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We have an unknown $m\times n$ matrix $X=(x_{ij})_{i=1,j=1}^{m,n}$. Assume we are given measurements of the differences

$$x_{i,j+1}-x_{i,j}$$

and

$$x_{i+1,j}-x_{i,j}$$ for all $(i,j)\in \{1,\dots,m\}\times\{1,\dots,n\}$ where $x_{i,n+1}=x_{i,1}$ and $x_{m+1,j}=x_{1,j}$. Assume the measurements are independent and have all the same standard deviation. Now we would like to reconstruct a difference $x_{i,j}-x_{1,1}$. For a given $m,n$ one can consider the least squares problem and find a linear combination of measurements such that the random variable has the smallest standard deviation. My question now is if there is an explicit expression for this linear combination and the standard deviation of the reconstruction. If not is the asymptotics $m,n\rightarrow \infty$ known?

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Given

  • $\mathrm R \in \mathbb R^{m \times n}$, whose $m$ rows are the differences of neighboring rows of $\mathrm X \in \mathbb R^{m \times n}$

  • $\mathrm C \in \mathbb R^{m \times n}$, whose $n$ columns are the differences of neighboring columns of $\mathrm X \in \mathbb R^{m \times n}$

we would like to determine the unknown matrix $\mathrm X \in \mathbb R^{m \times n}$. Do note that we consider the 1st and $m$-th rows neighbors, and that we also consider the 1st and $n$-th columns neighbors.


The subtraction matrix

We define the $d \times d$ circulant matrix

$$\mathrm S_d := \begin{bmatrix} -1 & 1 & 0 & \dots & 0 & 0\\ 0 & -1 & 1 & \dots & 0 & 0\\ 0 & 0 & -1 & \dots & 0 & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & \dots & -1 & 1\\ 1 & 0 & 0 & \dots & 0 & -1\end{bmatrix}$$

Let

$$w_d := \exp \left( i \frac{2 \pi}{d} \right)$$

From results on circulant matrices [0], we know that the spectrum of $\mathrm S_d$ is

$$\{ w_d^0 - 1, w_d^1 - 1, \dots, w_d^{d-1} - 1 \} = \{ 0, w_d - 1, \dots, w_d^{d-1} - 1 \}$$

In other words, the $d$ eigenvalues of $\mathrm S_d$ are on the unit circle centered at $-1 + i 0$. Hence, we can conclude that $\mbox{rank} (\mathrm S_d) = d-1$. Since $\mathrm S_d 1_d = 0_d$, the (right) null space of $\mathrm S_d$ is $\{ \gamma 1_d \mid \gamma \in \mathbb R \}$.


A linear system

We have two linear matrix equations in $\mathrm X$

$$\mathrm S_m \mathrm X = \mathrm R \qquad\qquad\qquad \mathrm X \mathrm S_n^{\top} = \mathrm C$$

which is a linear system of $2 m n$ scalar equations in $m n$ unknowns. Note that

  • if $\bar{\mathrm X}$ is a particular solution of $\mathrm S_m \mathrm X = \mathrm R$, then so is $\bar{\mathrm X} + \gamma \, 1_m \mathrm v^{\top}$, where $\gamma \in \mathbb R$ and $\mathrm v \in \mathbb R^n$, as

$$\mathrm S_m \left( \bar{\mathrm X} + \gamma \, 1_m \mathrm v^{\top} \right) = \underbrace{\mathrm S_m \bar{\mathrm X}}_{= \mathrm R} + \gamma \, \underbrace{\mathrm S_m 1_m}_{= 0_m} \mathrm v^{\top} = \mathrm R + \mathrm O_{m \times n} = \mathrm R$$

  • if $\bar{\mathrm X}$ is a particular solution of $\mathrm X \mathrm S_n^{\top} = \mathrm C$, then so is $\bar{\mathrm X} + \gamma \, \mathrm u 1_n^{\top}$, where $\gamma \in \mathbb R$ and $\mathrm u \in \mathbb R^m$, as

$$\left( \bar{\mathrm X} + \gamma \, \mathrm u 1_n^{\top} \right) \mathrm S_n^{\top} = \underbrace{\bar{\mathrm X} \mathrm S_n^{\top}}_{= \mathrm C} + \gamma \, \mathrm u \underbrace{1_n^{\top} \mathrm S_n^{\top}}_{= \mathrm 0_n^{\top}} = \mathrm C + \mathrm O_{m \times n} = \mathrm C$$

Thus, if $\bar{\mathrm X}$ is a particular solution of both $\mathrm S_m \mathrm X = \mathrm R$ and $\mathrm X \mathrm S_n^{\top} = \mathrm C$, then so is

$$\bar{\mathrm X} + \gamma \, 1_m 1_n^{\top}$$

where $\gamma \in \mathbb R$. Since there are infinitely many solutions, more information is needed to reconstruct $\mathrm X$.

