Polynomial graph filters of multiple shifts and distributed implementation of inverse filtering

Abstract

Polynomial graph filters and their inverses play important roles in graph signal processing. In this paper, we introduce the concept of multiple commutative graph shifts and polynomial graph filters, which could play similar roles in graph signal processing as the one-order delay and finite impulse response filters in classical multi-dimensional signal processing. We implement the filtering procedure associated with a polynomial graph filter of multiple shifts at the vertex level in a distributed network on which each vertex is equipped with a data processing subsystem for limited computation power and data storage, and a communication subsystem for direct data exchange only to its adjacent vertices. In this paper, we also consider the implementation of inverse filtering procedure associated with a polynomial graph filter of multiple shifts, and we propose two iterative approximation algorithms applicable in a distributed network and in a central facility. We also demonstrate the effectiveness of the proposed algorithms to implement the inverse filtering procedure on denoising time-varying graph signals and a dataset of US hourly temperature at 218 locations.

Appendix A Commutative graph shifts

Graph shifts are building blocks of a polynomial filter and the concept of commutative graph shifts ${{\mathbf {S}}}_1, \ldots , {{\mathbf {S}}}_d$ is similar to the one-order delay $z_1^{-1}, \ldots , z_d^{-1}$ in classical multi-dimensional signal processing. In Appendices A.1 and A.2, we introduce two illustrative families of commutative graph shifts on circulant/Cayley graphs and product graphs respectively, see also Sects. 5.2 and 5.3 for commutative graph shifts with specific features. For commutative graph shifts ${{\mathbf {S}}}_1, \ldots , {{\mathbf {S}}}_d$, we define their joint spectrum (A.5) in Appendix A.3, which is crucial for us to develop the IOPA and ICPA algorithms in Sect. 4.

Commutativity of multiple graph shifts is essential to design polynomial graph filters with certain spectral characteristic. If a graph filter $\mathbf{H}$ is a polynomial of commutative multiple graph shifts $\mathbf{S}_1,\ldots ,\mathbf{S}_d$, then it commutes with $\mathbf{S}_k, 1\le k\le d$, i.e., commutators $[\mathbf{H}, \mathbf{S}_k]{:=}{} \mathbf{H}{} \mathbf{S}_k-\mathbf{S}_k \mathbf{H}$ between $\mathbf{H}$ and $\mathbf{S}_k, 1\le k\le d$ are always the zero matrix,

$$\begin{aligned}{}[\mathbf{H}, \mathbf{S}_k]=\mathbf{0},\ 1\le k\le d. \end{aligned}$$

(A.1)

The above necessary condition for a graph filter $\mathbf{H}$ to be a polynomial of $\mathbf{S}_1,\ldots ,\mathbf{S}_d$ is not sufficient in general. For instance, one may verify that any filter $\mathbf{H}$ satisfies (A.1) with $d=1$ and $\mathbf{S}_1=\mathbf{I}$, while $\mathbf{H}$ is not necessarily a polynomial $h(\mathbf{I})= h(1)\mathbf{I}$ of the identity matrix $\mathbf{I}$. For $d=1$, it is shown in [36, Theorem 1] that any filter satisfying (A.1) is a polynomial filter if the graph shift has distinct eigenvalues. In Theorem A.3 of Appendix A.4, we show that the necessary condition (A.1) is also sufficient under the additional assumption that elements in the joint spectrum of multiple graph shifts ${{\mathbf {S}}}_1, \ldots , {{\mathbf {S}}}_d$ are distinct.

