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 K-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.


Index Terms
Interference Alignment, MIMO Interference Channel, Polynomial Equations, Algebraic Geometry Editorial Area

Communications I. INTRODUCTION
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 [1], [2], 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 [3]. This idea has been applied in different forms (e.g., ergodic interference alignment [4], signal space alignment [1], or signal scale alignment [5], [6]), and adapted to various wireless networks such as interference networks [1], X channels [2], downlink broadcast channels in cellular communications [7] and, more recently, to two-hop relay-aided networks in the form of interference neutralization [8].
In this paper we consider the linear IA problem (i.e., signal space alignment by means of linear beamforming) for the K-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 been the preferred option for recent experimental studies on IA [9], [10], [11].
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 [12], [13], [14], [15], [16]. Combining algebraic geometry tools with differential topology ones, it has been recently proved in [17], 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 V (the solution variety, whose elements are the triplets formed by the channels, decoders and precoders satisfying the IA equations) to the tangent space of H (the complex space of MIMO interference channels) at some element of V is surjective. Note that this implies in particular that the dimension of V must be larger than or equal to the dimension of H. Exploiting this result, a general IA feasibility test with polynomial complexity has also been proposed in [17], [18]. 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 study the problem of how many different alignment solutions exist for a feasible IA problem. While the number of solutions is known for some particular cases, a general result is not available yet. For instance, it can be trivially shown that feasible systems for which the algebraic dimension of the solution variety is larger than that of the input space have 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 of the polynomial system. These scenarios typically represent cases where not all available DoF are achieved (for instance, we might have more antennas than strictly needed to achieve a certain DoF tuple) and therefore they do not receive further consideration in this paper. Much more interesting and challenging is the case where the dimensions of V and H are exactly the same (identical number of variables and equations) and the problem is feasible, because in this situation the number of IA solutions is finite and constant out of a zero measure set of H as proved in [17]. Following the nomenclature recently introduced in [19] we refer to these systems as tightly feasible, stressing the fact that removing a single antenna from the network turns the IA problem infeasible.
For tightly feasible single-beam (i.e., when all users wish to transmit d = 1 stream of data) MIMO networks, and elaborating on classic results from algebraic geometry, it was shown in [12] that 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 [20] can be very demanding, therefore 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 [12], [21]. 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, receive antennas and equal number of streams per user).
In addition to having a theoretical interest, the results proved in this work might also have some practical implications. For instance, to find scaling laws for the number of solutions with respect to the number of users could have interest to analyze the asymptotic performance of linear IA, as discussed in [21]. Also, for moderate-size networks for which the total number of solutions is not very high, the results of this paper also open the possibility to provide a systematic way to compute all (or practically all) interference alignment solutions for a channel realization. This idea is also briefly explored in the paper.
The rest of 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 [17]. The main results of the paper are presented in Section III, where two integral formulas for the number of IA solutions are provided, one valid for arbitrary networks and the other for symmetric multi-beam scenarios (all users with the same number of antennas at both sides of the link and transmitting the same number of streams).
Although these integrals, in general, cannot be computed in closed form, they can easily be estimated using Monte Carlo integration. 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.

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.

