1

Let $D$ be a given Euclidean Distance Matrix, where the entries are the non-squared distances.

The Gram Matrix of the corresponding points can be computed as follows : $$G = \frac{1}{2} (D_{0i}^2 + D_{0j}^2 - D_{ij}^2)$$

The embedding dimension is then $k = \operatorname{rank}G$.

Then, from the eigen-decomposition $G = OSO^T$ we can build the matrix $X = O\sqrt S$.

Another way to proceed is to compute :

$$F = -\frac{1}{2} (I - \mathbf{e}.\mathbf{x}^T).D.(I-\mathbf{x}.\mathbf{e}^T)$$ where $\mathbf{e}$ is the vector $(1,1,....,1)$ and $\mathbf{x}$ satisfies $\mathbf{x}^T\mathbf{e} = 1$ (one can simply take $\mathbf{x} = \mathbf{e}/n$). Then the matrix of coordinates is given by the decomposition $F = X^TX$. Again, $k= \operatorname{rank}F$.

Removing the $n-k$ null columns of $X$ we obtain the coordinates of the set of points described by $D$, in $k$-dimensional Euclidean space (up to an orthogonal transformation).

Question

Is it possible to exhibit some constraints over $D$, or over $F$ or $G$, as to bound the corresponding vectors in $\mathbb{R}^k$ inside the hypercube $[-L,L]^k$ ?

Clarification : I am looking for something like : If $G_{ij} \leq r$ then we will have $x_i \in [-f(r),f(r)]^d$ for some known function $f$.

Community
  • 1
Toool
  • 307
  • I dont understand the question :(.... – Brethlosze Aug 21 '17 at 17:55
  • @hyprfrcb What exactly you do not understand ? – Toool Aug 21 '17 at 19:29
  • You have a $D$ matrix (ok) and then you are calculating it for $de=I-e\cdot x^T$(why?), this is not the Gramian of $x$ but for $de$ (why?), $x$ satisfy $x^Te=1$ (so?), the range is $k$ (ok), now you want a constraint (of what?) to bound the image of $D$, inside an hypercube (the spanned space inside an hypercube? or the base of the spanned space? what is $L$?) – Brethlosze Aug 21 '17 at 19:37
  • $G$ is the Gramian of $X$ when $D_{ij} = ||x_i - x_j||^2$ (see the paper by (J.C. Gower 'Properties of Euclidean an non-Euclidean Distance Matrices')[http://www.sciencedirect.com/science/article/pii/0024379585901879] ). What I want is a constraint over $D$ that would bound the vectors $x_i$ inside a hypercube of side $2L$ in the Euclidean space of dimension $k$, here $L$ is a real finite number. I added another way to compute the Gramian Matrix of $X$ in the post. – Toool Aug 21 '17 at 20:02
  • So why are you taking it from $I$??, why not simply $x^TDx$??. And why not a sphere but a cube?? Why not a unit cube?? – Brethlosze Aug 21 '17 at 20:04
  • @hyprfrcb See the post, I've edited it for clarity. The reasons of why this works is not the question here, pleqse see the paper by Gower for an extensive derivation of the results. The reason I want to bound the point coordinates in a hypercube of side $2L$ are not for discussion here, it is my question, that's all :)

    (Bound the points in a hypersphere is trivial)

     – Toool Aug 21 '17 at 20:10
  • More answerable would be to constraint the diatance points into a cube!. And i dont see both those expressions as equivalent!! The question is turning unanswerable – Brethlosze Aug 21 '17 at 20:12
  • I see several questions: 1. How the expressions are the point distances (could be simplified), 2. How to find a closed solution for constraining into a cube (could be solved), 3. The orthogonal transformation (this can be anything). I am still unclear, which kind of solution do you want? – Brethlosze Aug 21 '17 at 20:19
  • Let's assume the matrix $D$ is Euclidean i.e. it represents a set of points, so 1. is assumed. The question is how to bound the points inside a hypercube *up to an orthogonal transformation* from their distance matrix. You can see it as : how to bound the points inside a rotation/translation of the hypercube. – Toool Aug 21 '17 at 20:26
  • An answerable question would be: Let be $x$ a matrix column of vectors, with $G=x'Dx$. Let be the eigen descomposition of $G=OSO'$ and $y=O\sqrt S$. Is there any condition $Ax\le b$ making $y_{ij} \le 1$? – Brethlosze Aug 21 '17 at 20:35
  • @hyprfrcb As far as I know, the matrix $G$ you've defined is not the Gramian matrix of the points described by the distance matrix $D$, so this formulation does not work (Otherwise prove it?) See https://math.stackexchange.com/questions/156161/finding-the-coordinates-of-points-from-distance-matrix for how to recover the Gramian matrix. – Toool Aug 21 '17 at 20:42

0 Answers0