1. Introduction
The
p-variate Complex Gaussian Distribution (CGD) is defined by [
1] to be the image under the complex affine transformation
of a standard CGD. In the Cartesian representation this class of distributions is a proper subset of the general
-variate Gaussian distribution and its elements are also called rotational invariant Gaussian Distribution. Statistical properties and applications are discussed in [
2].
As it is in the real case, CGD’s are characterised by a complex covariance matrix , which an Hermitian operator, . In some cases, we assume to be positive definite, if . This assumption is equivalent to the existence of a density. We assume zero mean everywhere and we use the notation .
When the complex field is identified with it becomes a -vector space, and the monomials must be replaced by complex monomials. The object of this paper is the computation of the moments of a CGD, i.e., the expected values of the complex monomials under the distribution .
The computation of Gaussian moments is a classical subject that relies on a result usually called Wick’s (or Isserlis’) theorem, see ([
3], Ch. 1). The real and the complex cases are similar, but the complex case has a peculiar combinatorics. Actually, from many perspectives, the complex case is simpler, as observed in
Section 2.3.
The paper is organised as follows. In
Section 2 we offer a concise but complete overview of the basic results concerning the CGD. In particular we give a proof of Wick’s theorem based on a version of the Faà di Bruno formula. In
Section 3 we present recurrence relations for the moments and apply them to derive a new closed-form equation for the moments. Other results are sufficient conditions for the nullity of a moment, which is a feature where the complex case is different from the real case. The presentation is supported by a small running example. In
Section 4 we present conditions on the moment of interest and on the sparsity of the correlation matrix that ensure the factorisation of that moment into a product of lower degree moments. Both the results on the closed form of the moment formula and the factorisation are expected to produce computational algorithm of interest. This issue is discussed in the final
Section 5. The standard version of the Wick’s theorem reduces the computation of the moments to the computation of the permanent of a matrix for which optimised algorithms have been designed. We have not performed this optimisation for the algorithms suggested by our results so that a full comparison is not possible by now. Some technical combinatorial computations are presented in the
Appendix A.
2. The Multivariate Complex Gaussian Distribution and Its Moments
The identification turns into a 2-dimensional real vector space with inner product . The image of the Lebesgue measure is denoted .
If
is seen as a real vector space of dimension 2, then the space of linear operators has dimension 4. It is easy to verify that a generic linear operator
A has the form
Notice that the linear operator
is just a special case. A generic
-multilinear operator is of the form
A general complex monomial is obtained from a suitable multilinear form by identifying some of the variables e.g., , with , and .
The set of p-variate complex monomials characterise probability distributions on .
2.1. Calculus on
If
is differentiable, the derivative at
z in the direction
h is expressed in the form in Equation (
1) as
and the derivative operators
and
are related with the Cartesian derivatives by the equations
The operators appearing in the equation above are sometimes called Wirtinger derivatives and denoted by
and
. The Wirtinger derivatives act on complex monomials as follows
If
f is
k-times differentiable, the the
k-th derivative at
z in the direction
is
where
and
.
We are going to use the following form of the Faà di Bruno formula, see [
4].
Proposition 1. Let and . For each set of commuting derivative operators ,where is the set of partitions of , is defined in Equation (4) and . Proof. The proof easily follows by induction on k by using the fact that each partition of is derived by a partition of by adding the singleton as a new element of the k-partition, and by adding k to each element of the -partition. ☐
Remark 1. The formula applies either when each derivation is a different partial derivation or when , for some . In case of repeated derivations, some terms in the RHS appear more than once. If equal factors are collected, combinatorial counts appear. In the following, first we use the basic formula, then we switch, in the general case, to a different approach, based on the explicit use of recursion formulæ instead of closed form equations.
2.2. Complex Gaussian
The p-variate Complex Gaussian Distribution (CGD) is, when expressed in the real space , a special case of a -variate Gaussian Distribution. The definition is given below.
The univariate CGD is the distribution of a complex random variable
were the real and imaginary part form a couple of independent identically distributed centered Gaussian
X and
Y. As
, the density of
Z is
. The complex variance is
and we write
. Notice that the standard Complex Gaussian has
that is,
. The complex moments of
are
hence 0 if
, otherwise
If and , then it is easy to prove that . The univariate complex Gaussian is sometimes called “circularly symmetric” because . Moreover, .
