On the Number of Interference Alignment Solutions for the K-User MIMO Channel with Constant Coefficients
In this paper, we study the number of different interference alignment (IA) solutions in a -user multiple-input multiple-output (MIMO) interference channel, when the alignment is performed via beamforming and no symbol extensions are allowed. We focus on the case where the number of IA equations matches the number of variables. In this situation, the number of IA solutions is finite and constant for any channel realization out of a zero-measure set and, as we prove in the paper, it is given by an integral formula that can be numerically approximated using Monte Carlo integration methods. More precisely, the number of alignment solutions is the scaled average of the determinant of a certain Hermitian matrix related to the geometry of the problem. Interestingly, while the value of this determinant at an arbitrary point can be used to check the feasibility of the IA problem, its average (properly scaled) gives the number of solutions. For single-beam systems the asymptotic growth rate of the number of solutions is analyzed and some connections with classical combinatorial problems are presented. Nonetheless, our results can be applied to arbitrary interference MIMO networks, with any number of users, antennas and streams per user.
Interference alignment (IA) has received a lot of attention in recent years as a key technique to achieve the maximum degrees of freedom (DoF) of wireless networks in the presence of interference. Originally proposed in ,, the basic idea of IA consists of designing the transmitted signals in such a way that the interference at each receiver falls within a lower-dimensional subspace, therefore leaving a subspace free of interference for the desired signal . This idea has been applied in different forms (e.g., ergodic interference alignment , signal space alignment , or signal scale alignment ,), and adapted to various wireless networks such as interference networks , X channels , downlink broadcast channels in cellular communications  and, more recently, to two-hop relay-aided networks in the form of interference neutralization .
In this paper we consider the linear IA problem (i.e., signal space alignment by means of linear beamforming) for the -user multiple-input multiple-output (MIMO) interference channel with constant channel coefficients. Moreover, the MIMO channels are considered to be generic, without any particular structure, which happens, for instance, when the channel matrices have independent entries drawn from a continuous distribution. This setup has also been the preferred option for recent experimental studies on IA ,,.
The feasibility of linear IA for MIMO interference networks, which amounts to study the solvability of a set of polynomial equations, has been an active research topic during the last years ,,,,. Combining algebraic geometry tools with differential topology, it has been recently proved in  that an IA problem with any number of users, antennas and streams per user, is feasible iff the linear mapping given by the projection from the tangent space of (the solution variety, whose elements are the triplets formed by the channels, decoders and precoders satisfying the IA equations) to the tangent space of (the complex space of MIMO interference channels) at some element of is surjective. Note that this implies, in particular, that the dimension of must be larger than or equal to the dimension of ,.
Exploiting this result, a general IA feasibility test with polynomial complexity has also been proposed in . This test reduces to check whether the determinant of a given square Hermitian matrix is zero (meaning infeasible almost surely) or not (feasible).
In this paper we build on the results in  to study the problem of how many different alignment solutions exist for a given IA scenario. While the number of solutions is known for some particular cases (e.g the 3-user interference channel ), a general result is not available yet. In  it was proved that systems for which the algebraic dimension of the solution variety is strictly larger than that of the input space can have either zero or an infinite number of alignment solutions. In plain words, these are MIMO interference networks for which the number of variables is larger than the number of equations in the polynomial system. On the other hand, systems with less variables than equations are always infeasible [13, 14, 17]. Herein we will focus on the case in between, where the dimensions of and are exactly the same (identical number of variables and equations), and consequently, the number of IA solutions is finite (it may be even zero) and constant out of a zero measure set of as also proved in . In summary, rather than just characterizing feasible or infeasible system configurations, we seek to provide a more refined answer to the feasibility problem.
