A) Notation

Throughout the paper, we consider signals residing on an undirected, connected, and weighted graph ${\cal G} = \lcub {\cal V}\comma \; {\cal E}\comma \; {\bf W\rcub }$, where ${\cal V}$ is a finite set of N vertices ($\vert {\cal V}\vert = N$), ${\cal E}$ is a finite set of edges, and W is the weight or adjacency matrix. The entry W _ij of W represents the weight of an edge connecting vertices i and j. We denote the complement of a set S by S ^c. A graph signal $f\colon {\cal V} \rightarrow {\open C}$ is a function assigning one value to each vertex. Such a signal f can be written as a vector of size N with the n ^th component representing the signal value at the n ^th vertex. The generalization of Fourier analysis to the graph setting requires a graph Fourier basis $\lcub u_{\ell}\rcub _{\ell \in \lcub 0\comma 1\comma \ldots\comma N-1\rcub }$. The most commonly used graph Fourier bases are the eigenvectors of the combinatorial (or non-normalized) graph Laplacian, which is defined as ${\cal L} = {\bf D} - {\bf W}$, where D is the diagonal degree matrix with diagonal entries ${\bf D}_{ii}=\sum_{j=1}^{N} W_{ij}$, and $i \in {\cal V}$, or the eigenvectors of the normalized graph Laplacian $\tilde{\cal L}={\bf D}^{-\lpar {1}/{2}\rpar } {\cal L} {\bf D}^{-\lpar {1}/{2}\rpar }$. However, the eigenbases (or Jordan eigenbases) of other matrices such as the adjacency matrix have also been used as graph Fourier bases [Reference Sandryhaila and Moura2,Reference Sandryhaila and Moura49]. All of our results in this paper hold for any choice of the graph Fourier basis. For concreteness, we use the combinatorial Laplacian, which has a complete set of orthonormal eigenvectors $\lcub u_{l}\rcub _{l \in \lcub 0\comma 1\comma \ldots\comma N-1\rcub }$ associated with the real eigenvalues $0=\lambda_{0} \lt \lambda_{1} \leq \lambda_{2} \leq \cdots \leq \lambda_{N-1} = \lambda_{\rm max}$. We denote the entire Laplacian spectrum by $\sigma\lpar \L\rpar =\lcub \lambda_{0}\comma \; \ldots\comma \; \lambda_{N-1}\rcub $. The graph Fourier transform $\hat{f}\in {\open C}^{N}$ of a function $f \in {\open C}^{N}$ defined on a graph $\cal G$ is the projection of the signal onto the orthonormal graph Fourier basis $\lcub u_{\ell}\rcub _{\ell=0\comma 1\comma \ldots\comma N-1}$, which we take to be the eigenvectors of the graph Laplacian associated with ${\cal G}$:

(3)$$\eqalign{ &\hat{f}\lpar \lambda_\ell\rpar =\langle f\comma \; u_{\ell} \rangle= \sum_{n=1}^N f\lpar n\rpar \overline{u_{\ell}\lpar n\rpar }\comma \; \cr & \quad \quad \ell \in \lcub 0\comma \; 1\comma \; \ldots\comma \; N-1\rcub .}$$

See, for example, [Reference Chung50] for more details on spectral graph theory, and [Reference Shuman, Narang, Frossard, Ortega and Vandergheynst1] for more details on signal processing on graphs.

B) Concentration measures

In order to discuss uncertainty principles, we must first introduce some concentration/sparsity measures. Throughout the paper, we use the terms sparsity and concentration somewhat interchangeably, but we reserve the term spread to describe the spread of a function around some mean or center point, as discussed in the first family of uncertainty principles in Section I. The first concentration measure is the support measure of f, denoted ‖f‖₀, which counts the number of non-zero elements of f. The second concentration measure is the Shannon entropy, which is used often in information theory and physics:

$$H\lpar f\rpar =-\displaystyle\sum_{n} \vert f\lpar n\rpar \vert ^2\ln\vert f\lpar n\rpar \vert ^2\comma \;$$

where the variable n has values in {1,2,…,N} for functions on graphs and $\lcub 0\comma \; 1\comma \; \ldots\comma \; N-1\rcub $ in the graph Fourier representation. Another class of concentration measures is the ℓ^p-norms, with $p\in\lsqb 1\comma \; \infty\rsqb $. For p ≠ 2, the sparsity of f may be measured using the following quantity:

$$s_p\lpar f\rpar = \left\{\matrix{\displaystyle{\Vert f\Vert_2 \over \Vert f\Vert_p}\comma \; &\hbox{if } 1 \leq p \leq 2 \cr \displaystyle{\Vert f\Vert_p \over \Vert f\Vert_2}\comma \; &\hbox{if } 2 \lt p \leq \infty }\right..$$

For any vector $f \in {\open C}^{N}$ and any $p \in \lsqb 1\comma \; \infty\rsqb $, $s_{p}\lpar f\rpar \in \lsqb N^{-\vert \lpar {1}/{p}\rpar -\lpar {1}/{2}\rpar \vert }\comma \; 1\rsqb $. If s _p(f) is high (close to 1), then f is sparse, and if s _p(f) is low, then f is not concentrated. Figure 2 uses some basic signals to illustrate this notion of concentration, for different values of p. In addition to sparsity, one can also relate ℓ^p-norms to the Shannon entropy via Renyi entropies (see, e.g., [Reference Rényi51,Reference Ricaud and Torrésani52] for more details).

C) Concentration of the graph Laplacian eigenvectors

The spectrum of the graph Laplacian replaces the frequencies as coordinates in the Fourier domain. For the special case of shift-invariant graphs with circulant graph Laplacians [Reference Grady and Polimeni47, Section 5.1], the Fourier eigenvectors can still be viewed as pure oscillations. However, for more general graphs (i.e., all but the most highly structured), the oscillatory behavior of the Fourier eigenvectors must be interpreted more broadly. For example, [1, Fig. 3] displays the number of zero crossings of each eigenvector; that is, for each eigenvector, the number of pairs of connected vertices where the signs of the values of the eigenvector at the connected vertices are opposite. It is generally the case that the graph Laplacian eigenvectors associated with larger eigenvalues contain more zero crossings, yielding a notion of frequency to the graph Laplacian eigenvalues. However, despite this broader notion of frequency, the graph Laplacian eigenvectors are not always globally-supported, pure oscillations like the complex exponentials. In particular, they can feature sharp peaks, meaning that some of the Fourier basis elements can be much more similar to an element of the canonical basis of Kronecker deltas on the vertices of the graph. As we will see, uncertainty principles for signals on graphs are highly affected by this phenomenon.

Fig. 3. Coherence between the graph Fourier basis and the canonical basis for the graphs described in Example 1. Top left: Comet graphs with k = 6 and k = 12 branches, all of length one except for one of length ten. Top right: Evolution of the graph Fourier coherence $\mu_{\cal G}$ with respect to k. Bottom left: Example of a modified path graph with 10 nodes. Bottom right: Evolution of the coherence of the modified path graph with respect to the distance between nodes 1 and 2. As the degree of the comet's center vertex increases or the first node of the modified path is pulled away, the coherence $\mu_{\cal G}$ tends to the limit value $\sqrt{\lpar \lpar {N-1}\rpar /{N}\rpar }$.

One way to compare a graph Fourier basis to the canonical basis is to compute the coherence between these two representations.

Definition 1 (Graph Fourier Coherence $\mu_{\cal G}$)

Let ${\cal G}$ be a graph of N vertices. Let $\lcub \delta_{i}\rcub _{i\in \lcub 1\comma 2\comma \ldots\comma N\rcub }$ denote the canonical basis of $\ell^{2}\lpar {\open C}^{N}\rpar $ of Kronecker deltas and let $\lcub u_{\ell}\rcub _{\ell\in \lcub 0\comma 1\comma \ldots\comma N-1\rcub }$ be the orthonormal basis of eigenvectors of the graph Laplacian of ${\cal G}$. The graph Fourier coherence is defined as:

$$\mu_{\cal G}=\mathop {\max }\limits_{i\comma \ell }\vert \langle \delta_i\comma \; u_{\ell}\rangle\vert =\mathop {\max }\limits_{i\comma \ell }\vert u_{\ell}\lpar i\rpar \vert =\mathop {\max }\limits_{\ell } s_{\infty}\lpar u_{\ell}\rpar .$$

This quantity measures the similarity between the two sets of vectors. If the sets possess a common vector, then $\mu_{\cal G}=1$ (the maximum possible value for $\mu_{\cal G}$). If the two sets are maximally incoherent, such as the canonical and Fourier bases in the standard discrete setting, then $\mu_{\cal G}=1/\sqrt{N}$ (the minimum possible value).

