised: Let us for completeness give a proof (we emphasize again that this result is essentially taken from [24]).

Original: Sketch of proof of Lemma 1.
Revised: Sketch of proof of Lemma 1.

Original: Note that it is sufﬁcient to show that the maps h R,a are volume preserving for ﬁxed t so we ignore the time argument.
Revised: Note that it is sufﬁcient to show that the maps h R,a are volume preserving for ﬁxed t so we ignore the time argument.

Original: It is easy to see that h R,a is bijective (the inverse isgiven h Q,a where Q ( t, r )
Revised: It is easy to see that h R,a is bijective (the inverse is given h Q,a where Q ( t, r )

Original: =
Revised: =

Original: R ( t, r ) − 1 ).
Revised: R ( t, r ) − 1 ).

Original: Then we only need to show that Det Dh R,a ( s ) = 1 for alls .
Revised: Then we only need to show that Det Dh R,a ( s ) = 1 for all s .

Original: We calculate (denoting r = | s − a | )( Dh R,a ( s )) ij = ∂
Revised: We calculate (denoting r = | s − a | ) [Equation]

Original: j ( h R,a ) i = R ( r ) ij + (cid:88) k
Revised: We conclude (writing R (cid:48) =

Original: ( ∂ j R )
Revised: ∂ r R )

Original: ik ( | s − a | )( s − a ) k
Revised: Dh R,

Original: = ∂
Revised: [Equation]

Original: j ( h R,a )i = 
Revised: ( s ) = R ( r ) +

Original: R ( r ) ij + (cid:88) k
Revised: [Equation]

Original: (∂ r R ) ik ( r )( s − a ) k
Revised: ( r [Equation] [Equation]

Original: ∂ j | s − a |.
Revised: | [Equation] |

Original: We conclude (writing R (cid:48) = ∂ r R ) Dh R,a ( s ) =
Revised: [Equation] [Equation]

Original: R ( r ) +R (cid:48) ( r )( s − a ) ⊗ ∇| s − a |
Revised: +

Original: = R ( r ) + | s − a | R (cid:48) ( r )( s − a ) ⊗ ( s − a ) .
Revised: [Equation]

Original: Then we obtain, using the matrix determinant lemma for rank 1 updates ( Det( A + u ⊗ v ) =(
Revised: Then we obtain, using the matrix determinant lemma for rank 1 updates ( Det( A + u ⊗ v ) = (1 + u · A − 1 v ) Det A Det Dh R

Original: [Equation]
Revised: [Equation]

Revised: a ( s

Original: [Equation]
Revised: [Equation]

Revised: = (cid:18)

Revised: [Equation]

Revised: (cid:19)

Revised: [Equation]

Revised: [Equation]

Original: Now we use that R ( r ) ∈
Revised: Now we use that R ( r ) ∈

Original: O( d )
Revised: O( d )

Original: so Det( R ( r ))
Revised: so Det( R ( r ))

Original: = 1 and R ( r ) − 1 =
Revised: = 1 and R ( r ) − 1 =

Original: R ( r ) (cid:62) .
Revised: R ( r ) (cid:62) .

Original: Differentiating
Revised: Differentiating

Original: R ( r ) (cid:62) R ( r ) =
Revised: R ( r ) (cid:62) R ( r ) =

Original: Id d with respect to r we conclude that R ( r ) (cid:62) R (cid:48)
Revised: Id d with respect to r we conclude that R ( r ) (cid:62) R (cid:48)

Original: ( r ) is skew which implies
Revised: ( r ) is skew which implies

Original: [Equation]
Revised: [Equation]

Original: We have therefore shown Det Dh R,a ( s ) = 1 , completing the proof.
Revised: We have therefore shown Det Dh R,a ( s ) = 1 , completing the proof.

Original: Now we give another construction that also establishes Fact 2 based on suitable divergence free vectorﬁelds.
Revised: Now we give another construction that also establishes Fact 2 based on suitable divergence free vector ﬁelds.

