On Consistency of Compressive Spectral Clustering

TL;DR

This paper analyzes the consistency of spectral clustering via graph filtering on the stochastic block model, focusing on how sparsity, dimensionality, and approximation errors affect community detection accuracy.

Contribution

It provides a theoretical analysis of spectral clustering with graph filtering, highlighting conditions for consistent community recovery in large graphs.

Findings

Sparsity impacts the accuracy of spectral clustering.

Dimensionality reduction influences clustering consistency.

Approximation errors in graph filtering affect community detection performance.

Abstract

Spectral clustering is one of the most popular methods for community detection in graphs. A key step in spectral clustering algorithms is the eigen decomposition of the $n{\times}n$ graph Laplacian matrix to extract its $k$ leading eigenvectors, where $k$ is the desired number of clusters among $n$ objects. This is prohibitively complex to implement for very large datasets. However, it has recently been shown that it is possible to bypass the eigen decomposition by computing an approximate spectral embedding through graph filtering of random signals. In this paper, we analyze the working of spectral clustering performed via graph filtering on the stochastic block model. Specifically, we characterize the effects of sparsity, dimensionality and filter approximation error on the consistency of the algorithm in recovering planted clusters.

Equations

$$\lVert X^{(n)}-\mathcal{X}^{(n)}O^{(n)}\rVert_{F}^{2}=o\bigg{(}\frac{(\log n)^{2}}{n\bar{\lambda}_{k_{n}}^{4}\tau_{n}^{4}}\bigg{)}.$$

$$h_{{\lambda }_k}(\lambda )=\{\begin{array}{l}1if|\lambda |\ge |{\lambda }_k|\\ 0otherwise.\end{array}$$

$$d_n>\frac{4+2\beta }{{ϵ}_1^2/2-{ϵ}_1^3/3}\log (n+k_n),$$

$$\displaystyle\lVert X^{(n)}_{R}-\mathcal{X}^{(n)}_{R}\rVert_{F}^{2}=o\bigg{(}\frac{(\log n)^{2}}{n\bar{\lambda}_{k_{n}}^{4}\tau_{n}^{4}}\bigg{)},$$

$$\Vert {{\mathcal{X}}_R}_{i\ast }^{(n)}-{{\mathcal{X}}_R}_{j\ast }^{(n)}{\Vert }_2\ge (1-{ϵ}_1)\sqrt{2/P_n}$$

$$\Vert {\tilde{X}}_R^{(n)}-X_R^{(n)}{\Vert }_F^2\le (1+{ϵ}_2)n^2{e}_n^2.$$

$$(1-{ϵ}_2)k_n-2(1+{ϵ}_2)k_n{e}_n\le \frac{1}{n}\Vert {\tilde{X}}_R^{(n)}{\Vert }_F^2\le (1+{ϵ}_2)(k_n+2k_n{e}_n+n{e}_n^2).$$

$$\Vert c_i^{(n)}-{\mathcal{X}}_{R_{i\ast }}^{(n)}{\Vert }_2<(1-{ϵ}_1)\frac{1}{\sqrt{2P_n}}\Rightarrow \Vert c_i^{(n)}-{\mathcal{X}}_{R_{i\ast }}^{(n)}{\Vert }_2<\Vert c_i^{(n)}-{\mathcal{X}}_{R_{j\ast }}^{(n)}{\Vert }_2.$$

$$\displaystyle\mathscr{M}=\Big{\{}i:\lVert c_{i}^{(n)}-\mathcal{X}_{R_{i*}}^{(n)}\rVert_{2}\geq(1-\epsilon_{1})\frac{1}{\sqrt{2P_{n}}}\Big{\}}$$

$$C={\mathrm{argmin}}_{M\in {\mathcal{C}}_{n,k}}\Vert M-{\tilde{X}}_R{\Vert }_F^2.$$

$$\displaystyle|\mathscr{M}|=o\bigg{(}P_{n}\big{(}\frac{(\log n)^{2}}{n\bar{\lambda}_{k_{n}}^{4}\tau_{n}^{4}}+n^{2}e_{n}^{2}\big{)}\bigg{)}$$