Because the graph Laplacian matrix encodes the weights of the edges of the graph, the coherence $\mu_{\cal G}$ clearly depends on the structure of the underlying graph. It remains an open question exactly how structural properties of weighted graphs such as the regularity, clustering, modularity, and other spectral properties can be linked to the concentration of the graph Laplacian eigenvectors. For certain classes of random graphs [Reference Dekel, Lee and Linial53–Reference Tran, Vu and Wang55] or large regular graphs [Reference Brooks and Lindenstrauss56], the eigenvectors have been shown to be non-localized, globally oscillating functions (i.e., $\mu_{\cal G}$ is low). Yet, empirical studies such as [Reference McGraw and Menzinger34] show that graph Laplacian eigenvectors can be highly concentrated (i.e., $\mu_{\cal G}$ can be close to 1), particularly when the degree of a vertex is much higher or lower than the degrees of other vertices in the graph. The following example illustrates how $\mu_{\cal G}$ can be influenced by the graph structure.

Example 1

In this example, we discuss two classes of graphs that can have high graph Fourier coherences. The first, called comet graphs, are studied in [Reference Saito and Woei35,Reference Nakatsukasa, Saito and Woei57]. They are composed of a star with k vertices connected to a center vertex, and a single branch of length greater than one extending from one neighbor of the center vertex (see Fig. 3, top). If we fix the length of the longest branch (it has length 10 in Fig. 3), and increase k, the number of neighbors of the center vertex, the graph Laplacian eigenvector associated with the largest eigenvalue approaches a Kronecker delta centered at the center vertex of the star. As a consequence, the coherence between the graph Fourier and the canonical bases approaches 1 as k increases.

The second class are the modified path graphs, which we use several times in this contribution. We start with a standard path graph of 10 nodes equally spaced (all edge weights are equal to one) and we move the first node out to the left; i.e., we reduce the weight between the first two nodes (see Fig. 3, bottom). The weight is related to the distance by W₁₂ = 1/d(1, 2) with d(1, 2) being the distance between nodes 1 and 2. When the weight between nodes 1 and 2 decreases, the eigenvector associated with the largest eigenvalue of the Laplacian becomes more concentrated, which increases the coherence $\mu_{\cal G}$. These two examples of simple families of graphs illustrate that the topology of the graph can impact the graph Fourier coherence, and, in turn, uncertainty principles that depend on the coherence.

In Fig. 4, we display the eigenvector associated with the largest graph Laplacian eigenvalue for a modified path graph of 100 nodes, for several values of the weight W₁₂. Observe that the shape of the eigenvector has a sharp local change at node 1.

Fig. 4. Eigenvectors associated with the largest graph Laplacian eigenvalue of the modified path graph with 100 nodes, for different values of W ₁₂. As the distance between the first two nodes increases, the eigenvector becomes sharply peaked.

Example 1 demonstrates an important point to keep in mind. A small local change in the graph structure can greatly affect the behavior of one eigenvector, and, in turn, a global quantity such as $\mu_{\cal G}$. However, intuitively, a small local change in the graph should not drastically change the processing of signal values far away, for example in a denoising or inpainting task. For this reason, in Section VI, we introduce a notion of local uncertainty that depicts how the graph is behaving locally.

Note that not only special classes of graphs or pathological graphs yield highly localized graph Laplacian eigenvectors. Rather, graphs arising in applications such as sensor or transportation networks, or graphs constructed from sampled manifolds (such as the graph sampled from manifold 2 in Fig. 1) can also have graph Fourier coherences close to 1 (see, e.g., [Reference Shuman, Ricaud and Vandergheynst23, Section 3.2] for further examples).

A) Direct applications of uncertainty principles for discrete signals

We start by applying five known uncertainty principles for discrete signals to the graph setting.

The first uncertainty principle is given by a direct application of the Elad–Bruckstein inequality [Reference Elad and Bruckstein28]. It states that the sparsity of a function in one representation limits the sparsity in a second representation. As displayed in (1), the work of [Reference Elad and Bruckstein28] holds for representations in any two bases. As we have seen, if we focus on the canonical basis $\lcub \delta_{i}\rcub _{i=1\comma \dots\comma N}$ and the graph Fourier basis $\lcub u_{\ell}\rcub _{\ell=0\comma \dots\comma N-1}$, the coherence $\mu_{\cal G}$ depends on the graph topology. For the ring graph, $\mu_{\cal G}={1}/{\sqrt{N}}$, and we recover the result from the standard discrete case (regular sampling, periodic boundary conditions). However, for graphs where $\mu_{\cal G}$ is closer to 1, the uncertainty principle (4) is much weaker and therefore less informative. For example, $\Vert \hat{f}\Vert _{0} \Vert f\Vert _{0} \geq \lpar {1}/{\mu_{\cal G}^{2}}\rpar \approx 1$ is trivially true of non-zero signals. The same caveat applies to (5), (6), and (8), the first two of which follow directly from [Reference Ricaud and Torrésani32,Reference Maassen and Uffink45], respectively, by once again specifying the canonical and graph Fourier bases.

The inequality (7) is an adaptation of [36, Eq. (4.1)] to the graph setting, using the Hausdorff–Young inequality of Theorem 2 (see next section). It states that the energy of a function in a subset of the domain is bounded from above by the size of the selected subset and the sparsity of the function in the Fourier domain. If the subset ${\cal V}_S$ is small and the function is sparse in the graph Fourier domain, this uncertainty principle limits the amount of energy of f that fits inside of the subset of $\cal V_S$. Because $\cal V_S$ can be chosen to be a local region of the domain (the graph vertex domain in our case), Folland and Sitaram [Reference Folland and Sitaram36] refer to such principles as “local uncertainty inequalities”. However, the term $\mu_{\cal G}$ in the uncertainty bound is not local in the sense that it depends on the whole graph structure and not just on the topology of the subgraph containing vertices in $\cal V_S$. The last inequality (8), a direct application of the Ghobber–Jaming inequality [31, Theorem A], also limits the extent to which a signal can be simultaneously compressed in two different bases; specifically, if a graph signal's energy is concentrated heavily enough on vertices $\cal V_S$ in the vertex domain and frequencies Λ_T in the spectral domain, then these sets cannot both be small.

The following example illustrates the relation between the graph, the concentration of a specific graph signal, and one of the uncertainty principles from Theorem 1. We return to this example in Section III-C to discuss further the limitations of these uncertainty principles featuring $\mu_{\cal G}$.

Example 2

Figure 5 shows the computation of the quantities involved in (5), with p = 1 and different ${\cal G}$'s taken to be the modified path graphs of Example 1, with different distances between the first two vertices. We show the left-hand side of (5) for two different Kronecker deltas, one centered at vertex 1, and one centered at vertex 10. We have seen in Fig. 3 that as the distance between the first two vertices increases, the coherence increases, and therefore the lower bound on the right-hand side of (5) decreases. For δ₁, the uncertainty quantity on the left-hand side of (5) follows a similar pattern. The intuition behind this is that as the weight between the first two vertices decreases, a few of the eigenvectors start to have local jumps around the first vertex (see Fig. 4). As a result, we can sparsely represent δ₁ as a linear combination of those eigenvectors and $\Vert \widehat{\delta_{1}}\Vert _{1}$ is reduced. However, since there are not any eigenvectors that are localized around the last vertex in the path graph, we cannot find a sparse linear combination of the graph Laplacian eigenvectors to represent δ₁₀. Therefore, its uncertainty quantity on the left-hand side of (5) does not follow the behavior of the lower bound.

Fig. 5. Numerical illustration of the ℓ^p-norm uncertainty principle on a sequence of modified path graphs with different mutual coherences between the canonical basis of deltas and the graph Laplacian eigenvectors. For each modified path graph, the weight W ₁₂ of the edge between the first two vertices is the reciprocal of the distance shown on the horizontal axis. The black crosses show the lower bound on the right-hand side of (5), with p = 1. The blue and red lines show the corresponding uncertainty quantity on the left-hand side of (5), for the graph signals δ₁ and δ₁₀, respectively.

B) The Hausdorff–Young inequalities for signals on graphs

The classical Hausdorff–Young inequality [Reference Reed and Simon46, Section IX.4] is a fundamental harmonic analysis result behind the intuition that a high degree of concentration of a signal in one domain (time or frequency) implies a low degree of concentration in the other domain. This relation is used in the proofs of the entropy and ℓ^p-norm uncertainty principles in the continuous setting. In this section, as we continue to explore the role of $\mu_{\cal G}$ and the differences between the Euclidean and graph settings, we extend the Hausdorff–Young inequality to graph signals.

Theorem 2

Let $\mu_{\cal G}$ be the coherence between the graph Fourier and canonical bases of a graph $\cal G$. Let p, q > 0 be such that $\lpar {1}/{p}\rpar +\lpar {1}/{q}\rpar =1$. For any signal $f \in {\open C}^{N}$ defined on $\cal G$ and 1 ≤ p ≤ 2, we have

(9)$$\Vert \hat f \Vert _q \leq \mu_{\cal G}^{1-\lpar {2}/{q}\rpar } \Vert f \Vert _p.$$

Conversely, for $2 \leq p \leq \infty$, we have

(10)$$\Vert \hat f \Vert _q \geq \mu_{\cal G}^{1-\lpar {2}/{q}\rpar } \Vert f \Vert _p.$$

The proof of Theorem 2, given in the Appendix, is an extension of the classical proof using the Riesz–Thorin interpolation theorem. In the classical (infinite dimensional) setting, the inequality only depends on p and q [Reference Beckner58]. On a finite graph, it depends on $\mu_{\cal G}$ and hence on the structure of the graph. On a ring graph with N vertices, substituting $\mu_{\cal G}={1}/{\sqrt{N}}$ into (9) coincides with the bound on the norm of the DFT that is calculated by Gilbert and Rzeszotnik in [Reference Gilbert and Rzeszotnik59].