Consider
d independent standard Complex Gaussian random variables,
,
and let
be a
complex matrix. As in the real case, the distribution of
,
, is a multivariate Complex Gaussian
, with covariance matrix
. In the special case of a non singular covariance matrix
, the density exists and is obtained by performing the change of variable
, to get
see [
2].
Our aim is to discuss various methods to compute the value of a given complex moment of a normal random vector
namely,
where
are non-negative integers. In the case of independent standard CG’s, i.e., identity covariance matrix, we have
which is zero unless
for all
. In the general case, each component
is a
-linear combination of independent standard
’s, so that each moment is the result of
-multilinear and
-anti-multilinear computations. The result will be a sum of products hence, one should expect numerically hard computations.
Various approaches are available and their respective merits depend largely on the setting of the problem: number of variables, total degree of the complex monomial, sparsity of the covariance matrix. All this issues will be discussed in the following.
2.3. Wick’s Theorem
Let us first consider the case where all the exponents in the complex monomial are 1. The general case is a special case of that one, where some of the variables are equal. The unity case is solved by the classical Wick’s theorem (or Isserlis’ theorem). In real case, for example in ([
3], [Th. 1.28]), if
is a centered (real) Gaussian vector with covariance matrix
, then, if
q is even
where
is the set of all partitions of
into subsets of two elements. If
q is odd, then
.
The moment can be zero even in special cases depending on the sparsity of the covariance matrix. For example,
and, if
, then
, even if the variables are not independent.
A similar equation applies to the complex case, see e.g., ([
5], [Lemma 4.5]). For sake of completeness, we give here a proof based on the Faà di Bruno formula.
Let us recall that the permanent of the
complex matrix
is
where
is the symmetric group of permutations on
. The properties of the permanent are discussed in [
6].
Theorem 1 (Wick’s theorem)
. Given a -variate complex Gaussian , then Proof. Let us consider first the case where
and
. There are two standard independent
such that
and
. From straightforward algebra, the independence of
and Equation (
6), we get
Second, we apply a typical Gaussian argument. For each real
and
, we define the jointly complex Gaussian random variables
and
to get
The LHS
is an homogeneous polynomial in
and
namely,
We have proved that for exponents such that
it holds
Finally, we can use the Faà di Bruno formula to compute the derivative in the RHS. Assume all exponents are equal 1,
. The
k-derivative of the power
is
, so that, if we write
and
The factor is zero unless the partition is of the form . In such a case, the factor is equal to for some permutation . Cancellation of concludes the proof. ☐
Remark 2. We note that the same argument shows that unequal lengths of the real and complex part give zero. In case of repeated components, i.e., non-unit exponent, the condition for nullity is the fact the sum of the two blocks of exponent are different.
Remark 3. We observe that the complex case can be considered, in some perspective, more simpler than the real one. In fact, for example, when summing over Wick pairings, the former case considers only pairings matching to variables; while the real case considers all pairings (and thus the sums involved are more complicated).
3. Computation of the Moments via Recurrence Relations
In the previous section we have offered a compact review of Wick’s theorem which is a tool for the computation of the moments of the CGD.
The case were there exponents in the complex moment are not all equal 1 is reduced to the case of the theorem by considering repeated components. However, repeated components lead to identities between terms in the expansion of the permanent, which is always an homogeneous polynomial in the covariances. In this section we derive expressions of the permanent where all the monomials appear once, presented as a system of recurrence relations among the moments of a . In conclusion, we present an explicit formula for the moments, which is an homogeneous polynomial in the elements of in standard form.
Let us first introduce some definitions.
Definition 1. Let be a multi-index, let , and let be the α-moment of Z.
- 1.
The sets and are the supporting sets of the multi-index α.
- 2.
Each element of the matrix Σ is an elementary moment, , with , and the canonical basis of .
- 3.
Given and , is the multi-index : .
We derive the recurrence relations among the moments, explicitly computing the integrals by part. Our proof of Proposition 2 requires the following lemma. Recall that
and
are the Wirtinger derivatives, as in Equation (
3).
Lemma 1. Let us assume that the covariance matrix Σ
is not degenerate and let φ be the associated density of Equation (7). For each bounded function with Wirtinger derivatives bounded by a polynomial, the following relations hold: - 1.