Original: All we need to construct is divergence free vector ﬁelds with compact support.
Revised: All we need to construct is divergence free vector ﬁelds with compact support.

Original: Consider anysmooth function ϕ : R d → R such that its support is contained in Ω .
Revised: Consider any smooth function ϕ : R d → R such that its support is contained in Ω .

Original: Then we consider the vectorﬁelds X ij : R d → R d for 1 ≤ i < j ≤ d given by[Equation] 0 for
Revised: Then we consider the vector ﬁelds X ij : R d → R d for 1 ≤ i < j ≤ d given by

Original: [Equation]
Revised: [Equation]

Original: Then we get Div X ij = ∂
Revised: Then we get Div X ij = ∂

Original: i ∂ j ϕ − ∂ j ∂ i ϕ = 0 .
Revised: i ∂ j ϕ − ∂ j ∂ i ϕ = 0 .

Original: So those vector ﬁelds are divergence free and weconclude that the space
Revised: So those vector ﬁelds are divergence free and we conclude that the space

Original: [Equation]
Revised: [Equation]

Original: supp [Equation] (152) is inﬁnite dimensional.
Revised: is inﬁnite dimensional.

Original: Every X ∈ X generates a ﬂow Φ t deﬁned by
Revised: Every X ∈ X generates a ﬂow Φ t deﬁned by

Original: [Equation]
Revised: [Equation]

Original: = [Equation] Using equation (15) we conclude that (Φ t ) ∗ ν
Revised: Using equation (16) we conclude that (Φ t ) ∗ ν

Original: = ν because ν has a constant density and the supportcondition of X ensures that Φ t ((0 , 1) d ) = (0 , 1) d .
Revised: = ν because ν has a constant density and the support condition of X ensures that Φ t ((0 , 1) d ) = (0 , 1) d .

Original: Then the family f t = f ◦ Φ t has the property that( f t ) ∗ ν = f ∗ ν .
Revised: Then the family f t = f ◦ Φ t has the property that ( f t ) ∗ ν = f ∗ ν .

Original: Note that this construction can be easily generalised to source distributions P withdifferentiable density p .
Revised: Note that this construction can be easily generalised to source distributions P with differentiable density p .

Original: In this case the condition Φ t P
Revised: In this case the condition Φ t P

Original: = P is satisﬁed when Div( pX ) = 0 .
Revised: = P is satisﬁed when Div( pX ) = 0 .

Original: Clearlyit is sufﬁcient to consider X
Revised: Clearly it is sufﬁcient to consider X

Original: = Y/p where Y ∈ X (assuming that p > c for some c > 0 on Ω ).
Revised: = Y/p where Y ∈ X (assuming that p > c for some c > 0 on Ω ).

Original: G Proofs for the result on volume preserving maps
Revised: G Proofs for the result on volume preserving maps

Original: Next we show that this construction can be generalised to volume preserving transformations and weprove Theorem 6.
Revised: Next we show that this construction can be generalised to volume preserving transformations and we prove Theorem 6.

Original: Note that in the special case that the distribution of s is ν the construction abovealready works.
Revised: Note that in the special case that the distribution of s is ν the construction above already works.

Original: This is a special case because the condition ( f t ) ∗ ν = f ∗ ν already implies that f t isvolume preserving as soon as f is volume preserving as the density of ν is constant.
Revised: This is a special case because the condition ( f t ) ∗ ν = f ∗ ν already implies that f t is volume preserving as soon as f is volume preserving as the density of ν is constant.

Original: So in this casethe condition that f t is volume preserving and ( f t ) ∗ ν = f ∗ ν essentially agree which is not the casefor general base measures.
Revised: So in this case the condition that f t is volume preserving and ( f t ) ∗ ν = f ∗ ν essentially agree which is not the case for general base measures.

Original: Proof of Theorem 6.
Revised: Proof of Theorem 6.