Dividing both sides of each inequality in Theorem 2 by |f|₂ leads to bounds on the concentrations (or sparsity levels) of a graph signal and its graph Fourier transform.

Corollary 1

Let p, q > 0 be such that $\lpar {1}/{p}\rpar +\lpar {1}/{q}\rpar =1$. For any signal $f \in {\open C}^{N}$ defined on the graph $\cal G$, we have

$$\eqalign{s_p\lpar f\rpar s_q\lpar \hat{f}\rpar \leq {\mu_{\cal G}^{\vert 1-\lpar {2}/{q}\rpar \vert }}.}$$

Theorem 2 and Corollary 1 assert that the concentration or sparsity level of a graph signal in one domain (vertex or graph spectral) limits the concentration or sparsity level in the other domain. However, once again, if the coherence $\mu_{\cal G}$ is close to 1, the result is not particularly informative as $s_{p}\lpar f\rpar s_{q}\lpar \hat{f}\rpar $ is trivially upper bounded by 1. The following numerical experiment illustrates the quantities involved in the Hausdorff–Young inequalities for graph signals. We again see that as the graph Fourier coherence increases, signals may be simultaneously concentrated in both the vertex domain and the graph spectral domain.

Example 3

Continuing with the modified path graphs of Examples 1 and 2, we illustrate the bounds of the Hausdorff–Young inequalities for graph signals in Fig. 6. For this example, we take the signal f to be δ₁, a Kronecker delta centered on the first node of the modified path graph. As a consequence, $\Vert \delta_{1}\Vert _{p}=1$ for all p, which makes it easier to compare the quantities involved in the inequalities. For this example, the bounds of Theorem 2 are fairly close to the actual values of $\Vert \hat{\delta_{1}}\Vert _{q}$.

Fig. 6. Illustration of the bounds of the Hausdorff–Young inequalities for graph signals on the modified path graphs with f = δ₁. (a) The quantities in (9) and (10) for q = 1, ${4 \over 3}$, 4, and ∞. (b) The quantities in Corollary 1 for the same values of q.

Finally, we briefly examine the sharpness of these graph Hausdorff–Young inequalities. For p = q = 2, (9) and (10) becomes equalities. Moreover, for p = 1 or p = ∞, there is always at least one signal for which the inequalities (9) and (10) become equalities, respectively. Let i ₁ and i ₁ satisfy $\mu_{\cal G}=\max_{i\comma \ell}\vert{u_{\ell}\lpar i\rpar }\vert=\vert{u_{\ell_{1}}\lpar i_{1}\rpar }\vert$. For p = 1, let $f=\delta_{i_{1}}$. Then ||f||₁ = 1, and ${\Vert{\hat f}\Vert}_{\infty}=\max_{\ell} \vert\langle \delta_{i_{1}}\comma \; u_{\ell}\rangle\vert=\mu_{\cal G}$, and thus (9) is tight. For p = ∞, let $f=u_{\ell_{1}}$. Then $\Vert{f}\Vert_{\infty}=\mu_{\cal G}$, ${\Vert{\hat f}\Vert}_{1}=\Vert\widehat{u_{\ell_{1}}}\Vert_{1}=1$, and thus (10) is tight. The red curve and its bound in Fig. 6 show the tight case for p = 1 and q = ∞.

C) Limitations of global concentration-based uncertainty principles in the graph setting

The motivation for this section was twofold. First, we wanted to derive the uncertainty principles for graph signals analogous to some of those that are so fundamental for signal processing on Euclidean domains. However, we also want to highlight the limitations of this approach (the second family of uncertainty principles described in Section I) in the graph setting. The graph Fourier coherence is a global parameter that depends on the topology of the entire graph. Hence, it may be greatly influenced by a small localized change in the graph structure. For example, in the modified path graph examples above, a change in a single edge weight leads to an increased coherence, and in turn significantly weakens the uncertainty principles characterizing the concentrations of the graph signal in the vertex and spectral domains. Such examples call into question the ability of such global uncertainty principles for graph signals to accurately describe phenomena in inhomogeneous graphs. This is the primary motivation for our investigation into local uncertainty principles in Section VI. However, before getting there, we consider global uncertainty principles from the third family of uncertainty principles described in Section I that bound the concentration of the analysis coefficients of a graph signal in a time-frequency transform domain.

A) Discrete version of Lieb's uncertainty principle

The following discrete version of Lieb's uncertainty principle is partially presented in [66, Proposition 2].

Theorem 3

Define the discrete Fourier transform (DFT) as

$$\hat{f}\lsqb k\rsqb =\displaystyle{1 \over \sqrt{N}}\displaystyle\sum_{n=0}^{N-1}f\lsqb n\rsqb \exp\bigg({-i2\pi k n \over N}\bigg)\comma \;$$

and the discrete windowed Fourier transform (or discrete cross-ambiguity function) as (see, e.g., [Reference Mallat37, Section 4.2.3])

$${{\cal A}_{\cal D}}_{DWFT} f\lsqb u\comma \; k\rsqb = \sum\limits_{n = 0}^{N - 1} f \lsqb n\rsqb \overline {g\lsqb n - u\rsqb } \exp \left({\displaystyle{{ - i2\pi kn} \over N}} \right).$$

For two discrete signals of period N, we have for $2 \le p \lt \infty$

(14)$$\eqalign{ \Vert {{\cal A}_{\cal D}}_{DWFT} f \Vert _p &= \left(\sum_{u=1}^N \sum_{k=0}^{N-1} \vert {{\cal A}_{\cal D}}_{DWFT} f \lsqb u\comma \; k\rsqb \vert^p\right)^{{1 \over p}} \cr &\leq N^{1 \over p}\Vert f \Vert _2 \Vert g\Vert _2\comma \; }$$

and for 1 ≤ p ≤ 2

(15)$$\eqalign{ \Vert {{\cal A}_{\cal D}}_{DWFT} f \Vert _p &= \left(\sum_{u=1}^N \sum_{k=0}^{N-1} \vert {{\cal A}_{\cal D}}_{DWFT} f \lsqb u\comma \; k\rsqb \vert^p\right)^{{1 \over p}} \cr &\geq N^{1}/{p}\Vert f \Vert _2 \Vert g\Vert _2. }$$

These inequalities are proven in the Appendix. Note that the minimizers of this uncertainty principle are the so-called “picket fence” signals, trains of regularly spaced diracs.

B) Generalization of Lieb's uncertainty principle to frames

Theorem 4

Let ${\cal D}=\lcub g_{i\comma k}\rcub $ be a frame of atoms in ${\open C}^{N}$, with lower and upper frame bounds A and B, respectively. For any signal $f \in {\open C}^{N}$ and any p ≥ 2, we have

(16)$$\Vert {\cal A}_{\cal D}f\Vert _{p}\leq B^{{1}/{p}}\lpar \max_{i\comma k}\Vert g_{i\comma k}\Vert_{2}\rpar ^{1-\lpar {2}/{p}\rpar }\Vert f\Vert _{2}.$$

For any signal $f \in {\open C}^{N}$ and any 1 ≤ p ≤ 2, we have

(17)$$\Vert {\cal A}_{\cal D}f\Vert _{p}\geq A^{{1}/{p}}\lpar \max_{i\comma k}\Vert g_{i\comma k}\Vert_{2}\rpar ^{1-\lpar {2}/{p}\rpar }\Vert f\Vert _{2}.$$

Combining (16) and (17), for any $p\in \lsqb 1\comma \; \infty\rsqb $, we have

(18)$$s_p\lpar {\cal A}_{\cal D}f\rpar \leq{B^{\min\lcub \lpar {1}/{2}\rpar \comma \lpar {1}/{p}\rpar \rcub } \over A^{\max\lcub {1}/{2}\comma {1}/{p}\rcub }}\lpar \max_{i\comma k}\Vert g_{i\comma k}\Vert_{2}\rpar ^{\vert 1-\lpar {2}/{p}\rpar \vert }.$$

When $\cal D$ is a tight frame with frame bound A, (18) reduces to

$$s_p\lpar {\cal A}_{\cal D}f\rpar \leq A^{-\vert \displaystyle\lpar {1}/{2}\rpar -\lpar {1}/{p}\rpar \vert }\lpar \max_{i\comma k}\Vert g_{i\comma k}\Vert _{2}\rpar ^{\vert 1-\lpar {2}/{p}\rpar \vert }.$$

A proof is included in the Appendix. The proof of Theorem 3 in the Appendix also demonstrates that this uncertainty principle is indeed a generalization of the discrete periodic variant of Lieb's uncertainty principle.

