Unitarily invariant errors-in-variables estimation

Abstract

Linear relations, containing measurement errors in the input and output data, are considered. Parameters of these errors-in-variables (EIV) models can be estimated by minimizing the total least squares (TLS) of the input–output disturbances, i.e., penalizing the orthogonal squared misfit. This approach corresponds to minimizing the Frobenius norm of the error matrix. An extension of the traditional TLS estimator in the EIV model—the EIV estimator—is proposed in the way that a general unitarily invariant norm of the error matrix is minimized. Such an estimator is highly non-linear. Regardless of the chosen unitarily invariant matrix norm, the corresponding EIV estimator is shown to coincide with the TLS estimator. Its existence and uniqueness is discussed. Moreover, the EIV estimator is proved to be scale invariant, interchange, direction, and rotation equivariant.

Appendices

Auxiliary definitions and theorems

Definition 8

(Unitary matrix) A unitary matrix $\mathbf {A}$ is a square matrix satisfying $\mathbf {A}^{\top }\mathbf {A}=\mathbf {A}\mathbf {A}^{\top }=\mathbf {I}$.

Definition 9

(Permutation matrix) For a permutation $\pi :\,\{1,\ldots ,p\}\rightarrow \{1,\ldots ,p\}$, a permutation matrix is a square matrix $[\mathbf {e}_{\pi (1)},\ldots ,\mathbf {e}_{\pi (p)}]^{\top }$, where its rows are permuted canonical vectors.

Definition 10

(Rotation matrix) A rotation matrix is a unitary matrix whose determinant is equal to one.

Theorem 4

(Sturm interlacing property) Let $n\ge p$ and the singular values of $\mathbf{A}\in \mathbb {R}^{n\times p}$ are $\sigma _1\ge \ldots \ge \sigma _p$. If $\mathbf{B}$ results from $\mathbf{A}$ by deleting one column of $\mathbf{A}$ and $\mathbf{B}$ has singular values $\sigma _1'\ge \ldots \ge \sigma _{p-1}'$, then

$$\begin{aligned} \sigma _1\ge \sigma _1'\ge \sigma _2\ge \sigma _2'\ge \ldots \ge \sigma _{p-1}'\ge \sigma _p\ge 0. \end{aligned}$$

(14)

Theorem 5

(Eckart–Young–Schmidt–Mirsky matrix approximation) Suppose the SVD of $\mathbf{A}\in \mathbb {R}^{n\times p}$ is given by $\mathbf{A}=\sum _{i=1}^r\sigma _i \mathbf{u}_i \mathbf{v}_i^{\top }$ with $rank(\mathbf{A})=r$. If $k<r$ and $\mathbf{A}_k=\sum _{i=1}^k\sigma _i \mathbf{u}_i \mathbf{v}_i^{\top }$, then for any unitarily invariant matrix norm $\Vert \cdot \Vert$ holds

$$\begin{aligned} \min _{rank(\mathbf{B})=k}\left\| \mathbf{A}-\mathbf{B}\right\| =\left\| \mathbf{A}-\mathbf{A}_k\right\| . \end{aligned}$$

Lemma 1

(Equivariantness) Suppose $\mathbf {J}\in \mathbb {R}^{p\times p}$ is a unitary matrix and $\Vert \cdot \Vert$ is a unitarily invariant matrix norm. If $\{\widehat{{\varvec{\beta }}},[\widehat{{\varvec{\Theta }}},\widehat{\varepsilon }]\}$ is a solution to optimization problem (3), then $\{\mathbf {J}^{\top }\widehat{{\varvec{\beta }}},[\widehat{{\varvec{\Theta }}},\widehat{\varepsilon }]\}$ is a solution to the optimization problem

$$\begin{aligned} \min _{{\varvec{\beta }}\in \mathbb {R}^p,[{\varvec{\Theta }},\varepsilon ]\in \mathbb {R}^{n\times (p+1)}}\Vert [{\varvec{\Theta }},\varepsilon ]\widetilde{\mathbf {J}}\Vert \quad \text{ s.t. }\quad \mathbf {Y}-{\varvec{\varepsilon }}=(\mathbf {X}-{\varvec{\Theta }})\mathbf {J}{\varvec{\beta }}, \end{aligned}$$