A. Linear IA
We consider the K-user MIMO interference channel with transmitter k having M k ≥ 1 antennas and receiver k having N k ≥ 1 antennas. Each user k wishes to send d k ≥ 0 streams or messages. We adhere to the notation used in [12] and denote this (fully connected) asymmetric interference channel . The symmetric case in which all users transmit d streams and are equipped with M transmit and N receive antennas is denoted as (M × N, d) K .
In the square symmetric case all users have the same number of antennas M = N .
The MIMO channel from transmitter l to receiver k is denoted as H kl and assumed to be flat-fading and constant over time. Each H kl is an N k × M l complex matrix with independent entries drawn from a continuous distribution. We denote the set of users as Υ = {1, . . . , K} and the set of interfering links as Φ = {(k, l) ∈ Υ × Υ : k = l}. Also, (Φ) denotes the cardinality of Φ, that is the number of elements in the finite set Φ. In this paper we focus on fully connected interference channels and, consequently, User j encodes its message using an M j × d j precoding matrix V j and the received signal is given by where x j is the d j × 1 transmitted signal and n j is the zero mean unit variance circularly symmetric additive white Gaussian noise vector. The first term in (1) is the desired signal, while the second term represents the interference space. The receiver j applies a linear decoder U j of dimensions N j × d j , i.e., where superscript T denotes transpose.
The interference alignment (IA) problem is to find the decoders and precoders, V j and U j , 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 d k , that is B. Feasibility of IA: a brief review The IA feasibility problem amounts to study the relationship between d j , M j , N j , K such that the linear alignment problem is feasible. If the problem is feasible, the tuple (d 1 , . . . , d K ) 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 K-user MIMO interference channel have been found in [22] and [23], respectively. In this work we make the following assumptions: and which are necessary conditions for feasibility derived, respectively, for point-to-point MIMO links and for the 2-user MIMO channel.
The IA feasibility problem has also been deeply investigated in [12]- [16]. In the following we make a short review of the main feasibility result presented in [17], [18], which forms the starting point of this work.
We start by describing the three main algebraic sets involved in the feasibility problem.
• Input space formed by the MIMO matrices, which is formally defined as where holds for Cartesian product, and M Nk×Ml (C) is the set of N k × M l complex matrices.
Note that in [17], [18], we let H 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 G a,b is the Grassmannian formed by the linear subspaces of (complex) dimension a in C b .
• The solution variety, which is given by where H is the collection of all matrices H kl and, similarly, U and V denote the set of U k and V l , respectively. The set V is given by certain polynomial equations, linear in each of the H kl , U k , V l and therefore is an algebraic subvariety of the product space H × S. Let us remind here that the IA equations given by (3) hold or do not hold independently of the particular chosen affine representatives of U, V .
Once the main algebraic sets have been defined, it is interesting to consider the following diagram where the sets and the main projections involved in the feasibility problem are depicted. Note that, given H ∈ H, the set π −1 1 (H) is a copy of the set of U, V such that (3) holds, that is the solution set of the linear interference alignment problem. On the other hand, given (U, V ) ∈ S, the set π −1 2 (U, V ) is a copy of the set of H ∈ H such that (3) holds.
The feasibility question can then be restated as, is π −1 1 (H) = ∅ for a generic H? The question was solved in [17], basically stating that the problem is feasible if and only if two conditions are fulfilled: 1) The algebraic dimension of V must be larger than or equal to the dimension of H, i.e., In other words this condition means that, for the problem of polynomial equations to have a solution, the number of variables must be larger than or equal to the number of equations. This condition was already established in [12], hereby classifying interference channels as proper (s ≥ 0) or improper (s < 0). More recently, in [13] it was rigorously proved that improper systems are always infeasible.
2) For some element (H, U, V ) ∈ V, the linear mapping is surjective, i.e., it has maximal rank equal to (k,l)∈Φ d k d l . 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 (H, U, V ) ∈ V.

A. Preliminaries
As it was shown in [17], [18], the surjectivity of the mapping θ in (12) can easily checked by a polynomial-complexity test that can be applied to arbitrary K-user MIMO interference networks. The test basically consists of two main steps: i) to find and arbitrary point in the solution variety and ii) to check the rank of a matrix constructed from that point. To find an arbitrary point in the solution variety, in [17] we generated a set of random precoders and decoders, and then solved a linear underdetermined problem to get a set of channel matrices satisfying the IA equations (3): this was called inverse IA problem in [17], [18]. In this paper we choose an even simpler (trivial) solution satisfying the IA equations. Specifically, we take structured matrices given by with precoders and decoders given by which trivially satisfy U T k H kl V l = 0 and therefore belong to the solution variety. We claim that essentially all the useful information about V can be obtained from the subset of V consisting on triples (H kl , U k , V l ) of the form (13) and (14). The reason is that given any other element (H kl ,Ũ k ,Ṽ l ) ∈ V, one can easily find sets of orthogonal matrices P k and Q l satisfying where the superscript * denotes Hermitian. That is, the transformed channels H kl = (P * k ) TH kl Q * l have the form (13), and the transformed precoders V l and decoders U k have the form (14). Thus, we have just described an isometry which sends (H kl ,Ũ k ,Ṽ l ) to (H kl , U k , V l ). 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, V can be thought of as moving the set of triples of the form (13) and (14), and keeping track of the visited places. Technically, V 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 summary, the main idea is that, for the purpose of checking feasibility or counting solutions, we can replace the set of arbitrary complex matrices H by the set of structured matrices The mapping θ in (12) has a more simple form for triples of the form (13) and (14), and can be replaced by a new mapping Ψ defined as We will be interested in the function det(ΨΨ * ), which depends on the channel realization H only through the blocks A kl and B kl . The vectorization of the mapping (15) where ⊗ denotes Kronecker product and K m,n is the mn × mn commutation matrix which is defined as the matrix that transforms the vectorized form of an m × n matrix into the vectorized form of its In the particular case of s = 0, Ψ is a square matrix of size k =l d k d l .
Notice that Ψ has the same structure as the incidence matrix of the network connectivity graph. Taking the 3-user interference channel as an example, Ψ is constructed as follows kl are given by (16).

