1 Introduction
Let us have a set of objects, and distances between every pair of them. The distance between object and is denoted , or . This defines a metric space with . Among metric spaces, Euclidean spaces play a special role, because they establish a link with geometry. Moreover, the geometry of Euclidean spaces is well understood. Hence, even if many other metric spaces exist, like Riemanian manifolds with distances along a geodesic or graphs with shortest distance between vertices, many efforts have been devoted to specify those metric spaces for which an isometry exists with a Euclidean space, and if such an isometry exists, to build it. In such a case, any metric problem in
can be translated into a problem in Euclidean geometry, and, when lucky, solved. For example, supervised learning by discriminant analysis in a discrete metric space can be translated into the same problem in a Euclidean space and solved by Support Vector Machine approaches (see
[2]).The conditions for existence of an isometry between a discrete metric space and a subset of a Euclidean space are known, and are given by classical multidimensional scaling, proposed in [18] (see [12, 1] for classical presentation of MDS, and [5] for a recent presentation). If there is an isometry between and a subset of points in a Euclidean space , then the Gram matrix of vectors is definite positive. It is known that the Gram matrix can be computed from pairwise distances only. A set of points such that
(1) 
can be computed from the Singular Value Decomposition of the Gram matrix as a second step. If the Gram matrix has non positive eigenvalues
^{1}^{1}1In such a case, strictly speaking, the matrix built from the pairwise distances is not a Gram matrix., there is no isometry on any subset of , regardless of the dimension. In such a case, for a given dimension , one defines the cost of a map by(2) 
If there is an isometry between and a subset of , then there is a map for which . If not, Least Square Scaling (LSS) finds a map with minimal cost for a given dimension by solving
Given  a discrete metric space 
a dimension  
find  a map 
such that  is minimal 
Least square scaling has been pioneered in [7]. See as well [1] for a presentation and a comparison with classical MDS^{2}^{2}2It is unfortunate that leastsquare scaling has been proposed with the same name MDS than classical MDS. However, classical texts like [12, 1, 5] are clear on this matter and agree on setting the vocabulary..
In some situations, one is interested in relative error/distorsion between the distances and . Therefore, Sammon has developed in [15] what he called Non Linear Mapping (NLM) in which each term in the cost function is weighted by the inverse of the distance:
(3) 
(Sammon introduces a normalizing constant , that we do not mention here). It is natural to extend this towards
(4) 
where is a weight function, which can be as in Sammon’s seminal paper, or other classical maps such as or . Nonlinear mapping is minimizing the cost function over all possible point sets. It is clear that a solution is not unique, and is given up to an isometry in a Euclidean space, i.e. the composition of a translation, a rotation and a reflection.
Let us now consider a slightly different situation, where there exists an unknown point set of points in a Euclidean space and where distances are known for a subset only of pairs of points. Here, it is known as part of the problem that the distances are taken between points living in a Euclidean space, whereas in NLM, such an hypothesis is not required. Let us consider the graph where is the set of pairs for which is known. The aim is to find a mapping
(5) 
such that
(6) 
This is known as Distance Geometry Problem (DGP, see [11], equation ). A recent and thorough survey of this problem with historical background on how it grew over decades and in different guises is [11]. See also [14]. Here, the dimension is given, and the distances are known accurately. In real world problems, such as determination of protein structures, the distances are known up to a given precision only. DGP has been studied as well as embeddability problem for graphs. It has been proved in [16] that the embeddability problem is NPcomplete and the embeddability problem is NPhard for .
A link between both problems has been established in [13], where the DGP problem is recast as finding a map
such that
(7) 
is minimal, where is the matrix with in row and are weights. Let us note different choices for the exponents of the quantities to be compared: in equation (7) and in equation (4). Clearly, if DGP has a solution, the minimum of is zero, and a set of rows of where is a solution of the DGP problem. A known difficulty is that has in general many local minima. The technique used in [13] is to progressively smooth by a convolution (see [11, section 3.2.2] for a general presentation of smoothingbased methods and of DGSOL which implements it).
If all weights are equal to 1 in (4), one recovers leastsquare scaling (see [1]). If one has
one recovers DGP. There is a sort of continuity between leastsquare scaling, nonlinear mapping and DGP when the weights vary smoothly from 1 to 0 on pairs of items outside .
Here, we study whether this continuity can be given a sound basis in a relevant topology, and whether it can be translated into a continuity of numerical solutions between NLM and DGP when one or a set of parameters vary.
2 Continuity between LSS, NLM and DGP
Let be a discrete metric space, with . We denote with . Let be a weight function on distances. being known, this yields a weight matrix of general term with enabling to run a nonlinear mapping of in a Euclidean space , where is the image of . The set of points is denoted as a matrix , with being row . The set of real matrices is denoted . We define
(8) 
and consider the NLM problem
(9) 
conditioned by . This problem encompasses the problems mentioned above. For example, one can see that

