Note: This is a working paper which will be expanded/updated frequently. The directory deleeuwpdx.net/pubfolders/rstressdiff has a pdf copy of this article and the complete Rmd file.

# 1 Problem

We study differentiability of the multidimensional scaling loss function rStress ((???)), defined as $\begin{equation} \sigma_r(x):=\sum_{i=1}^n w_i(\delta_i-(x'A_ix)^r)^2 \end{equation}$

for some $$r>0$$. Here the $$w_i$$ are positive weights and the $$\delta_i$$ are positive dissimilarities. The matrices $$A_i$$ are positive semi-definite, and the quantities $$x'A_ix$$ are squared distances.

Clearly if $$x'A_ix>0$$ for all $$i$$ the loss function is differentiable. De Leeuw (1984) proves directional differentiability for $$r=\frac12$$ and he shows that at a local minimum we generally have $$x'A_ix>0$$. We investigate if and how this results generalizes to $$\sigma_r$$.

# 2 Directional Derivatives

Define the directional derivative $d\sigma_r(x,y):=\lim_{\epsilon\downarrow 0}\frac{\sigma_r(x+\epsilon y)-\sigma_r(x)}{\epsilon}.$ For our computations we need \begin{align*} I_+(x)&:=\{i\mid x'A_ix>0\},\\ I_0(x)&:=\{i\mid x'A_ix=0\}. \end{align*} Then $\begin{multline*} \frac{\sigma_r(x+\epsilon y)-\sigma_r(x)}{\epsilon}=-4r\sum_{i\in I_+}w_i(\delta_i-(x'A_ix)^r)(x'A_ix)^{r-1}y'A_ix\\ -2\epsilon^{2r-1}\sum_{i\in I_0}w_i\delta_i(y'A_iy)^r+\epsilon^{4r-1}\sum_{i\in I_0}w_i(y'A_iy)^{2r} +\frac{o(\epsilon)}{\epsilon}, \end{multline*}$

and thus $d\sigma_r(x,y)= \begin{cases} -4r\sum_{i=1}^nw_i(\delta_i-(x'A_ix)^r)(x'A_ix)^{r-1}y'A_ix&\text { if }r>\frac12,\\ -4r\sum_{i\in I_+}w_i(\delta_i-(x'A_ix)^r)(x'A_ix)^{r-1}y'A_ix-2\sum_{i\in I_0}w_i\delta_i(y'A_iy)^r&\text { if }r=\frac12,\\ +\infty&\text{ if }r<\frac12. \end{cases}$

# 3 Results

From our computations we derive the following results.

Theorem 1: If $$r>\frac12$$ then $$\sigma_r$$ is differentiable at $$x$$. If $$\sigma_r$$ has a local minimum at $$x$$ then $\sum_{i=1}^nw_i\delta_i(x'A_ix)^{r-1}A_ix=\sum_{i=1}^nw_i(x'A_ix)^{2r-1}A_ix.$

Theorem 2: If $$r=\frac12$$ then $$\sigma_r$$ is directionally differentiable at $$x$$ in every direction $$y$$. If $$\sigma_r$$ has a local minimum at $$x$$ then $\sum_{i\in I_+(x)}w_i\delta_i(x'A_ix)^{r-1}A_ix=\sum_{i\in I_+(x)}w_i(x'A_ix)^{2r-1}A_ix.$ and $$I_0(x)=\emptyset$$.

Theorem 3: If $$r<\frac12$$ then $$\sigma_r$$ is directionally differentiable only in those directions $$y$$ with $$y'A_iy=0$$ for all $$i\in I_0(x)$$.

Thus for $$r=\frac12$$ we have non-zero distances and differentiability at local minima, for $$r>\frac12$$ it is quite possible that local minima with zero distances exist, and for $$r>\frac12$$ rStress is not even directionally differentiable at points with zero distances.

# 4 Local Maximum

We can also generalize a result of De Leeuw (1993) to rStress.

Theorem 4: $$\sigma_r$$ has a local maximum at $$x$$ if and only if $$x=0$$.

Proof: If $$x=0$$ then $\sigma_r(x+\epsilon y)-\sigma_r(x)=-2\epsilon^{2r}\left\{\sum_{i=1}^nw_i\delta_i(y'Ay)^r-\frac12\epsilon^{2r}\sum_{i=1}^nw_i(y'A_iy)^{2r}\right\}.$ It follows that if $\frac12\epsilon^{2r}\leq\frac{\sum_{i=1}^nw_i\delta_i(y'Ay)^r}{\sum_{i=1}^nw_i(y'A_iy)^{2r}}$ we have $$\sigma(x+\epsilon y)-\sigma(x)\leq 0$$. So, although $$\sigma_r$$ may not even directionally differentiable at $$x=0$$, it does decrease in all directions and is thus a local minimum.

Converse, suppose $$\sigma_r$$ has a local maximum at $$x\not= 0$$. Then $\sigma_r(\epsilon x)=\sum_{i=1}^nw_i\delta_i^2-2\theta\sum_{i=1}^nw_i\delta_i(x'Ax)^r+\theta^2\sum_{i=1}^nw_i(x'A_ix)^{2r},$ with $$\theta:=\epsilon^{2r}$$. Thus $$\sigma_r$$ is a convex quadratic in $$\theta$$ and it cannot have a local maximum on the ray through $$x$$. QED