B. Main results
We use the following notation: given a Riemannian manifold X with total finite volume denoted as V ol(X) (the volume of the manifolds used in this paper are reviewed in Appendix A), let be the average value of a integrable (or measurable and nonnegative) function f : X→R. Fix d j , M j , N j and Φ satisfying (5) and (6) and let s ≥ 0 be defined as in (11). The main results of the paper are Theorems 1, 2 and 3 below, which give integral expressions for the number of IA solutions when s = 0 December 12, 2013 DRAFT and the system is tightly feasible: this number is denoted as (π −1 1 (H 0 )), which is the same for all channel realizations out of some zero-measure set.
Theorem 1: Assume that s = 0, and let H ⊆ H be any open set such that the following holds: if (We may just say that H is invariant under unitary transformations). Then, for every H 0 ∈ H out of some zero-measure set, we have: where with S being the output space (Cartesian product of Grasmannians) in Eq. (8).
Proof: See Appendix B.
If we take H to be the set and we let → 0 we get: Theorem 2: For a tightly feasible (s = 0) fully connected interference channel, and for every H 0 ∈ H out of some zero-measure set, we have: Proof: See Appendix C.
Remark 1: As proved in [17] (see also [18]), if the system is infeasible then det(ΨΨ * ) = 0 for every choice of H, U, V and hence Theorem 1 still holds. Moreover, if the system is feasible and s > 0 then there is a continuous of solutions for almost every H kl 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 H , which in particular implies that the right-hand side of (17) has the same value for all such H (recall that we proved in [17] that almost all channel realizations in H have the same number of solutions).
Theorem 2 can be used to approximate the number of solutions of a given MIMO system using Montecarlo integration (see Section III-C below). However, the convergence of the integral is quite slow in general. In the square symmetric case when all the d k and all the N k and M k are equal ∀k and greater than 2d, that is N = M ≥ 2d, which holds automatically when s = 0 and K ≥ 3; we can write another integral which has faster convergence in practice: Theorem 3: Let us consider a tightly feasible (s = 0) square interference channel (N k = M k = N and d k = d, ∀k). Assuming additionally that K ≥ 3, then for every H 0 ∈ H out of some zero-measure set, we have: where Ψ is again defined by (15) and the input space of MIMO channels where we have to integrate are whose blocks, A kl and B kl , are matrices in the complex Stiefel manifold, denoted as U (N −d)×d , and formed by all orthonormal d-dimensional vectors in C (N −d) . On the other hand, U a denotes the unitary group of dimension a, whose volumen can be found in Appendix A.

Remark 2:
If the problem is fully connected, the value of the constant preceding the integral in Theorem Additionally, if N = 2d (which implies K = 3) then this constant is exactly equal to 1.
Proof: See Appendix D.
In the next section we discuss how the results in Theorems 1 and 2 can be used to get approximations to the number of IA solutions for a given interference network.

C. Estimating the number of solutions by Monte Carlo integration
The integrals in Theorems 2 and 3 are too difficult to be computed analytically, but one can certainly try to compute them approximately using Monte Carlo integration. Our main reference here is [24,Sec. 5]. The Crude Monte Carlo method for computing the average of a function f defined on a finite-volume manifold X consists just in choosing many points at random, say x 1 , . . . , x n for n >> 1, uniformly distributed in X, and approximating The most reasonable way to implement this in a computer program is to write down an iteration that The unique point to be decided is how many such x j we must choose to get a reasonable approximation of the integral. A usual tool for measuring that is the standard deviation, that can be approximated by If we stop the iteration when Σ n E n < ε, then, with a probability of 0.95 on the set of random sequences of n terms, the relative error satisfies For example, if we stop the iteration when then we can expect to be making an error of about 10 percent in our calculation of − x∈X f (x) dx.
The whole procedure for a general system is illustrated in Algorithm 1 which follows Theorem 2. Its particularization to square systems is shown in Algorithm 2 and follows Theorem 3.