if , i.e. the matrix with zeros only, or , any point cloud is a solution

if is a symmetric boolean matrix, i.e. , problem (9) is DGP.
Let be the graph such that if . Then, in DGP vocabulary, is a realization of . (see [11, sec. 1.1.4]). Let us mention that here and in the following is undetermined, and one can select or . This can be summarized as follows:
minimum of  minimum of  

Least square scaling  Isometry with a Euclidean space  
= Classical MDS  
Nonlinear mapping  
if  Distance Geometry Problem 
This raises the question of the continuity of the solutions of (9) depending on . The definition of continuity is not straightforward as there is a set of matrices which are solutions of (9). We begin by an intuitive notion of continuity, and make it rigorous in section 3. A solution is said continuous at if, being a solution at , whatever the neighborhood of , there exists a neighborhood of such that for each in it there is a solution in , or^{3}^{3}3Rigorously, one should replace by: there exists an in the set of solutions for such that .
One can observe that the solution is not continuous at . Indeed, any point cloud is a solution for . Let us take an which has a "nice" solution, i.e. the set of solutions is an orbit of the group of isometries acting on , and denote a solution. Whatever ,
and for any , the set of solutions for still is the set of solutions for . Let us take a point cloud distant from any in this set of solutions, i.e. for some whatever the point cloud in the set of solutions for . is a solution for . Whatever the neighborhood of , there is a such that is in this neighborhood. And, for this , there is no point cloud in the neighborhood of that is a solution for . Then, the solution of (9) is not continuous at . Let us note that and are incongruent in the sense of [11, sect. 1.1.4.3], as a congruency class is an orbit of the action of the group of isometries in .
Our objective is to study the continuity of the solution of (9), when varies within the space of matrices with non negative elements. The motivation for this is that each problem has a specific approach to build a solution:

DGP builds it explicitly (often with an optimization scheme)