$$\displaystyle\frac{|\mathscr{M}|}{n}=o\Big{(}\frac{k^{3}}{n}(\log n)^{2}+\frac{n^{2}}{k}e_{n}^{2}\Big{)}.$$

$$\frac{4+2\beta }{{ϵ}_1^2/2-{ϵ}_1^3/3}\log (n+k),$$

$$(1-{ϵ}_1)\Vert {\mathcal{X}}_{i\ast }O{X}^T-{\mathcal{X}}_{j\ast }O{X}^T{\Vert }_2\le \Vert {{\mathcal{X}}_R}_{i\ast }-{{\mathcal{X}}_R}_{j\ast }{\Vert }_2\le (1+{ϵ}_1)\Vert {\mathcal{X}}_{i\ast }O{X}^T-{\mathcal{X}}_{j\ast }O{X}^T{\Vert }_2$$

$$(1-{ϵ}_1)\Vert X_{i\ast }{X}^T-X_{j\ast }{X}^T{\Vert }_2\le \Vert {{\mathcal{X}}_R}_{i\ast }-{{\mathcal{X}}_R}_{j\ast }{\Vert }_2\le (1+{ϵ}_1)\Vert X_{i\ast }{X}^T-X_{j\ast }{X}^T{\Vert }_2$$

$$(1-{ϵ}_1)\Vert X_{i\ast }O{X}^T-{\mathcal{X}}_{j\ast }{X}^T{\Vert }_2\le \Vert {X_R}_{i\ast }-{{\mathcal{X}}_R}_{j\ast }{\Vert }_2\le (1+{ϵ}_1)\Vert X_{i\ast }O{X}^T-{\mathcal{X}}_{j\ast }{X}^T{\Vert }_2$$

$$\Vert {{\mathcal{X}}_R}_{i\ast }-{{\mathcal{X}}_R}_{j\ast }{\Vert }_2$$

$$=(1-{ϵ}_1)\Vert ({\mathcal{X}}_{i\ast }-{\mathcal{X}}_{j\ast })O{X}^T{\Vert }_2$$

$$=(1-{ϵ}_1)\Vert {\mathcal{X}}_{i\ast }-{\mathcal{X}}_{j\ast }{\Vert }_2$$

$$\ge (1-ϵ)\sqrt{2/P_n}$$

$$\Vert X_R-{\mathcal{X}}_R{\Vert }_F^2=\sum _{i=1}^n\Vert {X_R}_{i\ast }-{{\mathcal{X}}_R}_{i\ast }{\Vert }_2^2$$

$$=(1+{ϵ}_1{)}^2\sum _{i=1}^n\Vert (X_{i\ast }O-{\mathcal{X}}_{j\ast })X^T{\Vert }_2^2$$

$$=(1+{ϵ}_1{)}^2\sum _{i=1}^n\Vert X_{i\ast }O-{\mathcal{X}}_{j\ast }{\Vert }_2^2$$

$$=(1+{ϵ}_1{)}^2\Vert X_{i\ast }O-{\mathcal{X}}_{j\ast }{\Vert }_F^2$$

$$\displaystyle=o\bigg{(}\frac{(\log n)^{2}}{n\bar{\lambda}_{k_{n}}^{4}\tau_{n}^{4}}\bigg{)}$$

$$\displaystyle\mathsf{Pr}\Big{(}\Big{|}\frac{1}{n}\lVert R^{(n)}\rVert_{F}^{2}-1\Big{|}>\epsilon_{2}\Big{)}\leq e^{-nd_{n}(\epsilon_{2}^{2}-\epsilon_{2}^{3})/4}$$

$$\Vert U({\tilde{h}}_{{\lambda }_{k_n}}(\Lambda )-h_{{\lambda }_{k_n}}(\Lambda ))U^T{\Vert }_F^2=\Vert {\tilde{h}}_{{\lambda }_{k_n}}(\Lambda )-h_{{\lambda }_{k_n}}(\Lambda ){\Vert }_F^2$$