(15)

where

$$\begin{aligned} \widetilde{\mathbf {J}}=\left[ \begin{array}{cc}\mathbf {J}&{}\quad \mathbf {0}\\ \mathbf {0}&{}\quad 1\end{array}\right] . \end{aligned}$$

Proofs

Proof

(of Theorem 1) See von Neumann (1937). $\square$

Proof

(of Theorem 2) (a) Firstly, we show that $v_{p+1,p+1}\ne 0$. By contradiction, let us suppose $v_{p+1,p+1}=0$. Then, there exist $\mathbf {0}\ne \mathbf{w}\in \mathbb {R}^p$ such that

$$\begin{aligned} \left[ \mathbf{w}^{\top },0\right] \left[ \mathbf {X},\mathbf {Y}\right] ^{\top }\left[ \mathbf {X},\mathbf {Y}\right] \left[ \begin{array}{c} \mathbf{w}\\ 0\\ \end{array}\right] =\sigma _{p+1}^2, \end{aligned}$$

which yields $\mathbf{w}^{\top }{} \mathbf{X}^{\top }{} \mathbf{X}\mathbf{w}=\sigma _{p+1}^2$. But this contradicts to the assumption $\sigma _p'>\sigma _{p+1}$, since $\sigma _p'^2$ is the smallest eigenvalue of $\mathbf{X}^{\top }{} \mathbf{X}$.

Moreover, the Sturm interlacing property (14) from Theorem 4 and assumption $\sigma _p'>\sigma _{p+1}$ provide $\sigma _p>\sigma _{p+1}$. Therefore, $\sigma _{p+1}$ is not a repeated singular value of $\left[ \mathbf {X},\mathbf {Y}\right]$ and $\sigma _p>0$.

If $\sigma _{p+1}\ne 0$, then $rank([\mathbf {X},\mathbf {Y}])=p+1$. We want to find $[\widehat{\mathbf {X}},\widehat{\mathbf {Y}}]$ such that $\Vert [\mathbf {X},\mathbf {Y}]-[\widehat{\mathbf {X}},\widehat{\mathbf {Y}}]\Vert$ is minimal and $[\widehat{\mathbf {X}},\widehat{\mathbf {Y}}][\widehat{\varvec{\beta }}^{\top },-1]^{\top }={\varvec{0}}$ for some $\widehat{\varvec{\beta }}$. Thus, $rank([\widehat{\mathbf {X}},\widehat{\mathbf {Y}}])=p$. Applying the Eckart–Young–Schmidt–Mirsky Theorem 5, one may easily obtain the SVD of $[\widehat{\mathbf {X}},\widehat{\mathbf {Y}}]$ as in (6) and the correction matrix from (5), which must have rank one. The EIV estimator is given by the last column of $\mathbf{V}$. Hence, a solution to (3) is found. Its uniqueness needs to be proved. Since $[\mathbf {v}_1,\ldots ,\mathbf {v}_{p+1}]$ is an orthogonal basis in $\mathbb {R}^{p+1}$, let $[\widehat{{\varvec{\beta }}},-1]^{\top }=\mathbf {v}\Vert [\widehat{{\varvec{\beta }}},-1]^{\top }\Vert _E$, where $\mathbf {v}=\sum _{i=1}^{p+1}\alpha _i\mathbf {v}_i$ and $\Vert \mathbf {v}\Vert _E=1$. Theorem II.3.9 by Stewart and Sun (1990, Eq. (3.6) together with $[\widehat{\mathbf {X}},\widehat{\mathbf {Y}}]\mathbf {v}=\mathbf {0}$ give

because unit norm of $\mathbf {v}$ implies $\sum _{i=1}^{p+1}\alpha _i^2=1$ and $\sigma _{p+1}$ is the uniquely smallest singular value. The latter inequality becomes equality if and only if $\alpha _{p+1}=1$ and $\alpha _i=0$ for $i=1,\ldots ,p$. Thus, $\mathbf {v}=\mathbf {v}_{p+1}$ and $\widehat{{\varvec{\beta }}}=-[v_{1,p+1},\ldots ,v_{p,p+1}]^{\top }/v_{p+1,p+1}$ is unique.