D. The single-beam case
Although the results of Theorems 1, 2 and 3 are general and can be applied to arbitrary systems, for the particular case of single-beam MIMO networks (d k = 1, k ∈ Υ) it is possible to develop specific, much more efficient, techniques to count the exact number of alignment solutions. This subsection is devoted Build channel matrices {H kl } according to (13).
Normalize every channel matrix H kl such that H kl F = 1.
Calculate E n and Σ n according to (18) and (19), respectively, where f (x j ) is now D j .
to this particular case. First, we should mention that, from a theoretical point of view, the single-beam case was solved in [12], where it was shown that the number of IA solutions for single-beam feasible systems coincides with the mixed volume of the Newton polytopes that support each equation of the system 1 . However, from a practical point of view, the computation of the mixed volume of a set of bilinear equations using the available software tools [20] can be very demanding. In consequence, the exact number of IA solutions is only known for some particular cases [12], [21].
The main idea that allows us to count efficiently the number of IA solution for single-beam MIMO networks is that, as first discussed in [25], for single-beam MIMO networks the mixed volume does not change if we consider rank-one MIMO channels instead of full-rank channels. The proof of this fact is straightforward by taking into account that the same monomials are present in both systems of equations and, thus, the Newton polytopes that support each equation are identical in both cases. Therefore, for the purpose of counting the number of alignment solutions in single-beam feasible systems, we can simplify our problem by considering rank-one channels without loss of generality. Assuming rank-one MIMO channels, H kl = f kl g * kl , the set of alignment equations (3) can be rewritten as where now v l and u k are column vectors representing the precoders and decoders for the particular case of d k = 1 ∀k. We notice that there are K(K − 1) equations, each one being the product of two linear factors L(u k ) and L(v l ) in the entries of u k and v l , respectively, as indicated in (20). Finding a solution to this system reduces to choose from every equation exactly one factor and force it to be zero, i.e., either u T k f kl = 0 or g * kl v l = 0. Now the questions is how many different solutions exist for such a system. As a first approach, one may think that the total number of solutions would be 2 K(K−1) since we A tighter bound would be obtained by considering that we can design v l (of size M l × 1)) to lie in, at most, M l − 1 non-intersecting nullspaces. In other words, for a given l, L(v l ) = g * kl v l = 0 can be satisfied for a maximum of M l − 1 values of k. This observation would allow us to upper bound the number of solutions by K(K−1) l (Ml−1) , or equivalently, K(K−1) k (Nk−1) . Although this bound is much tighter than 2 K(K−1) for small values of K, they are asymptotically equivalent since the rate of growth with K of the latter is also exponential.
Due to the fact that all the equations in the system are strongly coupled and they cannot be solved independently, the combinatorics of finding the exact number of solutions is much more complicated than this last approach and forces us to design a counting routine. In order to explain how this computational routine works, we will use the (2 × 3, 1)(3 × 2, 1)(2 × 4, 1)(2 × 2, 1) system as an example. The proposed routine proceeds as follows: 1) We start from a K × K table. Each cell in the table corresponds to a link of the interference channel. Cells in the main diagonal represent direct links and they are ruled out since they do not play any role in the IA problem. All other cells correspond to interfering links. The table for the (2 × 3, 1)(3 × 2, 1)(2 × 4, 1)(2 × 2, 1) system (or any 4-user system) would be as follows. 2) We will now fill the cells according to some rules. The value in the cell (k, l) indicates how the equation corresponding to the (k, l) link has been satisfied. If it has been satisfied by forcing L(v l ) = 0 it will contain a one. Otherwise, it will contain a zero, meaning that it has been satisfied by setting L(u k ) = 0. We recall that given l, L(v l ) = 0 can be satisfied for a maximum of M l − 1 values of k and given k, L(u k ) = 0 can be satisfied for a maximum of N k − 1 values of l. We also recall that when s = 0, l (M l − 1) + k (N k − 1) = K(K − 1). Thus, for any valid solution, the l-th column of the table must contain exactly M l − 1 ones whereas the k-th row must contain exactly N k − 1 zeros. On the other hand, all cells must contain either a zero or a one.
3) The approach to fill the table for an arbitrary single-beam network is a recursive tree search approach, commonly known as backtracking procedure [26] which is widely used to solve combinatorial enumeration problems. We first start with an all-zeros table and try to build up our solution cell by cell, filling it with ones, starting from the upper left corner; first right, then bottom. We can keep track of the approaches that we explored so far by maintaining a backtracking tree whose root is the all-zeros board and where each level corresponds to the number of ones we have placed so far. Figure 2 shows the backtracking tree for our example system which was constructed according to Algorithm 3.
In general this procedure is much more efficient than resorting to general software packages to compute the mixed volume since it exploits the specific structure of the IA bilinear equations. As When this matrix is seen as the adjacency matrix of a graph (or the biadjacency matrix of a bipartite graph) some connections to graph theory problems arise. Most of these problems have been of historical interest and hence a lot of research has been done on them. It is natural, then, to find out that the number of solutions for some scenarios have already been computed in this field. We mention a few of them in the following. • The number of solutions for (2 × (K − 1), 1) K scenarios is given by the number of derangements (permutations of K elements with no fixed points), rencontres numbers or subfactorial. It is also the number of labeled 1-regular digraphs with K nodes. Interestingly, as found in [27], [28, p.195], they are equal to • The number of solutions for (3 × (K − 2), 1) K systems matches the number of labeled 2-regular digraphs with K nodes. In this case, a closed-form expression is also available [27]: • In general, for (M × (K − M + 1), 1) K scenarios, closed-form solutions do not exist and most of them have not even been studied. It is clear that this problem matches that of counting the number  return list of candidate cells to store the next 1 of labeled (M − 1)-regular digraphs with K nodes but, as far as we know, no closed-form solution has been found yet. Further details can be found in Section IV.