Let ${{\mathcal {A}}}$ be a Banach algebra of graph filters with its norm denoted by $\Vert \cdot \Vert _{{\mathcal {A}}}$. Our representative examples are the algebra of graph filters with Frobenius norm $\Vert \cdot \Vert _F$, operator algebras ${{\mathcal {B}}}(\ell ^p), 1\le p\le \infty$, on the space $\ell ^p$ of all p-summable graph signals, Gröchenig-Schur algebras, Wiener algebra, Beurling algebras, Jaffard algebras and Baskakov-Gohberg-Sjöstrand algebras, see [15, 25, 42,43,44] for historical remarks and various applications. Denote the set of polynomials of commutative graph shifts $\mathbf{S}_1,\ldots ,\mathbf{S}_d$ by ${{\mathcal {P}}}{:=}{{\mathcal {P}}}(\mathbf{S}_1, \ldots , \mathbf{S}_d)$. Under the assumption that ${{\mathbf {S}}}_k\in {{\mathcal {A}}}, 1\le k\le d$, one may verify that all polynomials of $\mathbf{S}_1,\ldots ,\mathbf{S}_d$ reside in the Banach algebra ${{\mathcal {A}}}$ too, i.e., ${{\mathcal {P}}}\subset {{\mathcal {A}}}$. For any filter ${{\mathbf {H}}}\in {{\mathcal {A}}}$, define its distance to the polynomial set ${{\mathcal {P}}}$ of graph shifts $\mathbf{S}_1,\ldots ,\mathbf{S}_d$ by

$$\begin{aligned} \mathrm{dist}(\mathbf{H}, {{\mathcal {P}}})=\inf _{\mathbf{P}\in {{\mathcal {P}}}} \Vert \mathbf{H}-\mathbf{P}\Vert _{{\mathcal {A}}}. \end{aligned}$$

(A.2)

Under the assumption that the elements of joint spectrum of multiple graph shifts ${{\mathbf {S}}}_1, \ldots , {{\mathbf {S}}}_d$ are distinct, we obtain from Theorem A.3 that $\mathrm{dist}(\mathbf{H}, {{\mathcal {P}}})=0$ for any filter $\mathbf{H}\in {{\mathcal {A}}}$ satisfying (A.1). In Theorem A.4 of Appendix A.5, we establish some quantitative estimates to the distance $\mathrm{dist}(\mathbf{H}, {{\mathcal {P}}}), {{\mathbf {H}}}\in {{\mathcal {A}}}$, in terms of norms of commutators $[\mathbf{H}, \mathbf{S}_k], 1\le k\le d$, on an unweighted and undirected finite graph.

1.1 A.1 Commutative graph shifts on circulant graphs and Cayley graphs

Let ${{\mathcal {C}}}(N, Q)=(V_N, E_N(Q))$ be the circulant graph of order N generated by $Q=\{q_1, \ldots , q_M\}$, where $1\le q_1<\ldots<q_M< N/2$, see (5.1) and Fig. 2. Observe that

$$\begin{aligned} E_N(Q)=\cup _{1\le k\le d}\big \{(i,i\pm q_k\ \mathrm{mod}\ N), i\in V_N\big \}. \end{aligned}$$

Then the circulant graph ${{\mathcal {C}}}(N, Q)$ can be decomposed into a family of circulant graphs ${{\mathcal {C}}}(N, Q_k)$ generated by $Q_k=\{q_k\}, 1\le k\le d$, and the symmetric normalized Laplacian matrix $\mathbf{L}_{{{\mathcal {C}}}(N,Q)}^{\mathrm{sym}}$ on ${{\mathcal {C}}}(N, Q)$ is the average of symmetric normalized Laplacian matrices $\mathbf{L}_{{{\mathcal {C}}}(N,Q_k)}^{\mathrm{sym}}$ on ${{\mathcal {C}}}(N, Q_k), 1\le k\le d$, i.e.,

$$\begin{aligned} \mathbf{L}_{{{\mathcal {C}}}(N, Q)}^{\mathrm{sym}} = \frac{1}{d} \sum _{k=1}^d \mathbf{L}_{{{\mathcal {C}}}(N,Q_k)}^{\mathrm{sym}}, \end{aligned}$$

where $Q_k=\{q_k\}, 1\le k\le d$. In the following proposition, we establish the commutativity of $\mathbf{L}_{{{\mathcal {C}}}(N,Q_k)}^{\mathrm{sym}}, 1\le k\le d$.