The number of solutions for single-beam MIMO networks (i.e., all users wish to transmit a single stream of data) follows directly from a classical result from algebraic geometry, Bernstein’s Theorem, as shown in . More specifically, the number of alignment solutions coincides with the mixed volume of the Newton polytopes that support each equation of the polynomial system. Although this solves theoretically the problem for single-beam networks, in practice the computation of the mixed volume of a set of IA equations using the available software tools  can be very demanding. As a consequence, only a few cases have been solved so far. For single-beam networks, some upper bounds on the number of solutions using Bezout’s Theorem have also been proposed in,. For multi-beam scenarios, however, the genericity of the polynomials system of equations is lost and it is not possible to resort to mixed volume calculations to find the number of solutions. Furthermore, the existing bounds in multi-beam cases are very loose.
The main contribution of this paper is an integral formula for the number of IA solutions for arbitrary feasible networks. More specifically, we prove that while the feasibility problem is solved by checking the determinant of a certain Hermitian matrix, the number of IA solutions is given by the integral of the same determinant over a subset of the solution variety scaled by an appropriate constant. Although the integral, in general, is hard to compute analytically, it can be easily estimated using Monte Carlo integration. To speed up the convergence of the Monte Carlo integration method, we specialize the general integral formula for square symmetric multi-beam cases (i.e., equal number of transmit and receive antennas and equal number of streams per user). Analogously, in the particular case of single-beam networks, we provide a combinatorial counting procedure that allows us to compute the exact number of solutions and analyze its asymptotic growth rate.
In addition to being of theoretical interest, the results proved in this work might also have some practical implications. For instance, finding scaling laws for the number of solutions with respect to the number of users could serve to analyze the asymptotic performance of linear IA, as discussed in , where information about the number of solutions is used to predict system performance when the best solution (or the best out of N) solutions is picked. Recent results  also suggest that the number of solutions is related to the computational complexity of designing the precoders and decoders satisfying the IA conditions.
The paper is organized as follows. In Section II, the system model and the IA feasibility problem are briefly reviewed, paying special attention to the feasibility test in  which is the starting point of this work. The main results of the paper are presented in Section III, where an integral formula, valid for arbitrary networks, for the number of IA solutions is given. Two special cases, square symmetric and single-beam networks, are analyzed in Section IV. A short review on Riemmanian manifolds and other mathematical results that will also be used during the derivations as well as the proofs of the main theorems in Section III are relegated to appendices. Numerical results are included in Section V.
Ii System model and background material
In this section we describe the system model considered in the paper, introduce the notation, define the main algebraic sets used throughout the paper, and briefly review the feasibility conditions of linear IA problems for arbitrary wireless networks.
Ii-a Linear IA
We consider the -user MIMO interference channel with transmitter having antennas and receiver having antennas. Each user wishes to send streams or messages. We adhere to the notation used in  and denote this (fully connected) asymmetric interference channel as . The symmetric case in which all users transmit streams and are equipped with transmit and receive antennas is denoted as . In the square symmetric case all users have the same number of antennas at both sides of the link . In this paper we focus on the fully connected interference channel and, consequently, the number of interfering links will be .
User encodes its message using an precoding matrix and the received signal is given by
where is the transmitted signal and is the zero mean unit variance circularly symmetric additive white Gaussian noise vector. The MIMO channel from transmitter to receiver is denoted as and assumed to be flat-fading and constant over time. Each is an complex matrix with independent entries drawn from a continuous distribution. The first term in (1) is the desired signal, while the second term represents the interference space. The receiver applies a linear decoder of dimensions , i.e.,
where superscript denotes transpose.
The interference alignment (IA) problem is to find the decoders and precoders, and , in such a way that the interfering signals at each receiver fall into a reduced-dimensional subspace and the receivers can then extract the projection of the desired signal that lies in the interference-free subspace. To this end it is required that the polynomial equations
are satisfied, while the signal subspace for each user must be linearly independent of the interference subspace and must have dimension , that is
We recall that all matrices (including direct link matrices, ) are generic, that is, their entries are independently drawn from a continuous probability distribution. Consequently, once (3) holds, (4) is satisfied almost surely if and are of maximal rank.
