DISTRIBUTIONS OF PROJECTIONS OF UNIFORMLY DISTRIBUTED K-FRAMES
Stephen D. Howard1 Songsri Sirianunpiboon1 Douglas Cochran2
1Defence Science and Technology Organisation PO Box 1500, Edinburgh 5111, Australia 2School of Mathematical and Statistical Sciences Arizona State University, Tempe AZ 85287-5706 USA
ABSTRACT II. MATHEMATICAL FRAMEWORK N A geometrical perspective is introduced that enables unification Consider a set of M complex vectors x1, x2,..., xM ∈ C and generalization of several results regarding the distributions with M ≤ N. The Gram matrix of this set of vectors, denoted by of quantities that arise in connection with an important class of G(x1, ··· , xM ), is the M × M positive semi-definite hermitian † multiple-channel detectors. Standard models on sets of normalized matrix whose elements are gij = hxi, xj i = x xi, where † denotes vectors following from joint gaussian assumptions in this context j conjugate transpose. Denoting by X the N ×M matrix whose mth are relaxed to the geometrically appealing model of uniform column is x , the Gram matrix can be written as G = XX†. distributions on the Stiefel manifold of K-frames in N-dimensional m space. In addition to bolstering geometric insight, several prior re- The normalized Gram matrix Gb is obtained by normalizing the sults are subsumed and strengthened by results obtained under this vectors xj to unit length; i.e., formulation. Additionally, a generalization of a classical theorem xi xj hxi, xj i of W. G. Cochran is enabled by this framework. gˆij = , = . (1) kxik kxj k kxikkxj k Index Terms— Coherence, Multiple-channel detection, Stiefel ˆ manifold, Cochran’s theorem The elements on the main diagonal of G are gˆii = 1, and its determinant is |G| I. INTRODUCTION |Gb| = kx k2 · · · kx k2 A broad class of multiple-channel detectors are formulated under 1 M gaussian assumptions on the data under the null hypothesis. Early In [15] and [16], this structure is extended to the context of K×N examples include the two-channel magnitude-squared coherence vector-valued time series. In this generalization, Xj ∈ C for † † † † detector [1], [2] and its M-channel counterpart, the generalized j = 1,...,M. Denoting X = (X1 ,X2 ,...,XM ) . The Gram coherence detector [3], [4]. Inspired in part by recent application matrix associated with X is G = XX†, which is an M ×M block interest in spectrum sensing [5], [6], [7] and passive radar [8], † matrix with block elements Gij = XiXj . It is always possible to [9], [10], work on multiple-channel detectors for signals having decompose Xj uniquely as known rank [6], [11], [12], [13], multiple-channel estimators of signal rank [5], [12], [14], and multi-channel detection of spatially Xj = Rj Xbj (2) correlated signals [15], [16] have received considerable attention in † the research literature over the past few years. where Xbj is semi-unitary; i.e., Xbj Xbj = IK , the K × K identity K×K The geometrical nature of detection and estimation problems matrix, and Rj ∈ C is an upper triangular matrix with positive † † in this context is well recognized in the literature, and it has been diagonal elements. Note that Rj R = Xj X . The block elements exploited in connection with invariances of detection statistics (e.g., j j G [17], [18], [16], [19]) and Bayesian formulations [20]. Recently, of the normalized Gram matrix b are geometrical insight has been strengthened through formulations in −1 † −1† Gbij = (R )XiX R . terms of Grassmannian and Stiefel manifolds, which arise naturally i j j when considering collections of subspaces having given dimension When K = 1, this reduces to (1). The normalized Gram matrix in a vector space of higher dimension. has the properties that Gbii = IK , the K × K identity matrix, and This paper advances the geometrical insight that has character- |G| ized most of the work on multiple-channel detection and estimation |Gb | = , in which normalized gram matrices play a central role. After |G11| · · · |GMM | establishing the necessary mathematical framework in Section II, which is the detection statistic defined and analyzed in [15], [16]. Section III proceeds to show how the gaussian null hypothesis Analysis of the statistics of these detectors under the null hy- assumed in much of the preceding work in this vein can be relaxed pothesis involves knowledge of various properties of the probability to one that assumes a uniform distribution on a Stiefel manifold. distribution of XPb Xb † or, more generally, the joint distribution This model strengthens geometric insight and leads to a derivation † † of the distribution of the normalized gram matrix under the null of {XPb 1Xb , ··· , XPb M Xb }, where the Pj are orthonormal pro- N PM hypothesis that subsumes several previously published results as jectors on C such that j=1 Pj = IM . These quantities also special cases. In Section IV, this perspective leads to a generaliza- arise in the analysis of the effect of compression on detection and tion of a classical theorem of W. G. Cochran [21] regarding the estimation [22]. When K = 1 and X is a gaussian random vector joint distribution of quantities obtained when a random vector is with iid component, then Cochran’s Theorem [21], [23] shows that † 2 projected into a collection of mutually orthogonal subspaces that the random variables XPj X are independent and χ distributed partition a vector space. Some concluding remarks are given in with the degrees of freedom of the distribution depending only Section V. on the dimensions of the subspaces associate with the orthogonal † PL projections Pj . Furthermore, each of the quantities XPb j Xb is beta where Mj ≥ K for j = 1, ··· ,L, j=1 Mj = N, and distributed. Cochran’s Theorem can be extended to K > 1. If B1(M1,M2) is the usual beta function. K > 1 and the matrix X has iid gaussian components, then the As an example of this geometric view, suppose that † quantities XPj X are independently Wishart distributed [24], [25], {xˆ1, ··· , xˆK } is a random orthonormal K-frame, uniformly dis- † N [26] and each of the random matrices XPb j Xb are matrix beta tributed on the Stiefel manifold VK,N . Let S ⊂ C be a subspace distributed. of dimension L > K and PS be the orthogonal projector on to V . This paper has two goals. The first is to weaken the gaussian The properties of gram determinants imply that null hypothesis to one that assumes only that the matrix Xb is † uniformly distributed on the Stiefel manifold. The usual assumption |XPS X | = |G(PS xˆ1, ··· ,PS xˆK )| (5) that there exists a gaussian X associated with X is an unnecessary 2 b = |G(PS xˆ1, ··· ,PS xˆK−1)|kPWK−1 xˆK k , artifice, and deriving results directly from uniform distributions on ⊥ the Stiefel manifold gives greater geometric insight. The proof that where WK−1 = S ∩ < xˆ1, ··· , xˆK−1 > and PWK−1 is † XPb j Xb given here that is matrix beta distributed with parameters the orthogonal projector onto this subspace. In this expression, depending only on the rank of Pj follows directly from the < xˆ1, ··· , xˆK−1 > denotes the subspace spanned by the vectors ⊥ uniformity of the distribution of Xb. This proof involves a judicious {xˆ1, ··· , xˆK−1} and denotes orthogonal complement. Now xˆK ⊥ choice of the matrix Y the construction of the invariant measure is a uniformly distributed unit vector in < xˆ1, ··· , xˆK−1 > and ⊥ on the Stiefel manifold (3). WK−1 is an L-dimensional subspace of < xˆ1, ··· , xˆK−1 > , The second goal of this paper is to give a simple geometric except on a set of measure zero. Theorem 1, for K = 1 (see [30, † † 2 proof that the joint distribution of {XPb 1Xb , ··· , XPb M Xb } is Theorem 2]), implies that kPWK−1 xˆK k ∼ B(N−L−(K−1),L), matrix Dirichlet distributed with parameters depending only on and this distribution depends only on the subspace WK−1 only the ranks of the orthogonal projectors P1, ··· ,PM . This has been through its dimension. Thus the two factors in (5) are independently shown by Tan [27] (see also [28]), using an argument involving distributed. Continuing in this way yields gaussian random matrices. Theorem 2 is a step towards a version K of Cochran’s Theorem for uniformly distributed K-frame. Y 2 N |G(P xˆ , ··· ,P xˆ )| = kP xˆ k , For a subspace S ⊂ C with dim S > K, the Stiefel manifold S 1 S K Wk−1 k k=1 VK (S) is the space of all sets of K orthonormal vectors in S. Such a set of K orthonormal vectors is called an orthonormal K- where the factors on the right-hand side are independently beta frame or just a K-frame when no confusion arises. A K-frame distributed as † can be regarded as an N × K matrix X, satisfying X X = IK , 2 so VK (S) can be regarded the space of all such matrices X. kPWk−1 xˆkk ∼ B(N − L − (k − 1),L). Note that here and in the remainder of this paper the notation b ⊥ denoting orthonormality is dropped. VK (S) is a smooth manifold Here Wk−1 = S ∩ < xˆ1, ··· , xˆk−1 > and W0 = S. The 2 N of dimension 2(dim S)K −K and is a sub-manifold of VK (C ). main result in [16] is consequence of the above result and some The invariant measure on VK (S) can be constructed following elementary properties of gram matrices. James [29]. For each X ∈ V (S) choose a K × (dim S − K) K X III. UNIFORMLY DISTRIBUTED K-FRAMES matrix Y such that Y is unitary and Y is a smooth function This section begins with some