Proposition A.1

The symmetric normalized Laplacian matrices $\mathbf{L}_{{{\mathcal {C}}}(N,Q_k)}^{\mathrm{sym}}$ of the circulant graphs ${{\mathcal {C}}}(N, Q_k), 1\le k\le d$, are commutative graph shifts on the circulant graph ${{\mathcal {C}}}(N, Q)$.

Proof

Clearly $\mathbf{L}_{{{\mathcal {C}}}(N,Q_k)}^{\mathrm{sym}}, 1\le k\le d$, are graph shifts on the circulant graph ${{\mathcal {C}}}(N, Q)$. Define

$$\begin{aligned} \mathbf{B}=(b({i-j\ \mathrm{mod}\ N}))_{1\le i,j\le N}, \end{aligned}$$

where $b(0)=\cdots =b({N-2})=0$ and $b({N-1})=1$. Then one may verify that

$$\begin{aligned} \mathbf{L}_{{{\mathcal {C}}}(N,Q_k)}^{\mathrm{sym}}= \mathbf{I}-\frac{1}{2}(\mathbf{B}^{q_k}+\mathbf{B}^{-q_k})= -\frac{1}{2}{} \mathbf{B}^{-q_k} (\mathbf{B}^{q_k}-\mathbf{I})^2, \end{aligned}$$

where $1\le k\le d$. Therefore for $1\le k, k'\le d$,

$$\begin{aligned} \mathbf{L}_{{{\mathcal {C}}}(N,Q_{k'})}^{\mathrm{sym}} \mathbf{L}_{{{\mathcal {C}}}(N,Q_k)}^{\mathrm{sym}} = \frac{1}{4} \mathbf{B}^{-q_k-q_{k'}} (\mathbf{B}^{q_k}-\mathbf{I})^2 (\mathbf{B}^{q_{k'}}-\mathbf{I})^2 = \mathbf{L}_{{{\mathcal {C}}}(N,Q_k)}^{\mathrm{sym}} \mathbf{L}_{{{\mathcal {C}}}(N,Q_{k'})}^{\mathrm{sym}}. \end{aligned}$$

This completes the proof. $\square$

Following the proof of Proposition A.1, we have that the adjacent matrices $2(\mathbf{I}- \mathbf{L}_{{{\mathcal {C}}}(N,Q_{k})}^{\mathrm{sym}})=\mathbf{B}^{q_k}+\mathbf{B}^{-q_k}$ of the circulant graphs ${{\mathcal {C}}}(N, Q_k), 1\le k\le d$, (and their linear combinations) are commutative graph shifts on the circulant graph ${{\mathcal {C}}}(N, Q)$.

Connected circulant graphs are regular undirected Cayley graphs of finite cyclic groups. In general, for an Abelian group G generated by a finite set S of non-identity elements and a color assignment $c_s$ to each element $s\in S$, the Cayley graph ${{\mathcal {G}}}$ is defined to have elements in the group G as its vertices and directed edges of color $c_s$ between vertices g to gs in G. For the case that the generator S is symmetric (i.e., $S^{-1}= S$) and the same color is assigned for any element in the generator S and its inverse (i.e., $c_s=c_{s^{-1}}, s\in S$), one may verify that the Cayley graph ${{\mathcal {G}}}$ is a regular undirected graph, it can be decomposed into a family of regular subgraphs ${{\mathcal {G}}}_s, s\in S_1$ with the same colored edges,

$$\begin{aligned} {{\mathcal {G}}}=\cup _{s\in S_1} {{\mathcal {G}}}_s, \end{aligned}$$

and the adjacent matrix of the Cayley graph is the summation of the adjacent matrices associated with the regular subgraphs ${{\mathcal {G}}}_s, s\in S_1$, where the subset $S_1\subset S$ is chosen so that different colors are assigned for distinct elements in $S_1$ and all colors $c_s, s\in S$ are represented in $S_1$. Furthermore, similar to the commutativity for symmetric normalized Laplacian matrices in Proposition A.1, we can show that the adjacent matrices, Laplacian matrices, symmetric normalized Laplacian matrices associated with the regular subgraphs ${{\mathcal {G}}}_s, s\in S_1$, are commutative graph shifts of the Cayley graph ${{\mathcal {G}}}$.