C) Lieb's uncertainty principle for localized spectral graph filter frames

Lemma 1 implies that

$$\max_{i\comma k} {\Vert T_i g_k\Vert}_2 \leq \sqrt{N} \mu_{\cal G} \max_k {\Vert\widehat{g_k}\Vert}_2.$$

Therefore the following is a corollary to Theorem 4 for the case of localized spectral graph filter frames.

Theorem 5

Let ${\cal D}_{\rm g}=\lcub g_{i\comma k}\rcub =\lcub T_{i} g_{k}\rcub $ be a localized spectral graph filter frame of atoms on a graph $\cal G$ generated from the sequence of filters $\hbox{g}=\lcub \widehat{g_{0}}\lpar{\cdot}\rpar \comma \; \widehat{g_{1}}\lpar{\cdot}\rpar \comma \; \ldots\comma \; \widehat{g_{K-1}}\lpar{\cdot}\rpar \rcub $. For any signal $f \in {\open C}^{N}$ on $\cal G$ and for any $p\in \lsqb 1\comma \; \infty\rsqb $, we have

(19)$$\eqalign{ s_p\lpar {\cal A}_{\rm g}f\rpar & \leq {B^{\min\lcub \lpar {1}/{2}\rpar \comma \lpar {1}/{p}\rpar \rcub } \over A^{\max\lcub \lpar {1}/{2}\rpar \comma \lpar {1}/{p}\rpar \rcub }}\lpar \max_{i\comma k}\Vert g_{i\comma k}\Vert_{2}\rpar ^{\vert 1-\lpar {2}/{p}\rpar \vert } \cr &\leq {B^{\min\lcub \lpar {1}/{2}\rpar \comma \lpar {1}/{p}\rpar \rcub } \over A^{\max\lcub \lpar {1}/{2}\rpar \comma \lpar {1}/{p}\rpar \rcub }}\left(\sqrt{N}\mu_{\cal G} \max_{k}\Vert \widehat{g_k}\Vert_{2}\right)^{\vert 1-\lpar {2}/{p}\rpar \vert }\comma \; }$$

where $A=\min_{\lambda \in \sigma\lpar \cal L\rpar } G\lpar \lambda\rpar $ is the lower frame bound and $B=\max_{\lambda \in \sigma\lpar \L\rpar } G\lpar \lambda\rpar $ is the upper frame bound. When $\cal D$ is a tight frame with frame bound A, (19) reduces to

(20)$$\eqalign{ s_p\lpar {\cal A}_{\rm g}f\rpar &\leq A^{-\vert \lpar {1}/{2}\rpar -\lpar {1}/{p}\rpar \vert }\lpar \max_{i\comma k}\Vert g_{i\comma k}\Vert_{2}\rpar ^{\vert 1-\lpar {2}/{p}\rpar \vert } \cr &\leq A^{-\vert \lpar {1}/{2}\rpar -\lpar {1}/{p}\rpar \vert }\left(\sqrt{N}\mu_{\cal G} \max_{k}\Vert \widehat{g_k}\Vert_{2}\right)^{\vert 1-\lpar {2}/{p}\rpar \vert }. }$$

The bounds depend on the frame bounds A and B, which are fixed with the design of the filter bank. However, in the tight frame case, we can choose the filters in a manner such that the bound A does not depend on the graph structure. For example, if the ${\hat g}_k$ are defined continuously on the interval $\lsqb 0\comma \; \lambda_{\rm max}\rsqb $ and $\sum_{k=0}^{M-1}\vert {\hat g}_{k}\lpar \lambda\rpar \vert ^{2}$ is equal to a constant for all λ, A is not affected by a change in the values of the Laplacian eigenvalues, e.g., from a change in the graph structure. The second quantity, $\max_{i\comma k}\Vert g_{i\comma k}\Vert _{2}$, reveals the influence of the graph. The maximum ℓ₂-norm of the atoms depends on the filter design, but also, as discussed previously in Section IV, on the graph topology. However, the bound is not local as it depends on the maximum |g _{i, k}|₂ over all localizations i and filters k, which takes into account the entire graph structure.

The second bounds in (19) and (20) also suggest how the filters can be designed so as to improve the uncertainty bound. The quantity $\Vert \widehat{g_{k}}\Vert _{2} = \lpar \sum_{\ell} \vert {\hat g}_{k}\lpar \lambda_{\ell}\rpar \vert ^{2} \rpar $ depends on the distribution of the eigenvalues $\lambda_{\ell}$, and, as a consequence, on the graph structure. However, the distribution of the eigenvalues can be taken into account when designing the filters in order to reduce or cancel this dependency [Reference Shuman, Wiesmeyr, Holighaus and Vandergheynst22].

In the following example, we compute the first uncertainty bound in (20) for different types of graphs and filters. It provides some insight on the influence of the graph topology and filter bank design on the uncertainty bound.

Example 5

We use the techniques of [Reference Shuman, Wiesmeyr, Holighaus and Vandergheynst22] to construct four tight localized spectral graph filter frames for each of eight different graphs. Figure 8 shows an examples of the four sets of filters for a 64 node sensor network. For each graph, two of the sets of filters (b and d in Fig. 8) are adapted via warping to the distribution of the graph Laplacian eigenvalues so that each filter contains an appropriate number of eigenvalues (roughly equal in the case of translates and roughly logarithmic in the case of wavelets). The warping avoids filters containing zero or very few eigenvalues at which the filter has a non-zero value. These tight frames are designed such that A = N, and thus Theorem 5 yields

Fig. 8. Four different filter bank designs of [22], shown for a random sensor network with 64 nodes. Each colored curve is a filter defined continuously on $\lsqb 0\comma \; \lambda_{\rm max}\rsqb $, and each filter bank has 16 such filters. They are designed such that $G\lpar \lambda\rpar =1$ for all λ (black line), and thus all four designs yield tight localized spectral graph filter frames. The frame bounds here are A = B = N.

$$\eqalign{ s_{\infty}\lpar {\cal A}_{\rm g}f\rpar ={\Vert {\cal A}_{\rm g}f\Vert _\infty \over \Vert {\cal A}_{\rm g}f\Vert _2} & \leq { N^{-\lpar {1}/{2}\rpar }} \max_{i\comma k}\Vert T_ig_k\Vert _2 \cr &\leq \mu_{\cal G}\max_k \Vert \widehat{g_k}\Vert _2.}$$

Table 1 displays the values of the first concentration bound $\max_{i\comma k}\Vert T_{i}g_{k}\Vert _{2}$ for each graph and frame pair. The uncertainty bound is largest when the graph is far from a regular lattice (ring or path). As expected, the worst cases are for highly inhomogeneous graphs like the comet graph or a modified path graph with one isolated vertex. {Note also that the coherence $\mu_{\cal G}$ is very large (0.90) for the random sensor network. Because of randomness, there is a high probability that one node will be isolated, hence creating a large coherence. The choice of the filter bank may also decrease or increase the bound, depending on the graph.

The uncertainty principle in Theorem 5 bounds the concentration of the graph Gabor transform coefficients. In the next example, we examine these coefficients for a series of signals with different vertex and spectral domain localization properties.

Example 6 (Concentration of the graph Gabor coefficients for signals with varying vertex and spectral domain concentrations.)

In Fig. 9, we analyze a series of signals on a random sensor network of 100 vertices. Each signal is created by localizing a kernel $\widehat{h_{\tau}}\lpar \lambda\rpar = e^{-\lpar {\lambda^{2}}/{\lambda_{\rm max}^{2}} \rpar \tau^{2}}$ to be centered at vertex 1 (circled in black). To generate the four different signals, we vary the value of the parameter τ in the heat kernel. We plot the four localized kernels in the graph spectral and vertex domains in the first two columns, respectively. The more we “compress” ĥ in the graph spectral domain (i.e. we reduce its spectral spreading by increasing τ), the less concentrated the localized atom becomes in the vertex domain. The joint vertex-frequency representation $\vert {\cal A}_{\rm g} T_{1}h_{\tau}\lpar i\comma \; k\rpar \vert $ of each signal is shown in the third column, which illustrates the trade-off between concentration in the vertex and the spectral domains. The concentration of these graph Gabor transform coefficients is the quantity bounded by the uncertainty principle presented in Theorem 5. In the last row of the Fig. 9, τ = ∞ which leads to a Kronecker delta for the kernel and a constant on the vertex domain. On the contrary, when the kernel is constant, with τ = 0 (top row), the energy of the graph Gabor coefficients stays concentrated around one vertex but spreads along all frequencies.