$$\le \sum _{i=1}^n{e}_n^2=n{e}_n^2.$$

$$\Vert {\tilde{X}}_R^{(n)}-X_R^{(n)}{\Vert }_F^2=\Vert {\tilde{H}}_{{\lambda }_{k_n}}R-H_{{\lambda }_{k_n}}{R}^{(n)}{\Vert }_F^2$$

$$\le \Vert U({\tilde{h}}_{{\lambda }_{k_n}}(\Lambda )-h_{{\lambda }_{k_n}}(\Lambda ))U^T{\Vert }_F^2\Vert R^{(n)}{\Vert }_F^2$$

$$\le (1+{ϵ}_2)n^2{e}_n^2$$

$$\Vert X_R{\Vert }_F^2=\Vert X{X^{(n)}}^{T}R{\Vert }_F^2=k_n\Vert R{\Vert }_F^2$$

$$(1-{ϵ}_2)k_n\le \frac{1}{n}\Vert X_R{\Vert }_F^2\le (1+{ϵ}_2)k_n.$$

$$\displaystyle\lVert\widetilde{X}_{R}\rVert_{F}^{2}=\mathop{tr}\big{(}\widetilde{X}_{R}^{T}\widetilde{X}_{R}\big{)}$$

$$\displaystyle=\mathop{tr}\big{(}R^{T}U(\widetilde{h}_{\lambda_{k}}(\Lambda))^{2}U^{T}R\big{)}$$

$$\displaystyle=\mathop{tr}\big{(}(\widetilde{h}_{\lambda_{k}}(\Lambda))^{2}U^{T}RR^{T}U\big{)}$$

$$\displaystyle\leq\mathop{tr}\big{(}(\widetilde{h}_{\lambda_{k}}(\Lambda))^{2}\big{)}\mathop{tr}\big{(}U^{T}RR^{T}U\big{)}$$

Full text

Keywords: spectral methods, clustering, stochastic block model

1 Introduction

Detecting communities, or clusters in networks is an important problem in many fields of science (Fortunato, 2010; Jain et al., 1999). Spectral clustering is a widely used algorithm for community detection in networks (Von Luxburg, 2007) because of its strong theoretical grounding (Ng et al., 2002; Shi and Malik, 2000) and recently established consistency results (Rohe et al., 2011; Lei et al., 2015). Spectral clustering works by relaxing the NP-hard discrete optimization problem of graph partitioning, into a continuous optimization problem. As a first step, one computes the the $k$ leading eigenvectors of the graph Laplacian matrix, that gives a $k$ dimensional ’spectral’ embedding for each vertex of the graph. In the second step, one performs $k$-means on the embedding to retrieve the graph clusters.

However, computing the leading eigenvectors of the graph Laplacian requires eigen decomposition, which is very hard to compute for large datasets. Several approximate algorithms have been proposed to overcome this problem via Nyström sampling (Fowlkes et al., 2004; Li et al., 2011; Choromanska et al., 2013). While these methods do not skip the eigen decomposition, they reduce its complexity via column sampling of the Laplacian. Another class of methods use random projections to reduce the dimensionality of the dataset while obtaining an approximate spectral embedding (Sakai and Imiya, 2009; Gittens et al., 2013). On the other hand, with the emergence of signal processing on graphs (Shuman et al., 2013), there has been the development of techniques based on graph filtering that can side-step the eigen decomposition altogether (Ramasamy and Madhow, 2015; Tremblay et al., 2016b, a). While many of these approaches have been shown to work fairly well on real and synthetic datasets, a rigorous mathematical analysis is still lacking.

In this paper, we consider a variant of the compressive spectral clustering algorithm that uses graph filtering of random signals to compute an approximate spectral embedding of the graph nodes (Tremblay et al., 2016b). For a graph with $n$ nodes and $k$ clusters, the algorithm proceeds by calculating a $d$ dimensional embedding for the graph nodes, where $d$ is of the order of $\log(n)$. This compressed embedding acts as a substitute for the $k$ dimensional spectral embedding of the spectral clustering algorithm and does not need the eigen decomposition of the Laplacian. Instead, the embedding is obtained by filtering out the top $k$ frequencies for $d$ number of random graph signals using fast graph filtering.