LSS or NLM uses an optimization scheme
We study here whether a continuity between solutions of NLM, when a parameter varies in a family and the limit corresponds to a DGP problem (some distances have zero weight), can lead to a numerical solution of DGP. On the other hand, efficient optimization schemes have been derived for solving DGP which are close to some used for NLM (DC programming, see [8] and references 6 & 7 therein).
3 A topology on the set of solutions
Let us define
For a given , if , we have
We will define a topology in two spaces, in which: an element is a point cloud and an element is a set of point clouds. The latter will enable to define the neighborhood of a set for a given and study continuity of . If the topology is associated to a distance, this will read
For this, we must define a distance . A distance can be defined between compact subsets of a metric set: the Hausdorff distance (see e.g. [6, p. 34]). Let us briefly recall its definition. Let be two closed sets in a metric space where the distance between points is denoted . One defines
It is easy to show that . But it is not symmetric. So one defines
This is defined if and are compact subsets, and the triangular identity is fulfilled. We denote the Hausdorff distance between and . We first define a distance between two points clouds and as the Hausdorff distance between them. For this to hold, and must be compact. As they are obviously closed, they must be bounded. Therefore, we show a first lemma which gives the condition under which is bounded. Let be the graph associated to and defined by
Then
Lemma 1.
The set is bounded if, and only if, is connected.
Proof.
connected is bounded. We show first that is bounded for any pair if . If , we have
Hence,
and
Let and . Then,
Now, let with . As is connected, there is a path linking with . By triangular inequality, (with and ). As is finite, it has a diameter , and . If there is a such that for any pair and , then is bounded.
not connected is not bounded. If is not connected, there are at least two subsets of vertices without connection between and . We have
Let us now set
We still have and, , . If , is unbounded
As a consequence, is bounded if, and only if, is connected.
∎
Then, the set of point clouds solution of (9) for a weight matrix with a connected associated graph is a metric space, with Hausdorff distance. We then consider the subsets , running over the matrices fulfilling connectedness condition. We call a solution of (9), and the set of solutions. is closed as it is the preimage of , the minimum of . But it is not bounded. Indeed, is invariant by any isometry in . A translation is an isometry, and the distance between two solutions such that with is . Then, is unbounded. Therefore, we impose on each solution that it is centered, and consider
Then is bounded. Being a closed and bounded subset of a metric space, it is a compact set. We can use the Hausdorff distance between and , and continuity of the solution of (9) is defined as
(10) 
It is reasonable to impose that for any pair of points . The set of such weight matrices is homeomorphic to the hypercube with . The weight matrices associated to DGP are boolean. They correspond to vertices of the hypercube.
4 Continuity and rigidity
Let us denote by the set of weight matrices such that the associated graph is connected, and by the set of all centered point clouds with points. Then, is a compact subset of .
Let be the weight matrix of a DGP, i.e. there is a connected graph such that if and if . We show here that the continuity of at is linked with the rigidity of the framework associated to the graph . A framework is a set of points in solution of (6). A framework is rigid if it is defined up to an isometry (i.e. the set of known distances is sufficient to derive all other distances in a unique way). Otherwise, it is said flexible (see [11, sect. 1.1.4] for the definitions, and section 4.2 of same reference for a thorough discussion of those notions). We show here
Lemma 2.
The set of weight matrices for which any realization in is a rigid framework is not closed.
Proof.
For this, it suffices to exhibit a sequence for which for any DGP solution is a rigid framework which converges to a weight matrix for which the solution is flexible. Let us consider and (i.e. 3 points in ) and
where is the pairwise distance matrix between points. A realization is
If , the realization is defined up to an isometry: the lengths of the edges of the triangle made by are known, and this fixes the triangle up to an isometry. Whereas if , is the set of points such that , which is isomorphic to where is the group of rotations in ( and are on the circle of center and radius 1). At , is flexible. ∎
As a consequence, is not continuous at . Indeed,
and there is a constant (the calculation is easy, and omitted here) such that
whereas . and are incongruent.
5 Convergence to a Heaviside function
We first succinctly give a motivation for the next short casestudy of a given family of weight matrices. Molecular based taxonomy consists in assigning specimens to species based on similarities between some relevant DNA sequences, called markers [4]. Distances between sequences are computed with classical algorithms (see [3]). This dictionary between morphological based and molecular based taxonomy works up to a given threshold of distance (beyond this threshold, computed genetic distances are highly likely to be blurred). Thus, as a simplified setting, distances up to a given threshold only are taken into account. We have a set of pairwise distances, and we wish to build an Euclidean image of it that is as accurate as possible, ignoring the distances beyond a given threshold. Provided a dimension is given, this can be formalized as a DGP problem, where in distances below a given threshold only are given, i.e. is defined by
The weights are given by a Heaviside function . Such a function has a discontinuity at . We can consider the weight function
(11) 
as well because distances are blurred progressively. We have
with the topology of uniform convergence
Let us assume that is fixed. We are given a converging family of weight matrices satisfying
Then, if is continuous at , one can define
As is the weight matrix of a DGP, i.e. there is a graph such that for and for , this means that the solution of DGP can be found as the limit of a family of solutions of NLM, when .
Let us make a few remarks on possible frameworks to address such a problem. We call Euclidean image a solution of problem (9).

the dimension is not fixed by the problem itself. Hence, this problem can be rephrased as EMDCP.

