Oguzhan Teke

Extending Classical Multirate Signal Processing Theory to Graphs—Part II: M-Channel Filter Banks

This paper builds upon the basic theory of multirate systems for graph signals developed in the companion paper (Part I) and studies M-channel polynomial filter banks on graphs. The behavior of such graph filter banks differs from that of classical filter banks in many ways, the precise details depending on the eigenstructure of the adjacency matrix A. It is shown that graph filter banks represent, (linear and) periodically, shift-variant systems only when A satisfies the noble identity conditions developed in Part I. It is then shown that perfect reconstruction graph filter banks can always be developed when A satisfies the eigenvector structure satisfied by M-block cyclic graphs and has distinct eigenvalues (further restrictions on eigenvalues being unnecessary for this). If A is actually M-block cyclic then these PR filter banks indeed become practical, i.e., arbitrary filter polynomial orders are possible, and there are robustness advantages. In this case, the PR condition is identical to PR in classical filter banks-any classical PR example can be converted to a graph PR filter bank on an M-block cyclic graph. It is shown that for M-block cyclic graphs with all eigenvalues on the unit circle, the frequency responses of filters have meaningful correspondence with classical filter banks. Polyphase representations are then developed for graph filter banks and utilized to develop alternate conditions for alias cancellation and perfect reconstruction, again for graphs with specific eigenstructures. It is then shown that the eigenvector condition on the graph can be relaxed by using similarity transforms.

Extending Classical Multirate Signal Processing Theory to Graphs—Part I: Fundamentals

Signal processing on graphs finds applications in many areas. In recent years, renewed interest on this topic was kindled by two groups of researchers. Narang and Ortega constructed two-channel filter banks on bipartitie graphs described by Laplacians. Sandryhaila and Moura developed the theory of linear systems, filtering, and frequency responses for the case of graphs with arbitrary adjacency matrices, and showed applications in signal compression, prediction, etc. Inspired by these contributions, this paper extends classical multirate signal processing ideas to graphs. The graphs are assumed to be general with a possibly nonsymmetric and complex adjacency matrix. The paper revisits ideas, such as noble identities, aliasing, and polyphase decompositions in graph multirate systems. Drawing such a parallel to classical systems allows one to design filter banks with polynomial filters, with lower complexity than arbitrary graph filters. It is shown that the extension of classical multirate theory to graphs is nontrivial, and requires certain mathematical restrictions on the graph. Thus, classical noble identities cannot be taken for granted. Similarly, one cannot claim that the so-called delay chain system is a perfect reconstruction system (as in classical filter banks). It will also be shown that M-partite extensions of the bipartite filter bank results will not work for M-channel filter banks, but a more restrictive condition called M-block cyclic property should be imposed. Such graphs are studied in detail. A detailed theory for M-channel filter banks is developed in a companion paper.

Graph filter banks with M-channels, maximal decimation, and perfect reconstruction

Signal processing on graphs finds applications in many areas. Motivated by recent developments, this paper studies the concept of spectrum folding (aliasing) for graph signals under the downsample-then-upsample operation. In this development, we use a special eigenvector structure that is unique to the adjacency matrix of M-block cyclic matrices. We then introduce M-channel maximally decimated filter banks. Manipulating the characteristics of the aliasing effect, we construct polynomial filter banks with perfect reconstruction property. Later we describe how we can remove the eigenvector condition by using a generalized decimator. In this study graphs are assumed to be general with a possibly non-symmetric and complex adjacency matrix.

Fundamentals of multirate graph signal processing

In this work, the fundamental blocks of multirate signal processing on graphs are analyzed. First the decimator is defined, and expander is solved accordingly. Then, noble identities and lazy filter bank for graph signals are constructed. After decimation, the length of the signal changes and the original adjacency matrix is not applicable. Therefore, such equations on graphs do not exist in general. For noble identities to exist and lazy filter bank to provide perfect reconstruction, the necessary and sufficient conditions on the graph are derived. Some graph examples, on which the conditions are satisfied, are also provided.

Uncertainty Principles and Sparse Eigenvectors of Graphs

Analysis of signals defined over graphs has been of interest in the recent years. In this regard, many concepts from the classical signal processing theory have been extended to the graph case, including uncertainty principles that study the concentration of a signal on a graph and in its graph Fourier basis (GFB). This paper advances a new way to formulate the uncertainty principle for signals defined over graphs, by using a nonlocal measure based on the notion of sparsity. To be specific, the total number of nonzero elements of a graph signal and its corresponding graph Fourier transform (GFT) is considered. A theoretical lower bound for this total number is derived, and it is shown that a nonzero graph signal and its GFT cannot be arbitrarily sparse simultaneously. When the graph has repeated eigenvalues, the GFB is not unique. Since the derived lower bound depends on the selected GFB, a method that constructs a GFB with the minimal uncertainty bound is provided. In order to find signals that achieve the derived lower bound (i.e., the most compact on the graph and in the GFB), sparse eigenvectors of the graph are investigated. It is shown that a connected graph has a 2-sparse eigenvector (of the graph Laplacian) when there exist two nodes with the same neighbors. In this case, the uncertainty bound is very low, tight, and independent of the global structure of the graph. For several examples of classical and real-world graphs, it is shown that 2-sparse eigenvectors, in fact, exist.