1.2 A.2 Commutative graph shifts on Cartesian product graphs

Let ${{\mathcal {G}}}_1=(V_1, E_1)$ and ${{\mathcal {G}}}_2=(V_2, E_2)$ be two finite graphs with adjacency matrices $\mathbf{A}_1$ and $\mathbf{A}_2$. Their Cartesian product graph ${{\mathcal {G}}}_1\times {{\mathcal {G}}}_2$ has vertex set $V_1\times V_2$ and adjacency matrix given by $\mathbf{A}=\mathbf{A}_1\otimes \mathbf{I}_{\# V_2}+ \mathbf{I}_{\# V_1}\otimes \mathbf{A} _2$ [14, 28]. Shown in Fig. 7 is an illustrative example of product graphs and the time-varying graph signal $\mathbf{X}$ can be considered as a signal on the Cartesian product graph ${{\mathcal {T}}} \times {{\mathcal {G}}}$, see Sect. 5.2.

Fig. 7

Cartesian product ${{\mathcal {T}}}\times {{\mathcal {G}}}$ of a line graph $\mathcal {T}$ and an undirected graph $\mathcal {G}$

Denote symmetric normalized Laplacian matrices and orders of the graph ${{\mathcal {G}}}_i, i=1, 2$ by $\mathbf{L}_i^{\mathrm{sym}}$ and $N_i$ respectively. By the mixed-product property

$$\begin{aligned} (\mathbf{A}\otimes \mathbf{B}) (\mathbf{C}\otimes \mathbf{D})=(\mathbf{AC})\otimes (\mathbf{BD}) \end{aligned}$$

(A.3)

for Kronecker product of matrices $\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D}$ of appropriate sizes [27], one may verify that $\mathbf{L}_1^{\mathrm{sym}}\otimes \mathbf{I}_{N_2}$ and $\mathbf{I}_{N_1}\otimes \mathbf{L}_2^{\mathrm{sym}}$ are graph filters of the Cartesian product graph ${{\mathcal {G}}}_1\times {{\mathcal {G}}}_2$. In the following proposition, we show that they are commutative.

Proposition A.2

Let ${{\mathcal {G}}}_1=(V_1, E_1)$ and ${{\mathcal {G}}}_2=(V_2, E_2)$ be two finite graphs with normalized Laplacian matrices $\mathbf{L}_1^{\mathrm{sym}}$ and $\mathbf{L}_2^{\mathrm{sym}}$ respectively. Then $\mathbf{L}_1^{\mathrm{sym}}\otimes \mathbf{I}_{\#V_2}$ and $\mathbf{I}_{\#V_1}\otimes \mathbf{L}_2^{\mathrm{sym}}$ are commutative graph shifts of the Cartesian product graph ${{\mathcal {G}}}_1\times {{\mathcal {G}}}_2$.

Proof

Let $N_i=\# V_i, i=1, 2$ and set $\mathbf{C}_1=\mathbf{L}_1^{\mathrm{sym}}\otimes \mathbf{I}_{N_2}$ and $\mathbf{C}_2=\mathbf{I}_{N_1}\otimes \mathbf{L}_2^{\mathrm{sym}}$. By the mixed-product property (A.3) of Kronecker product, then

$$\begin{aligned} \mathbf{C_1}{} \mathbf{C}_2 = \mathbf{L}_1^{\mathrm{sym}}\otimes \mathbf{L}_2^{\mathrm{sym}}= \mathbf{C}_2 \mathbf{C}_1. \end{aligned}$$