Original: We deﬁne a suitable vector ﬁeld explicitly.
Revised: We deﬁne a suitable vector ﬁeld explicitly.

Original: Consider X ij : R d → R d for1 ≤ i < j ≤ d deﬁned by
Revised: Consider X ij : R d → R d for 1 ≤ i < j ≤ d deﬁned by

Original: [Equation]
Revised: [Equation]

Original: An illustration of this vector ﬁeld is given in Figure 6.
Revised: An illustration of this vector ﬁeld is given in Figure 6.

Original: We consider the family of functions f t = f ◦ Φ ijtwhere the ﬂow Φ ijt is deﬁned by Φ ij 0 ( s ) =
Revised: We consider the family of functions f t = f ◦ Φ ijt where the ﬂow Φ ijt is deﬁned by Φ ij 0 ( s ) =

Original: s and ∂ t Φ ijt ( s )
Revised: s and ∂ t Φ ijt ( s )

Original: =
Revised: =

Original: X (Φ ijt ( s )) .
Revised: X (Φ ijt ( s )) .

Original: Note that boundednessof ∇ p and p ∈ C 2 imply that Φ t exists globally and deﬁnes a diffeomorphism.
Revised: Note that boundedness of ∇ p and p ∈ C 2 imply that Φ t exists globally and deﬁnes a diffeomorphism.

Original: We claim that Φ ijtsatisﬁes Det Φ ijt ( s ) = [-
-] for all s and (Φ ijt ) ∗ P = P .
Revised: We claim that Φ ijt satisﬁes Det Φ ijt ( s ) = [-
-]1 for all s and (Φ ijt ) ∗ P = P .

Original: The former condition means that Φ ij preservesthe standard volume (Lebesgue-measure) which is the case if Div( X ij ) = 0 while the second relationis satisﬁed if Div( pX ij ) = 0{+
+} by equation (15).
Revised: The former condition means that Φ ij preserves the standard volume (Lebesgue-measure) which is the case if Div( X ij ) = 0 while the second relation is satisﬁed if Div( pX ij ) = {+
+} by equation (16).

Original: We calculate
Revised: We calculate

Original: [Equation]
Revised: [Equation]

Original: We also ﬁnd
Revised: We also ﬁnd

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: This ends the proof.
Revised: This ends the proof.

Original: To give an example, we consider d = 2 and P with rotation invariant density p ( s ) =
Revised: To give an example, we consider d = 2 and P with rotation invariant density p ( s ) =

Original: p ( | s | ) .
Revised: p ( | s | ) .

Original: Then
Revised: Then

Original: X ( s ) =
Revised: X ( s ) =

Original: f ( | s | ) s ⊥ where s ⊥ =
Revised: f ( | s | ) s ⊥ where s ⊥ =

Original: ( s 2 , − s 1 ) (cid:62) and the ﬂow lines are circles around the origin where thespeed depends on the radius through the derivative of p ( | s | ) .
Revised: ( s 2 , − s 1 ) (cid:62) and the ﬂow lines are circles around the origin where the speed depends on the radius through the derivative of p ( | s | ) .

Original: Let us add some remarks concerningthis result.
Revised: Let us add some remarks concerning this result.

Original: Remark 1 .
Revised: Remark 1 .

Original: The constructed ﬂows are non-trivial, i.e., not constant because the probabilitydensity cannot be constant (as we assumed it to be C 2 ) and thus X ij is not identicallyvanishing.2.
Revised: The constructed ﬂows are non-trivial, i.e., not constant because the probability density cannot be constant (as we assumed it to be C 2 ) and thus X ij is not identically vanishing.

Original: It is easy to see (e.g., through the example above) that the ﬂows Φ ij will, in general, mix thecoordinates i and j thus this really shows that ICA is not identiﬁable for volume preservingmaps.3.
Revised: It is easy to see (e.g., through the example above) that the ﬂows Φ ij will, in general, mix the coordinates i and j thus this really shows that ICA is not identiﬁable for volume preserving maps.