Contributions

In this paper, we analyze the spectral clustering algorithm performed via graph filtering (Algorithm 2) using the stochastic block model (SBM). We derive a bound on the number of vertices that would be incorrectly clustered with the algorithm, and prove that the algorithm can consistently recover planted clusters from SBM under mild assumptions on the sparsity of the graph and the filter approximation used to compute the spectral embedding. For our analysis, we specifically consider the high-dimensional stochastic block model that allows for the number of clusters $k$ to grow faster than $\log(n)$. This is very important considering that the computational gains of compressive spectral algorithm is more apparent in the high-dimensional case. In proving the weak consistency of Algorithm 2, we primarily use the proof techniques from Rohe et al. (2011), which were originally used to analyze the spectral clustering algorithm under the high-dimensional SBM. Finally, we analyze our consistency result in some special cases of the block model and validate our findings with accompanying experiments.

2 Preliminaries

2.1 Notation

We use capital letters to denote matrices, and specifically their formal script versions for random matrices. We use the superscript (n) to denote matrices corresponding to a graph of $n$ nodes. We use $\lVert\cdot\rVert_{2}$ for the Euclidean norm of a vector and the spectral norm of a matrix. We use $\lVert\cdot\rVert_{F}$ for the Frobenius norm of a matrix. For a matrix $M$, we use $M_{i*}$ and $M_{*j}$ to denote the $i$th row and $j$th column respectively. We also use the standard notation $o(\cdot)$, $O(\cdot)$ and $\Omega(\cdot)$ to describe the limiting behavior of functions.

2.2 Stochastic Block Model

We consider an undirected, unweighted graph $G$ with $n$ nodes. Under SBM, each node of the graph $G$ is assigned to one of $k$ clusters or blocks via the membership matrix $Z\in\{0,1\}^{n\times k}$. $Z_{ig}=1$ if and only if the node $i$ belongs to block $g$. The SBM adjacency matrix is defined as $\mathscr{W}=ZBZ^{T}$ where $B\in[0,1]^{k\times k}$ is the block matrix, whose entry $B_{gh}$ gives the probability of an edge between nodes of cluster $g$ and cluster $h$. $B$ is full rank and symmetric. The diagonal entries of $\mathscr{W}$ are set to zero to prevent self edges. From $\mathscr{W}$, we define the degree matrix $\mathscr{D}$ such that $\mathscr{D}_{ii}=\sum_{k}\mathscr{W}_{ik}$ and the normalized Laplacian matrix $\mathscr{L}=\mathscr{D}^{-1/2}\mathscr{W}\mathscr{D}^{-1/2}$. We define $\tau_{n}=\min_{1\leq i\leq n}\mathscr{D}^{(n)}_{ii}/n$ to indicate the level of sparsity in the graph.

To generate a random graph with SBM, we sample a random adjacency matrix $W$ from it’s population version, $\mathscr{W}$. Let $D$ and $L$ represent the corresponding degree matrix and the normalized Laplacian for the sampled graph. Using Davis-Kahan theorem, it can be shown that the eigenvectors of $L$ and $\mathscr{L}$ converge asymptotically as $n$ becomes large. This is important because the spectral clustering algorithm relies on the eigenvectors of the sampled graph Laplacian $L$ to estimate the node membership $Z$.

Now, we borrow a result from Rohe et al. (2011) that shows the conditions for convergence of the leading $k$ eigenvectors of $L$ and $\mathscr{L}$.