$\square$

1.3 A.3 Joint spectrum of commutative shifts

Let $\mathbf{S}_1,\ldots ,\mathbf{S}_d$ be commutative graph shifts. An important property in [18, Theorem 2.3.3] is that they can be upper-triangularized simultaneously over ${{\mathbb {C}}}$, i.e.,

$$\begin{aligned} \widehat{\mathbf{S}}_k=\mathbf{U}{} \mathbf{S}_k\mathbf{U}^{\mathrm{H}},\ 1\le k\le d, \end{aligned}$$

(A.4)

are upper triangular matrices for some unitary matrix $\mathbf{U}$. Write $\widehat{\mathbf{S}}_k=({\widehat{S}}_{k}(i,j))_{1\le i, j\le N}, 1\le k\le d$, and set

$$\begin{aligned} \Lambda =\big \{\pmb \lambda _i=\big (\widehat{S}_1(i,i), \ldots , \widehat{ S}_d(i,i)\big ), 1\le i\le N\big \} \subset {{\mathbb {C}}}^d. \end{aligned}$$

(A.5)

As $\widehat{S}_k(i, i), 1\le i\le N$, are complex eigenvalues of $\mathbf{S}_k,\ 1\le k\le d$, we call $\Lambda$ as the joint spectrum of $\mathbf{S}_1, \ldots , \mathbf{S}_d$. The joint spectrum $\Lambda$ of commutative shifts $\mathbf{S}_1, \ldots , \mathbf{S}_d$ plays an essential role in Sect. 4 in the construction of optimal polynomial approximation filters and Chebyshev polynomial approximation filters to the inverse filter of a polynomial filter of $\mathbf{S}_1,\ldots ,\mathbf{S}_d$.

We remark that a sufficient condition for the graph shifts $\mathbf{S}_1, \ldots , \mathbf{S}_d$ to be commutative is that they can be diagonalized simultaneously, i.e., there exists a nonsingular matrix $\mathbf{P}$ such that $\mathbf{P}^{-1} \mathbf{S}_k\mathbf{P}, 1\le k\le d$, are diagonal matrices, which a necessary condition is that they can be upper-triangularized simultaneously, see (A.4).

1.4 A.4 Polynomial graph filters of commutative filters

Let $\mathbf{S}_1,\ldots ,\mathbf{S}_d$ be commutative graph shifts. In the following theorem, we show that the necessary condition (A.1) for a filter to be a polynomial of multiple graph shifts is also sufficient under the additional assumption that the joint eigenvalues $\pmb \lambda _i, 1\le i\le N$, in the joint spectrum $\Lambda$ in (A.5) are distinct.

Theorem A.3

Let $\mathbf{S}_1,\ldots , \mathbf{S}_d$ be commutative graph filters, and the joint spectrum ${\Lambda }$ be as in (A.5). If all elements ${\pmb \lambda }_i, 1\le i\le N$, in the set ${ \Lambda }$ are distinct, then any graph filter $\mathbf{H}$ satisfying (A.1) is a polynomial of $\mathbf{S}_1,\ldots ,\mathbf{S}_d$, i.e., $\mathbf{H}=h(\mathbf{S}_1,\ldots ,\mathbf{S}_d)$ for some polynomial h.

Proof

Let $\mathbf{U}$ be the unitary matrix in (A.4), $\widehat{\mathbf{S}}_1, \ldots , \widehat{\mathbf{S}}_d$ be upper triangular matrices in (A.4), and $\widehat{\mathbf{H}}=\mathbf{U}{} \mathbf{H}{} \mathbf{U}^{\mathrm{H}}= ({\widehat{H}}(i,j))_{1\le i, j\le N}$. By the assumption on the set $\Lambda$, there exist an interpolating polynomial h such that

$$\begin{aligned} h(\widehat{ S}_1(i,i), \ldots , \widehat{ S}_d(i,i))={\widehat{H}}(i,i), \ 1\le i\le N, \end{aligned}$$

(A.6)

see [4, Theorem 1 on p. 58]. Set

$$\begin{aligned} \mathbf{F}= \mathbf{U}\big (\mathbf{H}-h(\mathbf{S}_1,\ldots ,\mathbf{S}_d)\big ) \mathbf{U}^{\mathrm{H}}= \widehat{\mathbf{H}}-h(\widehat{\mathbf{S}}_1, \ldots , \widehat{\mathbf{S}}_d). \end{aligned}$$

(A.7)

Then it suffices to prove that $\mathbf{F}$ is the zero matrix.

