v2007.11.26 - Convex Optimization

342 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.4.3.3 Inner-product form, discussionWe deduce that knowledge of interpoint distance is equivalent to knowledgeof distance and angle from the perspective of one point, x 1 in our chosencase. The total amount of information N(N −1)/2 in Θ T Θ is unchanged 5.21with respect to EDM D .5.5 InvarianceWhen D is an EDM, there exist an infinite number of corresponding N-pointlists X (66) in Euclidean space. All those lists are related by isometrictransformation: rotation, reflection, and translation (offset or shift).5.5.1 TranslationAny translation common among all the points x l in a list will be cancelled inthe formation of each d ij . Proof follows directly from (730). Knowing thattranslation α in advance, we may remove it from the list constituting thecolumns of X by subtracting α1 T . Then it stands to reason by list-formdefinition (734) of an EDM, for any translation α∈ R nD(X − α1 T ) = D(X) (812)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 (801)and so(D(X −x 1 1 T ) = D(X −Xe 1 1 T ) = D X[0 √ ])2V N= D(X) (813)5.21 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 reduce therequired information but is antithetical. In the particular case n = 2, for example, orderingall points x l (in a length-N list) by increasing angle of vector x l − x 1 with respect tox 2 − x 1 , θ i1j becomes equivalent to j−1 ∑θ k,1,k+1 ≤ 2π and the amount of information isreduced to 2N −3; rather, O(N).k=i