Theorem 2.2.1** (Convergence of Eigenvalues and Eigenvectors)**

Let $W^{(n)}\in\{0,1\}^{n\times n}$ be a sequence of adjacency matrices sampled from the SBM with population matrices $\mathscr{W}^{(n)}$. Let $L^{(n)}$ and $\mathscr{L}^{(n)}$ be the corresponding graph Laplacians. Let $X^{(n)},\mathcal{X}^{(n)}\in\mathbb{R}^{n\times k_{n}}$ be the matrices that contain the eigenvectors corresponding to the leading $k_{n}$ eigenvalues of $L^{(n)}$ and $\mathscr{L}^{(n)}$ in absolute sense, respectively. Let $\bar{\lambda}_{k_{n}}$ be the least non-zero eigenvalue of $\mathscr{L}^{(n)}$.

Assumption 1** (Eigengap)**

$n^{-1/2}(\log n)^{2}=\mathcal{O}(\bar{\lambda}_{k_{n}}^{2})$**

Assumption 2** (Sparsity)**

$\tau_{n}^{2}>2/\log n$**

Under Assumptions 1 and 2, for some sequence of orthonormal matrices $O^{(n)}$,

[TABLE]

**Proof ** Theorem 2.2.1 is a special case of Theorem 2.2 from Rohe et al. (2011). The result follows by setting $S_{n}=[\bar{\lambda^{2}_{k_{n}}}/2,1]$ and $\delta_{n}=\delta^{\prime}_{n}=\bar{\lambda}^{2}_{k_{n}}/2$.  

While Assumption 1 ensures that the eigengap of $\mathscr{L}$ is high enough to enable the separability of the $k$ clusters, Assumption 2 puts a lower bound on the sparsity level of the graph. Under these two assumptions, Theorem 2.2.1 bounds the Frobenius norm of the difference between the top $k$ eigenvectors of the population and sampled versions of the graph Laplacian.

2.3 Spectral Clustering

Spectral clustering operates on the $k$ leading eigenvectors of $L$ i.e. the matrix $X$ in Theorem 2.2.1. Each row of $X$ is taken as the $k$-dimensional spectral embedding of the corresponding node, and $k$-means is performed on the new data points to retrieve the cluster membership matrix $Z$. The spectral clustering algorithm we consider is listed in Algorithm 1.

Note that the $k$-means is performed on the rows of the matrix $X$ in Algorithm 1. For this to result in the $k$ distinct clusters of the SBM, the rows belonging to nodes in different clusters must be ‘well-separated’ while the rows belonging to nodes in the same cluster must be closely spaced. This property of $X$ becomes evident from Theorem 2.3.1 that follows from the work of Rohe et al. (2011).

Theorem 2.3.1** (Separability of Clusters)**

Consider a SBM with $k$ blocks. Let $\mathscr{L}$ be the population version of the graph Laplacian. Let $\mathcal{X}\in\mathbb{R}^{n\times k}$ be the matrix containing the eigenvectors corresponding to $k$ nonzero eigenvalues of $\mathscr{L}$. Let $P$ be the number of nodes in the largest block i.e. $P=max_{1\leq j\leq k}(Z^{T}Z)_{jj}$ Then the following statements are true.

There exists a matrix $\mu\in\mathbb{R}^{k\times k}$ such that $Z\mu=\mathcal{X}$. 2. 2.

$\mathcal{X}_{i*}=\mathcal{X}_{j*}\Leftrightarrow Z_{i*}=Z_{j*}$* i.e. $\mu$ is invertible.* 3. 3.

$\lVert\mathcal{X}_{i*}-\mathcal{X}_{j*}\rVert_{2}\geq\sqrt{2/P}$* for any $Z_{i*}\neq Z_{j*}$.*

**Proof ** Statements 1 and 2 of Theorem 2.3.1 follow from Lemma 3.1 from Rohe et al. (2011). Statement 3 is equivalent to Statement D.3 from the proof of Lemma 3.2 in Rohe et al. (2011).

From Theorem 2.3.1, it is evident that performing $k$-means on the rows of $\mathcal{X}$ would retrieve the block membership of all the nodes in the graph exactly. However, the matrix $\mathcal{X}$ is hidden, and only its sampled version, $X$ can be accessed. But by theorem 2.2.1, we have that $X$ is a close approximation of $\mathcal{X}$ for large $n$. As Algorithm 1 performs $k$-means on $X$, the estimated membership matrix $\hat{Z}$ should be close to the true membership matrix $Z$.

