Shift-enabled graphs: Graphs where shift-invariant filters are representable as polynomials of shift operations

Shift-enabled graphs: Graphs where shift-invariant filters are representable as polynomials of shift operations

Liyan Chen, Samuel Cheng,  Vlandimir Stankovic,  and Lina Stankovic,  L. Chen is with the Department of Computer Science and Technology, Tongji University, Shanghai, 201804 China and the School of Mathematics, Physics & Information Science, Zhejiang Ocean University, Zhejiang, 316022, China(e-mail: chenliyan@tongji.edu.cn).S. Cheng is with the Department of Computer Science and Technology, Tongji University, Shanghai, 201804 China and the School of Electrical and Computer Engineering, University of Oklahoma, OK 74105, USA (email: samuel.cheng@ou.edu).V. Stankovic and L. Stankovic are with Department of Electronic and Electrical Engineering, University of Strathclye, Glasgow, G1 1XW U.K. (e-mail:{vladimir.stankovic, lina.stankovic}@strath.ac.uk). Corresponding author.Manuscript received Sept, 2017; revised 2017.
Abstract

In digital signal processing, shift-invariant filters can be represented as a polynomial expansion of a shift operation, that is, the -transform representation. When extended to graph signal processing (GSP), this would mean that a shift-invariant graph filter can be represented as a polynomial of the adjacency (shift) matrix of the graph. However, the characteristic and minimum polynomials of the adjacency matrix must be identical for the property to hold. While it has been suggested that this condition might be ignored as it is always possible to find a polynomial transform to represent the original adjacency matrix by another adjacency matrix that satisfies the condition, this letter shows that a filter that is shift invariant in terms of the original graph may not be shift invariant anymore under the modified graph and vice versa. We introduce the notion of “shift-enabled graph” for graphs that satisfy the aforementioned condition, and present a concrete example of a graph that is not “shift-enabled” and a shift-invariant filter that is not a polynomial of the shift operation matrix. The result provides a deeper understanding of shift-invariant filters when applied in GSP and shows that further investigation of shift-enabled graphs is needed to make it applicable to practical scenarios.

graph signal processing, shift invariant filter, polynomial, digital signal processing.

I Introduction

In digital signal processing (DSP), a filter is a system that takes a signal as an input and generates a new signal as an output. Most filters are time- or shift-invariant filters [1, 2, 3], for which the following property holds [4]:

 F1(F2(x))=F2(F1(x)), (1)

which guarantees that, for two filters and , the order of the filters does not change the output. An immediate consequence of (1) is that a shift-invariant filter can be represented as a polynomial expansion of a shift operation, that is, the -transform representation of the polynomials in  [3, 4, 5]:

 f(z−1)=N−1∑n=0fnz−n, (2)

where the coefficients are also known as the filter taps.

Graph signal processing (GSP) extends DSP to signals with inherent structures [6], by combining algebraic and spectral graph theory with DSP. Many DSP concepts can be extended to graph signals, including frequency analysis, signal convolution, and filtering [7]. Of particular interest to this study is that a shift-invariant graph filter can be represented as a polynomial of the adjacency matrix of the graph under the condition that the characteristic polynomial and minimal polynomial of the adjacency matrix are identical [7]. [7] presented it as a sufficient condition, but argued that this condition could be ignored in practice (see Theorem 2 in [7] and the discussion).

The objective of this letter is to highlight the importance of the aforementioned condition through a rigorous analysis with a concrete example, and prove that this condition is a sufficient and necessary condition for a shift-invariant graph filter to be representable as a polynomial in . In the next section, the basic concept of GSP is discussed and the notion of a “shift-enabled” graph introduced. Section III provides the theoretical guarantee for a shift-invariant graph filter to be representable as a polynomial in . Section IV presents a class of graphs that are not “shift-enabled” and an example of a shift-invariant filter that cannot be represented as a polynomial in . Conclusions and a possible extension are discussed in Section V.

Ii Basic concept on graph signal processing

In this section, we briefly review notation and concepts of GSP that are relevant to our discussion. For more details, see[4, 6, 7, 8, 9].

GSP studies signals on graphs, where a graph is determined by its set of vertices and its weighted adjacency matrix . Furthermore, each vertex corresponds to a continuous or discrete signal element and reflects the relationship between each element, such as similarity or dependency. A graph signal is a mapping from a set of vertices to the complex field and can be expressed conveniently as a vector :

 s=(s0,s1,⋯,sN−1)T, (3)

where for each , is indexed by a node .

The adjacency matrix is also called shift matrix or shift operation which can simply be considered as a graph filter [7]. Indeed,

 ~s=As=(sN−1,s0,⋯,sN−2)T,

shifts to the next signal as shown in Fig. 1(b). The adjacency matrix of Fig. 1(a) is

 AN×N=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝00⋯0110⋯00⋮⋮⋱⋮⋮00⋯10⎞⎟ ⎟ ⎟ ⎟ ⎟⎠,

where is a cyclic matrix.

