Matrix Rank-One Decomposition. and Applications

Transcription

1 Matrix rank-one decomposition and applications 1 Matrix Rank-One Decomposition and Applications Shuzhong Zhang Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong ICNPA, Beijing April 8, 2008

2 Matrix rank-one decomposition and applications 2 Outline Trust Region Subproblems Radar Code Selection The Matrix Rank-One Decomposition Theoretical Applications

3 Matrix rank-one decomposition and applications 3 Trust Region Subproblem The Trust-Region Subproblem: minimize x T Q 0 x 2b T 0 x subject to x δ.

4 Matrix rank-one decomposition and applications 4 The CDT Trust Region Subproblem The CDT (Celis, Dennis, Tapia, 1985) Trust-Region Subproblem: minimize x T Q 0 x 2b T 0 x subject to Ax b δ 1 x δ 2.

5 Matrix rank-one decomposition and applications 5 The Radar Code Selection Problem (Based on De Maio, De Nicola, Huang, Z., Farina, 2007) A radar system transmits a coherent burst of pulses s(t) = a t u(t) exp (i(2πf 0 t + φ)) a t is the transmit signal amplitude; u(t) = N 1 k=0 a(k)p(t kt r) is the signal s complex envelope; p(t) is the signature of the transmitted pulse, and T r is the Pulse Repetition Time (PRT); [a(0), a(1),..., a(n 1)] C N is the radar code (assumed without loss of generality with unit norm); f 0 is the carrier frequency, and φ is a random phase.

6 Matrix rank-one decomposition and applications 6 The Output The filter output is v(t) = α r e i2πf 0τ N 1 k=0 a(k)e i2πkf dt r χ p (t kt r τ, f d ) + w(t) where χ p (λ, f) is the pulse waveform ambiguity function χ p (λ, f) = + p(β)p (β λ)e i2πfβ dβ and w(t) is the down-converted and filtered disturbance component.

7 Matrix rank-one decomposition and applications 7 Sampling The signal v(t) is sampled at t k = τ + kt r, k = 0,..., N 1, the output becomes v(t k ) = αa(k)e i2πkf dt r χ p (0, f d ) + w(t k ), k = 0,..., N 1 where α = α r e i2πf 0τ. Denote c = [a(0), a(1),..., a(n 1)] T, p = [1, e i2πf dt r,..., e i2π(n 1)f dt r ] T (the temporal steering vector) w = [w(t 0 ), w(t 1 ),..., w(t N 1 )] T the backscattered signal can be written as v = αc p + w where denotes the Hadamard product.

8 Matrix rank-one decomposition and applications 8 Performance, Doppler Accuracy, and Similarity The Optimal Code Design Problem can be formulated as maxc c H Rc s.t. c H c = 1 c H R 1 c δ a c c 0 2 ɛ where R = Γ 1 (p H p) with Γ = E[ww H ], and R 1 = Γ 1 (pp H ) (uu H ) with u = [0, i2π,..., i2π(n 1)] T.

9 Matrix rank-one decomposition and applications 9 Commonalities Non-Convex Quadratically Constrained Quadratic Optimization (QCQP), in real and/or complex variables, with a few constraints.

10 Matrix rank-one decomposition and applications 10 Matrix Rank-One Decomposition Theorem (Sturm and Z.; 2003). Let A S n. Let X S n + with rank r. There exists a rank-one decomposition for X such that X = r x i x T i i=1 and x T i Ax i = A X r, i = 1,..., r.

11 Matrix rank-one decomposition and applications 11 Can we do more? It is easy to show by example that in general it is only possible to get a complete rank-one decomposition with respect to one matrix. But it is possible to get a partial decomposition for two: Theorem (Ai and Z.; 2006). Let A 1, A 2 S n and X S n +. If r := rank(x) 3 then one can find in polynomial-time (real-number sense) a rank-one decomposition for X, X = x 1 x T 1 + x 2 x T x r x T r, such that A 1 x i x T i A 2 x i x T i = A 1 X r, i = 1,..., r = A 2 X r, i = 1,..., r 2.

12 Matrix rank-one decomposition and applications 12 The Hermitian case Theorem (Huang and Z.; 2006). Let A 1, A 2 H n, and X H n + with rank r. There exists a rankone decomposition for X such that X = r x i x H i i=1 and x H i A kx i = A k X r, i = 1,..., r; k = 1, 2.