Write $\mathbf{F}=(F(i,j))_{1\le i, j\le N}$. By (A.1), we have that $\mathbf{F}\widehat{\mathbf{S}}_k=\widehat{\mathbf{S}}_k \mathbf{F}$ for all $1\le k\le d$. This together with the upper triangular property for $\widehat{\mathbf{S}}_k, 1\le k\le d$, implies that

$$\begin{aligned} \sum _{l=1}^j F(i,l) {\widehat{S}}_k(l,j)=\sum _{l=i}^N {\widehat{S}}_k(i, l) F(l, j),\ 1\le i, j\le N. \end{aligned}$$

(A.8)

By the assumption on $\Lambda$, we can find $1\le k(i,j)\le d$ for any $1\le i\ne j \le N$ such that

$$\begin{aligned} {\widehat{S}}_{k(i,j)}(i, i)\ne {\widehat{S}}_{k(i,j)}(j, j). \end{aligned}$$

(A.9)

Now we apply (A.8) and (A.9) to prove

$$\begin{aligned} { F}(i,j)=0 \end{aligned}$$

(A.10)

by induction on $j=1, \ldots , N$ and $i=N, \ldots , 1$.

For $i=N$ and $j=1$, applying (A.8) with k replaced by k(N, 1), we obtain

$$\begin{aligned} F(N,1) {\widehat{S}}_{k(N,1)}(1,1)= {\widehat{S}}_{k(N,1)}(N, N) F(N, 1), \end{aligned}$$

which together with (A.9) proves (A.10) for $(i,j)=(N, 1)$. Inductively we assume the conclusion (A.10) for all pairs (i, j) satisfying either $1\le j\le j_0$ and $i=i_0$, or $1\le j\le N$ and $i_0< i \le N$.

For the case that $j_0<i_0-1$, we have

$$\begin{aligned}&F(i_0,j_0+1) {\widehat{S}}_{k(i_0,j_0+1)}(j_0+1,j_0+1)= \sum _{l=1}^{j_0+1} F (i_0,l) {{\hat{S}}}_{k(i_0,j_0+1)}(l,j_0+1)\\&\quad = \sum _{l=i_0}^N {\widehat{S}}_{k(i_0, j_0+1)}(i_0, l) F(l, j_0+1)= {\widehat{S}}_{k(i_0, j_0+1)}(i_0, i_0) F(i_0, j_0+1), \end{aligned}$$

where the first and third equalities hold by the inductive hypothesis and the second equality is obtained from (A.8) with k replaced by $k(i_0,j_0+1)$. This together with (A.9) proves the conclusion (A.10) for $i=i_0$ and $j=j_0+1\le i_0-1$, and hence the inductive proof can proceed for the case that $j_0<i_0-1$.

For the case that $j_0=i_0-1$, it follows from the construction of the polynomial h and the upper triangular property for $\widehat{\mathbf{S}}_k, 1\le k\le d$, that the diagonal entries of $\mathbf{F}$ are

$$\begin{aligned} {\widehat{H}}(i, i)- h(\widehat{ S}_1(i,i), \ldots , \widehat{ S}_d(i,i))=0,\ 1\le i\le N \end{aligned}$$

by (A.6). Hence the conclusion (A.10) holds for $i=i_0$ and $j=j_0+1$, and hence the inductive proof can proceed for the case that $j_0=i_0-1$.

For the case that $i_0\le j_0\le N-1$, we can follow the argument used in the proof for the case that $j_0<i_0-1$ to establish the conclusion (A.10) for $i=i_0$ and $j=j_0+1\le N$, and hence the inductive proof can proceed for the case that $i_0\le j_0\le N-1$.