Original: While we construct a ﬁnite family of solutions they can be combined, e.g.,
Revised: While we construct a ﬁnite family of solutions they can be combined, e.g.,

Original: [Equation]
Revised: [Equation]

Original: to yield a large space of solutions.4.
Revised: to yield a large space of solutions.

Original: By choosing coordinates cleverly, it is possible to construct a vector ﬁeld X satisfying Div( X ) =
Revised: By choosing coordinates cleverly, it is possible to construct a vector ﬁeld X satisfying Div( X ) =

Original: Div( pX ) = 0 with compact support.
Revised: Div( pX ) = 0 with compact support.

Original: So even knowing f close to the boundaryof the support of P is not sufﬁcient to uniquely identify f .5.
Revised: So even knowing f close to the boundary of the support of P is not sufﬁcient to uniquely identify f . 5.

Original: While it is not possible to identify ICA using volume preserving transformations, it can bepossible to identify f ( s ) for certain values of f if P is known.
Revised: While it is not possible to identify ICA using volume preserving transformations, it can be possible to identify f ( s ) for certain values of f if P is known.

Original: If p has a unique maximum ats 0 then x 0 = f ( s 0 ) will be the point with the largest density of x because volume preservingtransformations transform the density trivially (see (13)).
Revised: If p has a unique maximum at s 0 then x 0 = f ( s 0 ) will be the point with the largest density of x because volume preserving transformations transform the density trivially (see (14)).

Original: Let us ﬁnally sketch a proof of Proposition 2.
Revised: Let us ﬁnally sketch a proof of Proposition 2.

Original: Proof of Proposition 2.
Revised: Proof of Proposition 2.

Original: We assume in addition that the line segment t i = { x i + λ ( y i − x i ) | λ ∈ [0 , 1] }does not contain any x j or y j for j (cid:54)
Revised: We assume in addition that the line segment t i = { x i + λ ( y i − x i ) | λ ∈ [0 , 1] } does not contain any x j or y j for j (cid:54)

Original: =
Revised: =

Original: i .
Revised: i .

Original: The generalisation to the general case is straightforward,e.g., by composing two diffeomorphisms as constructed here.
Revised: The generalisation to the general case is straightforward, e.g., by composing two diffeomorphisms as constructed here.

Original: It is clearly sufﬁcient to constructvolume preserving diffeomorphisms
Revised: It is clearly sufﬁcient to construct volume preserving diffeomorphisms

Original: h i such that h i ( x i ) =
Revised: h i such that h i ( x i ) =

Original: y i and h i ( x j ) =
Revised: y i and h i ( x j ) =

Original: x j , h i ( y j ) =
Revised: x j , h i ( y j ) =

Original: y j forj (cid:54) = i which can then be composed.
Revised: y j for j (cid:54) = i which can then be composed.

Original: We consider the vector ﬁeld X i =
Revised: We consider the vector ﬁeld X i =

Original: ( y i − x i ) ϕ where ϕ is a smooth cut-off function with ϕ ( x )
Revised: ( y i − x i ) ϕ where ϕ is a smooth cut-off function with ϕ ( x )

Original: = 1 for x ∈ t i and ϕ ( x j ) =
Revised: = 1 for x ∈ t i and ϕ ( x j ) =

Original: ϕ ( y j )
Revised: ϕ ( y j )

Original: = 

Revised: =

Original: for j (cid:54) = i .
Revised: for j (cid:54) = i .

Original: Using
Revised: Using

Original: Theorem 2 in [8] we conclude that there is Y i such that supp( Y i )
Revised: Theorem 2 in [8] we conclude that there is Y i such that supp( Y i )

Original: ⊂ supp( ∇ ϕ ) and Div( X i + Y i ) = 0 .
Revised: ⊂ supp( ∇ ϕ ) and Div( X i + Y i ) = 0 .