Fig. 9. Graph Gabor transform of four different signals $f_{\tau}=T_{1} h_{\tau}$, with each row corresponding to a signal with a different value of the parameter τ. Each of the signals is a kernel localized to vertex 1, with the kernel to be localized equal to $\widehat{h_{\tau}}\lpar \lambda\rpar =e^{-\lpar {\lambda^{2}}/{\lambda_{{\rm max}}^{2}\rpar } \tau^{2}}$. The underlying graph is a random sensor network of 100 vertices. First column: the kernel $h_{\tau}\lpar \lambda\rpar $ is shown in red and the localized kernel $\widehat{f_{\tau}}$ is shown in blue, both in the graph spectral domain. Second column: the signal f_τ in the vertex domain (the center vertex 1 is circled). Third column: $\vert {\cal A}_{\rm g} T_{1}h_{\tau} \lpar i\comma \; k\rpar \vert $, the absolute value of the Gabor transform coefficients for each vertex i and each of the 20 frequency bands k. Fourth column: since it is hard to see where on the graph the transform coefficients are concentrated when the nodes are placed on a line in the third column, we display the value $\sum_{k=0}^{19} \vert {\cal A}_{\rm g} T_{1}h_{\tau} \lpar i\comma \; k\rpar \vert $ on each vertex i in the network. This figure illustrates the tradeoff between the vertex and the frequency concentration.

A) Local uncertainty principle

In order to probe the local uncertainty of a graph, we take a set of localized kernels in the graph spectral domain and center them at different local regions of the graph in the vertex domain. The atoms resulting from this construction are jointly localized in both the vertex and graph spectral domains, where “localized” means that the values of the function are zero or close to zero away from some reference point. By ensuring that the atoms are localized or have support within a small region of the graph, we focus on the properties of the graph in that region. In order to get a local uncertainty principle, we apply the frame operator to these localized atoms, and analyze the concentration of the resulting coefficients. In doing so, we develop an uncertainty principle relating these concentrations to the local graph structure.

To prepare for the theorem, we first state a lemma that gives a hint to how the scalar product of two localized functions depends on the graph structure and properties. In the following, we multiply two kernels ĝ and ĥ in the graph spectral domain. For notation, we represent the product of these two kernels in vertex domain as g·h.

Lemma 3

For two kernels ĝ, ĥ and two nodes i, j, the localization operator satisfies

(21)$$\langle T_{i}g\comma \; T_{j}h\rangle = \sqrt{N}T_{i}\lpar g\cdot h\rpar \lpar j\rpar \comma$$

and

(22)$$\bigg(\sum_{i}\vert \langle T_{i}g\comma \; T_{j}h \rangle \vert ^{p}\bigg)^{{1}/{p}}=\sqrt{N}\Vert T_{j}\lpar g\cdot h\rpar \Vert _{p}.$$

Equation (21) shows more clearly the conditions on the kernels and nodes under which the scalar product is small. Let us take two examples. First, suppose ${\hat g}$ and ${\hat h}$ have a compact support on the spectrum and do not overlap (kernels localized in different places), then ${\hat g} \cdot\semicolon \; {\hat h}$ is zero everywhere on the spectrum, and therefore the scalar product on the left-hand side of (21) is also equal to zero. Second, assume i and j are distant from each other. Then $\vert T_{i}\lpar g\cdot h\rpar \lpar j\rpar \vert $ is small if ĝ and ĥ are reasonably smooth. In other words, the two atoms T _i g and T _j h must be localized both in the same area of graph in the vertex domain and the same spectral region in order for the scalar product to be large. This localization depends on the atoms, but also on the graph structure.

Proof of Lemma 3:

$$\eqalign{ &\langle T_{i}g\comma \; T_{j}h \rangle \cr &\quad= \langle \widehat{T_{i}g}\comma \; \widehat{T_{j}h} \rangle = N\sum_{\ell=0}^{N-1}\hat{g}\lpar \lambda_{\ell}\rpar u_{\ell}\lpar i\rpar \bar{\hat{h}}\lpar \lambda_{\ell}\rpar \bar{u}_{\ell}\lpar j\rpar \cr &\quad= N\sum_{\ell=0}^{N-1}\lpar \hat{g}\cdot \hat{h}\rpar \lpar \lambda_{\ell}\rpar u_{\ell}\lpar i\rpar \bar{u}_{\ell}\lpar j\rpar = \sqrt{N}T_{i}\lpar g\cdot h\rpar \lpar j\rpar .}$$

Moreover, a direct computation shows

$$\eqalign{&\bigg(\sum_{i}\vert \langle T_{i}g\comma \; T_{j}h \rangle \vert ^{p}\bigg)^{{1}/{p}} \cr &\quad=\bigg(\sum_{i}\left\vert \sqrt{N}T_{j}\lpar g\cdot h\rpar \lpar i\rpar \right\vert ^{p}\bigg)^{{1}/{p}}=N^{{1}/{2}}\Vert T_{j}\lpar g\cdot h\rpar \Vert _{p}. \square}$$

The inequalities in the following theorem constitute a local uncertainty principle. The local bound depends on the localization of the atom $T_{i_{0}}g_{k_{0}}$ in the vertex and spectral domains. The center vertex i ₀ and kernel $\hat{g}_{k_{0}}$ can be chosen to be any vertex and kernel; however, the locality property of the uncertainty principle appears when $T_{i_{0}}g_{k_{0}}$ is concentrated around node i ₀ in the vertex domain and around a small portion of the spectrum in the graph spectral domain. We again measure the concentration with ℓ^p-norms.

Theorem 6 (Local uncertainty)

Let $\lcub T_{i}g_{k} \rcub _{\lcub i\in\lsqb 1\comma N\rsqb \comma k\in\lsqb 0\comma M-1\rsqb \rcub }$ be a localized spectral graph filter frame with lower frame bound A and upper frame bound B. For any $i_{0}\in\lsqb 1\comma \; N\rsqb \comma \; k_{0}\in\lsqb 0\comma \; M-1\rsqb $ such that $\Vert T_{i_{0}}g_{k_{0}}\Vert _{2}>0$, the quantity

(23)$$\eqalign{ \Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{p} &= \bigg(\sum_{k=1}^{M}\sum_{i=1}^{N}\vert \langle T_{i}g_{k}\comma \; T_{i_{0}}g_{k_{0}} \rangle \vert ^{p}\bigg)^{{1}/{p}} \cr &= \sqrt{N}\bigg(\sum_{k=1}^{M}\Vert T_{i_0}\lpar g_{k_0}\cdot g_{k}\rpar \Vert _{p}^{p}\bigg)^{{1}/{p}}\comma \; }$$

satisfies for $p \in \lsqb 1\comma \; \infty\rsqb $

(24)$$\eqalign{ & s_p\lpar {\cal A}_{\rm g}T_{i_0}g_{k_0}\rpar \cr &\quad \leq{B^{\min\lcub \lpar {1}/{p}\rpar \comma 1-\lpar {1}/{p}\rpar \rcub }\Vert T_{\tilde{i}_{i_0\comma k_0}}g_{\tilde{k}_{i_0\comma k_0}}\Vert _{2}^{\vert 1-\lpar {2}/{p}\rpar \vert } \over A^{{1}/{2}}} \cr &\quad \leq { B^{\min\lcub \lpar {1}/{p}\rpar \comma 1-\lpar {1}/{p}\rpar \rcub } \lpar \sqrt{N}\nu_{\tilde{i}_{i_0\comma k_0}} \Vert g_{\tilde{k}_{i_0\comma k_0}} \Vert _2 \rpar ^{\vert 1-\lpar {2}/{p}\rpar \vert } \over A^{{1}/{2}}}\comma \; }$$

where ν_i is defined in Lemma 1,

$$\eqalign{\tilde{k}_{i_0\comma k_0}&=\arg\, \max_{k}\Vert T_{i_0}\lpar g_{k_0}\cdot g_{k}\rpar \Vert _{\infty}\comma \; \hbox{and} \cr \tilde{i}_{i_0\comma k_0}&=\arg\, \max_{i}\vert T_{i_0}\lpar g_{k_0}\cdot g_{\tilde{k}_{i_0\comma k_0}}\rpar \lpar i\rpar \vert .}$$

The bound in (24) is local, because we get a different bound for each i ₀, k ₀ pair. For each such pair, the bound depends on the quantities $\tilde{i}_{i_{0}\comma k_{0}}\comma \; \tilde{k}_{i_{0}\comma k_{0}}$, which are maximizers over a set of all vertices and kernels, respectively; however, as we discuss in Example 7 below, $\tilde{i}_{i_{0}\comma k_{0}}$ is typically close to i ₀, and $\tilde{k}_{i_{0}\comma k_{0}}$ is typically close to k ₀. For this reason, this bound typically depends only on local quantities.