, and analogously for .
- 2.
and .
Proof of 1. We prove the thesis component-wise, dropping the index
j.
☐
Proof of 2. Let
so that
. We prove that
and
, and so the thesis follows trivially. We have
and
, with
the canonical basis of
. As the directional derivative of
g in the direction
r is
, then
and we have for each
that
hence
. ☐
3.1. Recurrence Relations
We prove in the following proposition recurrence relations in which a moment is expressed as a linear combination of moments with total degree reduced by 2. The proof is based on Lemma 1, hence it assumes that the covariance matrix is non-degenerate.
Proposition 2 (Recurrence relations for the moments)
. Given the multi-index α with supporting sets N and M, there are recurrence relations for the moment As a consequence, the moment is zero unless .
Proof. Let
be the complex monomial with exponent
, and let
. We denote by
, the canonical basis of
.
By considering instead of , we can prove that for each .
Notice that such the proof holds without requiring any conditions on
and
. The stated sufficient condition for the nullity of the moment,
, is derived by considering a linear combination of the recurrence relations. In fact,
implies
hence,
if
. ☐
Remark 4. If , the recurrence relations in Proposition 2 coincide with the recurrence formula for computing the permanent of a matrix Γ, derived from Σ splitting the h-th row in copies and the k-th column in copies.
The recursive formula for the permanent of a matrix Γ
, developed with respect to the r-th row is, see [6]:where is obtained from Γ
deleting the r-th row and the j-th column. Suppose that the first rows and the first columns of Γ
are obtained repeating times the first row of Σ
and times the first colum of Σ
, respectively. If and , then , and sosince the matrices are all the same between them for . The matrix is associated to the multi-index . Then and so . The thesis follows by considering the sums associated to successive blocks of repeated columns. The nullity of a moment depends also on the sparsity of the covariance matrix. For example, here is a simple sufficient condition.
Corollary 1. If there exists such that for all or if there exists such that for all then .
Proof. In such cases the recurrence relations in Proposition 2 consist of null addends. ☐
3.2. Closed Form
The recurrence relations in Proposition 2 allow us to compute a complex moment as a linear function of lower degree moments. Hence, each complex moment is a polynomial of the elementary moments
,
,
. For example, in the simple case where
is proportional to a multi-index of an elementary moment,
,
,
,
,
, then each recurrence relation contains one term only, so that
In general, the value of
, with
such that
, can be obtained considering that
is generated by the vectors with integer coefficients:
The coefficient vector
is not uniquely determined. For instance, if
then
Considering all the possible integer coefficient vectors a that produce the same , leads to define the subset associated to each -moments.
Definition 2. Let and let be a multi-index. The set contains all integer vectors such that , , where the bounds are defined by Some elements of are uniquely determined, as shown in the following Proposition.
Proposition 3. Let be a multi-index.
- 1.
The free elements of the vector a are . In fact, for each , the elements , and are: - 2.
If there exists an index r such that , then , for . Analogously, if there exists an index s such that , then , for .
For simplicity
and
a are omitted. This proof is based on Proposition A1 in
Appendix A that states
.
Proof of 1. We show, by induction, the thesis for .
Base step: and so .
Induction step: assume
for
, that is
. We obtain
and so, since
, we conclude that
.
The proof for is analogous. Finally, from and , we obtain, by direct computation, . ☐
Proof of 2. We have
and so,
,
. Since
, we have
.
Analogously for . ☐
The following theorem gives an explicit expression of the
-moment of
. The proof is based on several propositions given in the
Appendix A.
Theorem 2. Let α be a multi-index with N and M supporting sets. Assume that . Then the α-moment of isby setting also when . The set is as in Definition 2. Proof. We denote by
the product
First, we show that is the elementary moment when . Let . Since for each and , Item 2 of Proposition 3 implies that , for , and , that is is the unique element of the vector a different from zero. Furthermore since and and the thesis follows.
Now, we show that
satisfies the recurrence relations for a general
. Let us consider
, with
. Let
be the vector
a containing the value
instead of
:
. From Equation (
12), for
, we have
Let
. From Equations (
12) and (
13) it follows
In Proposition A3 we have shown that
. Then
and so, since Proposition 3 shows
, we obtain
. The thesis follows because the function
satisfies the recurrence relations and coincides with the elementary moments, so that
.