Original: Considering the ﬂow of Z i
Revised: Considering the ﬂow of Z i

Original: =
Revised: =

Original: X i
Revised: X i

Original: + Y
Revised: + Y

Original: i up to time 1 we obtain a function h
Revised: i up to time 1 we obtain a function h

Original: i as desired.
Revised: i as desired.

Original: Indeed,
Revised: Indeed,

Original: Z i ( x j )
Revised: Z i ( x j )

Original: =
Revised: =

Original: Z i ( y j ) = 0 for j (cid:54) = i because x j and y j are by construction of ϕ outside the supportof X 
Revised: Z i ( y j ) = 0 for j (cid:54) = i because x j and y j are by construction of ϕ outside the support of X i

Original: i

Original: so X i ( x j ) =
Revised: so X i ( x j ) =

Original: X i ( y j )
Revised: X i ( y j )

Original: =
Revised: =

Original: Y i ( x j ) = Y i ( y j ) = 

Revised: Y i ( x j ) = Y i ( y j ) =

Original: for i (cid:54) =
Revised: for i (cid:54) =

Original: j .
Revised: j .

Original: Moreover, for x ∈ t i we get
Revised: Moreover, for x ∈ t i we get

Original: Z i ( x ) =
Revised: Z i ( x ) =

Original: X i ( x )
Revised: X i ( x )

Original: + Y i ( x )
Revised: + Y i ( x )

Original: =
Revised: =

Original: y i − x i .
Revised: y i − x i .

Original: This implies that the ﬂow Φ it of Z satisﬁes Φ it ( x i ) =
Revised: This implies that the ﬂow Φ it of Z satisﬁes Φ it ( x i ) =

Original: x i + t
Revised: x i + t

Original: ( y i − x i )for t ∈ [0 , 1] .
Revised: ( y i − x i ) for t ∈ [0 , 1] .

Original: In particular, h i =
Revised: In particular, h i =

Original: Φ it is as desired.
Revised: Φ it is as desired.

Original: H Experimental illustration of local identiﬁability
Revised: H Experimental illustration of local identiﬁability

Original: [Figure]

Original: We provide a toy experiment to illustrate the meaning and signiﬁcance of Theorem 4.
Revised: We provide a toy experiment to illustrate the meaning and signiﬁcance of Theorem 4 4 .

Original: Note thatthis shall just underpin this speciﬁc setting, for general experiments concerning the usefulness oforthogonal coordinate transforms we refer to [16].
Revised: Note that this shall just underpin this speciﬁc setting, for general experiments concerning the usefulness of orthogonal coordinate transforms we refer to [16].

Original: We consider a function f 0 ∈ F OCT and thenassume that there is a smooth time-dependent transformation f t such that f t ∈ F OCT ( f t is a smoothinvariant deformation in the language of the paper).
Revised: We consider a function f 0 ∈ F OCT and then assume that there is a smooth time-dependent transformation f t such that f t ∈ F OCT ( f t is a smooth invariant deformation in the language of the paper).

Original: We only observe the changing output distributionbut the latent sources are unobserved.
Revised: We only observe the changing output distribution but the latent sources are unobserved.

Original: Suppose, however, that we know the initial mixing f 0 , i.e., we trained an initial model Φ 0 , [-s.t.,Φ 
-] = f 0 .
Revised: Suppose, however, that we know the initial mixing f 0 , i.e., we trained an initial model Φ 0 , [-s.t.,Φ 
-]s.t., Φ 0 = f 0 .

Original: Then we train Φ t starting from Φ 0 such that Φ t ( s ) D
Revised: Then we train Φ t starting from Φ 0 such that Φ t ( s ) D

Original: = f t ( s ) where s is distributed accordingto some base distribution µ of the latent variables.
Revised: = f t ( s ) where s is distributed according to some base distribution µ of the latent variables.