Proof [Proof of Theorem 6] For notational brevity in this proof, we omit the indices i ₀, k ₀ for the quantities ĩ and $\tilde{k}$. First, note that

$$\eqalign{\Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{\infty} &= \max_{k}\sqrt{N}\Vert T_{i_0}\lpar g_{k_0}\cdot g_{k}\rpar \Vert _{\infty} \cr &\leq\Vert T_{\tilde{i}}g_{\tilde{k}}\Vert _{2}\Vert T_{i_0}g_{k_0}\Vert _{2}\comma \; }$$

where $\tilde{k}_{i_{0}\comma k_{0}}\, =\, \hbox{arg max}_{k}\, \Vert T_{i_{0}}\lpar g_{k_{0}}\, {\cdot}\, g_{k}\rpar \Vert _{\infty}$ and $\tilde{i}_{i_{0}\comma k_{0}}\, = \hbox{arg min}_{i} \vert T_{i_{0}}\lpar g_{k_{0}}\cdot g_{\tilde{k}}\rpar \lpar i\rpar \vert $. Let us then interpolate the two following expressions:

(25)$$\Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{2}\leq B^{{1 \over 2}}\Vert T_{i_0}g_{k_0}\Vert _{2}$$

(26)$$\hbox{and} \quad \Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{\infty}\le\Vert T_{\tilde{i}}g_{\tilde{k}}\Vert _{2}\Vert T_{i_0}g_{k_0}\Vert _{2}.$$

We use the Riesz–Thorin Theorem (Theorem 8) with $p_{1}=q_{1}=p_{2}=2$, q ₂ = ∞, $M_{p}=B^{{1 \over 2}}$ and $M_{q}=\Vert T_{\tilde{i}}g_{\tilde{k}}\Vert _{2}$. Note that ${\cal A}_{\rm g}$ is a bounded operator from the Hilbert space spanned by $T_{i_{0}}g_{k_{0}}$ (isomorphic to a one-dimensional Hilbert space) to the one spanned by $\lcub T_{i_{0}}g_{k_{0}}\rcub _{i\comma k}$. We take t = 2/r ₂ and find r ₁ = 2, leading to

$$\Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{r_{2}}\le B^{{1}/{r_{2}}}\Vert T_{\tilde{i}}g_{\tilde{k}}\Vert _{2}^{1-\lpar {2}/{r_{2}\rpar }}\Vert T_{i_0}g_{k_0}\Vert _{2}.$$

Since ${\cal A}_{\rm g}$ is a frame, we also have $\Vert {\cal A}_{\rm g}T_{i_{0}}g_{k_{0}}\Vert _{2}\ge A^{{1 \over 2}}\Vert T_{i_{0}} g_{k_{0}}\Vert _{2}$, which yields:

$${\Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{2} \over \Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{p}}\ge{A^{{1}/{2}} \over B^{{1 \over p}}\Vert T_{\tilde{i}}g_{\tilde{k}}\Vert _{2}^{1-\lpar {2}/{p}\rpar }}.$$

Finally, thanks to Hölder's inequality, we have for p ≤ 2 and $\lpar {1}/{p}\rpar +\lpar {1}/{q}\rpar =1$

$$\eqalign{{\Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{2} \over \Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{p}} &\leq {\Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{q} \over \Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{2}} \cr &\leq {B^{{1}/{q}}\Vert T_{\tilde{i}}g_{\tilde{k}}\Vert _{2}^{1-\lpar {2}/{p}\rpar } \over A^{{1}/{2}}} \cr &\leq {B^{1-\lpar {1}/{p}\rpar }\Vert T_{\tilde{i}}g_{\tilde{k}}\Vert _{2}^{\lpar {2}/{p}\rpar -1} \over A^{{1 \over 2}}} \cr &\leq { B^{1-\lpar {1}/{p}\rpar } \lpar \sqrt{N}\nu_{\tilde{i}} \Vert g_{\tilde{k}} \Vert _2 \rpar ^{\lpar {2}/{p}\rpar -1} \over A^{{1}/{2}}}. \square}$$

The next corollary shows that in many cases, the local uncertainty inequality (24) is sharp (becomes an equality). To obtain this, we require that the frame ${\cal A}_{\rm g}$ is tight and $\vert \langle T_{i}g_{k}\comma \; T_{i_{0}}g_{k_{0}} \rangle\vert $ is maximized when k = k ₀ and i = i ₀.

Corollary 2

Under the assumptions of Theorem 6 and, assuming additionally

1. (i) ${\cal A}_{\rm g}$ is a tight frame with frame-bound A,

2. (ii) $k_{0} = \hbox{arg max}_{k} \Vert T_{i_{0}}\lpar g_{k}\cdot g_{k_{0}}\rpar \Vert _{\infty}$, and

3. (iii) $i_{0} = \hbox{arg max}_{j} \vert T_{i_{0}}g_{k_{0}}^{2}\lpar j\rpar \vert $,

we have

(27)$$s_\infty\lpar {\cal A}_{\rm g}T_{i_0}g_{k_0}\rpar = {\Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{\infty} \over \Vert {\cal A}_{\rm g}T_{i_0}g_{k_0} \Vert _{2}} = {\Vert T_{i_0} g_{k_0} \Vert _{2} \over A^{{1 \over 2}}}.$$

Proof The proof follows directly from the two following equalities. For the denominators, since the frame is tight, we have:

$$\Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{2} = A^{{1 \over 2}}\Vert T_{i_0}g_{k_0}\Vert _{2}.$$

For the numerators, we have

(28)$$\eqalign{ {\rm \Vert }A_{\rm g}T_{i_0}g_{k_0}{\rm \Vert }_\infty &= \max _{i\comma k}\vert \langle{T_ig_kT_{i_0}g_{k_0}} \rangle \vert \cr &= \sqrt N \max _{i\comma k}\vert T_{i_0}\lpar g_k\cdot g_{k_0}\rpar \lpar i\rpar \vert }$$

(29)$$\eqalign{ &= \sqrt{N}\max_{k} \Vert T_{i_0}\lpar g_{k}\cdot g_{k_0}\rpar \Vert _\infty \cr &= \sqrt{N}\Vert T_{i_0} g_{k_0}^2\Vert _\infty }$$

(30)$$= \sqrt{N}\vert T_{i_0}g_{k_0}^2\lpar i_0\rpar \vert$$

(31)$$\eqalign{ &= \langle{T_{i_0}g_{k_0} }\comma \; {T_{i_0} g_{k_0} }\rangle \cr &= \Vert {T_{i_0}g_{k_0}\Vert _{2}^{2} }}$$

where (28) and (31) follow from (21), (29) follows from the second hypothesis, and (30) follows from the third hypothesis.□

Corollary 3

Under the assumptions of Theorem 6, we have

(32)$$\eqalign{ s_\infty\lpar {\cal A}_{\rm g}T_{i_{0}}g_{k_{0}}\rpar = {\Vert {\cal A}_{\rm g}T_{i_{0}}g_{k_{0}}\Vert _{\infty} \over \Vert {\cal A}_{\rm g}T_{i_{0}}g_{k_{0}}\Vert _{2}}\geq {\Vert T_{i_{0}}g_{k_{0}}\Vert _{2} \over B^{{1 \over 2}}} \comma \; }$$

which is a lower bound on the concentration measure.

Proof We have

(33)$$\eqalign{ \Vert {\cal A}_{\rm g}T_{i_{0}}g_{k_{0}}\Vert _{\infty}&= \max_{i\comma k} \vert \langle{T_{i}g_{k}}{T_{i_0}g_{k_0}}\rangle \vert \cr & \geq \vert \langle{T_{i_{0}}g_{k_{0}}}{T_{i_{0}}g_{k_{0}}}\rangle\vert = \Vert T_{i_{0}}g_{k_{0}}\Vert _{2}^{2}. }$$

Additionally, because $\lcub T_{i} g_{k}\rcub _{i=1\comma 2\comma \ldots\comma N\semicolon k=0\comma 1\comma \ldots\comma M-1}$ is a frame, we have

(34)$$\eqalign{ \Vert {\cal A}_{\rm g}T_{i_{0}}g_{k_{0}}\Vert _{2}\leq B^{{1}/{2}}\Vert T_{i_{0}}g_{k_{0}}\Vert _{2}. }$$

Combining (33) and (34) yields the desired inequality in (32). □

Together, Theorem 6 and Corollary 3 yield lower and upper bounds on the local sparsity levels $s_{\infty}\lpar {\cal A}_{\rm g}T_{i_{0}}g_{k_{0}}\rpar $:

$$\eqalign{\displaystyle{{\Vert T_{\tilde{i}}g_{\tilde{k}}\Vert _{2}}\over{A^{\displaystyle{{1}\over{2}}}}} & \ge s_\infty\lpar {\cal A}_{\rm g}T_{i_{0}}g_{k_{0}}\rpar \cr & = \displaystyle{{\Vert {\cal A}_{\rm g}T_{i_{0}}g_{k_{0}}\Vert _{\infty}}\over{\Vert {\cal A}_{\rm g}T_{i_{0}}g_{k_{0}}\Vert _{2}}}\ge \displaystyle{{\Vert T_{i_{0}}g_{k_{0}}\Vert _{2}}\over{B^{\displaystyle{{1}\over{2}}}}}.}$$

B) Illustrative examples

In order to better understand this local uncertainty principle, we illustrate it with some examples.