Analogously if we consider . ☐
Remark 5. For each elementary moment , if , the corresponding addend of the sum on is null. Then, in the sum on are present only the addends such that , since we assume .
Remark 6. It should be noted that Equation (11) of Theorem 2 contains the multinomial coefficientsthat are related to the cardinality of the special permutations of equal terms in the permanent. This remark prompts for a purely combinatorial proof of the equation for the moments. However, it should be noted that the specific value of α induces on each the constrains provided by the limits and that are stated in Definition 2 and Proposition 3. We do not follow here this line of investigation. We thank an anonymous referee for suggesting this remark. Example 2. Let us consider the case with and .
If and , then , where If and , then , where is shown in Example 1 and
Example 3 (Running example)
. Let , with Σ
such that Let , where , so that the condition of Theorem 2 is satisfied. Here and .
Since , from Proposition 3 it follows that , . Denoting by and by , with and , the set is defined by: The moment is:where the null elementary moments are highlighted. As specified in Remark 5, we set when . In this example we obtain and the other and so the moment is 4. Factorisation
In this section we present a factorisation of the complex moments which depends on the null elements of and on the supporting sets N and M of the given moment. The argument we use is not based on independence assumptions. Such a factorisation is possible when there exists a non trivial partition of the supporting sets induced by the non null elements of covariance matrix .
Definition 3. Let N and M be the supporting sets of a multi-index α. The graph induced by α is the bipartite graph , where the edges are defined by . Let be the connected components of , . The connected components of define a partitions of N and M, that we call the partition induced by Σ.
Notice that the partitions of M and N are uniquely defined and they can be the trivial partition.
Theorem 3. Let and be the partition induced by Σ
. Let be the multi-index α restricted to , . The moment is given by Proof. We use the notations of Theorem 2. Let
. There are no edges in
outside each connected component,
if
. According to the argument in Remark 5, each
is zero unless
. By cancelling trivial factors, we have
It follows that
so that
☐
The factorisation of the previous theorem reduces the computational complexity for computing
, since each factor is a moment of lower order. The computation of the connected components of a graph requires linear time, in terms of the numbers of its nodes and its edges, see [
7].
In presence of a non-trivial induced partition of the supporting sets, the following corollary shows necessary conditions for the nullity of the moment .
Corollary 2. Under the hypothesis of Theorem 3, the moment is null if there exists such that Proof. If there exists such that , then is null and the thesis follows. ☐
Remark 7. If there exists an such that for each or if there exists an such that for each , then Corollary 1 shows that . This is a degenerate case, where the bipartite graph has a connected component or , for some r. It follows that, for example, if there exists a connected component , then and since does not belong to N. In such a case, we have .
Example 4 (Running example—continued)
. As in Example 3, let with Σ such that .
Let and . Then there exist the non trivial induced partition of N and M, given by and , and . In such a case the matrix is The permuted matrix highlights the induced partitions. The conditions of Corollary 2 for the nullity of are We compute the moments with two different α.
- 1.
Let . In such a case even if . In fact .
- 2.
Let . From Theorem 3 the moment factorises in , where and .
We compute, separately, and applying the restrictions on and specified in Proposition 3 and in Remark 5, and the formulæ for and of Example 2:
5. Discussion
This piece of research about the moments of the CGD was originally motivated by the desire to evaluate the approximation error of a cubature formula with nodes on a suitable subset of the complex roots of the unity, first introduced in [
8].
In this paper, we discussed particular cases of the Wick’s theorem for the CGD. When the exponent of the complex moment is not 0–1 valued, the permanent has repeated terms in the sum. By collecting them, one obtains the form of Theorem 2, which is an homogeneous polynomial in the covariances. By the way of this expression, cases of factorisation of the moments have been derived in Theorem 3.
The relevance of Theorem 2 is mainly theoretical. On one side, it allows for the analysis of the moment’s behaviour as function of the elementary moments . On the other side, our the proof of the factorisation depends on it. The factorisation, if any, reduces the computational complexity of .
