DIAGONALIZABLE SHIFT AND FILTERS FOR DIRECTED GRAPHS BASED ON THE JORDAN-CHEVALLEY DECOMPOSITION Panagiotis Misiakos∗ Chris Wendler, Markus Püschel Electrical and Computer Engineering Department of Computer Science NTU Athens, Greece ETH Zürich, Switzerland the shift) are diagonalizable, have one-dimensional frequency responses, and Parseval’s theorem holds. Using these GSP ABSTRACT tools many applications for graph signals have been devel- Graph signal processing on directed graphs poses theoretical oped, e.g., for compression, sampling, denoising, label prop- challenges since an eigendecomposition of filters is in gen- agation, outlier detection and alias-free filtering [4, 6, 7, 8]. eral not available. Instead, Fourier analysis requires a Jor- In addition, graph convolutions are the foundation of graph dan decomposition and the frequency response is given by convolutional neural networks that have been applied to su- the Jordan normal form, whose computation is numerically pervised [9] and semisupervised learning tasks [10]. unstable for large sizes. In this paper, we propose to replace a Directed graphs. Unfortunately, for directed graphs (di- given adjacency shift A by a diagonalizable shift AD obtained graphs), the GSP theory does not translate as well into prac- via the Jordan-Chevalley decomposition. This means, as we tice. The reason is that the Fourier basis, given by subspaces show, that AD generates the subalgebra of all diagonalizable that are invariant under filtering, is now determined by Jordan filters and is itself a polynomial in A (i.e., a filter). For several subspaces and the frequency response by the Jordan normal synthetic and real-world graphs, we show how AD adds and form. This results in various challenges for GSP theory and removes edges compared to A. applications including: Index Terms— graph signal processing, digraphs, Jordan 1. Frequency components are no longer one-dimensional. normal form, algebraic signal processing, diagonalizable fil- 2. The Fourier basis and transform are not unitary. ters 3. The computation of the Jordan decomposition is nu- merically unstable [11, 12]. 1. INTRODUCTION There have been various attempts to overcome these prob- lems. Reference [13] replaces the Jordan basis with the ba- There is a plethora of data that is, or can be viewed as, in- sis corresponding to the block-diagonal Schur factorization, dexed by the vertices of graphs. Examples include biological which factorizes a matrix A into a block-diagonal matrix networks, social networks, or communication networks such T = F AF −1. Similar to the Jordan basis, this basis de- as the internet [1, 2]. To bring signal processing (SP) tools composes the signal space into filtering invariant subspaces, to such graph data, fundamental SP concepts including shift, but, not necessarily the irreducible ones. Reference [14] filters, Fourier transform, and frequency response, have been introduces a Hermitian Laplacian operator based on a gen- generalized to the graph domain [3, 4] and build the founda- eralization of the Hermitian adjacency matrix [15]. The tion of graph signal processing (GSP). There are two basic Hermitian Laplacian is as the name suggests Hermitian and, variants of GSP. The framework in [4] builds on algebraic by construction, captures the directions of the edges of the signal processing (ASP) [5] to derive these concepts from the underlying graph. The work in [16, 17] defines the directed definition of the shift, given by the adjacency matrix. In con- graph Fourier transform as the orthonormal basis with either trast, [3] defines the eigenbasis of the graph Laplacian as the minimal directed total variation or maximum spread, respec- graph Fourier basis. In ASP terms, it chooses the Laplacian tively. Further, [18] addresses the ambiguity in the choice of matrix as shift operator. Jordan base vectors and proposes a basis-free computation of Undirected graphs. Both approaches yield a satisfying spectral components. GSP framework for undirected graphs. Namely, since the Contributions. In this work, we stay within the GSP shift operator is symmetric, a unitary Fourier basis exists. As framework of [4] and make use of the Jordan-Chevalley de- a consequence, the shift, and thus all filters (polynomials in composition [19, 20] to derive a diagonalizable shift AD from ∗The first author conducted the research as a Summer Research Fellow at a given digraph shift (adjacency matrix) A. We show that AD ETH Zürich is a polynomial in A (i.e., a valid filter) and that it generates Copyright 2020 IEEE. Published in the IEEE 2020 International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2020), scheduled for 4-9 May, 2020, in Barcelona, Spain. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from the IEEE. Contact: Manager, Copyrights and Permissions / IEEE Service Center / 445 Hoes Lane / P.O. Box 1331 / Piscataway, NJ 08855-1331, USA. Telephone: + Intl. 908-562-3966. 00 0 2 0 0 1 01 00 1 0 0 0 0 0 1 00 0 2 0 0 1 01 the set of all diagonalizable filters. More precisely, the di- 1 B1 0 1 0 0 1 0C B0 0 1 0 0 0 0 C B1 0 1 0 0 0C B C B C B 2 C agonalizable polynomials in A are precisely the polynomials B1 0 0 0 0 0 0C B0 0 0 0 0 0 0 C B1 0 0 0 0 0 0C B C B C B 1 1 C B0 0 0 0 0 1 1C B0 0 0 0 0 0 0 C B 2 0 1 0 0 2 0C A B C B C B 1 C in D. We present prototypical experiments with synthetic B0 0 1 0 0 1 1C B0 0 0 0 0 0 0 C B 0 2 0 0 1 0C B C B p C B 2 C and real-world graphs. They show that AD often differs by a @0 0 0 0 0 0 0A @0 0 0 0 0 − 2 p0 A @0 0 0 0 0 0 0A 1 0 1 0 0 1 0 0 0 0 0 0 0 2 1 0 1 0 0 1 0 relatively small number of edges from A. This suggests that 2 −1 it might be possible to amend a graph given by A to AD to (a) A (b) J = F AF (c) AD = p(A) overcome the problems with Jordan bases. Fig. 1: (a) The adjacency matrix of our example, (b) the asso- ciated Jordan normal form of A, and (c) an associated diago- 2. SIGNAL PROCESSING ON DIRECTED GRAPHS nalizable shift derived in this paper. We briefly review graph signal processing for directed graphs (digraphs) as introduced in [4]. Let G be a weighted digraph n Each Sij denotes the subspace of C spanned by the Jordan with vertices V = fv1; : : : ; vng, edges E, and an adjacency chain corresponding to the j-th eigenvector for the i-th eigen- matrix A 2 n×n containing the weights of the edges. C value λi. The geometric multiplicity gi is the number of such Graph signal. A graph signal on G is a signal indexed by chains, i.e., the dimension of the eigenspace for λi. Invariance its vertices means that for s 2 Sij and H 2 A, we have Hs 2 Sij. s : V ! C; v 7! sv: (1) Frequency response. The frequency response of a fil- For mathematical convenience, we fix a vertex ordering and ter H = h(A) captures its action on the pure frequencies (= T columns of F −1). Thus, it is given by write the signal as column vector s = (sv1 ; : : : ; svn ) . Graph shift. GSP [4] is an instantiation of the algebraic −1 signal processing theory [5] to graphs. Hence, convolution, FHF = h(J): (4) filters and Fourier transform are derived from the definition Example 2. The frequency response of the graph shift A in of a graph shift (that we also denote with A): Fig. 1a) is given by its JNF in Fig. 1b. n n A : C ! C ; s 7! As: (2) It is worth mentioning that the standard cyclic shift used for 3. DIAGONALIZABLE DIGRAPH FILTERS finite time series is the graph shift on the directed circle graph. Filters. The corresponding graph filters are linear, shift In this section we present our main contribution. For a given invariant mappings given by polynomials in A of the form digraph shift A, we constructively derive an associated diag- onalizable shift AD. The new shift AD is a polynomial in A k X (i.e., a filter) and generates the algebra of all diagonalizable H : n ! n; s 7! h Ais: (3) C C i filters. Further, as we see later in the experiments, if AD is i=0 again interpreted as graph, it often differs from A by only a Pk i small number of edges. The matrix associated with H is i=0 hiA , which implies shift-invariance: H(As) = AH(s). The set of all such fil- ters is closed under polynomial addition and multiplication 3.1. Diagonalizable Digraph Shift and thus forms an algebra A. The filter algebra A is isomor- We use the Jordan-Chevalley decomposition of algebras [19, phic to the polynomial algebra C[x]=mA(x), where mA(x) 20] imported to the GSP setting, i.e., algebras of the form denotes the minimal polynomial of A. We write the minimal C[x]=mA(x) generated by a matrix A. Qk di polynomial of A as mA(x) = i=1(x − λi) , where the λi denote A’s distinct eigenvalues and the di the associated Theorem 1.