IV. NUMERICAL EXPERIMENTS
In this section we present some results obtained be means of the integral formulae in Theorem 2 (for arbitrary interference channels) and Theorem 3 (for square symmetric interference channels). We first evaluate the accuracy provided by the approximation of the integrals by Monte Carlo methods. To this end, we focus initially on single-beam systems, for which the procedure described in Section III-D allows us to efficiently obtain the exact number of IA solutions for a given scenario. The true number of solutions can thus be used as a benchmark to assess the accuracy of the approximation. Tables I and II show the number of solutions given by the exact and the approximate procedures, respectively. To simplify the analysis, we have considered (M × (K − M + 1), 1) K symmetric singlebeam networks for increasing values of M and K. As shown in Section III-D1, counting IA solutions for this scenario is equivalent to the well-studied graph theory problem of counting labeled (M − 1)regular digraphs with K nodes. Thus, additional terms and further information can be retrieved from integer sequences databases such as [27] from its corresponding A-number given in the last row of Table   I. Percentages in Table II represent the upper bound for the relative error, 2ε · 100, obtained in each scenario (see Section III-C).   allowing us to get smaller relative errors. For the sake of completeness,    Although these results have mainly a theoretical interest, they might also have some important practical implications. For instance, knowing the rate of increase of the number of solutions with K could have interest to analyze the asymptotic performance of linear IA, as discussed in [21]. Also, for moderatesize networks for which the total number of solutions is not very high, the results of this paper also open the possibility to provide a systematic way to compute all (or practically all) interference alignment solutions for a channel realization. Although all IA solutions are asymptotically equivalent, their sum-rate performance in low or moderate SNRs behavior may differ significantly [21], [29]. The main idea here is that if we are able to obtain all or almost all IA solutions for a particular channel realization, we can get all or almost all IA solutions for any other channel realization by using a homotopy-continuation based method such as that described in [25]. This idea is illustrated in Figure 4, which shows in grey the sum-rate curves of 973 different solutions for the (4 × 6, 2) 4 network. The maximum sum-rate solution is plotted in a thicker solid line, while the average sum-rate of all solutions is represented with a dashed line. The relative performance improvement provided by the maximum sum-rate solution over the average is always above 10 % for SNR values below 40 dB, and is more than 20 % for SNR=20 dB. We note that this improvement is comparable to the one provided by sum-rate optimization algorithms which take into account additional information in the optimization procedure such as direct channels and noise variance.