Ii-B Feasibility of IA: a brief review
The IA feasibility problem amounts to study the relationship between such that the linear alignment problem is feasible. If the problem is feasible, the tuple defines the degrees of freedom (DoF) of the system, that is the maximum number of independent data streams that can be transmitted without interference in the channel. The IA feasibility problem and the closely related problem of finding the maximum DoF of a given network have attracted a lot of research over the last years. For instance, the DoF for the 2-user and, under some conditions, for the symmetric -user MIMO interference channel have been found in  and , respectively. In this work we make the following assumptions:
which are necessary conditions for feasibility which arise from the fact that two users of an interference channel cannot reach their point-to-point bounds simultaneously since they have to leave at least a one-dimensional subspace for the interference.
The IA feasibility problem has also been intensively investigated in [12, 13, 14, 15, 16]. In the following we make a short review of the main feasibility result presented in , which forms the starting point of this work.
We start by describing the three main algebraic sets involved in the feasibility problem which were first introduced in :
Input space formed by the MIMO matrices, which is formally defined as
where holds for Cartesian product, and is the set of complex matrices. Note that in [24, 17], we let be the product of projective spaces instead of the product of affine spaces. The use of affine spaces is more convenient for the purposes of root counting.
Output space of precoders and decoders (i.e., the set where the possible outputs exist)
where is the Grassmannian formed by the linear subspaces of (complex) dimension in .
The solution variety, which is given by
where is the collection of all matrices and, similarly, and denote the set of and , respectively. The set is given by certain polynomial equations, linear in each of the and therefore is an algebraic subvariety of the product space . Let us remind here that the IA equations given by (3) hold or do not hold independently of the particular chosen affine representatives of .
The following diagram, illustrating the sets and the main projections involved in the feasibility problem, was considered in :
Note that, given , the set is a copy of the set of such that (3) holds, that is the solution set of the linear interference alignment problem. On the other hand, given , the set is a copy of the set of such that (3) holds.
The feasibility question can then be restated as, is for a generic ? Following this formulation, the problem was first tackled in  and  where some necessary and sufficient conditions were given. Analytical expressions were limited to some symmetric scenarios of interest. In , a solution to this problem was given by proposing a probabilistic polynomial time feasibility test for completely arbitrary interference channels. The test exploited the fact that system is feasible if and only if two conditions are fulfilled:
The algebraic dimension of must be larger than or equal to the dimension of , i.e.,
In other words this condition means that, for the problem of polynomial equations to have a solution, the total number of variables must be larger than or equal to the total number of equations (). We recall that a more general version of this condition was first established in . In that work, an interference channel was classified as proper when the number of variables was larger than or equal to the number of equations for every subset of equations. Otherwise, it was classified as improper. More recently, in  it was rigorously proved that improper systems are always infeasible which implies that a system with is infeasible.
For some element , the linear mapping
is surjective, i.e., it has maximal rank equal to . This condition amounts to saying that the projection from the tangent plane at an arbitrary point of the solution variety to the tangent plane of the input space must be surjective: that is, one tangent plane must cover the other. Moreover, in this case, the mapping (12) is surjective for almost every .
We recall that these conditions were essentially found in  and  by using different mathematical tools than the ones used in . In this paper we will build on the results in  using as a starting point the result stating that, when a system is feasible and , the number of IA solutions is finite and constant for almost all channel realizations. This is formally stated in the following lemma.
Lemma 1 (See Th. 1 in )
For a feasible scenario and for almost every , the solution set is a smooth complex algebraic submanifold of dimension . If , then there is constant such that for every choice of out of a proper algebraic subvariety (thus, for every choice out of a zero measure set) the system has exactly aligment solutions.