Original: If we in addition enforce Φ t ∈ F OCT Theorem 4essentially tells us that Φ t = f t while no such guarantee exists without the functional restriction.
Revised: If we in addition enforce Φ t ∈ F OCT Theoremessentially tells us that Φ t = f t while no such guarantee exists without the functional restriction.

Original: Here, we verify experimentally that this is indeed the case in a simple setting.
Revised: Here, we verify experimentally that this is indeed the case in a simple setting.

Original: Detailed Experimental Setting We work in dimension 2 and consider a standard normal basedistribution µ .
Revised: Detailed Experimental Setting We work in dimension 2 and consider a standard normal base distribution µ .

Original: We consider polar coordinates for f 
Revised: We consider polar coordinates for f f polar ( r, ϕ ) =

Original: [Equation]
Revised: ( r sin( ϕ ) , r cos( ϕ )) .

Original: We then deﬁne f pol t = f polar ◦ h t where h t is a coordinate-wise transformation deﬁned for 0 ≤ t ≤ 1by
Revised: We then deﬁne f pol t = f polar ◦ h t where h t is a coordinate-wise transformation deﬁned for 0 ≤ t ≤by

Original: [Equation]
Revised: [Equation]

Original: Note that we shift the ﬁrst coordinate and scale the angular coordinates to ensure that the map isinjective (except for the light tail of the Gaussian).
Revised: Note that we shift the ﬁrst coordinate and scale the angular coordinates to ensure that the map is injective (except for the light tail of the Gaussian).

Revised: The additional sin term makes the rescaling non-trivial, i.e., not just a time dependent shift.

Original: We also consider the setting where
Revised: We also consider the setting where

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: To model Φ t we use a normalizing ﬂow model [36, 35].
Revised: To model Φ t we use a normalizing ﬂow model [40, 38].

Original: while this is also convenient for theexperiments, it is important to not use models that implicitly promote orthogonal columns of the
Revised: while this is also convenient for the experiments, it is important to not use models that implicitly promote orthogonal columns of the

Original: Jacobian, e.g., VAEs (see end of Section 2) to extract the effect of enforcing Φ t ∈ F OCT .
Revised: Jacobian, e.g., VAEs (see end of Section 2) to extract the effect of enforcing Φ t ∈ F OCT .

Original: We writeΦ t ( s ) =
Revised: We write Φ t ( s ) =

Original: g ( θ t , s ) =
Revised: g ( θ t , s ) =

Original: g θ t ( s ) where θ denote the trainable parameters of the ﬂow which will varywith time.
Revised: g θ t ( s ) where θ denote the trainable parameters of the ﬂow which will vary with time.

Original: For our implementation we use nﬂows [12] and we use 5 masked afﬁne autoregressivetransformation layers with 15 hidden features followed by random permutations.
Revised: For our implementation we use nﬂows [12] and we use 5 masked afﬁne autoregressive transformation layers with 15 hidden features followed by random permutations.

Original: Then the followingprocedure is used.
Revised: Then the following procedure is used.

Original: We train the normalizing ﬂow such that f 0 = g
Revised: We train the normalizing ﬂow such that f 0 = g

Original: ( θ 0 , · ) .
Revised: ( θ 0 , · ) .

Original: We assume the basedistribution of the ﬂow is µ and denote the induced distribution ( g θ ) ∗ µ by p θ .
Revised: We assume the base distribution of the ﬂow is µ and denote the induced distribution ( g θ ) ∗ µ by p θ .

Original: We discretize the timeinterval in 10 intervals with endpoints t 1 to t 10 .
Revised: We discretize the time interval in 10 intervals with endpoints t 1 to t 10 .

Original: Iteratively we train g
Revised: Iteratively we train g

Original: θ t i , · ) =
Revised: θ t i , · ) =

Original: g ( θ i , · ) starting fromg ( θ i − 1 , · ) to maximize the likelihood of observations from x i ∼ ( f t i ) ∗ µ , i.e., we consider the loss 
Revised: g ( θ i , · ) starting from g ( θ i − 1 , · ) to maximize the likelihood of observations from x i ∼ ( f t i ) ∗ µ , i.e., we consider the loss [Equation]

Original: [Equation]
Revised: [Equation]

Original: We do this without regularisation and with a regularisation that promotes g θ t ∈ F OCT .
Revised: We do this without regularisation and with a regularisation that promotes g θ t ∈ F OCT .

