v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
298 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.4.1 −V T N D(X)V N convexity([ ])xiWe saw that EDM entries d ij are convex quadratic functions. Yetx j−D(X) (705) is not a quasiconvex function of matrix X ∈ R n×N because thesecond directional derivative (3.3)− d2dt 2 ∣∣∣∣t=0D(X+ t Y ) = 2 ( −δ(Y T Y )1 T − 1δ(Y T Y ) T + 2Y T Y ) (709)is indefinite for any Y ∈ R n×N since its main diagonal is 0. [109,4.2.8][149,7.1, prob.2] Hence −D(X) can neither be convex in X .The outcome is different when instead we consider−V T N D(X)V N = 2V T NX T XV N (710)where we introduce the full-rank skinny Schoenberg auxiliary matrix (B.4.2)⎡V ∆ N = √ 12 ⎢⎣(N(V N )=0) having range−1 −1 · · · −11 01. . .0 1⎤⎥⎦= 1 √2[ −1TI]∈ R N×N−1 (711)R(V N ) = N(1 T ) , V T N 1 = 0 (712)Matrix-valued function (710) meets the criterion for convexity in3.2.3.0.2over its domain that is all of R n×N ; videlicet, for any Y ∈ R n×N− d2dt 2 V T N D(X + t Y )V N = 4V T N Y T Y V N ≽ 0 (713)Quadratic matrix-valued function −VN TD(X)V N is therefore convex in Xachieving its minimum, with respect to a positive semidefinite cone (2.7.2.2),at X = 0. When the penultimate number of points exceeds the dimensionof the space n < N −1, strict convexity of the quadratic (710) becomesimpossible because (713) could not then be positive definite.
5.4. EDM DEFINITION 2995.4.2 Gram-form EDM definitionPositive semidefinite matrix X T X in (705), formed from inner product of thelist, is known as a Gram matrix; [181,3.6]⎡⎤‖x 1 ‖ 2 x T⎡ ⎤1x 2 x T 1x 3 · · · x T 1x Nx T 1 [x 1 · · · x N ]xG = ∆ X T ⎢ ⎥T 2x 1 ‖x 2 ‖ 2 x T 2x 3 · · · x T 2x NX = ⎣ . ⎦ =x T 3x 1 x T 3x 2 ‖x 3 ‖ 2 ... x T 3x N∈ S N +xNT ⎢⎥⎣ . ....... . ⎦xN Tx 1 xN Tx 2 xN Tx 3 · · · ‖x N ‖ 2⎡⎤⎛⎡⎤⎞1 cos ψ 12 cos ψ 13 · · · cos ψ 1N ⎛⎡⎤⎞‖x 1 ‖‖x ‖x 2 ‖cos ψ 12 1 cos ψ 23 · · · cos ψ 2N1 ‖= δ⎜⎢⎥⎟⎝⎣. ⎦⎠cos ψ 13 cos ψ 23 1... cos ψ ‖x 2 ‖3Nδ⎜⎢⎥⎟⎢⎥ ⎝⎣. ⎦⎠‖x N ‖⎣ . ....... . ⎦‖x N ‖cos ψ 1N cosψ 2N cos ψ 3N · · · 1∆= √ δ 2 (G) Ψ √ δ 2 (G) (714)where ψ ij (734) is angle between vectors x i and x j , and where δ 2 denotesa diagonal matrix in this case. Positive semidefiniteness of interpoint anglematrix Ψ implies positive semidefiniteness of Gram matrix G ; [46,8.3]G ≽ 0 ⇐ Ψ ≽ 0 (715)When δ 2 (G) is nonsingular, then G ≽ 0 ⇔ Ψ ≽ 0. (A.3.1.0.5)Distance-square d ij (701) is related to Gram matrix entries G T = G ∆ = [g ij ]d ij = g ii + g jj − 2g ij= 〈Φ ij , G〉(716)where Φ ij is defined in (703). Hence the linear EDM definition}D(G) = ∆ δ(G)1 T + 1δ(G) T − 2G ∈ EDM N⇐ G ≽ 0 (717)= [〈Φ ij , G〉 , i,j=1... N]The EDM cone may be described, (confer (794)(800))EDM N = { }D(G) | G ∈ S N +(718)
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5.4. EDM DEFINITION 2995.4.2 Gram-form EDM definitionPositive semidefinite matrix X T X in (705), formed from inner product of thelist, is known as a Gram matrix; [181,3.6]⎡⎤‖x 1 ‖ 2 x T⎡ ⎤1x 2 x T 1x 3 · · · x T 1x Nx T 1 [x 1 · · · x N ]xG = ∆ X T ⎢ ⎥T 2x 1 ‖x 2 ‖ 2 x T 2x 3 · · · x T 2x NX = ⎣ . ⎦ =x T 3x 1 x T 3x 2 ‖x 3 ‖ 2 ... x T 3x N∈ S N +xNT ⎢⎥⎣ . ....... . ⎦xN Tx 1 xN Tx 2 xN Tx 3 · · · ‖x N ‖ 2⎡⎤⎛⎡⎤⎞1 cos ψ 12 cos ψ 13 · · · cos ψ 1N ⎛⎡⎤⎞‖x 1 ‖‖x ‖x 2 ‖cos ψ 12 1 cos ψ 23 · · · cos ψ 2N1 ‖= δ⎜⎢⎥⎟⎝⎣. ⎦⎠cos ψ 13 cos ψ 23 1... cos ψ ‖x 2 ‖3Nδ⎜⎢⎥⎟⎢⎥ ⎝⎣. ⎦⎠‖x N ‖⎣ . ....... . ⎦‖x N ‖cos ψ 1N cosψ 2N cos ψ 3N · · · 1∆= √ δ 2 (G) Ψ √ δ 2 (G) (714)where ψ ij (734) is angle between vectors x i and x j , and where δ 2 denotesa diagonal matrix in this case. Positive semidefiniteness of interpoint anglematrix Ψ implies positive semidefiniteness of Gram matrix G ; [46,8.3]G ≽ 0 ⇐ Ψ ≽ 0 (715)When δ 2 (G) is nonsingular, then G ≽ 0 ⇔ Ψ ≽ 0. (A.3.1.0.5)Distance-square d ij (701) is related to Gram matrix entries G T = G ∆ = [g ij ]d ij = g ii + g jj − 2g ij= 〈Φ ij , G〉(716)where Φ ij is defined in (703). Hence the linear EDM definition}D(G) = ∆ δ(G)1 T + 1δ(G) T − 2G ∈ EDM N⇐ G ≽ 0 (717)= [〈Φ ij , G〉 , i,j=1... N]The EDM cone may be described, (confer (794)(800))EDM N = { }D(G) | G ∈ S N +(718)