If $\sigma _{p+1}=0$, then $\mathbf{v}_{p+1}\in Ker([\mathbf {X},\mathbf {Y}])$ and $[\mathbf {X},\mathbf {Y}][{\varvec{\beta }}^{\top },-1]^{\top }={\varvec{0}}$. Hence, no approximation is needed, the overdetermined system (1) is compatible, and the exact EIV solution to the EIV problem (3) is given by (4). The uniqueness of this EIV solution follows from the fact that $[{\varvec{\beta }}^{\top },-1]^{\top }\perp Range([\mathbf {X},\mathbf {Y}]^{\top })$.

Furthermore, $\Vert \sigma _{p+1}{} \mathbf{u}_{p+1}\mathbf{v}_{p+1}^{\top }\Vert =\sigma _{p+1}$ for the arbitrary unitarily invariant matrix norm $\Vert \cdot \Vert$ due to the orthonormality of vectors $\mathbf{u}_{p+1}$ and $\mathbf{v}_{p+1}$.

A closed-form expression of the EIV estimator (7) has to be derived. If $\sigma _p'>\sigma _{p+1}$, the existence and the uniqueness of the EIV estimator has already been shown. Since the singular vectors $\mathbf{v}_i$ from (6) are the eigenvectors of $\left[ \mathbf {X},\mathbf {Y}\right] ^{\top }\left[ \mathbf {X},\mathbf {Y}\right]$, the EIV estimator $\widehat{\varvec{\beta }}$ also satisfies

$$\begin{aligned} \left[ \mathbf {X},\mathbf {Y}\right] ^{\top }\left[ \mathbf {X},\mathbf {Y}\right] \left[ \begin{array}{l} {\widehat{\varvec{\beta }}}\\ -1\\ \end{array}\right] =\left[ \begin{array}{ll} \mathbf{X}^{\top }{} \mathbf{X}&{}\quad \mathbf{X}^{\top }{} \mathbf{Y}\\ \mathbf{Y}^{\top }{} \mathbf{X}&{}\quad \mathbf{Y}^{\top }{} \mathbf{Y}\\ \end{array}\right] \left[ \begin{array}{l} {\widehat{\varvec{\beta }}}\\ -1\\ \end{array}\right] =\sigma _{p+1}^2 \left[ \begin{array}{l} {\widehat{\varvec{\beta }}}\\ -1\\ \end{array}\right] \end{aligned}$$

(16)

and, hence,

$$\begin{aligned} \widehat{{\varvec{\beta }}}=\big (\mathbf{X}^{\top }\mathbf{X}-\sigma _{p+1}^2\mathbf{I}\big )^{-1}{} \mathbf{X}^{\top }{} \mathbf{Y}. \end{aligned}$$

(17)

Finally, the smallest eigenvalue of the positive semidefinite matrix $[\mathbf {X},\mathbf {Y}]^{\top }[\mathbf {X},\mathbf {Y}]$ is the squared smallest singular value of matrix $[\mathbf {X},\mathbf {Y}]$ due to the SVD of $[\mathbf {X},\mathbf {Y}]$ and the eigen decomposition property of $[\mathbf {X},\mathbf {Y}]^{\top }[\mathbf {X},\mathbf {Y}]$. Combining this fact with (17) leads to (7).

(b) If $\sigma _q>\sigma _{q+1}=\ldots =\sigma _{p+1},\,q<p$, then

$$\begin{aligned}{}[\widehat{{\varvec{\Theta }}},\widehat{{\varvec{\varepsilon }}}]:=[\mathbf {X},\mathbf {Y}]\left[ \begin{array}{c} \mathbf {w}\\ \alpha \end{array}\right] [\mathbf {w}^{\top },\alpha ]/\Vert [\mathbf {w}^{\top },\alpha ]\Vert _E^2 \quad \text{ and }\quad \left[ \begin{array}{c} \mathbf {w}\\ \alpha \end{array}\right] \in Range([\mathbf {v}_{q+1},\ldots ,\mathbf {v}_{p+1}]) \end{aligned}$$

solve optimization problem (3) provided $\alpha \ne 0$. Indeed, suppose $[\mathbf {w}^{\top },\alpha ]^{\top }=\sum _{i=q+1}^{p+1}\eta _i\mathbf {v}_i$, then