For the case that $j_0=N$ and $i_0\ge 2$, we obtain

$$\begin{aligned}&F(i_0-1,1) {\widehat{S}}_{k(i_0-1,1)}(1,1) = \sum _{l=i_0-1}^N {\widehat{S}}_{k(i_0-1, l)}(i_0-1, l) F(l, 1) \\&\quad = {\widehat{S}}_{k(i_0-1, 1)}(i_0-1, i_0-1) F(i_0-1, 1), \end{aligned}$$

where the first equality follows from (A.8) with k replaced by $k(i_0-1,1)$ and the second equality holds by the inductive hypothesis. This together with (A.9) proves the conclusion (A.10) for $i=i_0-1$ and $j=1$, and hence the inductive proof can proceed for the case that $j_0=N$ and $i_0\ge 2$.

For the case that $j_0=N$ and $i_0=1$, the inductive proof of the zero matrix property for the matrix $\mathbf{F}$ is complete. This completes the inductive proof. $\square$

1.5 A.5 Distance between a graph filter and the set of polynomial of commutative graph shifts

Let ${{\mathcal {G}}}=(V, E)$ be a connected, unweighted and undirected finite graph, ${{\mathcal {A}}}$ be a Banach algebra of graph filters on the graph ${{\mathcal {G}}}$ with norm denoted by $\Vert \cdot \Vert _{{\mathcal {A}}}$, $\mathbf{S}_1, \ldots , \mathbf{S}_d$ be nonzero commutative graph shifts in ${{\mathcal {A}}}$, and ${{\mathcal {P}}}$ be the set of all polynomial filters of graph shifts. In this Appendix, we consider estimating the distance $\mathrm{dist}(\mathbf{H}, {{\mathcal {P}}})$ in (A.2) between a graph filter $\mathbf{H}$ and the set ${{\mathcal {P}}}$ of polynomial filters.

Theorem A.4

If the commutative graph shifts ${{\mathbf {S}}}_1, \ldots , \mathbf{S}_d$ can be diagonalized simultaneously by a unitary matrix and elements in their joint spectrum $\Lambda$ are distinct, then there exist positive constants $C_0$ and $C_1$ such that

$$\begin{aligned} C_0 \Big (\sum _{k=1}^d \Vert [\mathbf{H}, \mathbf{S}_k]\Vert _{{\mathcal {A}}}^2\Big )^{1/2} \le \mathrm{dist}(\mathbf{H}, {{\mathcal {P}}})\le C_1 \Big (\sum _{k=1}^d \Vert [\mathbf{H}, \mathbf{S}_k]\Vert _{{\mathcal {A}}}^2\Big )^{1/2}, \ \mathbf{H}\in {{\mathcal {A}}},\qquad \end{aligned}$$

(A.11)

where $[\mathbf{H}, \mathbf{S}_k]=\mathbf{H}{} \mathbf{S}_k-\mathbf{S}_k \mathbf{H}, 1\le k\le d$.

Proof

Take $\mathbf{H}\in {{\mathcal {A}}}$. For any $\mathbf{P}\in {{\mathcal {P}}}$, we have

$$\begin{aligned} \Vert [\mathbf{H}, \mathbf{S}_k]\Vert _{{\mathcal {A}}} \le \Vert (\mathbf{H}-{{\mathbf {P}}}) \mathbf{S}_k\Vert _{{\mathcal {A}}}+ \Vert \mathbf{S}_k(\mathbf{H}-{{\mathbf {P}}})\Vert _{{\mathcal {A}}}\le 2 \Vert \mathbf{S}_k\Vert _{{\mathcal {A}}} \Vert \mathbf{H}-\mathbf{P}\Vert _{{\mathcal {A}}}, \ 1\le k\le d. \end{aligned}$$

Therefore

$$\begin{aligned} \mathrm{dist}(\mathbf{H}, {{\mathcal {P}}}) \ge \max _{1\le k\le d} \frac{\Vert [\mathbf{H}, \mathbf{S}_k]\Vert _{{\mathcal {A}}}}{2 \Vert \mathbf{S}_k\Vert _{{\mathcal {A}}}}, \end{aligned}$$

and the first inequality in (A.11) follows.