13 Matrix rank-one decomposition and applications 13 Analog in the Hermitian case Theorem (Ai, Huang and Z.; 2007). Suppose that A 1, A 2, A 3 H n and X H n +. If r = rank(x) 3, then one can find in polynomial-time (real-number sense) a rankone decomposition for X, X = r x i x H i, i=1 such that A 1 x i x H i = δ 1 /r, A 2 x i x H i = δ 2 /r, for all i = 1,..., r; A 3 x i x H i = δ 3 /r, for i = 1,..., r 2.

14 Matrix rank-one decomposition and applications 14 Solving QP by Matrix Decomposition Quadratically Constrained Quadratic Programming (QCQP): (Q) minimize q 0 (x) = x H Q 0 x 2Re b H 0 x subject to q i (x) = x H Q i x 2Re b H i x + c i 0, i = 1,..., m.

15 Matrix rank-one decomposition and applications 15 SDP Relaxation Let M(q 0 ) := 0 bh 0 b 0 Q 0, M(q i ) := c i b H i b i Q i, for i = 1,..., m. Then, (Q) is equivalently written as (Q) min M(q 0 ) t x t x s.t. M(q i ) t t x x H H = x H Q 0 x 2Re b H 0 x t = x H Q i x 2b H i x t + c i t 2 0, i = 1,..., m t 2 = 1.

16 Matrix rank-one decomposition and applications 16 SDP Relaxation The so-called SDP relaxation of (Q) is where I 00 = (SP ) minimize M(q 0 ) X subject to M(q i ) X 0, i = 1,..., m I 00 X = 1 X 0 X rank one H n+1. The dual problem of (SP ) is: (SD) maximize y 0 subject to Z = M(q 0 ) y 0 I 00 + m i=1 y im(q i ) 0 y i 0, i = 1,..., m.

17 Matrix rank-one decomposition and applications 17 Complementary Slackness Under suitable conditions, (SP ) and (SD) have complementary optimal solutions, X and Z : X Z = 0. If we can decompose X into rank-one summations, evenly satisfying all the constraints, then each of the rank-one vectors will be optimal!

18 Matrix rank-one decomposition and applications 18 Consequences of the Matrix Decomposition Theorems Polynomially solvable cases of the nonconvex quadratic programs: Real quadratic program: m = 1 (m = 2 if homogeneous) = (Sturm & Z., 2003) Real quadratic program: m = 2 (m = 3 if h.) rank(x ) 3 = (Ai & Z., 2006) Complex quadratic program: m = 2 (m = 3 if h.) = (Huang & Z., 2005) Complex quadratic program: m = 3 (m = 4 if h.) rank(x ) 3 = (Ai, Huang & Z., 2007)

19 Matrix rank-one decomposition and applications 19 Further Theoretical Applications Field of Values of a Matrix Let A be any n n matrix, the field of values of A is given by F(A) := {z H Az z H z = 1} C. This set, like the spectrum set, contains a lot of information about the matrix A. The set is known to be convex. Reference: R.A. Horn and C.R. Johnson. Topics in Matrix analysis. Cambridge University Press, Cambridge, 1991.

20 Matrix rank-one decomposition and applications 20 Joint Numerical Ranges In general, the joint numerical range of matrices is defined to be z H A 1 z F(A 1,..., A m ) :=. z H z = 1, z C n. z H A m z Theorem (Hausdorff; 1919). If A 1 and A 2 are Hermitian, then F(A 1, A 2 ) is a convex set.

21 Matrix rank-one decomposition and applications 21 A Theorem of Brickman Theorem (Brickman; 1961). Suppose that A 1, A 2, A 3 are n n Hermitian matrices. Then z H A 1 z z H A 2 z z C n z H A 3 z is a convex set.

22 Matrix rank-one decomposition and applications 22 The S-Procedure It is often useful to consider the following implication G 1 (x) 0, G 2 (x) 0,..., G m (x) 0 = F (x) 0. A sufficient condition is: τ 1 0, τ 2 0,..., τ m 0 such that F (x) m τ i G i (x) 0 x. i=1 This procedure is called lossless if the above condition is also necessary.

