Dec 17, 2002 - A sequential Monte Carlo method is used for sampling from the time-varying posterior and the samples are utilized to estimate the underlying ...
Applied Probability Trust (17 December 2002)
BAYESIAN, GEOMETRIC SUBSPACE TRACKING ANUJ SRIVASTAVA,∗ Florida State University ERIC KLASSEN,∗∗ Florida State University
Abstract We address the problem of tracking the time-varying linear subspaces (of a larger system) under a Bayesian framework. Variations in subspaces are treated as a piecewise-geodesic process on a complex Grassmann manifold and a Markov prior is imposed on it. This prior model, together with an observation model, gives rise to a hidden Markov model on a Grassmann manifold, and admits Bayesian inferences. A sequential Monte Carlo method is used for sampling from the time-varying posterior and the samples are utilized to estimate the underlying process. Simulation results are presented for principal subspace tracking in array signal processing. Keywords: Subspace tracking, Grassmann manifold, Bayesian tracking, nonEuclidean filtering, sequential Monte Carlo. AMS 2000 Subject Classification: Primary 93E11 Secondary 94A11;65C35
1. Introduction Many applications in signal processing, image analysis, computational biology, and environmental statistics involve inferences on large-dimensional, time-varying systems. Since most sophisticated statistical procedures are limited to smaller spaces, a common strategy is to project the observations from larger spaces to low-dimensional subspaces, and then apply the statistical procedures. In view of their simplicity and computational efficiency, the linear projections are commonly used for this purpose. Examples include principal component analysis, independent components, and Fisher’s discriminants. Different linear projections result from different optimality criteria. For example, ∗
Postal address: Department of Statistics, Florida State University, Tallahassee, FL 32306 Postal address: Department of Mathematics, Florida State University, Tallahassee, FL 32306
∗∗
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maximizing the variance of the projected variables results in principal components [8], and minimizing the mutual information between them leads to independent components [2]. Since the original larger system is time-varying, one has to find an optimal projection for each time or, in other words, find a time sequence of optimal projections. Equivalently, one can estimate a time sequence of optimal subspaces using observations from the original system. This task of estimating a sequence of subspaces of a timevarying system is called subspace tracking. To illustrate the problem of tracking principal subspaces, consider a time-varying system taking values in Cn , for a large n. We are interested in tracking m-dimensional principal subspaces of Cn (m ≤ n) using the observed data. Let Yt = [yt,1 yt,2 . . . yt,p ], yt,i ∈ Cn be the set of p (p ≥ m) observations collected in a small interval around time t. The space spanned by the m-dominant eigenvectors of the sample covariance matrix Kt ∈ Cn×n estimates the principal subspace at time t. Here p
ˆt = K
p
1 X 1X (yt,i − y¯t ) (yt,i − y¯t )† , y¯t = yt,i , p − 1 i=1 p i=1
(1)
where † denotes conjugate transpose. Shown in the left panel of Figure 1 is a pictorial illustration of subspace tracking for n = 2 and m = 1. This panel shows observations for times t = 1 in dots and t = 2 in ’+’, and the corresponding estimated one-dimensional subspaces. The goal is to track the rotation of these subspaces as new observations are made at regular intervals. UH
β
H
U U(n)
X I
IP P
α t=1 P
α
∼ α
IP
Q
t=0
t=1 P2
P1
Q t=0
Figure 1: Left panel: estimation of subspaces at two separate times. Middle panel: Finding a geodesic path connecting Q to P on P by lifting it to a specific geodesic path in U (n). Last panel: Finding a geodesic path between P1 and P2 by rotating them to Q and some P ∈ P.
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As an application, we will consider the following problem in array signal processing. An array of electromagnetic sensors, arranged linearly on the ground at uniform spacing, records incident signals (at a known wavelength) from multiple, ground-based, mobile signal transmitters. The goal is to detect and track angular locations of the signal transmitters using data recorded by the array. An important intermediate goal is to track the signal subspace, the subspace spanned by the transmitters, using data collected in noisy environments (see [17] for details). Given a signal subspace, there are efficient algorithms for computing the transmitters’ locations. Assuming that the signal energies exceed the noise energy, and assuming m signal transmitters for n sensors in the array (m < n), m-dimensional principal subspace of the data estimates the signal subspace, and we are interested in tracking it. Pre-processing of signals makes the sensor recording at each time a complex number; the array output at each time is an element of Cn . A statistical model for array data is given later in Section 3.2. Several approaches have been presented in the literature for estimating and tracking subspaces, with applications to signal processing (see [20, 3] and the references therein). While most approaches rely on relating the observations across time (for example using windowing methods) and then estimating the principal subspaces, our focus is on investigating geometric representations and resulting procedures that can handle tracking of arbitrary subspaces. Our ideas are also applicable to tracking on quotient spaces of general (finite-dimensional) Lie groups. Through the introduction of a prior model on the evolution of subspaces, we will treat subspace tracking as a problem in Bayesian inference. The first issue is: On what space should the subspace tracking problem be posed? A natural way to represent subspaces is as elements of a Grassmann manifold, the set of all (fixed-dimensional) subspaces of a larger vector space ([5, 18]). This subspace tracking on Grassmann manifold extends the Bayesian framework introduced in [18] for estimating time-invariant subspaces. A distinct advantage of the Bayesian approach is its ability to estimate the trajectories of subspaces as a whole rather than estimating individual subspaces. This tackles an important issue in subspace tracking that, at any given time, there may not be enough observations for estimating each of the subspaces individually to a required precision, i.e. p (in the definition of Kt ) may be small. Often the observations are too noisy to provide a reliable estimate at each observation time. An obvious choice, in such problems relating to time-series
