Multidimensional scaling in dually flat spaces

Abstract

Formulations of multidimensional scaling (MDS) in dually flat spaces are proposed. First the space supposed in the classical MDS is extended to a tangent space around a specific point in a dually flat space. We see that Riemannian metric of the tangent point plays a key role in the extension. Next, in order to remove the restriction of symmetry in dissimilarities, the affine connection is incorporated. We pay attention to the fact that it is an affine connection term that causes an asymmetry in dissimilarities in infinitesimal space. To mitigate the difficulty in treating the affine connection term, an approximation is shown and we can see the effect of the affine connection term to modify the effective Riemannian metric. Finally a numerical example is shown.

Appendices

Appendix A: Additional examples of numerical calculation

In this appendix, additional dissimilarity data are numerically analyzed according to Sect. 6.

First the following circulant dissimilarity matrix is analyzed. Table 2 shows the result.

$$\begin{aligned} D= \left( \begin{array}{l@{\quad }l@{\quad }l} 0 &{} 0.10 &{} 0.12 \\ 0.12 &{} 0 &{} 0.10 \\ 0.10 &{} 0.12 &{} 0 \end{array} \right) \end{aligned}$$

(39)

Table 2 Result of the numerical calculation in the case of dissimilarity (39)

Next the following non-circulant dissimilarity matrix is analyzed. Table 3 shows the result.

$$\begin{aligned} D= \left( \begin{array}{l@{\quad }l@{\quad }l} 0 &{} 0.10 &{} 0.10 \\ 0.12 &{} 0 &{} 0.10 \\ 0.12 &{} 0.12 &{} 0 \end{array} \right) \end{aligned}$$

(40)

Table 3 Result of the numerical calculation in the case of dissimilarity (40)

Appendix B: An example of graphical representation

In addition to the decomposition of Riemannian metric

$$\begin{aligned} \tilde{g}_n^{ij} / 2 = \sum _{l=1}^q \mu _l u^i_l u^j_l \end{aligned}$$

(41)

through common eigenvalue decomposition, let us consider a symmetric tensor decomposition [38, 39]

$$\begin{aligned} - T_n^{ijk} / 12 = \sum _{l=1}^r \xi _l w^i_l w^j_l w^k_l \end{aligned}$$

(42)

where $\xi _1 \ge \cdots \ge \xi _r > 0$, $\sum _{l=1}^r (w_l^i)^2 = 1$. Then the Eq. (30) is written as

$$\begin{aligned} d_{\iota \kappa } = \sum _{l=1}^q (x_{\iota l} - x_{\kappa l})^2 + \sum _{l=1}^r (y_{\iota l} - y_{\kappa l})^3 \end{aligned}$$

(43)

where $x_{\iota l} = \sqrt{\mu _l} u^i_l \eta _{i\iota }, \ y_{\iota l} = \root 3 \of {\xi _l} w^i_l \eta _{i\iota }$.

The first term in the Eq. (43), the symmetric part of $d_{\iota \kappa }$, corresponds to the squared distance in classical MDS. The calculation of coordinates $x_{\iota l}$ and their graphical representation are the same as in classical MDS.

The remaining is the treatment of the second term in the Eq. (43), the antisymmetric part of $d_{\iota \kappa }$. In addition to $x_{\iota l}$, $y_{\iota l}$ stands for the coordinate of the object $\iota$ in the $l$-th dimension. Here, as an example, a two-dimensional representation is presented. Supposing that the coordinate of the object $n$ be zero, the antisymmetric part of $d_{\iota n}$ is written for each $\iota$ as follows:

$$\begin{aligned} ( d_{\iota n} - d_{n \iota } )/2 = y_{\iota 1}^3 + y_{\iota 2}^3. \end{aligned}$$

(44)

This is rewritten for graphical representation as

$$\begin{aligned} ( d_{\iota n} - d_{n \iota } )/2 = s_{\iota } (s_\iota ^2 + t_\iota ^2) /4 \end{aligned}$$

(45)

where $s_\iota = y_{\iota 1} + y_{\iota 2}, \ t_\iota = \sqrt{3} (y_{\iota 1} - y_{\iota 2})$. Thus we can represent the antisymmetric parts by plotting $(s_\iota , \ t_\iota )$ in a plane. For each object $\iota$, the antisymmetric part of $d_{\iota n}$ is represented as a quarter of the product of the value of $s_\iota$ and the squared distance between $\iota$ and $n$.