Proximity Problems

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by M. Bennani Dosse

Contents

Abstract

The aim of this short paper is to give an algebraic result that relate two criteria in multidimensional scaling...

Key-words
Euclidean distance, Multidimensional scaling, strain, sstress, comparing criteria.

Introduction

We consider an \,n\times n\, matrix \,D=[d_{ij}]\, defined as a real symmetric matrix that is hollow \,d_{ii}=0\, for \,i=1\ldots n\, and nonnegative \,d_{ij}\geqslant 0\, for all \,i,j\,.

D\, is said to be Euclidean distance matrix of dimension \,p\, if there exists a list of points \,\{z_1\ldots z_n\}\, in \,\Bbb R^p\, \,(p\leqslant n-1)\, such that

d_{ij}=\|z_i-z_j\|^2 \quad\forall\,i,j=1\ldots n

where \,\|~\|\, denotes Euclidean norm. Denote by \,\Bbb{EDM}^n(p)\, the set of all \,n\times n\, Euclidean distance matrices of dimension \,p\,.

A problem common to various sciences is to find the Euclidean distance matrix \,D\in\Bbb{EDM}^n(p)\, closest, in some sense, to a given predistance matrix \,\Delta=[\delta_{ij}]\, defined to be any symmetric hollow nonnegative real matrix. There are three statements of the closest-EDM problem prevalent in the literature, the multiplicity due primarily to choice of projection on the EDM versus positive semidefinite (PSD) cone and vacillation between the distance-square variable \,d_{ij}\, versus absolute distance \,\sqrt{d_{ij}}\,.

During the past two decades a large amount of work has been devoted to Euclidean distance matrices and approximation of predistances by an \,\Bbb{EDM}^n(p)\, in a series of works including Gower[6-8], Mathar..., Critchley..., Hayden et al..., etc.

Mathematical preliminaries

It is well known that \,D\in\Bbb{EDM}^n(p)\, if and only if the symmetric \,n\times n\, matrix

W_s(D)=-\frac{1}{2}(I-es^{\rm T})D(I-se^{\rm T})\qquad(1)

is positive semidefinite with \,\text{rank}(W_s(D))\leqslant p\,, where \,e\, is a vector of ones and \,s\, is any vector such that \,s^{\rm T}e=1\,.

This result was proved by Gower... as a generalization of an earlier result of Schoenberg... Later Gower considered the particular choices \,s=\frac{1}{n}e\, and \,s=e_i\, where \,e_i\, is the \,i^\text{th}\, vector from the standard basis. In what follows, when \,s=\frac{1}{n}e\, then matrix \,W_s(D)\, will be denoted by \,W(D)\,:

W(D)=-\frac{1}{2}(I-\frac{1}{n}ee^{\rm T})D(I-\frac{1}{n}ee^{\rm T})\qquad(2)

We see no compelling reason to prefer one particular \;s\, over another. Each has its own coherent interpretation. Neither can we say any particular problem formulation produces generally better results than another. Dattorro...

The aim of this short paper is to clarify that point...

We shall also use Wolkowicz' notation:

\begin{array}{rcl} J_s &=& I-es^{\rm T}\qquad(3)\\ J &=& I-\frac{1}{n}ee^{\rm T}\qquad(4) \end{array}

so the equations (2) and (3) can be written:

\begin{array}{rcl} W_s(D) &=& -\frac{1}{2}J_sDJ_s^{\rm T} \\ W(D) &=& -\frac{1}{2}JDJ \end{array}

It is easy to verify the following properties:

\begin{array}{c} J^{\rm T}=J,\;J^2=J,\;Je=0\\ J_s^2=J_s,\;J_se=0,\;s^{\rm T}J_s=0\\ JJ_s=J,\;J_sJ=J_s\\ W=JW_sJ\\ W_s=J_sWJ_s^{\rm T} \end{array}

Classical Multidimensional Scaling

Given \,p\leqslant n\,, let \,\Bbb S_+^n(p)\, denote the closed set of symmetric \,n\times n\, matrices that are positive semidefinite and have rank no greater than \,p\,.