V. CONCLUSION
In this paper we have provided two integral formulae to compute the finite number of IA solutions in tightly feasible problems, including multi-beam (d k > 1) networks. The first one can be applied to arbitrary K-user channels, whereas the second one solves the symmetric square case. Both integrals can be estimated by means of Monte Carlo methods. For single-beam networks, it is possible to obtain the exact number of solution resorting to more classic results on algebraic geometry.

APPENDIX A MATHEMATICAL PRELIMINARIES
To facilitate reading, in this section we recall the mathematical results used in this paper. Firstly, we provide a short review on mappings between Riemannian manifolds and the main mathematical result used to derive the number of IA solutions, which is the Coarea formula. Secondly, we review the volume of the complex Stiefel and Grassmanian manifolds and the volume of the unitary group, which are also used throughout the paper.

A. Riemannian manifolds and the Coarea formula
The following result is immediate from [30,Th. 9.23].
Theorem 4: Let X be a compact, embedded, (real) codimension c submanifold of the Riemannian manifold Y . Then, for sufficiently small > 0, Here, V ol(X) is the volume of X w.r.t. its natural Riemannian structure inherited from that of Y .
One of our main tools is the so-called Coarea Formula. The most general version we know may be found in [31], but for our purposes a smooth version as used in [32, p. 241] or [33] suffices. We first need a definition.
Definition A.1: Let X and Y be Riemannian manifolds, and let ϕ : X −→ Y be a C 1 surjective map.
Let k = dim(Y ) be the real dimension of Y . For every point x ∈ X such that the differential mapping Dϕ(x) is surjective, let v x 1 , . . . , v x k be an orthogonal basis of Ker(Dϕ(x)) ⊥ . Then, we define the Normal Jacobian of ϕ at x, NJϕ(x), as the volume in the tangent space T ϕ(x) Y of the parallelepiped spanned by Dϕ(x)(v x 1 ), . . . , Dϕ(x)(v x k ). In the case that Dϕ(x) is not surjective, we define NJϕ(x) = 0. Theorem 5 (Coarea formula): Let X, Y be two Riemannian manifolds of respective dimensions k 1 ≥ k 2 . Let ϕ : X −→ Y be a C ∞ surjective map, such that the differential mapping Dϕ(x) is surjective for almost all x ∈ X. Let ψ : X −→ R be an integrable mapping. Then, the following equality holds: Note that from the Preimage Theorem and Sard's Theorem (see [34,Ch. 1]), the set ϕ −1 (y) is a manifold of dimension equal to dim(X) − dim(Y ) for almost every y ∈ Y . Thus, the inner integral of (21) is well defined as an integral in a manifold. Moreover, if dim(X) = dim(Y ) then ϕ −1 (y) is a finite set for almost every y, and then the inner integral is just a sum with x ∈ ϕ −1 (y).
Theorem 6: Let X, Y and V ⊆ X × Y be smooth Riemannian manifolds, with dim(V) = dim(X) and Y compact. Assume that π 2 : V → Y is regular (i.e. Dπ 2 is everywhere surjective) and that Dπ 1 (x, y) is surjective for every (x, y) ∈ V out of some zero measure set. Then, for every open set U ⊆ X contained in some compact set K ⊆ X , where DET (x, y) = det(DG x,y (x)DG x,y (x) * ) and G x,y is the (locally defined) implicit function of π 1 near x = π 1 (x, y). That is, close to (x, y) the sets V and {(x, G x,y (x))} coincide.
Corollary 1: In addition to the hypotheses of Theorem 6, assume that there exists y 0 ∈ Y such that for every y ∈ Y there exists an isometry ϕ y : Y → Y with ϕ y (y) = y 0 and an associated isometry Proof: Let y ∈ Y and let ϕ y , χ y as in the hypotheses. Then, consider the mapping which is the restriction of an isometry, hence an isometry. Let G x be the local inverse of π 1 close to x ∈ X. The change of variables formula then implies: Note that the following diagram is commutative: and the composition rule for the derivative gives: Now, χ y , ϕ y and χ y × ϕ y are isometries of their respective spaces. Thus, we conclude: That is, the inner integral in the right-hand side term (22) is constant. The corollary follows.

