v2006.03.09 - Convex Optimization

248 CHAPTER 4. EUCLIDEAN DISTANCE MATRIX4.4.3.3 Inner-product form, discussionWe deduce that knowledge of interpoint distance is equivalent to knowledge ofdistance and angle from the perspective of one point, x 1 in our chosen case.The total amount of information in Θ T Θ , N(N −1)/2, is unchanged 4.15with respect to EDM D .4.5 InvarianceWhen D is an EDM, there exist an infinite number of corresponding N-pointlists X (64) in Euclidean space. All those lists are related by isometrictransformation: rotation, reflection, and translation (offset or shift).4.5.1 TranslationAny translation common among all the points x l in a list will be cancelledin the formation of each d ij . Proof follows directly from (456). Knowingthat translation α in advance, we may remove it from the list constitutingthe columns of X by subtracting α1 T . Then it stands to reason by list-formdefinition (460) of an EDM, for any translation α∈ R nD(X − α1 T ) = D(X) (528)In words, interpoint distances are unaffected by offset; EDM D is translationinvariant. When α = x 1 in particular,[x 2 −x 1 x 3 −x 1 · · · x N −x 1 ] = X √ 2V N ∈ R n×N−1 (517)and so(D(X −x 1 1 T ) = D(X −Xe 1 1 T ) = D X[0 √ ])2V N= D(X) (529)4.15 The reason for the amount O(N 2 ) information is because of the relative measurements.The use of a fixed reference in the measurement of angles and distances would reducethe required information but is antithetical. In the particular case n = 2, for example,ordering all points x l in a length-N list by increasing angle of vector x l −x 1 with respectto x 2 − x 1 , θ i1j becomes equivalent to j−1 ∑θ k,1,k+1 ≤ 2π and the amount of informationis reduced to 2N −3; rather, O(N).k=i