See [17, Section V]. \qed
Iii The number of solutions of feasible IA problems
As shown in [24, 17], the surjectivity of the mapping in (12) can easily checked by a polynomial-complexity test that can be applied to arbitrary -user MIMO interference channels. The test basically consists of two main steps: i) to find an arbitrary point in the solution variety and ii) to check the rank of a matrix constructed from that point. As a solution to the first step we follow [17, Sec. IV] and choose a simple solution to the IA equations. Specifically, we take structured channel matrices given by
with precoders and decoders given by
which trivially satisfy and therefore belong to the solution variety. We claim that essentially all the useful information about can be obtained from the subset of consisting of the triples where its elements have the form (13) and (14). In order to see this, we pick any other element . Without loss of generality we can assume and lie in the Stiefel manifold i.e. they satisfy and where the superscript denotes Hermitian (conjugate transpose). Now, we will show how this element of can be converted into one of the form (13) and (14). First, we compute a QR decomposition of and , that is
where and are unitary matrices. Then, the IA condition can be written as
It is now clear that the transformed channels have the form (13), and the transformed precoders and decoders have the form (14). We have just described an isometry that sends to . The situation is thus similar to that of a torus: every point can be sent to some predefined vertical circle through a rotation, thus the torus is essentially understood by “moving” a circumference and keeping track of the visited places. The same way, can be thought of as moving the set of triples of the form (13) and (14), and keeping track of the visited places. Technically, is the orbit of the set of triples of the form (13) and (14) under the isometric action of a product of unitary groups.
In  this idea is rigorously exploited, proving that, for the purpose of checking feasibility or counting solutions, we can replace the set of arbitrary complex matrices by the set of structured matrices
We remark that, since the mapping (16) is linear in both and , it can be represented by a matrix. With a slight abuse of notation we will use the symbol to refer to both the mapping and the matrix representing that mapping. In this paper, we will be interested in the function , which depends on the channel realization through the blocks and only. The dimensions of are . In the particular case of , the one of interest for this paper, is a square matrix of size and, therefore, . The interested reader can find additional details on the structure of the matrix in  and in the example in Section III-C below.
Iii-B Main results
We use the following notation: given a Riemannian manifold with total finite volume denoted as (the volume of the manifolds used in this paper are reviewed in Appendix A), let
be the average value of a integrable function . Fix and satisfying (5) and (6) and let be defined as in (11). The main results of the paper are Theorems 1 and 2 below, which give integral expressions for the number of IA solutions when which is denoted as . For the sake of rigorousness, we denote a generic channel realization as . Recall that the particular choice of is irrelevant since the number of solutions is the same for all channel realizations out of some zero-measure set.
Assume that , and let be any open set such that the following holds: if and , are unitary matrices of respective sizes , then
(We may just say that is invariant under unitary transformations). Then, for every out of some zero–measure set, we have:
See Appendix B. \qed
If we take to be the set
(with denoting Frobenius norm) and we let we get:
For an interference channel with , and for every out of some zero–measure set, we have:
See Appendix C. \qed
As proved in , if the system is infeasible then for every choice of and hence Theorem 1 still holds. On the other hand, if the system is feasible and then there is a continuous of solutions for almost every and hence it is meaningless to count them (the value of the integrals in our theorems is not related to the number of solutions in that case). Note also that the equality of Theorem 1 holds for every unitarily invariant open set , which from Lemma 1 implies that the right–hand side of (17) has the same value for all such .
Iii-C Example: the system
In this example we specialize Theorem 2 to the scenario. Although the number of IA solutions for this network is known to be from the seminal work , this example will serve to illustrate the main steps followed to find the solution of the integral equation, and the difficulties to extend this analysis to more complex scenarios.
