v2009.01.01 - Convex Optimization

382 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
5.4.3.3 Inner-product form, discussion
We deduce that knowledge of interpoint distance is equivalent to knowledge
of distance and angle from the perspective of one point, x 1 in our chosen
case. The total amount of information N(N −1)/2 in Θ T Θ is unchanged 5.21
with respect to EDM D .
5.5 Invariance
When D is an EDM, there exist an infinite number of corresponding N-point
lists X (68) in Euclidean space. All those lists are related by isometric
transformation: rotation, reflection, and translation (offset or shift).
5.5.1 Translation
Any translation common among all the points x l in a list will be cancelled in
the formation of each d ij . Proof follows directly from (794). Knowing that
translation α in advance, we may remove it from the list constituting the
columns of X by subtracting α1 T . Then it stands to reason by list-form
definition (798) of an EDM, for any translation α∈ R n
D(X − α1 T ) = D(X) (876)
In words, interpoint distances are unaffected by offset; EDM D is translation
invariant. 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 (865)
and so
(
D(X −x 1 1 T ) = D(X −Xe 1 1 T ) = D X
[0 √ ])
2V N
= D(X) (877)
5.21 The reason for amount O(N 2 ) information is because of the relative measurements.
Use of a fixed reference in measurement of angles and distances would reduce 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 respect to
x 2 − x 1 , θ i1j becomes equivalent to j−1 ∑
θ k,1,k+1 ≤ 2π and the amount of information is
reduced to 2N −3; rather, O(N).
k=i