Let \,\|~\|_{\rm F}\, denote the Frobenius norm and \,\Delta\, a given symmetric \,n\times n\, matrix of squared dissimilarities. Let \,W=W(\Delta)\, and \,W_s=W_s\,(\Delta).

Classical MDS can be defined by the optimization problem

\begin{array}{rl} \text{minimize}_B&\|W-B\|_{\rm F}^2\\ \text{subject to}&B\in\Bbb S_+^n(p) \end{array}~~~~~~~~~~~~~~~~~~~~~~~~\textbf{(P)}

Problem (P) can be viewed as a particular case of a more general optimization problem

\begin{array}{rl} \mbox{minimize}_B&\|W_s-B\|_{\rm F}^2\\ \text{subject to}&B\in\Bbb S_+^n(p) \end{array}~~~~~~~~~~~~~~~~~~~~~~~~(\textbf{P}_\textbf{s)}

The following explicit solution to problem (P) (respectively problem (Ps)) is well known: let \,\lambda_1\geqslant\ldots\geqslant\lambda_n\, denote the eigenvalues of \,W\, (respectively of \,W_s\,) and \,v_1\ldots v_n\, denote the corresponding eigenvectors.

Assume that the \,p\, largest eigenvalues are positive. Then

B^\star=\sum_{i=1}^p\lambda_iv_iv_i^{\rm T}

is a global minimum of problem (P) (respectively of problem (Ps)). Furthermore, the minimum value for problem (P) is

f=\sum_{i=p+1}^n\lambda_i^2(W)

and for problem (Ps)

f_s=\sum_{i=p+1}^n\lambda_i^2(W_s)

In Section 5, we will prove that for any squared dissimilarity matrix \,\Delta\, we have

f\leqslant f_s

that is, at the minimum, the strain criterion always gives smaller value than criterion (Ps). In order to show this result we shall use

Lemma

Let \,\lambda(C)\in\reals^n\, denote the eigenvalues of any symmetric \,n\times n\, matrix in nonincreasing order.

  • For all \,A,B\,

\lambda_i(A+B) \leqslant \lambda_i(A)+\lambda_1(B)

  • For all positive semidefinite \,A,B\,

\lambda_i(A\,B)\leqslant \lambda_i(A)\lambda_1(B)\qquad\diamond


Proof. see, for instance, Wilkinson...

Comparing strain and sstress

In this section we recall a result (see [2]) that relate the strain and sstress criteria. The sstress criterion is given by:

\begin{array}{rl}\text{minimize}&S(D)=\|\Delta-D\|^2\\ \text{subject to}&D\in\Bbb{EDM}^n(p) \end{array}

Result. The following inequality holds: Given \,p\leqslant n-1\,, for any \,B\in\Bbb S_+^n(p)\,, let \,D=\text{diag}(B)e^{\rm T}+e\;\text{diag}(B)^{\rm T}-2B\,. Then

\|\Delta-D\|^2 \geqslant 4\|W-B\|^2


Proof. Let \,B\in\Bbb S_+^n(p)\,; we have

\begin{array}{rcl} \delta_{ij} &=& w_{ii}+w_{jj}-2w_{ij}\\ d_{ij} &=& b_{ii}+b_{jj}-2b_{ij} \end{array}

Writing \,a_{ij}=w_{ij}-b_{ij}\, we get

\sum_i\sum_j (\delta_{ij}-d_{ij})^2=2n\,\sum_ia_{ii}^2+4\sum_i\sum_j a_{ij}^2\geqslant 4\sum_i\sum_ja_{ij}^2\qquad\square

Main result

In this section we show an inequality involving the criteria \,f\, in (12) and \,f_s\, in (13).

Theorem.