Example 7 [Local uncertainty on a sensor network]

Let us concentrate on the case where p = ∞. Theorem 6 tells us that

(35)$$\eqalign{\displaystyle{{\Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{\infty}}\over{\Vert {\cal A}_{\rm g}T_{i_0}g_{k_0}\Vert _{2}}} & \leq \displaystyle{{\Vert T_{\tilde{i}_{i_0\comma k_0}}g_{\tilde{k}_{i_0\comma k_0}}\Vert _{2}}\over{A^{\displaystyle{{1}\over{2}}}}} \cr & \leq \displaystyle{{ \lpar \sqrt{N}\nu_{\tilde{i}_{i_0\comma k_0}} \Vert g_{\tilde{k}_{i_0\comma k_0}} \Vert _2 \rpar }\over{A^{\displaystyle{{1}\over{2}}}}}\comma \; }$$

meaning that the concentration of ${\cal A}_{\rm g}T_{i_{0}}g_{k_{0}}$ is limited by ${1}/{\Vert T_{\tilde{i}}g_{\tilde{k}_{i_{0}\comma k_{0}}}\Vert _{2}}$. One question is to what extent this quantity is local or reflects the local behavior of the graph. As a general illustration for this discussion, we present in Fig. 10 quantities related to the local uncertainty of a random sensor network of 100 nodes evaluated for two different values of k (one in each column) and all nodes i.

Fig. 10. Illustration of Theorem 6 and related variables ĩ and $\tilde{k}$ for a random sensor graph of 100 nodes. Top figure: the 8 uniformly translated kernels $\lcub \widehat{g_{k}}\rcub _{k}$ (in 8 different colors) defined on the spectrum and giving a tight frame. Each row corresponds to quantities related to the local uncertainty principle. The first column concerns the kernel (filter) in blue on the top figure, the second is associated with the orange one. On a sensor graph, the local uncertainty level (inversely proportional to the local sparsity level plotted here) is far from constant from one node to another or from one frequency band to another.

The first row (not counting the top figure) shows the local sparsity levels of ${\cal A}_{\rm g}T_{i_{0}}g_{k_{0}}$ in terms of the ℓ^∞-norm (left hand side of (35)) at each node of the graph. The second row shows the values of the upper bound on local sparsity for each node of the graph (middle term of (35)). The values of both rows are strikingly close. Note that for this type of graph, local sparsity/concentration is lowest where the nodes are well connected.

We focus now on the values of $\tilde{k}$ and ĩ as they are crucial in Theorem 6. We also give insights that explain when a tight bound is obtained, as stated in Corollary 2. There is not a simple way to determine the value of $\tilde{k}$, because it depends not only on the node i₀ and the filters $\widehat{g_{k}}$, but also on the graph Fourier basis. However, the definition $\tilde{k} = {argmax}_{k} \Vert T_{i_{0}}\lpar g_{k}\cdot g_{k_{0}}\rpar \Vert _{\infty}$ implies that the two kernels $\widehat{g_{\tilde{k}}}$ and $\widehat{g_{k_{0}}}$ have to overlap “as much as possible” in the graph Fourier domain in order to maximize the infinity-norm. In the case of a Gabor filter bank like the one presented in the first line of Fig. 10, $k_0={\tilde{k}}$ for most of the nodes. This happens because the filters $\widehat{g_{k}}$ and $\widehat{g_{k_{0}}}$ do not overlap much if k ≠ k₀, i.e when

$$\eqalign{\Vert \widehat{g_{k_{0}}}^{2}\Vert _{2}^{2} & =\sum_{\ell}\lpar \widehat{g_{k_{0}}}^{2}\lpar \lambda_\ell\rpar \rpar ^{2} \cr & \gg \sum_{\ell}\lpar \widehat{g_{k_{0}}}\lpar \lambda_\ell\rpar \widehat{g_{k}}\lpar \lambda_\ell\rpar \rpar ^{2}=\Vert \widehat{g_{k}}\cdot \widehat{g_{k_{0}}}\Vert _{2}^{2}.}$$

In fact, in the case of Fig. 10, $\tilde{k}$ is bounded between k₀ − 1 and k₀ + 1 because there is no overlap with the other filters. In Fig. 10, we plot ${\tilde{k}}(i)$ for k₀ = 0 and k₀ = 1. For the first filter, we have $\tilde{k}_{i_{0}\comma k_{0}} = k_{0}$ for all vertices i₀. The second filter follows the same rule except for two nodes. The isolated node on the north east is less connected to the rest and there is a Laplacian eigenvector well localized on it. As a consequence, the localization on the graph is affected in a counter-intuitive manner.

Let us now concentrate on the second important variable: ĩ. Under the assumption that the kernels $\widehat{g_{k}}$ are smooth, the energy of localized atoms $T_{i_{0}}g_{k}$ reside inside a ball centered at i₀ [Reference Shuman, Ricaud and Vandergheynst23]. Thus, the node j maximizing $\vert T_{i_{0}}\lpar g_{k_{0}} g_{\tilde{k}}\rpar \lpar j\rpar \vert$ cannot be far from the node i₀. Let us define the hop distance $h_{\cal G}\lpar i\comma \; j\rpar $ as the length of the shortest pathFootnote ⁴ between nodes i and j. If the kernels $\widehat{g_{k}}$ are polynomial functions of order K, the localization operator $T_{i_{0}}$ concentrates all of the energy of $T_{i_{0}}g_{k}$ inside a K-radius ball centered in i₀. Since the resulting kernel $\widehat{g_{k_{0}}} \widehat{g_{\tilde{k}}}$ is a polynomial of order 2K, ĩ will be at a distance of at most of 2K hops from the node i₀. In general, ĩ is close to i₀. In fact, the distance $h_{\cal G}\lpar i\comma \; \tilde{i}\rpar$ is related to the smoothness of the kernel $\widehat{g_{k_{0}}}\widehat{g_{\tilde{k}}}$ [Reference Shuman, Ricaud and Vandergheynst23]. To illustrate this effect, we present in Fig. 11 the average and maximum hop distance $h_{\cal G}\lpar i\comma \; \tilde{i}\rpar $. In this example, we control the concentration of a kernel ${\hat g}$ with a dilation parameter a: $\widehat{g_{a}}\lpar x\rpar = {\hat g}\lpar ax\rpar $. Increasing the factor a compresses the kernel in the Fourier domain and increases the spread of the localized atoms in the vertex domain. Note that even for high spectral compression, the hop distance $h_{\cal G}\lpar i\comma \; \tilde{i}\rpar$ remains low. Additionally, we also compute the mean relative error between $\Vert T_{i} g^{2} \Vert _{\infty}$ and $\vert T_{i} g^{2}\lpar i\rpar \vert $. This quantity asserts how well $\Vert T_{i} g \Vert _{2}^{2}$ estimates $\Vert T_{i} g^{2} \Vert _{\infty}$.Footnote ⁵ Returning to Fig. 10, the fourth row shows the hop distance between i₀ and ĩ. It never exceeds 3 for both the first and the second filter, so $\tilde{i}_{k_{0}\comma i_{0}}$ is close to i₀.

Fig. 11. Localization experiment using the sensor graph of Fig. 10. The heat kernel (top) is defined as ${\hat g}\lpar ax\rpar = e^{-{{10 \cdot ax}\over{\lambda_{\rm max}}}}$ and the wavelet kernel (middle) is defined as ${\hat g}\lpar ax\rpar = \sqrt{40} \cdot ax \cdot e^{-{{40\cdot ax}\over{\lambda {\rm max}}}}$. For a smooth kernel ${\hat g}$, the hop distance $h_{\cal G}$ between i and $\tilde{i} =\hbox{arg max}_{j} \vert T_{i}g\lpar j\rpar \vert $ is small.

In practice we cannot always determine the values of $\tilde{k}$ and ĩ, but as we have seen, the quantity $B^{-{{1}\over{2}}}\Vert T_{i}g_{k_{0}}\Vert _{2}$ may still be a good estimate of the local sparsity level. Row 5 of Fig. 10 shows these estimates, and the last row shows the relative error between these estimates and the actual local sparsity levels. We observe that for the first kernel, the estimate gives a sufficiently rough approximation of the local sparsity levels. For the second kernel, the approximation error is low for most of the nodes, but not all.

In the next example, we compare the local and global uncertainty principles on a modified path graph.

Example 8

On a 64 node modified path graph (see Example 1 for details), we compute the graph Gabor transform of the signals f₁ = T₁ g₀ and f₂ = T₆₄g₀. In Fig. 12, we show the evolution of the graph Gabor transforms of the two signals with respect to the distance d = 1/W₁₂ from the first to the second vertex in the graph. As the first node is pulled away, a localized eigenvector appears centered on the isolated vertex. Because of this, as this distance increases, the signal f₁ becomes concentrated in both the vertex and graph spectral domains, leading to graph Gabor transform coefficients that are highly concentrated (see the top right plot in Fig. 12). However, since the graph modification is local, it does not drastically affect the graph Gabor transform coefficients of the signal f₂ (middle row of Fig. 12), whose energy is concentrated on the far end of the path graph.

Fig. 12. Graph Gabor transforms of f₁ = T₁g₀ and f₂ = T₆₄g₀ for 5 different distances between vertices 1 and 2 of the modified path graph. The distance d = 1/W₁₂ is the inverse of the weight of the edge connecting the first two vertices in the path. The node 64 is not affected by the change in the graph structure, because its energy is concentrated on the opposite side of the path graph. The graph Gabor coefficients of f₁, however, become highly concentrated as a graph Laplacian eigenvector becomes localized on vertex 1 as the distance increases. The bottom row shows that as the distance between the first two vertices increases, the atom T₁ g₀ also converges to a Kronecker delta centered on vertex 1.