2.4 Graph Filtering

As in Algorithm 1, extracting the top $k$ eigenvectors of the Laplacian is a key step in the spectral clustering algorithm. This can be viewed as extracting the $k$ lowest frequencies or Fourier modes of the graph Laplacian. This interpretation allows us to use the fast graph filtering approach (Tremblay et al., 2016b; Ramasamy and Madhow, 2015) to speed up the computation. We briefly describe this here.

A graph signal $y\in\mathbb{R}^{n}$ is a mapping from vertex set $V$ of a graph $G$ to $\mathbb{R}$. If the eigen decomposition of the graph Laplacian is $L=U\Lambda U^{T}$, then the graph Fourier transform of $y$ is $\hat{y}=U^{T}y$. The entries of $\hat{y}$ give the $n$ Fourier modes of the graph signal $y$. Assuming that the rows of $U$ are ordered in the decreasing order (in absolute value) of the corresponding eigenvalues, the top $k$ Fourier modes of $y$ can be obtained by $\hat{y}_{k}=X^{T}y$ where $X\in\mathbb{R}^{n\times k}$ is the matrix whose columns are the top $k$ eigenvectors of $L$.

A graph filter function $h$ is defined over $[-1,1]$, the range of eigenvalues of the normalized graph Laplacian. The filter operator in the graph domain, $h(\Lambda)$ is a diagonal matrix defined as $h(\Lambda):=\mathrm{diag}(h(\lambda_{1}),\ldots,h(\lambda_{n}))$ where $\lambda_{1},\ldots,\lambda_{n}$ are the eigenvalues of $L$ ordered in the decreasing order of absolute value. The equivalent filter operator in the spectral domain, $H\in\mathbb{R}^{n\times n}$ is defined as $H:=Uh(\Lambda)U^{T}$.

To extract the top $k$ Fourier modes of a graph signal, we use an ideal low-pass filter defined as

[TABLE]

The result of graph signal $y$ filtered through $h_{\lambda_{k}}$ is given by $y_{\lambda_{k}}=Uh_{\lambda_{k}}(\Lambda)U^{T}y=XX^{T}y$. Obviously, filtering a graph signal with the ideal filter in (1) needs the eigen decomposition of the graph Laplacian. Now we define $\widetilde{h}_{\lambda_{k}}(\lambda):=\sum_{{\ell}=0}^{p}\alpha_{\ell}\lambda^{\ell}$, an order $p$ polynomial, to be the non-ideal approximation of the filter $h_{\lambda_{k}}(\lambda)$. The filter operator in spectral domain, $\widetilde{H}_{\lambda_{k}}$ can be computed as $\widetilde{H}_{\lambda_{k}}=U\widetilde{h}_{\lambda_{k}}(\lambda)U^{T}=\sum_{{\ell}=0}^{p}\alpha_{\ell}L^{\ell}$. The signal $y$ filtered by $\widetilde{H}_{\lambda_{k}}$ can be computed as $\widetilde{y}_{k}=\sum_{{\ell}=0}^{p}\alpha_{\ell}L^{\ell}y$, which does not required the eigen decomposition of $L$. Moreover, it only involves computing $p$ matrix-vector multiplications.

The method that we use for our analysis is outlined in Algorithm 2.

3 SBM and Spectral Clustering via Graph Filtering

In this section, we lay down the building blocks that make up Algorithm 2. In 3.1 we shall see how a compressed spectral embedding can be computed with graph filtering and prove that the compressed embedding is still a close approximation of the SBM’s population version of graph Laplacian. In Section 3.2 we show the effect of using the fast graph filtering technique to compute the compressed embedding. In Section 3.3 we deal with the estimation of the $k^{th}$ eigenvalue of the graph Laplacian without resorting to eigen decomposition.

