**Appendix A**

**Theorem A1.** *The image of the unit hypersphere under any n* × *m matrix is a hyperellipsoid.*

**Proof.** Let **<sup>A</sup>** be an *<sup>n</sup>* <sup>×</sup> *<sup>m</sup>* matrix with rank *<sup>r</sup>*. Let **<sup>A</sup>** <sup>=</sup> **USV***<sup>T</sup>* be a singular-value decomposition of **A**. The left and right singular vectors of **A** are denoted as **u**1, **u**2, ... , **u***<sup>n</sup>* and **v**1, **v**2, ... , **v***m*, respectively. Since *rank*(*A*) = *r*, the singular values of **A** have the properties: *σ*<sup>1</sup> ≥ *σ*<sup>2</sup> ≥,..., ≥ *σ<sup>r</sup>* > 0 and *σr*+<sup>1</sup> = *σr*+<sup>2</sup> =,..., = *σ<sup>m</sup>* = 0. ⎛ ⎞

Let **x** = ⎜⎝ *x*1 . . . *xm* ⎟⎠ be an unit vector in R*m*. Because **V** is an orthogonal matrix and **V***<sup>T</sup>*

is also, we have that **<sup>V</sup>***T***<sup>x</sup>** is an unit vector (it is easy to see that **<sup>V</sup>***T***<sup>x</sup>** <sup>=</sup> **<sup>x</sup>**). Therefore, (**v***<sup>T</sup>* <sup>1</sup> **<sup>x</sup>**)<sup>2</sup> + (**v***<sup>T</sup>* <sup>2</sup> **<sup>x</sup>**)2+,..., +(**v***<sup>T</sup> <sup>m</sup>***x**)<sup>2</sup> = 1.

On the other hand, we have **A** = *σ*1**u**1**v***<sup>T</sup>* <sup>1</sup> + *<sup>σ</sup>*2**u**2**v***<sup>T</sup>* <sup>2</sup> +,..., +*σr***u***r***v***<sup>T</sup> <sup>r</sup>* . Therefore:

$$\begin{split} \mathbf{A}\mathbf{x} &= \sigma\_1 \mathbf{u}\_1 \mathbf{v}\_1^T \mathbf{x} + \sigma\_2 \mathbf{u}\_2 \mathbf{v}\_2^T \mathbf{x} + \dots , \dots + \sigma\_r \mathbf{u}\_r \mathbf{v}\_r^T \mathbf{x} \\ &= (\sigma\_1 \mathbf{v}\_1^T \mathbf{x}) \mathbf{u}\_1 + (\sigma\_2 \mathbf{v}\_2^T \mathbf{x}) \mathbf{u}\_2 + \dots , \dots + (\sigma\_r \mathbf{v}\_r^T \mathbf{x}) \mathbf{u}\_r \\ &= \mathbf{y}\_1 \mathbf{u}\_1 + \mathbf{y}\_2 \mathbf{u}\_2 + \dots , \dots + \mathbf{y}\_r \mathbf{u}\_r \\ &= \mathbf{U} \mathbf{y} \end{split} \tag{A1}$$

where **y***<sup>i</sup>* denotes the *σi***v***<sup>T</sup> <sup>i</sup>* **x** and **y** = ⎛ ⎜⎝ *y*1 . . . *yr* ⎞ ⎟⎠.

From (A1), we have: **Ax** = **Uy** = **y** (since **U** is an orthogonal matrix). Moreover, **y** has the following property:

$$\begin{aligned} (\frac{\mathcal{Y}\_1}{\sigma\_1})^2 + (\frac{\mathcal{Y}\_2}{\sigma\_2})^2 + \dots &+ (\frac{\mathcal{Y}\_r}{\sigma\_r})^2 = \\ = (\mathbf{v}\_1^T \mathbf{x})^2 + (\mathbf{v}\_2^T \mathbf{x})^2 + \dots &+ (\mathbf{v}\_r^T \mathbf{x})^2 \le 1 \end{aligned} \tag{A2}$$

Specifically:


#### **References**