Original: Here we usethe IMA contrast introduced in [16] which we will call C OCT for conciseness.
Revised: Here we use the IMA contrast introduced in [16] which we will call C OCT for conciseness.

Original: It is deﬁned by
Revised: It is deﬁned by

Original: [Equation]
Revised: [Equation]

Original: and we consider the total loss
Revised: and we consider the total loss

Original: [Equation]
Revised: [Equation]

Original: We will use λ = 0 and λ = 2 to train a regularized and an unregularized model.
Revised: We will use λ = 0 and λ = 2 to train a regularized and an unregularized model.

Original: Note that C OCT isnon-negative and vanishes exactly on OCTs (see Prop. 4.4 in [16]).
Revised: Note that C OCT is non-negative and vanishes exactly on OCTs (see Prop. 4.4 in [16]).

Original: Results To measure how well the model recovers the ground truth sources we consider
Revised: Results To measure how well the model recovers the ground truth sources we consider

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Note that more complicated measures like the Amari distance are not necessary because our initialization removed the permutation symmetries.
Revised: Note that more complicated measures like the Amari distance are not necessary because our initialization removed the permutation symmetries.

Original: We also consider C OCT ( g θ ) to approximate the distance to
Revised: We also consider C OCT ( g θ ) to approximate the distance to

Original: F OCT and the forward KL-divergence D KL ( q i || p
Revised: F OCT and the forward KL-divergence D KL ( q i || p

Original: ( θ i , · ))
Revised: ( θ i , · ))

Original: =
Revised: =

Original: E q i ( x ) (log( q i ( x ))
Revised: E q i ( x ) (log( q i ( x ))

Original: − log( p ( θ i , x ))whereq i =
Revised: − log( p ( θ i , x ))

Original:  ( f i ) ∗ µ denotes the observational distribution at time t i .
Revised: where q i = ( f i ) ∗ µ denotes the observational distribution at time t i .

Original: Figures 7 and 8 indicate the results forthe maps f pol t and f rot t , respectively.
Revised: Figures 7 and 8 indicate the results for the maps f pol t and f rot t , respectively.

Original: Note that in both cases, the regularized and the unregularizedmodel have small forward KL-divergence to the observational distribution, i.e., they both trackthe changing observational distribution.
Revised: Note that in both cases, the regularized and the unregularized model have small forward KL-divergence to the observational distribution, i.e., they both track the changing observational distribution.

Original: However, the regularized model recovers the ground truthlatent variables more faithfully, while the unregularized model evolves towards a spurious solution.
Revised: However, the regularized model recovers the ground truth latent variables more faithfully, while the unregularized model evolves towards a spurious solution.

Original: Finally, we note that the regularizer indeed ensures orthogonality of the columns of the Jacobian, i.e.,g θ ∈ F OCT (small C OCT ).
Revised: Finally, we note that the regularizer indeed ensures orthogonality of the columns of the Jacobian, i.e., g θ ∈ F OCT (small C OCT ).

Original: Training details For training we use stochastic gradient descend with the ADAM-optimizer andtrain for 1000 steps with a batch size of 256 where we generate i.i.d.
Revised: Training details For training we use stochastic gradient descend with the ADAM-optimizer [30] with default learning rate and train for 1000 steps with a batch size of 256 where we generate i.i.d.

Original: samples from the observationaldistribution in each step.
Revised: samples from the observational distribution in each step.

Original: We averaged our results over 10 runs.
Revised: We averaged our results over 10 runs.

Original: Total computetime was less than 24hon a workstation.
Revised: Total compute