background. Suppose the space of of X. The invariant measure on VK (S) is constructed by taking M-dimensional positive definite hermitian matrices P is param- the exterior product of the independent entries in the matrix of M eterized by the matrix eigenvalues λ1, ··· , λM and eigenvectors X † u , ··· , u , differential forms Y dX , which results in 1 M G = U †ΛU (6) dim S−K K Y Y † † This parameterization is redundant as it stands [31], since multi- dµVK (S)(X) = Re(yj dxi) Im(yj dxi) † plying U by a unitary matrix U0 such that U ΛU0 is still diagonal j=1 i=1 0 (3) gives an alternative decomposition for the same matrix. This redun- K K Y † Y † dancy can be removed by choose the phases of one column of U, × Re(xj dxi) Im(xj dxi). or of its diagonal, and by choosing a ordering for the eigenvalues. i U † − UdU † + Λ1/2V dV †Λ1/2 This section considers the joint distribution of the Gram matrices of the projected components of a uniformly distributed K-frame + (I − Λ)1/2W dW †(I − Λ)1/2U with respect to such an orthogonal decomposition. K×N N 1 Suppose X ∈ is a K-frame; i.e., X ∈ VK ( ). X can U † − (1 − Λ)−1/2Λ−1/2dΛ C C 2 be uniquely decomposed as † QdX = − (I − Λ)1/2V dV †Λ1/2 . (9) M 1/2 † 1/2 + Λ W dW (I − Λ) U X X = Aj † 1/2 j=1 YV dV Λ U † 1/2 with the rows of Aj in Sj . Applying a SVD to Aj for each j = YW dW (I − Λ) U 1, ··· ,M, X can be written as † † † Noting that V dV , W dW and UdU are skew-hermitian and that M the diagonal of UdU † is dependent on the off diagonal elements, X † 1/2 X = U Λ Vj the exterior product of the (ij)th element of the top block of (9) j j j=1 where U1, ··· ,UM ∈ U˜(K), Λ1, ··· , ΛM are non-negative and the result extends from the case of M − 1 orthogonal subspaces to diagonal, and Vj ∈ VK (Sm) for j = 1, ··· ,M. Since X is a that of M orthogonal subspaces. † PM K-frame, XX = IK and consequently j=1 Gj = IK , where † An immediate consequence of Theorem 2 is that Gj = U Λj Uj is the Gram matrix of the projection of X onto the j (G1, ··· ,GM−1), the set of gram matrices of the projections subspace Sj . We define the standard open M-simplex of K × K of X onto the components of the orthogonal decomposition non-negative definite matrices by N {S1, ··· ,SM } of C , is matrix Dirichlet-distributed; i.e., ( M ) (G1, ··· ,GM−1) ∼ DK (dim S1, ··· , dim SM ) with X ∆K = (G1, ··· ,GM )|G1, ··· ,GM > 0 and Gm = IK . dF (G1, ··· ,GM−1) m=1 PM−1 dim SM −K M−1 ! M−1 |I − Gj | j=1 Y dim Sj −K Y The following Theorem gives the joint distribution of the gram = |Gj | dGj BK (dim S1, ··· , dim SM ) matrices Gj . j=1 j=1 K×N Theorem 2. Let X ∈ C , K < N, be uniformly distributed on N Note that this distribution depends only on the dimensions of the the Stiefel manifold VK (C ). Let {S1, ··· ,SM } be an orthogonal N N subspaces Sj in the orthogonal decomposition of C and not on decomposition of C and decompose X as which particular subspaces are chosen. M X † 1/2 X = Uj Λj Vj (11) V. CONCLUSIONS j=1 An important class of multiple-channel detectors and estimators where Λj are diagonal matrices with elements in [0,1], Uj ∈ are formulated in terms of gram matrices containing normalized U˜(M) = U(N)/(TN × SN ), Vj ∈ VK (Sj ) for j = 1, ··· ,M. data vectors from the channels. Standard null-hypothesis models † on the sets of unit vectors comprising these matrices arise from Then the joint distribution of Gj = Uj Λj Uj , Vj , for j = 1, ··· ,M is gaussian assumptions on the data prior to normalization. This paper has relaxed this traditional model to the geometrically appealing dF (G1, ··· ,GM−1,V1, ··· ,VM ) case of uniform distributions on the Stiefel manifold. This not only PM−1 dim SM −K M−1 ! fosters further geometric insight, but it also leads to derivation of |I − Gj | j=1 Y dim Sj −K = |Gj | results regarding the distributions of quantities associated with this BK (dim S1, ··· , dim SM ) j=1 class of multiple-channel detectors and estimators that subsume and M−1 M generalize previously known results. A generalization of a classical Y Y theorem of W. G. Cochran that is enabled by this framework × dGj dµ (Vj ) VK (Sj ) was also presented. Finally, it is noted that the quantities treated j=1 j=1 here also arise in analysis of the effects of signal compression on PM−1 detection and estimation. for (G1, ··· ,GM−1,I − j=1 Gj ) ∈ ∆K , where dGj = QK d[G ] QK d Re[G ] d Im[G ] and dµ de- `=1 j `` i<`=1 j i` j i` VK (S) notes the normalized invariant measure on the Stiefel manifold ACKNOWLEDGMENT VK (S). This work was supported in part by the U.S. Air Force Office of Scientific Research under Grant Nos. FA9550-12-1-0225 and Proof. 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