$$\begin{aligned} \Vert [\widehat{{\varvec{\Theta }}},\widehat{{\varvec{\varepsilon }}}]\Vert ^2&=\left\| \sum _{i=q+1}^{p+1}\eta _i\sigma _i\mathbf {u}_i\Big [\mathbf {w}^{\top },\alpha \Big ]\Big / \Big \Vert \Big [\mathbf {w}^{\top },\alpha \Big ]\Big \Vert _E^2\right\| ^2\\&=\sigma _{p+1}^2\left\| \frac{\sum _{i=q+1}^{p+1}\eta _i\mathbf {u}_i}{\Vert [\mathbf {w}^{\top },\alpha ]\Vert _E}\frac{[\mathbf {w}^{\top },\alpha ]}{\Vert [\mathbf {w}^{\top },\alpha ]\Vert _E}\right\| ^2=\sigma _{q+1}^2 \end{aligned}$$

is minimal. This holds due to Theorem 5, where $\Vert \sigma _{p+1}\mathbf {u}_{p+1}\mathbf {v}_{p+1}^{\top }\Vert =\sigma _{q+1}$. Condition $[v_{q+1,p+1},\ldots ,v_{p+1,p+1}]^{\top }\ne \mathbf {0}$ gives infinitely many vectors $[\mathbf {w}^{\top },\alpha ]^{\top }$ such that $\alpha \ne 0$ and $\widehat{{\varvec{\beta }}}=-\mathbf {w}/\alpha$ satisfies

$$\begin{aligned}{}[\widehat{\mathbf {X}},\widehat{\mathbf {Y}}]\left[ \begin{array}{c} \widehat{{\varvec{\beta }}}\\ -1 \end{array}\right]&=-\frac{1}{\alpha }([\mathbf {X},\mathbf {Y}]-[\widehat{{\varvec{\Theta }}},\widehat{{\varvec{\varepsilon }}}])\left[ \begin{array}{c} \mathbf {w}\\ \alpha \end{array}\right] \\&=-\frac{1}{\alpha }[\mathbf {X},\mathbf {Y}]\left[ \begin{array}{c} \mathbf {w}\\ \alpha \end{array}\right] \left( 1-\frac{[\mathbf {w}^{\top },\alpha ]\left[ \begin{array}{c} \mathbf {w}\\ \alpha \end{array}\right] }{\Vert [\mathbf {w}^{\top },\alpha ]\Vert _E^2}\right) =\mathbf {0}. \end{aligned}$$

Thus, any $[\mathbf {w}^{\top },\alpha ]^{\top }\in Range([\mathbf {v}_{q+1},\ldots ,\mathbf {v}_{p+1}])$ with $\alpha \ne 0$ provides the EIV estimator $\widehat{{\varvec{\beta }}}$.

(c) Let us assume that there exists a solution to the optimizing problem (3), which gives the EIV estimator $\widehat{{\varvec{\beta }}}$ such that $[\widehat{{\varvec{\beta }}}^{\top },-1]^{\top }=\sum _{i=1}^{p+1}\alpha _i\mathbf {v}_i$ and $[\widehat{\mathbf {X}},\widehat{\mathbf {Y}}][\widehat{{\varvec{\beta }}}^{\top },-1]^{\top }=\mathbf {0}$. Theorem II.3.9 by Stewart and Sun (1990, Eq. (3.6) provides

$$\begin{aligned} \left\| [\widehat{{\varvec{\Theta }}},\widehat{{\varvec{\varepsilon }}} ]\right\|&=\left\| [\mathbf {X},\mathbf {Y}]-[\widehat{\mathbf {X}},\widehat{\mathbf {Y}}]\right\| \ge \left\| \big ([\mathbf {X},\mathbf {Y}]-\big [\widehat{\mathbf {X}},\widehat{\mathbf {Y}}\big ]\big )\big [\widehat{{\varvec{\beta }}}^{\top },-1\big ]^{\top }\right\| \Big /\left\| [[\widehat{{\varvec{\beta }}}^{\top },-1]^{\top }\right\| _E\\&=\left\| [\mathbf {X},\mathbf {Y}][\widehat{{\varvec{\beta }}}^{\top },-1]^{\top }\right\| \Big /\left\| [\widehat{{\varvec{\beta }}}^{\top },-1]^{\top }\right\| _E =\left\| \sum _{i=1}^{p+1}\alpha _i\sigma _i\mathbf {u}_i\right\| \Big /\left\| [\widehat{{\varvec{\beta }}}^{\top },-1]^{\top }\right\| _E\\&=\left\| \sum _{i=1}^{p+1}\alpha _i\sigma _i\mathbf {u}_i\right\| _E \Big /\left\| [\widehat{{\varvec{\beta }}}^{\top },-1]^{\top }\right\| _E=\frac{1}{\Vert [\widehat{{\varvec{\beta }}}^{\top },-1]^{\top }\Vert _E}\sqrt{\sum _{i=1}^{p+1}\alpha _i^2\sigma _i^2}\\&>\frac{1}{\Vert [\widehat{{\varvec{\beta }}}^{\top },-1]^{\top }\Vert _E}\min _{\sum _{i=1}^{p+1}\alpha _i^2=\Vert [\widehat{{\varvec{\beta }}}^{\top },-1]^{\top }\Vert _E}\sqrt{\sum _{i=1}^{p+1}\alpha _i^2\sigma _i^2}=\sigma _{q+1}. \end{aligned}$$