B. The volume of classical spaces
Some helpful formulas are collected here: is the volumen of the complex sphere of dimension a.
(cf. [35, p. 54 is the volumen of the unitary group of dimension a. Note that, as pointed out in [35, p. 55] there are other conventions for the volume of unitary groups. Our choice here is the only one possible for Theorem 4 to hold: the volume of U a is the one corresponding to its Riemannian metric inherited from the natural Frobenius metric in M a (C).
We finally recall the volume of the complex Grassmannian. Let 1 ≤ a ≤ b; then, APPENDIX B PROOF OF THEOREM 1 We will apply Corollary 1 to the double fibration given by (10). In the notations of Corollary 1, we consider X = H, Y = S, V the solution variety and Given any other element y = (U k , V k ) ∈ S, let P k and Q k be unitary matrices of respective sizes N k and M k such that Then consider the mapping which is an isometry of S and satisfies ϕ y (y) = y 0 as demanded by Corollary 1. We moreover have the associated mapping χ y : H → H given by which is an isometry of H. Moreover, χ y (H ) = H and χ y × ϕ y (V) = V. We can thus apply Corollary 1 which yields where H is any open subset of H and G is the local inverse of π 1 close to H at (H, y 0 ). We now On the other hand, from the defining equations (3) and considering H ∈ H I andḢ ∈ T H H as block A straight-forward computation shows that: Thus, writing Ψ = Ψ H , we have:  Let H be the product for (k, l) ∈ Φ of the sets From Theorem 4, each of these sets have volume equal to Thus, using (24), On the other hand, consider the smooth mapping f : Hkl F k,l and apply Theorem 5 to get whereĤ kl = H kl (1 + t kl ). Note that the function inside the inner integral is smooth and hence for any We have thus proved (using ≈ for equalities up to O( )): It is very easy to see that From Theorem 1 and taking limits we then have that for almost every H 0 ∈ H, .
Finally, S = k∈Υ G dk,Nk × l∈Υ G dl,Ml is a product of complex Grassmannians, and its volume is thus the product of the respective volumes, given in (26). That is, Putting these computations together, we get the value of C claimed in Theorem 2.
APPENDIX D

PROOF OF THEOREM 3
The proof of this theorem is quite long and nontrivial. We will apply Theorem 1 to the sets Then, because (17) holds for every , one can take limits and conclude that for almost every H 0 ∈ H, The claim of Theorem 3 will follow from the (difficult) computation of that limit. We organize the proof in several subsections.

A. Unitary matrices with some zeros
In this section we study the set of unitary matrices of size N ≥ 2d which have a principal d × d minor equal to 0, and the set of closeby matrices. For simplicity of the exposition, the notations of this section are inspired in, but different from, the notations of the rest of the paper. Let Note that T is a vector space of complex dimension N 2 − d 2 . Our three main results are:

Proposition 2:
The following equality holds: Proposition 3: Let Ψ : T → R be a smooth mapping defined in the T and such that Ψ(H) depends only on the A and B part of H, but not on the part C. Denote Ψ(H) = Ψ (A, B). Then,

1) Proof of Proposition 1:
Let We claim that ξ is surjective. Indeed, let where R satisfies U RV = C. Now, this implies that the matrix is unitary, which forces R 1 = 0, R 2 = 0, R 3 = 0 and R 4 unitary. That is and the surjectivity of ξ is proved. Moreover, this describes U N ∩ T as the orbit of J under the action in T given by Then, U N ∩ T is a smooth manifold diffeomorphic to the quotient space Then, On the other hand, dim(T ) = 2N 2 − 2d 2 and thus as claimed. We now apply the Coarea formula to ξ to compute the volume of U N ∩ T . Note that by unitary invariance the Normal Jacobian of ξ is constant, and so is V ol(ξ −1 (H)). We can easily compute

December 12, 2013 DRAFT
For the Normal Jacobian of ξ, note that It is a routine task to see that η * (L) = (L, L * ) which implies ηη * (L) = 2L, that is As we have pointed out above, the value of the Normal Jacobian of ξ is constant. Thus, for every U, V , The Coarea formula applied to ξ then yields: The value of V ol(U N ∩ T ) is thus as claimed in Proposition 1.
2) Some notations: Given a matrix of the form (α and σ are d × d diagonal matrices with real positive ordered entries) we denote byH the associated where U 0 is some unitary matrix which minimizes the distance from C 4 to U N −2d . Note that We also let Note that 3) Approximate distance to U N and U N ∩ T : In this section we prove that for small values, More precisely: Here, we are writing O( 2 ) for some function of the form c(d) 2 .
Before proving Proposition 4 we state the following intermediate result.