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estimation, is to utilize a temporal structure or a prior model to treat tracking as Bayesian estimation of an underlying stochastic process. We will follow the approach of [16] where, in the context of tracking airplanes using remote sensing, the Newtonian equations of motion provide a prior model for Bayesian tracking. Such a prior, in effect, imposes a smoothness constraint on the estimated trajectories. This effect has also been achieved directly in [9] by fitting smooth curves for a given number of points on a particular Grassmann manifold (CP1 ). How does one model a stochastic process on a Grassmann manifold? We will model the process as a piecewise geodesic curve, and will impose a Markov prior on the random variables associated with the individual pieces. In particular, geodesic curves help define the notion of velocities, and a stochastic difference equation involving the velocities provides the desired model. Our motivation for such a model is that a stochastic process with smooth sample paths can be easily modeled by a linear equation on its velocities. An observation model provides a likelihood function for completing the Bayesian formulation. In order to define geodesics on a Grassmann manifold, one has to study its intrinsic geometry. Let G be the Grassmann manifold of all m-dimensional complex subspaces of Cn . Any element of this manifold, i.e. any m-dimensional subspace of Cn , can either be represented by its projection operator (uniquely) or by an orthonormal basis (non-uniquely). Choosing the projection matrices to represent subspaces, we will construct geodesics between two arbitrary projection matrices. Since a Grassmannian is a quotient space of a larger unitary group, modulo a subgroup, this geodesic is made explicit by lifting it to a particular geodesic in the unitary group. Furthermore, the tangents to the lifted geodesic curve in the unitary group help define the velocities associated with a curve on Grassmann manifold. A similar construction of geodesics on the well-studied shape spaces is given in [13]. The next task is to develop algorithms for tracking. In view of the inherent nonlinearities present in the model, and the problem formulation on curved manifolds, the classical Kalman-filtering framework does not apply. For Euclidean systems, a number of solutions including the extended Kalman filters, interacting multiple models, multiple hypothesis tests, and their combinations, have been suggested. However, there is little discussion in the literature on non-Euclidean tracking. We take a Monte Carlo
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approach to subspace tracking. This procedure is based on the particle filtering or the sequential Monte Carlo method. It involves sampling from the prior and resampling them according to their likelihoods in order to generate samples from the posterior, at each observation time. These samples are then averaged to estimate the underlying subspaces. This technique is popular for tracking in Euclidean spaces [6, 15], with a related idea presented in [16]. The main contributions of this paper are: (i) Posing subspace tracking as a problem in estimating a stochastic process on a complex Grassmann manifold, (ii) imposing a Markovian prior on the process using geodesic paths and treating subspace tracking as a problem in Bayesian inference, and (iii) applying a sequential Monte Carlo algorithm to sample from the time-varying posterior. In addition to tracking subspaces, this sampling also allows for the estimation of expected errors and other posterior moments for performance diagnostics, as described in [19]. The paper is organized as follows. Section 2 utilizes the intrinsic geometry of a Grassmann manifold to define geodesics, and suggests a prior model on the subspace process. Section 3 presents a Bayesian formulation of subspace tracking and Section 4 describes a sequential Monte Carlo method for generating inferences. Some simulation results, illustrating the algorithm for a particular problem in array signal subspace tracking, are presented in Section 5.
2. Representation of Subspace Trajectories Subspace estimation and tracking are important to many applications. Even though these inference problems are naturally posed on Grassmann manifolds, the use of geometric techniques has only been recent [1, 5, 18] . In this section, we study the intrinsic geometry of a Grassmann manifold and specify geodesics between arbitrary points on this manifold.
A
0A
, 0B B are matrices of zeros with appropriate sizes. For example, 0A has same
For any two matrices A and B, we will use diag(A, B) to denote a matrix where 0A , 0B
number of columns as B andsame number of rows as A. Similarly, define cdiag(A, B) 0A A . This notation generalizes, e.g. diag(A, B, C) implies a to be the matrix B 0B
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block-diagonal matrix with diagonal blocks A, B, and C, and so on. 2.1. Geometry of a Grassmann Manifold Let V be an n-dimensional complex vector space equipped with a Hermitian inner product. Assuming 0 ≤ m ≤ n < ∞, denote by G the Grassmann manifold of all m-dimensional subspaces of V (please refer to [12] p. 133 Ex. 2.4 for a detailed introduction). By fixing m, n throughout the paper, we avoid adding suffixes to index the set G. Using an orthonormal basis (v1 , v2 , . . . , vn ) for V , identify V with Cn , the set of n × 1 column vectors over C. Each element of G can be identified with a unique n × n matrix of orthogonal projection onto that m-dimensional subspace of V . Let P be the set of Hermitian symmetric, idempotent n × n complex matrices of rank m. P is the set of all projection matrices and is diffeomorphic to G; they are compact manifolds of complex dimension m(n − m). The subspace spanned by the vectors (v1 , v2 , . . . vm ) is identified with the projection matrix Q = diag(Im , 0n−m,n−m ) ∈ P, where 0a,b is an a × b matrix of zeros and Im is the m × m identity matrix. Q is fixed throughout the paper. Let U (n) be the Lie group of all n × n complex unitary matrices. Its subgroup H = U (m) × U (n − m) is the set of all matrices of the form diag(Ua , Ub ), where Ua ∈ U (m) and Ub ∈ U (n − m). There is a one-to-one correspondence between the quotient space U (n)/H and the Grassmann manifold G (or P) (see for example [12] p.134). The left coset, containing a point U ∈ U (n), can be explicitly stated as ˜ :U ˜ ∈ H} ⊂ U (n). The correspondence between the left cosets (elements U H = {U U of U (n)/H) and the projection matrices (elements of P) is given by U H 7→ U QU † , for any U ∈ U (n), and the map from U (n) to P is Φ(U ) = U QU † . Under Φ, each left coset maps to a point in P. Denote this coset by Φ−1 (P ); Φ−1 (P ) = U H whenever U QU † = P . An element of Φ−1 (P ) is a unitary matrix, whose first m columns form an orthonormal basis of the subspace whose projection is P . For instance, In ∈ Φ−1 (Q) and Φ−1 (Q) = H. The group U (n) acts on the vector space V (from the left) by the usual matrixvector multiplication. U (n) acts transitively on P from the left, according to the mapping: P 7→ U · P ≡ U P U † , for U ∈ U (n), P ∈ P. The transitive nature of this group action implies that P = {U · Q|U ∈ U (n)}. Φ is invariant to the group ˜ ) = U · Φ(U ˜ ), for all U, U ˜ ∈ U (n). (The left dot denotes the group action: Φ(U · U
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action on U (n) while the right dot denotes the group action on P.) The tangent space of U (n) at identity is u(n), the space of n × n, Hermitian skew-symmetric matrices (see for example [21] p. 107). Let H be a subset of u(n) defined as: H = diag(Ya , Yb )| Ya ∈ Cm×m , Yb ∈ C(n−m)×(n−m) are Hermitian skew-symmetric . Let M be the orthogonal complement of H in u(n): o n M = cdiag(A, −A† ) ∈ Cn×n : A ∈ Cm(n−m) ⊂ u(n) .