3.1 Compressed Spectral Embedding

From Algorithm 1, it seems that we need the matrix $X$ containing the $k$ most significant eigenvectors of $L$, in order to retrieve the clusters. Since we only use the rows of $X$ as data points for the subsequent $k$-means step, we only need a distance preserving embedding of the rows of $X$. In this section, we see how such an embedding can be obtained through the result of filtering random graph signals. The technique used is similar to that of Tremblay et al. (2016b), except that we employ stricter assumptions to help in proving consistency results.

Consider the matrix $R\in\mathbb{R}^{n\times d}$ whose entries are independent Gaussian random variables with mean [math] and variance $1/d$. Define $X_{R}:=H_{\lambda_{k}}R=Uh_{\lambda_{k}}(\Lambda)U^{T}R=XX^{T}R$ whose $d$ columns contain the result of filtering the corresponding $d$ columns of $R$ using the filter $h_{\lambda_{k}}$. In Theorem 3.1.1, we show that the rows of $X_{R}$ form an $\epsilon$-approximate distance preserving embedding of the rows of $X$ for sufficiently large $d$. To analyze the effect of this embedding on the true cluster centers, i.e. the $k$ unique rows of $\mathcal{X}$, we define the matrix $\mathcal{X}_{R}:=\mathcal{X}OX^{T}R$ where $O$ is the orthonormal rotation matrix as in Theorem 2.2.1. We aim to show that the separability of the true cluster centers is still ensured under the compressed embedding.

Theorem 3.1.1** (Convergence and Separability under Compressed Spectral Embedding)**

For the sequence of adjacency matrices as defined in Theorem 2.2.1, define $P_{n}=max_{1\leq j\leq k_{n}}(Z^{T}Z)_{jj}$ to be the sequence of populations of the largest block. Let $X^{(n)}_{R},\mathcal{X}^{(n)}_{R}\in\mathbb{R}^{n\times d_{n}}$ be the compressed embeddings for $X^{(n)},\mathcal{X}^{(n)}\in\mathbb{R}^{n\times k_{n}}$ as defined in Theorem 2.2.1. For $\epsilon_{1}\in[0,1]$ and $\beta>0$, if

[TABLE]

then with probability at least $1-n^{-\beta}$, we have the following under the assumptions of Theorem 2.2.1.

[TABLE]

for any $Z_{i*}\neq Z_{j*}$, where $i,j\in\{1,\cdots,n\}$.

**Proof ** See Appendix A.  

Theorem 3.1.1 is analogous to the theorems on convergence (Theorem 2.2.1) and separability (Theorem 2.3.1) of the spectral clustering algorithm. It ensures that the approximate spectral embedding $X_{R}^{(n)}$ converges to the corresponding population version $\mathcal{X}^{(n)}_{R}$ while still ensuring that the true clusters remain separable.

3.2 Efficient Computation via Fast Graph Filtering

Now, we define an additional level of approximation for the spectral embedding using the fast graph filtering technique discussed in Section 2.4. Let $\widetilde{X}_{R}:=\widetilde{H}_{\lambda_{k}}R=\sum_{{\ell}=0}^{p}\alpha_{\ell}L^{\ell}R$ to be the output of approximate filtering of the columns of $R$ where $R\in\mathbb{R}^{n\times d}$ with entries drawn from $\mathcal{N}(0,\frac{1}{d})$. Lemma 3.2.1 bounds the difference between $\widetilde{X}_{R}$ and $X_{R}$, which result from approximate and ideal filtering respectively.

Lemma 3.2.1** (Bounding the Approximate Filtering Error)**

For the sequence of adjacency matrices as defined in Theorem 2.2.1, let $\widetilde{X}^{(n)}_{R}\in\mathbb{R}^{n\times d_{n}}$ be the approximation for ${X}^{(n)}_{R}$ obtained using the polynomial filter $\widetilde{h}_{\lambda_{k_{n}}}$ (instead of the ideal filter $h_{\lambda_{k_{n}}}$). Let $\sigma(L^{(n)})$ be the spectrum of the sampled graph Laplacian $L^{(n)}$. Define the maximum absolute error in the polynomial approximation as $e_{n}=\max_{\lambda\in\sigma(L^{(n)})}|\widetilde{h}_{\lambda_{k_{n}}}(\lambda)-h_{\lambda_{k_{n}}}(\lambda)|$. For $\epsilon_{2}\in[0,1]$, with probability at least $1-e^{-nd_{n}(\epsilon_{2}^{2}-\epsilon_{2}^{3})/4}$,