For any \,s\in\Bbb R^n\, such that \,s^{\rm T}e=1\, and for any \,p\, we have

\,f \leqslant f_s\qquad(18)


Proof. We show, for all \,i\,, that \,|\lambda_i(W)|\leqslant|\lambda_i(W_s)|\,. Toward that end, we consider two cases:

  • If \,W\, is PSD then \,W_s\, is PSD and the inequality becomes \,\lambda_i(W)\leqslant \lambda_i(W_s)\,. But

\lambda_i(W)=\lambda_i(JW_sJ)\leqslant \lambda_i(W_s)\lambda_1(J)=\lambda_i(W_s)

because \,\lambda_1(J)=1\,.

  • If \,W\, is not PSD then, using the definition of \,J\,:

\begin{array}{rcl} \lambda_i(W^2) &=& \lambda_i(W_sJW_s-\frac{1}{n}ee^{\rm T}W_sJW_s) \\ &\leqslant& \lambda_i(W_sJW_s)+\lambda_1(-\frac{1}{n}ee^{\rm T}W_sJW_s) \end{array}

But

\lambda_1(-\frac{1}{n}ee^{\rm T}W_sJW_s)= 0

because \,J\, and \,W_s^2\, are PSD we have

\lambda_i(W_sJW_s)\leqslant\lambda_i(W_s^2)\lambda_1(J)=\lambda_i(W_s^2)\qquad\square

Modified Gower problem

In this Section we consider the following problem: Given a nonEuclidean matrix, can we find an \,s\, that maximizes the total squared real distances from the points to the centroid given by \,s\, in the fitted configuration. What is this choice of \,s\,?

This problem can be written as an optimization problem in the following manner. First note that if \,\Delta\, is not Euclidean, then the number of negative eigenvalues of \,W_s=W_s(\Delta)\, does not depend on \,s\,; call that number \,p\,.

The total squared-real distances from the points to the centroid given by \,s\, in the fitted configuration can be written as

\sum_{i=1}^p\lambda_i(W_s)\qquad(19)

where \,\lambda_i(W_s)\, denotes the \,i^{\rm th}\, eigenvalue of \,W_s\,. But by a well known result... we have, for \,X\!\in\reals^{n\times p}\,

\sum_{i=1}^p\lambda_i(W_s)=\max_{X^{\rm T}X=I}\;\mathrm{tr}(X^{\rm T}W_sX)

So the final optimization problem can be written as

\max_{s^{\rm T}e=1,\,s\succeq 0}\max_{X^{\rm T}X=I}\;\mathrm{tr}(X^{\rm T}W_sX)\qquad(20)

where

\,X^{\rm T}W_sX=X^{\rm T}WX-X^{\rm T}Wse^{\rm T}X-X^{\rm T}es^{\rm T}WX+X^{\rm T}es^{\rm T}Wse^{\rm T}X\,

Question

Is it true that at the optimum, the problem (20) is equivalent to the problem...

\max_{X^{\rm T}X=I}\max_{s^{\rm T}e=1,\,s\succeq 0}\;\mathrm{tr}(X^{\rm T}W_sX)\qquad(21)

References

[1] Critchley, F., 1986. On certain linear mappings between inner-product and squared-distance matrices. Linear Algebra Appl. 105, 91-107.

[2] De Leeuw, J., Heiser, W., 1982. Theory of multidimensional scaling. Krishnaiah, P.R., Kanal, I.N.(Eds.), Handbook of Statistics, vol. 2. North-Holland, Amsterdam, pp. 285-316 (chapter 13).

[3] Gower, J.C., 1966. Some distance properties of latent root and vector methods in multivariate analysis. Biometrika 53, 315-328.

[4] Gower, J.C., 1982. Euclidean distance geometry, Math. Scientist 7, 1-14.

[5] Schoenberg, I.J., 1935. Remarks to Maurice Fréchet's article Sur la définition axiomatique d'une classe d'espaces distanciés vectoriellement applicable sur l'espace de Hilbert. Ann. of Math. 38, 724-738.

[6] Torgerson, W.S., 1952. Multidimensional scaling: I. Theory and method. Psychometrika 17, 401-419.

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