Lemma 1:
There is an 0 > 0 such that H − I N ≤ < 0 implies: Proof: We will use the concept of normal coordinates (see for example [30, p. 14]). Consider the exponential mapping in U N , which is given by the matrix exponential R k k! , which is an isometry from a neighborhood of 0 ∈ T I U N to a neighborhood of I ∈ U N and defines the normal coordinates. Thus, for sufficiently small 1 > 0 there exists 0 > 0 such that if U ∈ U N , U − I < 0 then there exists a skew-symmetric matrix R such that Let R ∈ M N (C) be a skew-Hermitian matrix such that Then, e R = I + R + S and If we denote a = e R − I = R + S and b = d UN (e R , I) = R , we have proved that Using that b < 1/2 and some arithmetic, this implies Now, In particular, S ≤ 9 2 . We conclude: We now solve the following elementary minimization problem: Then, H − (I + R) is minimized when R 1 = 0, R 5 = 0, R 9 = 0 and It is easily seen that the solutions to these problems are: We have then proved and the minimum is reached at Hence, and the first lower bound claimed in the lemma follows. For the upper bound let R be defined by (35) and note that (following a similar reasoning to the one above) Now, H − I N ≤ in particular implies C 1 2 + C 2 2 + C 3 2 ≤ 2 and then we have as wanted. Now, for the second claim of the lemma, the same argument is used but now R is such that

Now, from the equality
and arguing as above we have that where we denote byR the matrix resulting from letting R 2 = 0. Thus, We have then proved and as before we can easily see that the minimum is reached when R 1 = 0, R 2 = 0, R 5 = 0, R 9 = 0, The lemma is now proved.

Proof of Proposition 4
Let E be a matrix such that E ≤ and H = U + E for some unitary matrix U . Then, On the other hand, where the entries X are terms which we do not need to compute. In particular, we have C 1 σ ≤ 4 and which implies σ −2 = σ −2 − I + I ≤ √ d + 4 and hence A similar argument works for C 3 as well, and using a symmetric argument for H * H we get the same bound for C 2 and an equivalent bound for α to that of (36). Summarizing these bounds, we have: Moreover, we also have where the β j are the singular values of C 4 . In particular, and we conclude that Using (36), (37) and (38) above we get: where C(d) depends only on d. Let be small enough for C(d) to satisfy the hypotheses of Lemma 1.
The Proposition 4 follows from applying that lemma.

4)
How the sets of closeby matrices to U N and U N ∩ T compare: Our main result in this section is the following.
Proposition 5: Let α > 1. For sufficiently small > 0, we have: Before the proof we state two technical lemmas.
Lemma 2: Let σ, α be as in (33). Then, Proof: Let where A = (σ 0) and B T = (α 0). The claim of the lemma is that Indeed, consider the mapping which has Jacobian equal to √ 2 The change of variables theorem yields: The lemma follows from the fact that S 1 (ϕ(C)) = S 2 (C).
Lemma 3: Let α > 1 and let A, B be complex matrices of respective sizes d×(N −d) and (N −d)×d.
Then, for sufficiently small > 0 we have are singular value decompositions of A and B respectively. Then, where the last inequality follows from unitary invariance of the volume. Let H be as in (33). From Proposition 4, we conclude: In particular, for every α > 1 and for sufficiently small > 0 we have proved that = V ol(U N ∩ T )α N 2 V ol(x ∈ R N 2 : x ≤ 1).
We have thus proved that for every α > 1 we have This implies: The reverse inequality is proved the same way using the other inequality of Proposition 5.
6) Integrals of functions of the subset of matrices in T which are close to U N : We are now close to the proof of Proposition 3, but we still need some preparation. We state two lemmas.

B. Proof of Theorem 3
Recall that we have defined H in (29), and we want to compute the limit (30): The claim of the Theorem 3 follows.