Let us start by considering structured matrices of the form
whose entries, without loss of generality, can be taken as independent complex normal random variables with zero mean and variance 2: , and 111The real and imaginary parts of each entry are independent real Gaussian random variables with zero mean and variance 1. Each one of these random matrices is now normalized to get
The collection of matrices generated in this way is uniformly distributed on the set in Theorem 2. Therefore, the integral formula given in Theorem 2 yields:
Choosing a natural order in the image space, the matrix defining the mapping for the scenario is
It is easy to compute the determinant of this matrix expanding it along the first column:
The first of these quantities is the product of i.i.d. random variables, thus
because has the same distribution as . That is, the isometry changes the sign of the function inside the expectation symbol but the expectation is unchanged when multiplied by . Hence, the expectation is . We have thus proved that
We now compute the last term using the fact that , where denotes a beta-distributed random variable with shape parameters 1 and 2.
Iii-D Estimating the number of solutions via Monte Carlo integration
Given the complexity of analytically computing the integral in Theorem 2 for general scenarios (as illustrated with a simple example in Section III-C), we will provide, in this section, a method to approximate its value using Monte Carlo integration. Our main reference here is . The Crude Monte Carlo method for computing the average
of a function defined on a finite-volume manifold consists just in choosing many points at random, say for , uniformly distributed in , and approximating
The most reasonable way to implement this in a computer program is to write down an iteration that computes The key question to be decided is how many such we must choose to get a reasonably good approximation of the integral. To do so, we follow the ideas in [25, Sec. 5]: first note that the random variable approaches, by the Central Limit Theorem, a Normal distribution, that is the density function of can be approximated by
for some which is actually the standard deviation of , given by
Now note that
Namely, for any random variable following a normal distribution , we have with probability greater than . Note that the reasoning above is not a formal proof but a heuristic argument. First, is not exactly normal but, for a large , our approximation will still serve its purpose. Second, there exists no way to guarantee that the integral of a generic function is correctly computed by Monte Carlo methods, see [25, Sec. 5].
In order to get an estimate, we need to approximate . The unbiased estimator of is
We thus have that, with probability greater than ,
If we stop the iteration when , then, with a probability of on the set of random sequences of terms, the relative error satisfies
For example, if we stop the iteration when , then, we can expect to be making an error of about percent in our calculation of . The whole procedure for a general system is illustrated in Algorithm 1, which is based on Theorem 2.
Iv Algorithmic aspects and special cases
We have shown how Theorem 2 can be used to approximate the number of IA solutions of a given interference channel using Monte Carlo integration. Nevertheless, our numerical experiments demonstrate that the convergence of the integral is, in general, slow. In this section, with the aim of mitigating this problem, we provide specializations for two cases of interest: square symmetric and single-beam scenarios.
Iv-a The square symmetric case
The so-called square symmetric case is that in which all the and all the and are equal for all . Furthermore, we are restricted to (for the solution counting to be meaningful) and to (for IA to make sense); which implies . Under these assumptions, we can write another integral such that Monte Carlo integration has been experimentally observed to converge faster:
Let us consider a symmetric square interference channel ( and , ) with . Assuming additionally that , then for every out of some zero–measure set, we have:
where is again defined by (16) and the input space of MIMO channels where we have to integrate are now
whose blocks, and , are matrices in the complex Stiefel manifold, denoted as , and formed by all the (ordered) collections of orthonormal vectors in . On the other hand, denotes the unitary group of dimension , whose volume can be found in Appendix A.
See Appendix D. \qed
The value of the constant preceding the integral in Theorem 3 is (using that when ):
In this example we will use Theorem 3 to calculate the number of solutions for the scenario again. First, we calculate the value of the constant which happens to be equal to 1 and, consequently, the number of solutions is directly given by the average of the determinant. 222Indeed, for all systems whenever or, equivalently, .
Subsequent calculations are similar to those in the example in Section III-C. The main difference is that, in this case, and are restricted to be elements of the complex Stiefel manifold, in this case, the unit-circle. Then,
From Example 1 it is clear that Theorem 3 has remarkably simplified the calculation of the integral by reducing the dimensionality of the integration domain. However, for larger scenarios we may still need to resort to the Monte Carlo integration procedure in Section III-D to approximate the integral in Theorem 3. Algorithm 2 summarizes the proposed method.