For an arbitrary graph, the graph shift at vertex is a linear combination of the signals at ’s neighbors, i.e.,

 ~sn=∑m∈{m|An,m≠0}An,msm, (4)

and can be computed locally. is a two-step shift, and so on. Since shifting on a graph signal is not obvious as in classic DSP, other shift operators have also been considered in literature (see [10] and references therein), but the adjacency matrix is the most natural and popular choice [7], and will be assumed in this letter.

With the graph shift operation defined, the shift-invariant filters can be naturally extended from classic DSP to GSP, with the shift matrix in place of the shift operation . Indeed, a graph signal filter is shift-invariant if and only if the filter and shift matrix satisfy:

 A(Hs)=H(As). (5)

This means that applying a graph shift to the filtered signal is equivalent to a graph filter applied to the shifted signal[7], that is, in a shift-invariant system the output to an input signal does not depend on when filtering is applied as shown in Fig. 2.

Note that in classic DSP, Eq. (2) is an immediate consequence of shift invariance, and hence holds for all shift-invariant filters. Similarly to Eq. (2), a shift invariant filter in GSP can be written as a polynomial in , but under the condition that (see Theorem 1 in [7]). The question is if any shift-invariant graph filter can be represented as a polynomial in , regardless of , which would form a perfect analogy with classic DSP.

Iii The sufficient and necessary condition for shift-invariant filters to be polynomials of the shift operator

In this section, we formulate the sufficient and necessary condition for the representation of shift-invariant filters as polynomials of the shift operation.

Theorem 1.

The characteristic and minimal polynomials of are equivalent, i.e., if and only if every matrix commuting with is a polynomial in . In other words, there exists a matrix polynomial , such that

 H=h(A)=h0I+h1A+⋯+hLAL,hl∈C, (6)

where as a unity matrix.

According to Theorem 1 (proof of theorem can be found in Appendix A), the condition is not just a sufficient condition but also a necessary condition for any matrix, hence any shift-invariant filter , to be representable as a polynomial in . For convenience, we introduce the notion of a “shift-enabled” graphs as follows:

Definition 1.

A graph is shift-enabled if its corresponding shift matrix satisfies . We say that is shift-enabled when the above condition is satisfied.

Note that this definition differs from a shift-invariant graph, as introduced in [11].

The condition does not hold for any matrix . However, [7] introduced the following theorem in order to bypass this condition.

Theorem 2.

For any matrix , there exists a matrix and matrix polynomial , such that and  [7].

It is argued in [7] that, based on Theorem 2, for any graph filter in , there exists a polynomial such that , thus the condition can be ignored since and filters are equivalent and holds. However, based on Theorem 1, does not necessarily exist, that is, filter cannot be represented as a polynomial of since . For example, we may have111Assuming that is positive definite and exists. and , then , where is a polynomial of but not that of .

In the following section, a concrete example of a shift-invariant filter not representable as a polynomial of the shift operation is shown. It is the intention of the authors that this example shed some light on how common intuition in DSP cannot be applied to GSP.

Iv Examples of shift-invariant filters not representable as polynomial of shift operations

To illustrate the importance of the condition , a specific example is given and extended to a class of filters.

Iv-a Example filter

For the graph topology shown in Fig. 3, the shift matrix is The characteristic polynomial is not equivalent to the minimal polynomial . Thus, according to Theorem 1, the graph is not shift-enabled and there might exist a filter not representable as a polynomial of .

Consider the filter . It can readily be verified that and thus the filter is shift-invariant. Yet, it is impossible to find polynomial representation of in terms of .

Indeed, note that

and

In general,

and , for all .

If we compare individual elements in , all powers of have the structure222Note that denotes the -element of matrix . . Thus for any polynomial , we must have . But since , for any polynomial function .

Iv-B A class of filters

The above example can be extended to the following class of filters:

 H={αH+q(A)|α∈R,q(A) is a polynomial of A}, (7)

where and are as defined in the previous subsection.

Since apparently for any polynomial and as discussed above, any filter commutes with as well. Thus any filter in is shift invariant. However, since is not representable as a polynomial of , as discussed above, so does .

From the examples presented above, we note that when the condition is violated, we may in fact find an infinite number of shift-invariant filters that are not representable as polynomials of .

Iv-C Conversion of non-shift-enabled graphs to shift-enabled graph

Let us continue our example by converting into a shift-enabled graph characterized by a shift matrix using Theorem 2 as suggested in [7].

Since is a real symmetric matrix, the Jordan canonical form of is diagonal. Decomposing , we have

, .