(2)
As a compact Lie group, U (n) is equipped with a unique bi-invariant Riemannian metric, which is inherited by P.
On u(n), this metric is just the inner product
hY1 , Y2 i = trace(Y1 Y2† ). Since hU Y1 , U Y2 i = hY1 , Y2 i, for any U ∈ U (n), this metric is invariant to the left translation generated by the group action. 2.2. Geodesics on Grassmann Manifold We will represent a stochastic process on P as a piecewise-geodesic curve with random velocities at individual pieces. Therefore, we need an explicit description of geodesics on P. This is done in two steps: (i) first construct a geodesic between Q and any P ∈ P, and then (ii) construct a geodesic between any two P1 , P2 ∈ P. We start with a specification of geodesics passing through the point Q. Proposition 1. The geodesics in P passing through the point Q (at time t = 0) are of the type α : (−, ) 7→ P,
α(t) = exp(tX) · Q = exp(tX)Q exp(−tX), for some
X ∈ M, where the set M is specified in Eqn. 2. Proof: This proposition is identical to the exercise 2(i) on p.226 in [7]. We sketch a proof here. Let α be the geodesic in P connecting Q (at t = 0) with a point P (at t = 1). Geodesics in P can be made explicit via corresponding geodesics in U (n), since P is identified with the quotient space U (n)/H. The geodesics in U (n), passing through a point U ∈ U (n), are known to be the one-parameter subgroups of the type β(t) = exp(tX) · U for any X ∈ u(n). The geodesic β (in U (n)) projects down to a geodesic α (in P) if and only if β is orthogonal to each coset that it intersects in U (n). On the other hand, invariance of the metric (last line of Section 2.1) implies that if β is orthogonal to one coset, then it is orthogonal to each and every coset it intersects. In particular, if β passes through I (at t = 0), it should be orthogonal to the coset ˙ H (i.e. β(0) ⊥ H). For β(t) = exp(tX) · I, this condition implies that X belongs to
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the orthogonal complement of H in u(n), namely M. Finally, the projection of β to P gives α, using the invariance of Φ, α(t) = Φ(β(t)) = exp(tX)Q exp(−tX). Shown in the middle panel of Figure 1 is an illustration of defining geodesics in P by lifting to a geodesic in U (n) such that they are always orthogonal to cosets in U (n). In this picture, the cosets are denoted by vertical lines. According to Proposition 1, α is completely specified by an X ∈ M such that exp(X)Q exp(−X) = P . Therefore, the problem of finding α becomes: Problem 1: Given a point P ∈ P, find an X ∈ M such that exp(X)Q exp(−X) = P . Note that for this X, exp(X) in addition to being in Φ−1 (P ), is nearest to I among all elements of the set Φ−1 (P ). We start the solution by motivate the case n = 2, m = 1. Example 1: Consider a two-dimensional vector space V . Let (v1 , v2 ) be an ordered orthonormal basis for V and Q = diag(1, 0) be the projection matrix of the subspace spanned by v1 . Let P denote (the projection matrix of) the one-dimensional subspace spanned by cos(α)v1 + sin(α)v2 , for some α > 0. Then, P is given by U (α)QU (α)T or cos(α) − sin(α) cos2 (α) cos(α) sin(α) . , where U (α) = 2 sin(α) cos(α) cos(α) sin(α) sin (α) Eigen decomposition of Q − P is given by W ΣW † , where Σ = diag(sin(α), − sin(α)), and the columns of W are the corresponding eigenvectors. Since Q − P is Hermitian symmetric, W can be taken to be a unitary matrix. We require that Qw1 is a positive real multiple of Qw2 , where w1 , w2 are the two columns of W . This can be achieved simply by multiplying w1 by an appropriate unit complex number (see Remark 1 later). Returning to the task of finding X as per Problem 1, it follows that: X = W ΩW † , and exp(X) = W exp(Ω)W † , where Ω = cdiag(−α, α) .