It is however interesting to have a Euclidean image in a low dimensional space where shapes of point clouds can be better studied (e.g. )

In order to build such a solution, we can select a large
, and project it in an optimal way in a low dimensional space (by Principal Component Analysis)
Here, we fix a dimension (in our example, ) and study the convergence of NLM solutions to a solution of DGP. We focus on the numerical stability of such an approach. Indeed, for a given , the manifold in defined by
is far from having a unique minimum on . The set when is the minimum of is a minimal submanifold (all its points have a minimum "elevation" ). It is likely that many other local minima exist, as well as "flat valleys". We study here whether the ill behavior of may hamper to use the continuity of the solution between NLM and DGP as a way to obtain a numerical solution to DGP. We mention that there exist several efficient methods to solve DGP by numerical optimization, like Difference on Convex functions programming [8], Distance Continuation [13]
or Isomapbased heuristics
[10]. We are interested here in the continuity between solutions of NLM and DGP on continuous families of weight matrices .6 Numerical optimization schemes
We have implemented a numerical optimization scheme for solving problem (9) with function (11) for , with two methods known to be efficient for finding global minima: BFGS and basin hopping [19] in package scipy.optimize. As distances are unreliable beyond a given threshold, we have adopted the framework of isomapbased heuristics [10]. We have built a data set in silico, that we wish to recover (it should be the solution of DGP and NLM problems). The objective is to test the continuity of the numerical solution. The dataset consists of points in randomly (uniform law) distributed within the ring delimited by circles and centered at origin (see FIGURE 1 top left). We have selected the weight function defined in equation (11) with different values of stiffness coefficient and threshold . The numerical results for various values of and are presented in TABLE 1 for both BFGS and Basin Hoping. Each cell contains the value of the cost function for the result of the procedure for one value of , one of and one method. The starting point for NLM optimization phase could be the result of Multidimensional Scaling. But this requires the knowledge of all distances. In our case, we wish to avoid the use of distances larger than a given threshold (even if they have been measured). Therefore, the starting point is the point cloud built on distances as computed by Isomap on the partial distance matrix of distances . This yields the following observations:

The cost function for Isomap + BFGS is always equal to or lower than the cost function for Isomap + basin hoping. For this type of problem, Isomap + BFGS is recommended (we have tested other methods, results not shown, like simulated annealing, for which results were worse).

The cost function decreases when increases for a given , and is less sensitive to . If
is too small, there is a significative probability that the graph of partial distanes is not connected. When
and are low, the optimization step may not converge.
A picture of the datasets obtained by each method (Isomap, Isomap + BFGS, Isomap + basin hopping) is displayed in FIGURE 1. The eye can recognize a deformed ring in the bottom right graph, namely the best reconstruction with basin hopping. The optimization scheme has been trapped on this pattern. One is tempted to twist the outer small loop to recover a shape close to a ring. The fact that this twist (ouwards like here, or inwards in some other simulations) of a fraction of the ring is a trap can be heuristically understood: the main discrepancy between exact distances and reconstructed distances is for those points which have been twisted outward. They are much closer to the points on the opposite on the ring on the reconstruction than in the initial data set (top left). However, the stiffness of the decrease of the weights for lets the cost function be nearly insensitive to those discrepancies. The role of stiffness is then more important than the role of threshold as the weight of pairs of points separated by a large distance (like ) is annihilated by a very low weight ( for and ). Careful observation of many simulations lead to the observation that similar low cost function values may correspond to very different geometric settings for the solution.
7 Conclusion
We have set a framework to study continuity of the solution of (9) when the weight matrix varies. The main point to address is that a solution to is never a single point cloud, but always a set of point clouds, a union of orbits of the action of the group of isometries on . We have exhibited an example where the solution is not continuous at a weight matrix where the realization of the solution is not rigid. We expect that it is continuous when realizations are rigid, but this has not been shown here. This is deferred to further work on study of the topological structure of the solutions in relation with the weight matrices. We have linked DGP and NLM in a common framework using this continuity. We have studied whether the continuity of the solution can serve as a basis for ensuring the continuity of numerical solutions when a parameter varies in the weight matrix. Therefore, we have built a simple in silico dataset, and derived a procedure with the output of Isomap as initial point for optimization step in NLM, with two optimization schemes (BFGS and Basin Hopping). We have produced good hints to show convergence of NLM solution to DGP solution (a situation when ). We have shown as well that different geometric settings of the solution may correspond to similar very low values of the cost function. It is likely that this is due to the complicated shape of the manifold of cost function as function of coordinates of a point cloud, and motivates further studies.
BFGS  BH  BFGS  BH  BFGS  BH  BFGS  BH  

94.82  94.82  0.848  10.82  
55.42  55.42  1.018  2.212  
0.672  0.672 
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