Now we prove the second inequality in (A.11). Let $\mathbf{U}$ be the unitary matrix to diagonalize ${{\mathbf {S}}}_1, \ldots , \mathbf{S}_d$ simultaneously, i.e., (A.4) holds for some diagonal matrices $\widehat{\mathbf{S}}_k=\mathrm{diag} ({\widehat{S}}(i,i))_{i\in V}, 1\le k\le d$. Then one may verify that polynomial filters of graph shifts ${{\mathbf {S}}}_1, \ldots , \mathbf{S}_d$ can also be diagonalized by the unitary matrix $\mathbf{U}$. Moreover by the distinct assumption on elements in the joint spectrum $\Lambda$ of the graph shifts, we have

$$\begin{aligned} {{\mathcal {P}}}= \{ \mathbf{U}^{\mathrm{H}} \mathbf{D} \mathbf{U}, \ \mathbf{D}\ \mathrm{are\ diagonal\ matrices} \}. \end{aligned}$$

(A.12)

Set $\mathbf{U}{} \mathbf{H}{} \mathbf{U}^{\mathrm{H}}=({\widehat{H}}(i,j))_{i,j\in V}$, and denote the Frobenius norm of a matrix $\mathbf{A}$ by $\Vert \mathbf{A}\Vert _F$. Therefore it follows from (A.12) that

$$\begin{aligned}&\inf _{\mathbf{P}\in {{\mathcal {P}}}}\Vert \mathbf{H}-\mathbf{P}\Vert _F=\inf _{\mathbf{D}\ \mathrm{are\ diagonal\ matrices}}\Vert \mathbf{U}^{\mathrm{H}}{} \mathbf{H}\mathbf{U}-\mathbf{D}\Vert _F \nonumber \\&\quad =\Big (\sum _{i,j\in V, j\ne i} |{\widehat{H}}(i,j)|^2\Big )^{1/2}. \end{aligned}$$

(A.13)

On the other hand, we have

$$\begin{aligned} \mathbf{U} [\mathbf{H}, \mathbf{S}_k] \mathbf{U}^{\mathrm{H}}= \Big ({\widehat{H}}(i,j) ({\widehat{S}}_{k}(j,j)- {\widehat{S}}_{k}(i,i))\Big )_{i,j\in V},\ 1\le k\le d. \end{aligned}$$

This implies that

$$\begin{aligned}&\sum _{k=1}^d \Vert [\mathbf{H}, \mathbf{S}_k]\Vert _F^2 = \sum _{k=1}^d \Vert \mathbf{U}^{\mathrm{H}}|[\mathbf{H}, \mathbf{S}_k]\mathbf{U}\Vert _F^2 \nonumber \\&=\sum _{i,j\in V, j\ne i} |{\widehat{H}}(i,j)|^2 \Big (\sum _{k=1}^d \big |{\widehat{S}}_{k}(j,j)- {\widehat{S}}_{k}(i,i)\big |^2\Big )\nonumber \\&\ge \inf _{i,j\in V, j\ne i} \Big (\sum _{k=1}^d \big |{\widehat{S}}_{k}(j,j)- {\widehat{S}}_{k}(i,i)\big |^2\Big ) \times \inf _{\mathbf{P}\in {{\mathcal {P}}}}\Vert \mathbf{H}-\mathbf{P}\Vert _F^2, \end{aligned}$$

(A.14)

where the last inequality follows from (A.13). Then the second inequality in (A.11) follows from (A.14), the equivalence of norms on a finite-dimensional linear space and the distinct assumption on the joint spectrum $\Lambda$. $\square$

We believe that the estimate (A.11) should hold without the simultaneous diagonalization assumption on commutative graph shifts $\mathbf{S}_1, \ldots , \mathbf{S}_d$.