(3)
In this 2 × 2 case W ΩW † happens to be the same as Ω. This example suggests a role for the eigen decomposition of Q − P in finding X. Theorem 1. For a point P ∈ P, let B = W ΣW † be the eigen decomposition of B = Q − P such that W is a unitary matrix. Then, 1. the eigenvalues of B (or the diagonal entries of Σ) are either 0’s or occur in pairs of the form (λj , −λj ), where 0 < λj ≤ 1. Qwj and Qwj 0 can be chosen
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to be positive real multiples of each other, where wj , wj 0 are the columns of W corresponding to the eigenvalues λj and −λj , respectively, for all j’s. This can be accomplished using the procedure in Remark 2. 2. Let Ω be a n × n matrix derived from Σ in the following way: replace all the 2 × 2 blocks diag(λj , −λj ) by cdiag(− sin−1 (λj ), sin−1 (λj )), with the remaining entries staying zeros. Then, set X to be W ΩW †
∈ M.
˜ is formed from Σ by replacing: (i) ˜ † ∈ U (n), where Ω 3. Set exp(X) = W ΩW the zeros in the diagonal by ones, and (ii) the 2 × 2 blocks diag(λj , −λj ) by q 1 − λ2j −λj . q λj 1 − λ2j 4. A geodesic in P from Q to P is then given by exp(tX)Q exp(−tX) for 0 ≤ t ≤ 1. Proof: Please refer to the Appendix. Remark 1: We require that Qwj be a positive multiple of Qwj 0 for all js. If λj ’s are all distinct, this can be achieved by multiplying wj by the unit complex number: c˜ = c/|c|, where c =
wj 0 (1) wj (1) ,
and wk (1) is the first element of the vector wk . If several
λj ’s are the same, with the corresponding columns {w1 , . . . , ws }, we will alter the columns {w10 , . . . , ws0 } as follows. For each i = 1 . . . , s, there is a unique unit vector yi ∈ span{w10 , . . . , ws0 } with the property that Qyi is a positive real multiple of Qwi . For each such i = 1, . . . , s, replace wi0 by yi . Continue to call the resulting matrix W . This completes the necessary modification of W . We note that for certain points in P (analogous to diametrically opposite points on a sphere), the matrix X, and therefore, the resulting geodesic may not be unique. In the context of subspace tracking, these pairs occur in P × P with zero probability, and hence are ignored. Also, note that the computational cost of calculating X is essentially that of finding the eigen decomposition of Q − P . The next step is to find a geodesic between two arbitrary two points P1 , P2 in P. The basic idea is to rotate these points back to Q and P (for some P ∈ P), respectively and then apply Theorem 1. Let U ∈ Φ−1 (P1 ) (that is, P1 = U QU † ) and define P = U † P2 U . Then, using Theorem 1, we can find an X such that α(t) = exp(tX) · Q is a geodesic from Q to P . Define a shifted geodesic α ˜ according to
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α ˜ (t) = (U exp(tX)U † )P1 (U exp(tX † )U † ); α ˜ (t) is the desired geodesic in P such that ˜ = P2 . The right panel in Figure 1 shows this idea pictorially. X α ˜ (0) = P1 and α(1) is dependent on the choice of U but the matrix U exp(X)U † is not. In tracking, we make an arbitrary choice for U at the track initiation and the remaining U s are fixed accordingly. Recall that X, as described above, is an element of M (M is defined in Eqn. 2) and therefore has only m(n−m) complex degrees of freedom in the form of the submatrix A in the upper-right corner of X. We define A as the velocity that takes P1 to P2 in unit time. Conversely, for a point P1 ∈ P, and a given m × (n − m) complex matrix A, we can find the point P2 ∈ P, that is reached in unit time by following a geodesic starting at P1 and having the velocity A. This can be accomplished as follows. Let U be any element of Φ−1 (P1 ). First, form an n × n Hermitian skew-symmetric matrix according to X = cdiag(−A, A† ), compute V = U exp(X)U † , and then set P2 = V P1 V † . Using the structure of X, exp(X) can be computed efficiently in order O(nm2 ) computations. Also, note that U 0 = U exp(X) is an element of Φ−1 (P2 ).
3. Bayesian Formulation of Subspace Tracking In this section, we describe a prior model and an observation model to setup the Bayesian tracking problem. The prior is a Markov process on P generated by i.i.d increments and the observations are generated by multivariate complex normal distributions parameterized by the subspace process. 3.1. Prior Model on Subspace Process Now that we have tools for computing geodesics and velocities, we are ready to state a model that governs the evolution of subspaces on P. As stated earlier, we will interpolate smooth paths by piecewise-geodesic curves; each curve is completely specified by an initial point P1 ∈ P (with a choice of U1 ∈ Φ−1 (P1 )) and a sequence of velocities A1 , A2 , . . . , At ∈ Cm(n−m) (assuming equally-spaced observation times). Let {Pt : t = 1, . . . } be a discrete process in P. For each pair (Pt−1 , Pt ), t = 2, . . . , let At−1 be the corresponding velocity. We will adopt a constant velocity model: At = At−1 + Nt−1 , t = 2, 3, . . . ,