23 Matrix rank-one decomposition and applications 23 The S-Lemma Theorem (Jakubovic; 1971). Suppose that m = 1, and F, G 1 are real quadratic forms. Moreover, there is x 0 R n such that x T 0 G 1 x 0 > 0. Then the S-procedure is lossless. Theorem (Jakubovic; 1971). Suppose that m = 2, and F, G 1, G 2 are Hermitian quadratic forms. Moreover, there is x 0 C n such that x H 0 G i x 0 > 0, i = 1, 2. Then the S-procedure is lossless.

24 Matrix rank-one decomposition and applications 24 Proof of the S-Lemma: The Hermitian case We need only to show that the S-procedure is lossless in this case. Let G i (x) = x H A i x, i = 1, 2, and F (x) = x H A 3 x. Consider the following cone x H A 1 x x H A 2 x x H A 3 x x C n. It is a convex cone in R 3 by Brickman s theorem. Moreover, it does not intersect with R ++ R ++ R.

25 Matrix rank-one decomposition and applications 25 Proof of the S-Lemma (continued) By the separation theorem, there is (t 1, t 2, t 3 ) 0, such that t 1 x 1 + t 2 x 2 + t 3 x 3 0, x 1 > 0, x 2 > 0, x 3 < 0, and t 1 x H A 1 x + t 2 x H A 2 x + t 3 x H A 3 x 0, x C n. The first condition implies that t 1 0, t 2 0, and t 3 0. We see that t 3 > 0 in this case, and so A 3 t 1 t 3 A 1 t 2 t 3 A 2 0.

26 Matrix rank-one decomposition and applications 26 But how to prove Brickman s theorem? Clearly, it will be sufficient if we can show z H A 1 z z H A 2 z z C n = z H A 3 z A 1 Z A 2 Z A 3 Z Z 0

27 Matrix rank-one decomposition and applications 27 Proof of the Brickman Theorem Take any nonzero vector v 1 v 2 v 3 = A 1 Z A 2 Z A 3 Z. Suppose that v 3 0. Consider two matrix equations ( ) A 1 v 1 v 3 A 3 Z = 0 ( ) A 2 v 2 v 3 A 3 Z = 0

28 Matrix rank-one decomposition and applications 28 Proof of the Brickman Theorem (continued) Using our decomposition, there will be Z = z H i z H i r z i zi H i=1 ( ) A 1 v 1 v 3 A 3 z i = 0 ( ) A 2 v 2 v 3 A 3 z i = 0 such that for i = 1,..., r. Among these, there will be one vector such that z H i A 3z i has the same sign as A 3 Z. Let ρ := v 3 /z H i A 3z i, and z := ρz i. Then, z H A 3 z = ρ 2 z H i A 3 z i = v 3, z H A k z = v k v 3 z H A 3 z = v k, k = 1, 2.

29 Matrix rank-one decomposition and applications 29 A Result of Yuan Theorem (Yuan; 1990). Let A 1 and A 2 be in S n. If max{x T A 1 x, x T A 2 x} 0 x R n then there exist µ 1 0, µ 2 0, µ 1 + µ 2 = 1 such that µ 1 A 1 + µ 2 A 2 0.

30 Matrix rank-one decomposition and applications 30 Extension Theorem (Ai, Huang, Zhang; 2007). Let A 1, A 2, A 3 be in H n. If max{z H A 1 z, z T A 2 z, z T A 3 z} 0 z C n then there exist µ 1, µ 2, µ 3 0, µ 1 + µ 2 + µ 3 = 1 such that µ 1 A 1 + µ 2 A 2 + µ 3 A 3 0.

31 Matrix rank-one decomposition and applications 31 Further Extension Theorem (Ai, Huang, Zhang; 2007). Suppose that n 3, A i H n, i = 1, 2, 3, 4, and there are λ i R, i = 1, 2, 3, 4, such that λ 1 A 1 + λ 2 A 2 + λ 3 A 3 + λ 4 A 4 0. If max{z H A 1 z, z H A 2 z, z H A 3 z, z H A 4 z} 0, z C n then there are µ i 0, i = 1, 2, 3, 4, such that µ 1 + µ 2 + µ 3 + µ 4 = 1 such that µ 1 A 1 + µ 2 A 2 + µ 3 A 3 + µ 4 A 4 0.