Note that is a diagonal matrix with the diagonal composed of the eigenvalues of , and the columns of are the corresponding eigenvector of  [12].

Now, we “convert” into by introducing a polynomial such that , obtaining

 VJV−1=A=r(~A)=r(V~JV−1)=Vr(~J)V−1. (8)

Note that , where is a diagonal matrix and if and only if each diagonal element of is distinct according to the lemma below.

Lemma 1.

If is a real symmetric matrix, then if and only if all eigenvalues of are distinct.

Proof.

If is a real symmetric matrix, then the Jordan canonical form of is a diagonal matrix with diagonal composed of eigenvalues of . Therefore, each eigenvalue in the diagonal is a Jordan block. By Lemma 2 in Appendix A, if and only if, all eigenvalues of are distinct. ∎

Now, let us write

.

Recall that needs to be distinct to ensure that . We will show that , as defined in the previous subsection, is not a shift-invariant filter to shift , regardless of the choice of . Assume that , then

 (V−1HV)~J=~J(V−1HV). (9)

Substituting , and into (9), we immediately obtain ; this contradicts the condition that must be distinct. Therefore, for any shift-enabled converted by the procedure described in Theorem 2.

Let us consider a concrete example of . Let

then

.

Note that the graph corresponding to adjacency matrix is notably different from the original, as shown in Fig. 3. In fact, the converted graph is not even undirected (symmetric) anymore (see red edge in Fig. 3 (b)).

In summary, we demonstrated via the above examples, that for a non shift-enabled graph , there exists a shift-invariant filter , that cannot be represented as a polynomial in , and this filter is not shift invariant in any shift-enabled graph , obtained through the conversion proposed in Theorem 2 [7].

Remark 1.

It would be very desirable if we could find polynomial such that instead. In that case, that commutes with will automatically commute with also. And if is shift-enabled, we would be able to find polynomial representation of in terms of . Unfortunately, the construction step suggested in Theorem 2 and [7] does not provide a way to seek such . From our example above, we need in order to have . However, since and is just a polynomial function, we must have , which contradicts the fact that must have distinct diagonal elements.

V Conclusion

Shift-invariant filters should be representable as a combination of a finite number of shifted components. Therefore, any shift-invariant filter in classic DSP can naturally be represented as a polynomial of the shift operation . The same idea may be extended to GSP. Namely, a shift-invariant filter in may still be represented as the polynomial of shift operation matrix , as pointed out in [7], but only if condition holds. In [7], it was suggested that this condition could be ignored since one can always find another matrix , with polynomial , such that , and find a filter that is shift invariant in and equivalent to , i.e., . However, we showed that may not exist, and hence Theorem 2 [7] bears little practical relevance.

We provided a concrete example that a shift-invariant filter may not be representable as a polynomial of the shift operation even after polynomial transform , depicted in [7]. This demonstrates the importance of the condition and provides a deeper understanding of shift-invariant filters under the GSP umbrella. In fact, we conjecture that may have deeper implications, and these corresponding graphs (shift-enabled graphs) that demonstrate enhanced properties of shift invariant filters may have distinct characteristics and structures apart from graphs that do not satisfy the condition.

An apparent future direction is to study rules and structures that may be used to identify the shift-enabled graphs. Moreover, it is also interesting to see if one may still construct a reasonable basis for non-shift-enabled graphs such that a filter can be represented efficiently by that basis, other than powers of shift operations.

Appendix A

We will use the following lemmas to prove Theorem 1.

Lemma 2.

For a graph adjacency matrix , if and only if each Jordan block in the Jordan canonical form of is associated with distinct eigenvalue [13].

Proof.

Let be eigenvalues of adjacency matrix , then

 pA(x)=(x−λ1)q1(x−λ2)q2⋯(x−λk)qk, (10)
 mA(x)=(x−λ1)t1(x−λ2)t2⋯(x−λk)tk (11)

where and .

Then is the maximal order of Jordan blocks associated with the eigenvalue in the Jordan canonical form of , and the sum of the order of the Jordan blocks associated to is equal to . Hence, if and only if has only one Jordan block associated to . ∎

The following lemma (Proposition 12.4.1 of [13]) provides a condition for the eigenvalues of different Jordan blocks to be distinct. The proof is omitted because of space limitation and can be found in [13].

Lemma 3.

Each Jordan block in the Jordan canonical form of is associated with distinct eigenvalue if and only if every matrix commuting with is a polynomial in [12, 13].

It readily follows from Lemma 2 and Lemma 3 that Theorem 1 holds. Note that is not only sufficient but also necessary condition for any shift invariant to be representable as a polynomial in .

Acknowledgment

The authors would like to thank Bochao Zhao, Minxiang Ye and Dusan Jakovetic for stimulating discussions. This project has received funding from the European Unionâs Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 734331.

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