(4)
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where Nt−1 is a m × (n − m) matrix of i.i.d complex normals (real, imaginary parts are i.i.d normal with mean zero and variance σp2 ). σp is the deviation of At , away from a given value of At−1 and can be estimated from the past trajectories. In a Markovian time-series analysis, there is often a characterization of a timevarying posterior density, in a convenient recursive form. This characterization involves a hidden Markov chain, with given transition densities, and an observation sequence, with a given observation model. To obtain Markovity, we consider the chain on the joint space of subspaces and velocities. Define the subspace-velocity pair Jt = (Pt , At−1 ) ∈ (P × Cm(n−m) ), for each time t. Jt is a discrete-time Markov process. For the purpose of defining velocities At ’s, we will keep track of the corresponding Ut ’s in Φ−1 (Pt )’s. (Note that given any two of the Pt , Pt−1 , and At−1 , the remaining third is completely determined.) This setup leads to the following transition density: f (Jt |Jt−1 )
= f (At−1 |At−2 )f (Pt |At−1 , Pt−1 , At−2 ) = f (At−1 |At−2 )δPt0 (Pt )
† † and Vt−1 = Ut−1 exp(Xt−1 )Ut−1 , Xt−1 = cdiag(−At−1 , A†t−1 ) , where Pt0 = Vt−1 Pt−1 Vt−1
for any Ut−1 ∈ Φ−1 (Pt−1 ). The conditional density f (At−1 |At−2 ) follows from Eqn. 4 and δP1 (P2 ) denotes a delta dirac function on P centered at P2 . For the next time step, set Ut = Ut−1 exp(Xt−1 ) ∈ Φ−1 (Pt ). The following algorithm specifies a procedure to sample from this Markov model: (i)
Algorithm 1. For some t = 2, 3, . . . , we are given the values for Jt−1 and points (i)
(i)
Ut−1 ∈ Φ−1 (Pt−1 ). For i = 1, 2, . . . , M : (i)
(i)
1. Generate a sample of At−1 , given At−2 , according to Eqn. 4. (i)
(i)
(i)
(i)
(i)
2. For each sample of At−1 , set Xt−1 = cdiag(−At−1 , (At−1 )† ), and calculate Pt (i)
according to Pt
(i)
(i)
(i)
(i)
(i)
(i)
(i)
= Vt−1 Pt−1 (Vt−1 )† , for Vt−1 = Ut−1 exp(Xt−1 )(Ut−1 )† . (i)
3. Define the sampled subspace-velocity pair Jt
(i)
(i)
(i)
= (Pt , At−1 ). Set Ut
(i)
(i)
= Ut−1 exp(Xt−1 ).
3.2. Observation Model In principle, this framework allows for any density function relating the hidden Markov chain {Jt } to the observed data {Yt }. Different choices of densities will lead to different specifications of subspaces. In the case of array signal processing and
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principal subspace tracking, the sensor observations are modeled as superpositions of signals received from multiple transmitters and the ambient noise [17]. Let there be n sensors and m transmitters (m ≤ n), and let the angular locations of the transmitters be denoted by θt = [θ1,t , . . . , θm,t ] ∈ [0, π]m . Then, the observation vector is modeled by the equation: yt,i = D(θt )st + ct,i ∈ Cn , i = 1, 2, . . . , p, where D(θt ) = [d(θ1,t ), . . . , d(θm,t )] ∈ Cn×m , for d(θ) = [1 exp(−jφ) . . . exp(−j(n − 1)φ)]T , and φ = π cos(θ). st is the m-vector of signal amplitudes and ct,i is a vector of ˆ t is the sample covariance, as defined in Eqn. 1, i.i.d. complex normal noise. If K the likelihood function is given by f (Yt |Jt ) =
1 Lt
ˆ t Pt ))), where Lt is exp( σ12 (trace(K
the normalizer. Also note that the m-dimensional signal subspace at time t has the projection matrix given by: Pt = D(θt )(D(θt )† D(θt ))−1 D(θt )† ∈ P.
4. Bayesian Subspace Tracking Given a prior model on the sequence {Jt } and an observation model on {Yt }, we can pose the Bayesian subspace tracking problem and propose a solution. 4.1. Problem Formulation For discrete observation times t = 1, 2, . . . , let the trajectory of subspaces be P1 , P2 , · · · ∈ P, and let the observation sequence be Y1 , Y2 , · · · ∈ Cnp . We are interested in solving the following problem: Given the observation sequence Y1:t = {Y1 , . . . , Yt }, find the Bayes’ estimate of the sequence P1:t = {P1 , . . . , Pt } ∈ Pt . As t increases, the underlying parameter space (Pt ) grows and the joint posterior, on P1 , . . . , Pt , changes at each time as the new observation is recorded. Solving for the joint posterior mean at each time is difficult. Recent papers [6, 15, 4], describe an efficient procedure, called particle filtering or sequential Monte Carlo method, to solve such problems. This procedure is greedy in that it restricts estimation to only the last time t and utilizes a Monte Carlo technique to sample from the posterior on Pt . Previous estimates (Pˆ1 , . . . , Pˆt−1 ) remained unchanged in the estimation steps performed at time t. The Monte Carlo idea is to approximate the posterior density of Pt by a large number of samples drawn from it, and then estimate Pt using the sample means. There are at least two ways of defining a “mean” value on a Riemannian manifold.