The above strict inequality cannot become an equality, since $[v_{q+1,p+1},\ldots ,v_{p+1,p+1}]^{\top }=\mathbf {0}$. However, one can find matrix $[\widehat{{\varvec{\Theta }}}_{\epsilon },\widehat{{\varvec{\varepsilon }}}_{\epsilon }]$ and vector $\mathbf {v}_{\epsilon }$ satisfying $[\widehat{{\varvec{\Theta }}}_{\epsilon },\widehat{{\varvec{\varepsilon }}}_{\epsilon }]\mathbf {v}_{\epsilon }=\mathbf {0}$ such that $\Vert [\widehat{{\varvec{\Theta }}}_{\epsilon },\widehat{{\varvec{\varepsilon }}}_{\epsilon }]\Vert$ is as small as possible, but still strictly greater than $\sigma _{q+1}$. Indeed, for the considered unitarily invariant matrix norm $\Vert \cdot \Vert$, there exist a symmetric gauge function $\varsigma$ according to Theorem 1. Let us fix arbitrary $\epsilon >0$ and take $\mathbf {v}_{\epsilon }=\epsilon \mathbf {v}_q+\sqrt{1-\epsilon ^2}\mathbf {v}_{q+1}$. Realize that $\Vert \mathbf {v}_{\epsilon }\Vert =\varsigma ([\Vert \mathbf {v}_{\epsilon }\Vert _E,0,\ldots ,0]^{\top })=\Vert \mathbf {v}_{\epsilon }\Vert _E\varsigma ([1,0,\ldots ,0]^{\top })=1$. Matrix

$$\begin{aligned} \big [\widehat{{\varvec{\Theta }}}_{\epsilon },\widehat{{\varvec{\varepsilon }}}_{\epsilon }\big ]:=[\mathbf {X},\mathbf {Y}]\mathbf {v}_{\epsilon }\mathbf {v}_{\epsilon }^{\top }=\Big (\epsilon \sigma _q\mathbf {u}_q+\sqrt{1-\epsilon ^2}\sigma _{q+1}\mathbf {u}_{q+1}\Big )\Big (\epsilon \mathbf {v}_q+\sqrt{1-\epsilon ^2}\mathbf {v}_{q+1}\Big )^{\top } \end{aligned}$$

satisfies $\mathbf {Y}-\widehat{{\varvec{\varepsilon }}}_{\epsilon }\in Range(\mathbf {X}-\widehat{{\varvec{\Theta }}}_{\epsilon })$. Theorem II.3.9 by Stewart and Sun (1990, Eq. (3.6)and (3.8)) provides

$$\begin{aligned}&\left\| \epsilon \sigma _q\mathbf {u}_q+\sqrt{1-\epsilon ^2}\sigma _{q+1}\mathbf {u}_{q+1}\right\| \sigma _{min}\Big (\epsilon \mathbf {v}_q+\sqrt{1-\epsilon ^2}\mathbf {v}_{q+1}\Big )\nonumber \\&\quad \le \left\| [\widehat{{\varvec{\Theta }}}_{\epsilon },\widehat{{\varvec{\varepsilon }}}_{\epsilon }]\right\| \le \left\| \epsilon \sigma _q\mathbf {u}_q+\sqrt{1-\epsilon ^2}\sigma _{q+1}\mathbf {u}_{q+1}\right\| \left\| \epsilon \mathbf {v}_q+\sqrt{1-\epsilon ^2}\mathbf {v}_{q+1}\right\| _{\infty },\qquad \quad \end{aligned}$$