Iv-B The single-beam case
The results of Theorems 1, 2 and 3 are general and can be applied to systems where each user wishes to transmit an arbitrary number of streams. This subsection is devoted to specialize Theorem 2 to the particular case of single-beam MIMO networks (i.e. , ). First, we should mention that, from a theoretical point of view, the single-beam case was solved in , where it was shown that the number of IA solutions for single-beam feasible systems matches the mixed volume of the Newton polytopes that support each equation of the system333This is not true for multibeam cases because, in this case, the genericity of the system of equations is lost.. However, from a practical point of view, the computation of the mixed volume of a set of bilinear equations using the available software tools  can be very demanding. As a consequence, the exact number of IA solutions is only known for some particular cases [12, 20].
The number of IA solutions for an arbitrary single beam scenario with is given by
where is the matrix built by replacing the non-zero elements of by ones and denotes its permanent.
where , and denotes the number of elements in which is defined as the class of zero-trace binary matrices with row sums and column sums .
See Appendix E. \qed
In spite of its apparent simplicity, evaluating (23) may be very hard. In fact, computing the permanent is, in general, proven to be #P-complete  even for (0,1)-matrices where #P is defined as the class of functions that count the number of solutions in an NP problem.
On the other hand, (24) establishes an equivalence between the problem of computing the number of solutions of single-beam scenarios and the problem of counting the number of zero-trace binary matrices with prescribed rows and column sums. From a practical point of view, the result in (24) suggests that the IA problem can be interpreted as transmitters and receivers collaborating to cancel every single interfering link. A transmitter zero-forcing a link is encoded as a one in whereas a receiver zero-forcing a link is encoded as a zero. The total number of possible collaboration strategies gives the number of IA solutions.
Unfortunately, calculating is a non-trivial particular case of a problem which is also known to be #P-complete [27, Theorem 9.1]. For the interested reader, we have computed several exact values which are compiled in Table I. Our algorithm performs a recursive tree search, commonly known as backtracking  and is summarized in Algorithm 3.
Iv-B1 Connections with graph theory problems
For the particular case of symmetric scenarios, the IA solution counting problem can be restated as several well-studied combinatorial and graph theory problems. Most of these problems have been of historical interest and hence a lot of research has been done on them. Specifically, when the matrices in are seen as the adjacency matrix of a graph some connections to graph theory problems arise. It is natural, then, to find out that the number of solutions for some scenarios have already been computed in the literature:
The number of solutions for scenarios is given by the number of derangements (permutations of elements with no fixed points), also known as rencontres numbers or subfactorial. It is also the number of simple loop-free labeled -regular digraphs with nodes. Interestingly, as demonstrated in [30, p.195], a closed-form solution is available:
The number of solutions for systems matches the number of simple loop-free labeled -regular digraphs with nodes. In this case, a closed-form expression is also available :
In general, the number of solutions for the scenario matches the number of simple loop-free labeled -regular digraphs with nodes. However, as far as we are aware, additional closed-form expressions do not exist.
Iv-B2 Bounds and asymptotic rate of growth
In order to derive appropriate bounds for the number of solutions it is convenient to go back to (23) and apply some classical combinatorial results to bound the value of . Herein, we will focus on symmetric systems: . Bérgman’s Theorem [31, Theorem 7.4.5] gives an upper bound for the permanent of an arbitrary matrix as a function of its row sums, . In our case, every row (and column) sum is and the bound simplifies quite notably:
From the previous bounds and (23), the number of solutions is shown to be bounded above and below as follows:
Now, we study the growth rate of the number of solutions when the number of users increases. As a first step, we approximate every factorial in both bounds applying Stirling’s formula, i.e.