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1. Extrinsic Mean: First, called the extrinsic mean, involves embedding the manifold in a vector space, computing the Euclidean mean in that space, and then projecting it down to the manifold [19]. As described in [18, 19], the extrinsic mean of Pt is given ˆt Q(U ˆt )† , where by Pˆt = U ˆt Λ(U ˆt )† is singular value decomposition of Gt = Gt = U
Z P
Pt f (Pt |Y1:t )γ(dPt ) , (5)
and where f (Pt |Y1:t ) is the posterior. Here γ is the invariant measure on P. Using Monte Carlo samples one can approximate this integral and estimate extrinsic mean using SVD of the mean matrix. In this paper, we have used the extrinsic means to estimate subspaces. 2. Intrinsic Mean: A more intrinsic definition, called the Karcher mean, has been used in [10, 11]. Let d(P1 , P2 ) be the length of the geodesic curve from P1 to P2 . To define the Karcher mean under the posterior density f (Pt |Y1:t ), define the variance R function V : P → R by V (P ) = P d(P, Pt )2 f (Pt |Y1:t )γ(dPt ). Then, define the Karcher mean to be any point Pˆt ∈ P for which V (Pˆt ) is a local minimum. An iterative algorithm given in [14] can be used for finding the Karcher mean on P. A major strength of these Monte Carlo methods is that they approximate the whole density, and not just a few moments. As a consequence, using the samples, one can estimate any higher order statistics under the posterior density f (Pt |Y1:t ). Computational efficiency of the sequential Monte Carlo methods comes from the recursion that takes samples from the posterior density of Pt−1 and generates samples from the posterior density of Pt . Bayesian filtering equations are given by: Z f (Jt |Y1:t−1 )
=
f (Jt |Y1:t )
=
P×Cm(n−m)
f (Jt |Jt−1 )f (Jt−1 |Y1:t−1 )γ(dJt−1 ) ,
f (Yt |Jt )f (Jt |Y1:t−1 ) . f (Yt |Y1:t−1 )
(6) (7)
Eqn. 6 is called the prediction equation and Eqn. 7 is called the update equation. The denominator in Eqn. 7 is difficult to compute and, for a given observation set, is a constant. In a Monte Carlo approach the normalizing constant need not be explicitly evaluated. The relationship between Eqns. 6 and 7 suggests a recursive form for the solutions derived from the posteriors f (Jt−1 |Y1:t−1 ) and f (Jt |Y1:t ). Given samples from f (Jt−1 |Y1:t−1 ) it is possible to efficiently generate samples from f (Jt |Y1:t ), instead
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of directly sampling from f (Jt |Y1:t ), which may be complicated and computationally expensive. f (Jt |Jt− ) is given in Section 3.1 and f (Yt |Jt ) is given in Section 3.2. The algorithm will be based on: (i) samples from the prior f (Jt |Jt−1 ) for a given value of Jt−1 , and (ii) the functional form of the density function f (Yt |Jt ). 4.2. Sequential Monte Carlo Approach A recursive formulation, which takes samples from f (Jt−1 |Y1:t−1 ) and generates the samples from f (Jt |Y1:t ) in an efficient fashion, is desirable. Assume that, at the (i)
observation time t − 1, we have a set St−1 = {Jt−1 : i = 1, 2, . . . , M } ,
(i)
Jt−1 ∼
f (Jt−1 |Y1:t−1 ). Following are the steps to generate the set St . 1.
Prediction: The first step is to sample from f (Jt |Y1:t−1 ) given the samples
from f (Jt−1 |Y1:t−1 ). We take a compositional approach by treating f (Jt |Y1:t−1 ) as a mixture density. According to Eqn. 6, f (Jt |Y1:t−1 ) is the integral of the product of (i)
a marginal and a conditional density. This implies that, for each element Jt−1 ∈ St−1 , (i)
by generating a sample from the conditional, f (Jt |Jt−1 ), we can generate a sample (i)
from f (Jt |Y1:t−1 ) (see Algorithm 1). Now we have samples {J˜t } from f (Jt |Y1:t−1 ); analogous to Kalman-filtering these samples are called predictions. 2. Resampling: Given these predictions, the next step is to generate samples from the posterior f (Jt |Y1:t ). For this, we utilize importance sampling as follows. The samples from the prior (f (Jt |Y1:t−1 )) are resampled (see reference [15]) according to (i)
the probabilities that are proportional to the likelihoods f (Yt |J˜t ). Form a discrete (i)
probability mass function on the set {J˜t
: i = 1, 2, . . . , M }:
(i)
f (Yt |J˜t ) , and set βt = [βt,1 βt,2 . . . βt,M ] . βt,i = PM ˜(j) j=1 f (Yt |Jt )
(8)
(1) (2) (M ) Then, resample M values from the set {J˜t , J˜t , . . . , J˜t } according to probability
βt . These values are the desired samples from the posterior f (Jt |Y1:t ). Denote the (i)
resampled set by St = {Jt
: i = 1, 2, . . . , M },
(i)
Jt
∼ f (Jt |Y1:t ). It must be
remarked that after resampling, the indices (i) are renamed so that the sequence (i)
(i)
(i)
Jt−1 , Jt , Jt+1 , . . . , for the same i, may not be consistent anymore. In other words, (i)
(i)
(i)
it is possible that the velocity At−1 does not take Pt−1 to Pt
in a unit time. This
inconsistency has no bearing on the estimation procedure since the past samples are not used in estimating future parameters, only the current samples are used.