Sparse eigenvectors of graphs

In order to analyze signals defined over graphs, many concepts from the classical signal processing theory have been extended to the graph case. One of these concepts is the uncertainty principle, which studies the concentration of a signal on a graph and its graph Fourier basis (GFB). An eigenvector of a graph is the most localized signal in the GFB by definition, whereas it may not be localized in the vertex domain. However, if the eigenvector itself is sparse, then it is concentrated in both domains simultaneously. In this regard, this paper studies the necessary and sufficient conditions for the existence of 1, 2, and 3-sparse eigenvectors of the graph Laplacian. The provided conditions are purely algebraic and only use the adjacency information of the graph. Examples of both classical and real-world graphs with sparse eigenvectors are also presented.

Discrete uncertainty principles on graphs

This paper advances a new way to formulate the uncertainty principle for graphs, by using a non-local measure based on the notion of sparsity. The uncertainty principle is formulated based on the total number of nonzero elements in the signal and its corresponding graph Fourier transform (GFT). By providing a lower bound for this total number, it is shown that a nonzero graph signal and its GFT cannot be arbitrarily sparse simultaneously. The theoretical bound on total sparsity is derived. For several real-world graphs this bound can actually be achieved by choosing the graph signals to be appropriate eigenvectors of the graph.

Extending classical multirate signal processing theory to graphs

A variety of different areas consider signals that are defined over graphs. Motivated by the advancements in graph signal processing, this study first reviews some of the recent results on the extension of classical multirate signal processing to graphs. In these results, graphs are allowed to have directed edges. The possibly non-symmetric adjacency matrix A is treated as the graph operator. These results investigate the fundamental concepts for multirate processing of graph signals such as noble identities, aliasing, and perfect reconstruction (PR). It is shown that unless the graph satisfies some conditions, these concepts cannot be extended to graph signals in a simple manner. A structure called M-Block cyclic structure is shown to be sufficient to generalize the results for bipartite graphs on two-channels to M-channel filter banks. Many classical multirate ideas can be extended to graphs due to the unique eigenstructure of M-Block cyclic graphs. For example, the PR condition for filter banks on these graphs is identical to PR in classical theory, which allows the use of well-known filter bank design techniques. In order to utilize these results, the adjacency matrix of an M-Block cyclic graph should be given in the correct permutation. In the final part, this study proposes a spectral technique to identify the hidden M-Block cyclic structure from a graph with noisy edges whose adjacency matrix is given under a random permutation. Numerical simulation results show that the technique can recover the underlying M-Block structure in the presence of random addition and deletion of the edges.

On the Role of the Bounded Lemma in the SDP Formulation of Atomic Norm Problems

In problems involving the optimization of atomic norms, an upper bound on the dual atomic norm often arises as a constraint. For the special case of line spectral estimation, this upper bound on the dual atomic norm reduces to upper-bounding the magnitude response of a finite impulse response filter by a constant. It is well known that this can be rewritten as a semidefinite constraint, leading to an elegant semidefinite programming formulation of the atomic norm minimization problem. This result is a direct consequence of some classical results in system theory, well known for many decades. This is not detailed in the literature on atomic norms, quite understandably, because the emphasis therein is different. In fact, these connections can be found in the book by B. A. Dumitrescu, cited widely in the atomic norm literature. However, they are spread out among many different results and formulations. This letter makes the connection more clear by appealing to one simple result from system theory, thereby making it more transparent to wider audience.

Linear systems on graphs

Many interesting ideas relating to signal processing on graphs have evolved in recent years. This paper visits some basic properties in linear system theory that have not been addressed in the context of graphs. In classical discrete-time system theory, a linear system is shift-invariant if and only if it can be described using a “polynomial” transfer function H(z) (albeit of infinite order). For such a system the Fourier transform of the output, Y (ejw), at any frequency ω i does not depend on the Fourier transform of the input X(ejw) at other frequencies ωj,· ≠ ωi (alias-free property). For a linear system, this alias-free property is equivalent to shift invariance, which in turn is equivalent to the existence of a “polynomial” description (transfer function). For linear systems on graphs, however, these three properties are in general not equivalent. This paper establishes conditions under which such equivalence holds, and also places in evidence some situations where it does not.