(18)

where $\sigma _{min}(\cdot )$ denotes the smallest singular value. Since $\sigma _{min}(\epsilon \mathbf {v}_q+\sqrt{1-\epsilon ^2}\mathbf {v}_{q+1})=\Vert \epsilon \mathbf {v}_q+\sqrt{1-\epsilon ^2}\mathbf {v}_{q+1}\Vert _{\infty }=\Vert \epsilon \mathbf {v}_q+\sqrt{1-\epsilon ^2}\mathbf {v}_{q+1}\Vert _E=1$, relation (18) gives

$$\begin{aligned} \left\| [\widehat{{\varvec{\Theta }}}_{\epsilon },\widehat{{\varvec{\varepsilon }}}_{\epsilon }]\right\| =\left\| \epsilon \sigma _q\mathbf {u}_q+\sqrt{1-\epsilon ^2}\sigma _{q+1}\mathbf {u}_{q+1}\right\| =\sqrt{\epsilon ^2\sigma _q^2+(1-\epsilon ^2)\sigma _{q+1}^2}. \end{aligned}$$

Since $\sqrt{\epsilon ^2\sigma _q^2+(1-\epsilon ^2)\sigma _{q+1}^2}>\sigma _{q+1}$ for all $\epsilon >0$ (due to $\sigma _q>\sigma _{q+1}$) and

$$\begin{aligned} \lim _{\epsilon \rightarrow 0+}\sqrt{\epsilon ^2\sigma _q^2+(1-\epsilon ^2)\sigma _{q+1}^2}=\sigma _{q+1}, \end{aligned}$$

a solution to the optimization problem (3) does not exist. $\square$

Proof

(of Theorem 3) (i) Finding a new EIV estimator for the multiplied data $a[\mathbf {X},\mathbf {Y}]$ leads to solving the following optimization problem

$$\begin{aligned} \min _{{\varvec{\beta }}\in \mathbb {R}^p,[{\varvec{\Theta }},\varepsilon ]\in \mathbb {R}^{n\times (p+1)}}\left\| [a{\varvec{\Theta }},a\varepsilon ]\right\| \quad \text{ s.t. }\quad a\mathbf {Y}-a{\varvec{\varepsilon }}=(a\mathbf {X}-a{\varvec{\Theta }}){\varvec{\beta }}. \end{aligned}$$

(19)

However, the minimizing problem (19) is directly equivalent (i.e., provides the same solution) to the original optimization problem (3).

(ii), (iii), (iv) Since the permutation matrices, the diagonal matrices having $\pm 1$ on the diagonal, and the rotation matrices are unitary ones, Lemma 1 straightforwardly completes the proof. $\square$

Proof

(of Theorem 4) See Thompson (1972). $\square$

Proof

(of Theorem 5) See Mirsky (1960). $\square$

Proof

(of Lemma 1) Since $\mathbf {J}$ is unitary, $\mathbf {J}^{-1}=\mathbf {J}^{\top }$ and, moreover, $\widetilde{\mathbf {J}}$ has to be unitary as well. Due to the basic property of the unitarily invariant matrix norms, the optimization problem (3) is equivalent to

$$\begin{aligned} \min _{{\varvec{\beta }}\in \mathbb {R}^p,[{\varvec{\Theta }},\varepsilon ]\in \mathbb {R}^{n\times (p+1)}}\left\| [{\varvec{\Theta }},\varepsilon ]\widetilde{\mathbf {J}}\right\| \quad \text{ s.t. }\quad \mathbf {Y}-{\varvec{\varepsilon }}=(\mathbf {X}-{\varvec{\Theta }})\mathbf {J}\mathbf {J}^{\top }{\varvec{\beta }}. \end{aligned}$$

(20)

Once a solution $\{\widehat{{\varvec{\beta }}},[\widehat{{\varvec{\Theta }}},\widehat{\varepsilon }]\}$ to (20) is found, then $\{\mathbf {J}^{\top }\widehat{{\varvec{\beta }}},[\widehat{{\varvec{\Theta }}},\widehat{\varepsilon }]\}$ has to be a solution to (15). $\square$