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3. Mean Estimation: Now that we have M samples from the posterior f (Jt |Y1:t ), we can average them appropriately to approximate the posterior mean of Pt . As described in the paper [18], the extrinsic mean estimate of Pt is given by Eqn. 5. Using Monte Carlo sampling, we approximate Gt by M X (i) ˆ t,M = 1 P ∈ Cn×n , G M i=1 t
ˆ t,M to obtain Pˆt,M . and compute SVD of G
(9)
In numerous papers, the ergodic properties of sequential Monte Carlo samples have been studied. It has been shown that the elements of the set St are exact samples from the posterior and the ergodic property (that is, sample averages converge to the expected values as the sample size gets larger) holds. It should be noted that due to the resampling step, the resulting samples are not independent any more. Error Analysis: There are two sources of error: (i) a sampling error in estimating ˆ t,M (this error is estimated using the sample variance Gt by a finite sample mean G [19]), and (ii) there is a difference between the underlying true value Pt and its exact posterior mean (this error is quantified using Hilbert-Schmidt lower bounds [18] that are achieved by the estimator defined in Eqn. 5). 4.3. Subspace Tracking Algorithm (i)
Given the samples {Jt−1 : i = 1, 2, . . . , M } ∼ f (Jt−1 |Y1:t−1 ), the following steps generate samples from posterior at time t and estimate Pˆt,M . (i)
Algorithm 2. 1. Sample Conditional: Draw {J˜t , i = 1, . . . , M } from the conditional prior according to Algorithm 1. (i)
2. Importance Weights: Compute probabilities βt , i = 1, . . . , M , using Eqn. 8. (i)
3. Resampling: Generate M samples from the set {J˜t , i = 1, . . . , M } with proba(i)
(i)
bilities {βt , i = 1, . . . , M }. Denote these samples by {Jt , i = 1, . . . , M }. ˆ t,M according to Eqn. 9 and 4. Mean Estimation: Calculate the sample average G ˆ t,M . Set t ← t + 1 and go to step 1. compute the estimate Pˆt,M using the SVD of G
5. Simulation Results Now we present some experimental results on subspace tracking. Consider the problem of subspace estimation using a uniform linear array of sensors with model as
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stated in Section 3.2. For these experiments, the transmitter motion is generated using a simple auto-regressive model. The lower panels of Figure 2 show the trajectories of two signal transmitters (m = 2). For each θt = [θ1,t , θ2,t ], we generated yi,t for ˆt n = 4 according to the sensor model and computed the sample covariance matrix K according to Eqn. 1. The tracking algorithm then estimates Pˆt,M for M = 500 at each time t. In Figure 2, each top panel shows the estimation error kPt − Pˆ k for three different estimation procedures. First, the error associated with the instantaneous maximumˆ t , is shown in likelihood estimate (MLE), obtained by SVD of the covariance matrix K the broken line. The error resulting from an adaptive procedure, relying on the SVD ˆ t−1 , is shown in the dotted line (for γ = 0.3). Note of the matrix Rt = γKt + (1 − γ)K that this choice of γ is arbitrary and the literature provides techniques for selecting a better γ. Finally, the estimation error for tracking resulting from Algorithm 2 is plotted in bold. Since the prior is based on a velocity model, MLEs at t = 1, 2 are used to initialize the algorithm and Bayes’ inference starts at t = 3. As the theory suggests, a prior model on subspace motion improves tracking performance in the presence of intermittent noise. Algorithm 2 estimates only the current state and gains in speed by not improving upon the past estimates. A slower algorithm for joint estimation of all the states, using all the observations, is given in [16].
6. Summary In this paper, we have proposed a Bayesian approach to tracking principal subspaces using observations taken from time-varying systems. A prior model, on stochastic process on a Grassmann manifold, is used to formulate the Bayesian problem. A recursive, computational technique to sample from the posterior and to generate Bayesian estimates is described.
Acknowledgements This research was supported by ARO DAAG55-98-1-0102, ARO DAAD19-99-1-0267 and NSF DMS0101429. We thank the anonymous referee for helpful comments.
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Estimation Performance
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Figure 2: Upper panels plot the error (kPˆt − Pt k) versus t for: (i) MLE (broken line), (ii) adaptive tracking (dotted line), and (iii) Bayesian tracking (solid line). Lower panels show the corresponding transmitter trajectories.
References [1] Bucy, R. S. (1991). Geometry and multiple direction estimation. Information Sci. 57-58, 145–58.
[2] Comon, P. (1994). Independent component analysis, a new concept?
Signal
Processing, Special issue on higher-order statistics 36,.
[3] Delmas, J. P. and Cardoso, J. F. (1998). Performance analysis of an adaptive algorithm for tracking dominant subspaces. IEEE transactions on signal processing 46, 3045–3057.
[4] Doucet, E. A., de Freitas, N. and Gordon, N. (2001). Sequential Monte Carlo Methods in Practice. Springer.
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[5] Edelman, A., Arias, T. and Smith, S. T. (1998). The geometry of algorithms with orthogonality constraints. SIAM Journal of Matrix Analysis and Applications 20, 303–353. [6] Gordon, N. J., Salmon, D. J. and Smith, A. F. M. (1993). A novel approach to nonlinear/non-gaussian bayesian state estimation. IEEE Proceedings on Radar Signal Processing 140, 107–113. [7] Helgason, S. (1978). Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press. [8] Jolliffe, I. T. (1986). Principal component analysis. Springer series in statistics. Springer-Verlag. [9] Jupp, P. E. and Kent, J. T. (1987). Fitting smooth paths to spherical data. Applied Statistics 36, 34–46. [10] Karcher, H. (1977).
Riemann center of mass and mollifier smoothing.
Communications on Pure and Applied Mathematics 30, 509–541. [11] Kendall, W. S. (1990). Probability, convexity, and harmonic maps with small image I: Uniqueness and fine existence. Proceedings of the London Mathematical Society 61, 371–406. [12] Kobayashi, S. and Nomizu, K. (1969). Foundations of Differential Geometry, vol 2. Interscience Publishers. [13] Le, H. (1991). On geodesics in euclidean shape spaces. J. Lond. Math. Soc. 44, 360–372. [14] Le, H. (2001). Locating frechet means with application to shape spaces. Advances in Applied Probability 33, 324–338. [15] Liu, J. S. and Chen, R. (1998). Sequential monte carlo methods for dynamic systems. Journal of the American Statistical Association 93, 1032–44. [16] Miller, M. I., Srivastava, A. and Grenander, U. (1995). Conditionalexpectation estimation via jump-diffusion processes in multiple target tracking/recognition. IEEE Transactions on Signal Processing 43, 2678–2690.