In Fig. 13, we plot the evolution of the uncertainty bounds as well as the concentration of the Gabor transform coefficients of f₁ and f₂. The global uncertainty bound from Theorem 5 tells us that

Fig. 13. Concentration of the graph Gabor coefficients of f₁ = T₁g₀ and f₂ = T₆₄g₀ with respect to the distance between the first two vertices in the modified path graph, along with the upper bounds on this concentration from Theorem 5 (global uncertainty) and Theorem 6 (local uncertainty). Each bump of the global bound corresponds to a local bound of a given spectral band of node 1. For clarity, we plot only bands $\widehat{g_{0}}$ and $\widehat{g_{2}}$ for node 1. For node 64, the local bound is barely affected by the change in graph structure, and the sparsity levels of the graph Gabor transform coefficients of T₆₄g₀ also do not change much.

$$s_1\lpar {\cal A}_{g}f\rpar \le \hbox{max}_{i\comma k} \Vert T_i g_k\Vert _2\comma \; \hbox{for any signal }f.$$

The local uncertainty bound from Theorem 6 tells us that

$$s_1\lpar {\cal A}_{g} T_{i_0} g_{k_0}\rpar \leq \Vert T_{\tilde{i}_{i_0\comma k_0}}g_{\tilde{k}_{i_0\comma k_0}}\Vert _2\comma \; \hbox{for all}\; i_0\; \hbox{and}\; k_0.$$

Thus, we can view the global uncertainty bound as an upper bound on all of the local uncertainty bounds. In fact the bumps in the global uncertainty bound in Fig. 13 correspond to the local bound with i₀ = 1 and different frequency bands k₀. We plot the local bounds for i₀ = 1 and k₀ = 0 and k₀ = 2.

C) Single kernel analysis

Let us focus on the case where we analyze a single kernel ${\hat g}$. Such an analysis is relevant when we model the signal as a linear combination of different localizations of a single kernel:

$$f\lpar n\rpar = \displaystyle\sum_{i=1}^N w_i T_ig\lpar n\rpar$$

This model has been proposed in different contributions [Reference Perraudin and Vandergheynst67–Reference Zhang, Florêncio and Chou69], and has also been used as an interpolation model, e.g., in [Reference Pesenson70] and [Reference Shuman, Faraji and Vandergheynst24, Section V.C]. In this case, we could ask the following question. If we measure the signal value at node j, how much information do we get about w _j? We can answer this by looking at the overlap between the atom T _jg and the other atoms. When T _jg has a large overlap with the other atoms, the value of f(j) does not tell us much about w _j. However, in the case where T _jg has a very small overlap with the other atoms (an isolated node for example), knowing f(j) gives an excellent approximation for the value of w _j. The following theorem uses the sparsity level of g(ℓ)T _jg to analyze the overlap between the atom T _jg and the other atoms.

Theorem 7

For a kernel ${\hat g}$, the overlap between the atom localized to center vertex j and the other atoms satisfies

$$\eqalign{O_{p}\lpar j\rpar & = {{\lpar \sum_{i}\vert \langle T_{i}g\comma \; T_{j}g \rangle \vert ^{p}\rpar ^{{1}/{p}}}\over{\lpar \sum_{i}\vert \langle T_{i}g\comma \; T_{j}g \rangle \vert ^{2}\rpar ^{{1}/{2}}}} \cr & = {{\Vert g\lpar {\cal L}\rpar T_{j}g\Vert _{p}}\over{\Vert g\lpar {\cal L}\rpar T_{j}g\Vert _{2}}} = {{\Vert T_{j}g^{2}\Vert _{p}}\over{\Vert T_{j}g^{2}\Vert _{2}}}}$$

Proof This result follows directly from the application of (22) in Lemma 3. □

D) Application: non-uniform sampling

Example 9 (Non-uniform sampling for graph inpainting)

In order to motivate Theorem 7 from a practical signal processing point of view, we use it to optimize the sampling of a signal over a graph. To asses the quality of the sampling, we solve a small inpainting problem where only a part of a signal is measured and the goal is to reconstruct the entire signal. Assuming that the signal varies smoothly in the vertex domain, we can formulate the inverse problem as:

(36)$$\hbox{arg} \mathop {\min }\limits_{x}\; x^T {\cal L}x \quad \hbox{s. t.}\quad y = Mx\comma$$

where y is the observed signal, M the inpainting masking operator and $x^{T} {\cal L} x$ the graph Tikhonov regularizer (${\cal L}$ being the Laplacian). In order to generate the original signal, we filter Gaussian noise on the graph with a low pass kernel ${\hat h}$. The frequency content of the resulting signal will be close to the shape of the filter ${\hat h}$. For this example, we use the low pass kernel ${\hat h}\lpar x\rpar = {1}/{\lpar 1+\lpar {100}/{\lambda}_{\rm max}\rpar x\rpar }$ to generate the smooth signal.

For a given number of measurements, the traditional idea is to randomly sample the graph. Under that strategy, the measurements are distributed across the network. Alternatively, we can use our local uncertainty principles to create an adapted mask. The intuitive idea that nodes with less uncertainty (higher local sparsity values) should be sampled with higher probability because their value can be inferred less easily from other nodes. Another way to picture this fact is the following. Imagine that we want to infer a quantity over a random sensor network. In the more densely populated parts of the network, the measurements are more correlated and redundant. As result, a lower sampling rate is necessary. On the contrary, in the parts where there are fewer sensors, the information has less redundancy and a higher sampling rate is necessary. The heat kernel $\hat{g}\lpar x\rpar =e^{-\tau x}$ is a convenient choice to probe the local uncertainty of a graph, because $\widehat{g^{2}}\lpar x\rpar =e^{-2\tau x}$ is also a heat kernel, resulting in a sparsity level depending only on $\Vert T_{j}g^{2}\Vert _{2}$. Indeed we have $\Vert T_{j}g^{2}\Vert _{1}=\sqrt{N}$. The local uncertainty bound of Theorem 7 becomes:

$$O_{1}\lpar j\rpar ={{\Vert T_{j}g^{2}\Vert _{1}}\over{\Vert T_{j}g^{2}\Vert _{2}}} = {{\sqrt{N}}\over{\Vert T_{j}g^{2}\Vert _{2}}}.$$

Based on this measure, we design a second random sampled mask with a probability proportional to $\Vert T_{i}g^{2}\Vert _{2}$; that is, the higher the overlap level at vertex j, the smaller the probability that vertex j is chosen as a sampling point, and vice-versa. For each sampling ratio, we performed 100 experiments and averaged the results. For each experiment, we also randomly generated new graphs. The experiment was carried out using open-source code: the UNLocBoX [Reference Perraudin, Shuman, Puy and Vandergheynst71] and the GSPBox [Reference Perraudin, Paratte, Shuman, Kalofolias, Vandergheynst and Hammond72]. Figure 14 presents the result of this experiment for a sensor graph and a community graph. In the sensor graph, we observe that our local measure of uncertainty varies smoothly on the graph and is higher in the more dense part. Thus, the likelihood of sampling poorly connected vertices is higher than the likelihood of sampling well connected vertices. In the community graph, we observe that the uncertainty is highly related to the size of the community. The larger the community, the larger the uncertainty (or, equivalently, the smaller the local sparsity value). In both cases, the adapted, non-uniform random sampling performs better than random uniform sampling.

Fig. 14. Comparison of random uniform sampling and random non-uniform sampling according to a distribution based on the local sparsity values. Top row: (a)-(b) The random non-uniform sampling distribution is proportional to $\Vert T_{i}g^{2}\Vert _{2}$ (for different values of i), which is shown here for sensor and community graphs with 300 vertices. (c)-(d) the errors resulting from using the different sampling methods on each graph, with the reconstruction in (36). Bottom row: an example of a single inpainting experiment. (e) the smooth signal, (f)-(g) the locations selected randomly according to the uniform and non-uniform sampling distributions, (h)-(i) the reconstructions resulting from the two different sets of samples.

Other works are also starting to use uncertainty principles to develop sampling theory for signals on graphs. In [Reference Puy, Tremblay, Gribonval and Vandergheynst73] and in [74, Algorithm 2], the cumulative coherence is used to optimize the sampling distribution. This can be seen as sampling proportionally to $\Vert T_{i}g\Vert _{2}^{2}$, where ${\hat g}$ is a specific rectangular kernel, in order to minimize the cumulative coherence of band-limited signals. In [Reference Tsitsvero, Barbarossa and Di Lorenzo42], Tsitsvero et al. make a link between uncertainty and sampling to obtain a non-probabilistic sampling method. Non-uniform random sampling is only an illustrative example in this paper. However, for the curious reader, they exists many contributions addressing the slightly different problem of active sampling [Reference Anis, Gadde and Ortega75,Reference Chen, Varma, Singh and }ević76].