Suppose now that we are given not only matrices $\mathrm R$ and $\mathrm C$, but also $s \in \mathbb R$, which denotes the sum of the $m n$ entries of $\mathrm X$. Appending the equation $1_m^{\top} \mathrm X 1_n = s$ to the linear system

$$\mathrm S_m \mathrm X = \mathrm R \qquad\qquad\qquad \mathrm X \mathrm S_n^{\top} = \mathrm C \qquad\qquad\qquad 1_m^{\top} \mathrm X 1_n = s$$

and vectorizing, we obtain a linear system of $2 m n + 1$ equations in $m n$ unknowns

$$\begin{bmatrix} \mathrm I_n \otimes \mathrm S_m\\ \mathrm S_n \otimes \mathrm I_m\\ 1_{mn}^{\top}\end{bmatrix} \mbox{vec} (\mathrm X) = \begin{bmatrix} \mbox{vec} (\mathrm R)\\ \mbox{vec} (\mathrm C)\\ s\end{bmatrix}$$


Minimizing the Frobenius norm

Let us look for the solution that has the least Frobenius norm. We have the quadratic program (QP)

$$\begin{array}{ll} \text{minimize} & \| \mathrm X \|_{\text{F}}^2\\ \text{subject to} & \mathrm S_m \mathrm X = \mathrm R\\ & \mathrm X \mathrm S_n^{\top} = \mathrm C\end{array}$$

We define the Lagrangian

$$\mathcal L (\mathrm X, \Lambda_1, \Lambda_2) := \frac 12 \| \mathrm X \|_{\text{F}}^2 - \langle \Lambda_1, \mathrm S_m \mathrm X - \mathrm R \rangle - \langle \Lambda_2, \mathrm X \mathrm S_n^{\top} - \mathrm C \rangle$$

Taking the partial derivative with respect to $\mathrm X$ and finding where it vanishes, we obtain

$$\mathrm X = \mathrm S_m^{\top} \Lambda_1 + \Lambda_2 \mathrm S_n$$

Left-multiplying by $\mathrm S_m$ and right-multiplying by $\mathrm S_n^{\top}$, we obtain two coupled linear matrix equations in the Lagrange multipliers $\Lambda_1$ and $\Lambda_2$

$$\begin{array}{rl} \mathrm S_m \mathrm S_m^{\top} \Lambda_1 + \mathrm S_m \Lambda_2 \mathrm S_n &= \mathrm R\\ \mathrm S_m^{\top} \Lambda_1 \mathrm S_n^{\top} + \Lambda_2 \mathrm S_n \mathrm S_n^{\top} &= \mathrm C\end{array}$$

Vectorizing, we obtain a linear system of $2 m n$ equations in $2 m n$ unknowns

$$\begin{bmatrix} \mathrm I_n \otimes \mathrm S_m \mathrm S_m^{\top} & \mathrm S_n^{\top} \otimes \mathrm S_m\\ \mathrm S_n \otimes \mathrm S_m^{\top} & \mathrm S_n \mathrm S_n^{\top} \otimes \mathrm I_m\end{bmatrix} \begin{bmatrix} \mbox{vec} (\Lambda_1)\\ \mbox{vec} (\Lambda_2)\end{bmatrix} = \begin{bmatrix} \mbox{vec} (\mathrm R)\\ \mbox{vec} (\mathrm C)\end{bmatrix}$$

Solving this linear system and unvectorizing the solutions, we obtain the optimal Lagrange multipliers $\Lambda_1^*$ and $\Lambda_2^*$. The least-norm solution is then given by

$$\mathrm X_{\text{LN}} := \mathrm S_m^{\top} \Lambda_1^* + \Lambda_2^* \mathrm S_n$$


Reference

[0] Robert M. Gray, Toeplitz and Circulant Matrices: A Review.

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