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[17] Schmidt, R. (Nov. 1981). A signal subspace approach to multiple emitter location and spectral estimation. Ph.D. Dissertation of Stanford University, Palo Alto, CA. [18] Srivastava, A. (2000). A bayesian approach to geometric subspace estimation. IEEE Transactions on Signal Processing 48, 1390–1400. [19] Srivastava, A. and Klassen, E. (2001). Monte carlo extrinsic estimators for manifold-valued parameters. IEEE Trans. on Signal Processing 50, 299–308. [20] Tong, L. and Perreau, S. (1998).
Multichannel blind estimation: From
subspace to maximum likelihood methods. Proc. of the IEEE 86, 1951–1968. [21] Warner, F. W. (1994). Foundations of Differentiable Manifolds and Lie Groups. Springer-Verlag, New York.
A. Proof of Theorem 1 ˜ be two m-dimensional subspaces of V , with projection matrices P and Let P˜ and Q Q, respectively. We will prove the theorem in three steps: (i) prove the case n = 2 ˜ for a and m = 1, (ii) show that if there exists a basis of V such that P˜ = exp(Ω)Q specific Ω ∈ M, then the theorem is just an extension of the n = 2, m = 1 case, and (iii) for any given P˜ , show that there exists a basis of V such that the requirements of the second step are met. ˜ ⊥ P˜ and Q ˜ = P˜ since they are easy 1. Let n = 2 and m = 1, and rule out the cases Q ˜ (kv1 k = 1), we have P v1 ∈ P˜ . Let w1 be the unit vector in P˜ to handle. For v1 ∈ Q such that w1 · P v1 > 0, and let α1 be the (positive) angle between v1 and w1 . As shown in Example 1, Q − P can be decomposed as W diag(λ1 , −λ1 ) W † for λ1 = sin(α1 ). The resulting X ∈ M and exp(X) ∈ U (n) are given in Eqn. 3. 2. For arbitrary P ∈ P, we will essentially decompose V as orthogonal direct sum of ˜ to P˜ . Part 1 will apply two-dimensional subspaces to obtain the best rotation from Q independently to each two-dimensional component. Let there be an orthonormal basis of V of the form (u1 , . . . , uk , v1 , . . . , vr , w1 , . . . , wr , x1 , . . . , xp )
(10)
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Srivastava and Klassen
where k, r, p are three nonnegative integers such that k + 2r + p = n and k + r = m. ˜ For α1 , . . . , αr ∈ R+ , Also, let (u1 , . . . , uk , v1 , . . . , vr ) be an orthonormal basis of Q. define an element of u(n) by Ω = diag(0k,k , B, 0p,p ) ∈ Rn×n , where B = cdiag(−C, C) ˜ We can also write and C = diag(α1 , . . . , αr ). Define a subspace P˜ = exp(Ω)Q. P˜ = span{u1 , . . . , uk , cos(α1 )v1 + sin(α1 )w1 , . . . , cos(αm )vm + sin(αm )wm }. (u1 , . . . uk ) ˜ ⊥ ∩ P˜ ⊥ . With respect ˜ ∩ P˜ ), and (x1 , . . . , xp ) is a basis of the space Q is a basis of (Q ˜ to P˜ into a sequence to the basis given in Eqn. 10, we can factor the rotation from Q of 2 × 2 rotations in 2-planes orthogonal to each other. The planes are spanned by vj , wj and the rotation angles are αj ’s. The results from Part 1 apply to each 2 × 2 rotation independently. Therefore, the eigen decomposition of Q − P takes the form: W diag(0k,k , B, −B, 0p,p ) W † where B = diag(λ1 , . . . , λr ) and λj = sin(αj ). Using the 2 × 2 example, X = W ΩW † is the desired X ∈ M, and exp(X) = W exp(Ω)W † . ˜ and P˜ Therefore, if there exists a basis of V of the type given in Eqn. 10 such that Q can be written in these specific forms, the result follows from two-dimensional analysis. Note that Ω here is a permutation of the Ω stated in Theorem 1. This is not an issue as long as the columns of W are also permuted accordingly. 3. Next, we show that for any P ∈ P, there exists an orthonormal basis of V of the ˜ ∩ P˜ ) and p = dim(Q ˜ ⊥ ∩ P˜ ⊥ ). Choose any form given in Eqn. 10. Let k = dim(Q ˜ ∩ P˜ and {x1 , . . . , xp } for Q ˜ ⊥ ∩ P˜ ⊥ . orthonormal bases {u1 , . . . , uk } for Q ˜ P˜ )⊕(Q+ ˜ P˜ )⊥ , Since all rotation will take place in the orthogonal complement of (Q∩ ˜ ∩ P˜ ) ⊕ we now replace V by the orthogonal complement in V of the subspace (Q ˜ and Q ˜ by the orthogonal ˜ + P˜ )⊥ , P˜ by the orthogonal complement in P˜ of P˜ ∩ Q, (Q ˜ of P˜ ∩ Q. ˜ Now, dim(P˜ ) = dim(Q) ˜ = r and dim(V ) = 2r; P˜ ∩ Q ˜=0 complement in Q ˜ respectively and they span V . Let SP and SQ denote the unit spheres in P˜ and Q, and let = inf{|v − w| : v ∈ SQ and w ∈ SP }. Since SP and SQ are disjoint compact sets, > 0 and there exist vectors v ∈ SQ and w ∈ SP satisfying |v − w| = . We now ˜ and P˜ as follows: Let v1 = v, and let w1 be the unique unit choose basis elements of Q vector in the real span of {v, w} such that (1) w1 ⊥ v and (2) w1 · P v > 0. To construct ˜ and V by the orthogonal complement the rest of the basis, inductively replace P˜ , Q, of span(v, w) in each of them, and repeat this step to find (v2 , w2 ), . . . , (vr , wr ).