Let us discuss briefly the complexity of the computation of moments that do not factorise. We can compare the method presented in this paper, based on Equation (
11), with the method based on the permanent, see Equation (
9). For simplicity, we consider
, with
without null elements, and all the exponents of the moment equal to
n, i.e.,
.
The computation of the moment as the permanent of a
matrix requires
products using a raw algorithm and it requires
products using the Ryser algorithm, see [
6].
The optimisation of the algorithm for Equation (
11) is outside the scope of this paper. The raw implementation of such a formula requires first to compute, only once,
, for
. Then, for each
, the computation of
which requires
products, since
. A conservative upper bound of
is obtained by assuming that each
. It follows that
, and so the the complexity of Equation (
11) is less than
. Actually, the
is much smaller in most cases. For instance, if
,
then the effective
, while
.
Notice that the complexity of our formula,
, is much smaller than the complexity of the raw version of the permanent,
. Furthermore, comparing the complexity of the Ryser algorithm to the upper bound of our algorithm, we find, by direct computation, that
when
p is small and
n is large. For example our algorithm is less expensive for
and
. We expect that the Ryser optimisation techniques applied to Equation (
11) will lead to a further reduction in complexity.
Author Contributions
Conceptualization, C.F., G.P. and M.P.R.; Methodology, C.F., G.P. and M.P.R.; Writing, review and editing, C.F., G.P. and M.P.R.
Funding
The research was funded by FRA 2017, University of Genova.
Acknowledgments
The authors thank G. Peccati (Luxembourg University) for suggesting relevant references and E. Di Nardo (Università di Torino) for providing useful insights in the application of Faà di Bruno formula. G. Pistone acknowledges the support of de Castro Statistics and Collegio Carlo Alberto. C. Fassino is member of GNCS-INDAM, M.P. Rogantin and G. Pistone are members of GNAMPA-INDAM.
Conflicts of Interest
The authors declare no conflict of interest.
Appendix A. Properties of the bounds lij and Lij
The following proposition states that the definition of in Definition 2 is consistent.
Proposition A1. Let α, a, and be as in Definition 2. If the entries of the vector a, , satisfy the bounds , then ,
Proof. For simplicity and a are omitted. Obviously, for each , we have and so . If , and . The thesis follows straightforward. The case with is proved by induction.
Base steps. We prove that , by induction on j. The inequalities hold for . We assume .
We have , that is and so . We show that and so, since , we conclude . If , obviously, . The case also implies . In fact, from the inductive hypothesis, we have , that is . Analogously, the relation between and can be shown.
Induction step. We show that
, by assuming that
for
,
, so that
, and for
,
, so that
. It follows that
since the inequalities
Furthermore, from
and from
it follows
and so, by adding
and
to both sides of the first and of second relation, respectively,
We conclude that
. In fact
and
☐
Proposition A1 also holds when some values and are equal to zero.
Proposition A2. Let and let be a vector belonging to . If coincides with a bound, then some are uniquely determined.
- 1.
Let . Then for .
- 2.
Let . Then for .
- 3.
Let .
Then , and , for , .
Proof. For simplicity and a are omitted.
Let . Then . We show, by induction, that for .
Base step: . We have .
Induction step: let for . Then .
Analogously to previous Item.
Since
then
and so, for each
it holds
We show by induction that for each .
Base step: . By Equation (
A1) with
we have
that is
, and so the thesis follows since
.
Induction step: let
for
. By Equation (
A1)
and so
, and the thesis follows since
.
Analogously, we can show for .
Furthermore, since for , then from Item 1 it follows for and so, by varying we obtain the thesis.
☐
Proposition A3. Let α be a multi-index, let , and let be as in Definition (2). Let be as in Equation (12). Then Proof. We define the set . We consider the bounds for . Using and instead of and , by direct computation we obtain the following conditions.
Let
. Since
, that is
, we have
Let
and
. Since
and
, that is
, we have
Let
and
. Since
and
, i.e.,
, we have
Let
and
. Since
, we have
Let
or
. Analysing each case (
and
,
,
;
and
,
) as in the previous items, we have
It follows that the set
is strictly contained in
. The vectors
are such that
or at least a component coincides with a bound
or
. This latter condition implies that
. In fact, if
then
From Proposition A2, we have
, i.e.,
. We conclude that
since the coefficients
correspond to null addends, since
. ☐
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