[TABLE]

**Proof ** See Appendix A.

3.3 Estimation of $\lambda_{k}$

Lemma 3.2.1 shows that in order to achieve a fixed error bound between $X_{R}$ and $\widetilde{X}_{R}$, the polynomial approximation must be increasingly accurate as $n$ grows large. Designing such a polynomial would necessitate knowing the value of $\lambda_{k}$. In this section, we explain how that can be done without having to do the eigen decomposition of $L$. First, we state the following Lemma which bounds the output of fast graph filtering, $\widetilde{X}_{R}$.

Lemma 3.3.1** (Estimation of $\lambda_{k}$)**

For the $\widetilde{X}_{R}^{(n)}$, $e_{n}$ and $\epsilon_{2}$ given in Lemma 3.2.1, with probability at least $1-e^{-nd_{n}(\epsilon_{2}^{2}-\epsilon_{2}^{3})/4}$ we have

[TABLE]

**Proof ** See Appendix A.  

For $(2ke_{n}+ne_{n}^{2})=o(1)$, Lemma 3.3.1 shows that the output of fast graph filtering, $\widetilde{X}_{R}$ is tightly concentrated around $k$, upon normalization by $n$. This can be used to estimate $|\lambda_{k}|$ by a dichotomic search in the range $[0,1]$ as explained in Puy et al. (2016). The basic idea is to make a coarse initial guess on $|\lambda_{k}|$ in the interval $[0,1]$, compute $\widetilde{X}_{R}$ with the current estimate, and iteratively refine the estimate by comparing $\frac{1}{n}\lVert\widetilde{X}_{R}\rVert_{F}^{2}$ with $k$.

Before we move on to proving the consistency of Algorithm 2, let us summarise the results from the previous sections. We have a tractable way to estimate $|\lambda_{k}|$ without the eigen decomposition of $L$. Through Lemma 3.2.1, we know that the resultant approximate embedding will be close to the ideal compressed embedding, for reasonably accurate polynomial approximation of the ideal filter. Through Theorem 3.1.1, we showed that a compressed embedding of the $k$ leading eigenvectors of $L$ converge to the corresponding embedding on $\mathscr{L}$. We also showed that the data points corresponding to different clusters are still separable under such an embedding.

4 Consistency of Algorithm SC-GF

4.1 Deriving the Error Bound

Once we get the approximate spectral embedding of the $n$ nodes of the graph in the form of $\widetilde{X}_{R}$, we perform $k$-means with the rows of $\widetilde{X}_{R}$ as data points in $\mathbb{R}^{d}$. Let $c_{1},\cdots,c_{n}\in\mathbb{R}^{d}$ be the centroids corresponding to the $n$ rows of $\widetilde{X}_{R}$, out of which only $k$ are unique. The $k$ unique centroids correspond to the centers of the $k$ clusters. Note that the true cluster centers correspond to the rows of $\mathcal{X}_{R}$, and Theorem 3.1.1 ensures that they are separable from each other. Hence, we say that a node $i$ is correctly clustered if its $k$-means cluster center $c_{i}$ is closer to its true cluster center $\mathcal{X}_{R_{i*}}$ than it is to any other center $\mathcal{X}_{R_{j*}}$, for $j\neq i$. In the following Lemma, we lay down the sufficient condition for correctly clustering a node $i$.

Lemma 4.1.1** (Sufficient Condition for Correct Clustering)**

Let $c_{1}^{(n)},\cdots,c_{n}^{(n)}\in\mathbb{R}^{d_{n}}$ be the centroids resulting from performing $k_{n}$-means on the rows of $\widetilde{X}_{R}^{(n)}$. For $P_{n}$ and $\epsilon_{1}$ as defined in Theorem 3.1.1,

[TABLE]

for any $z_{i}\neq z_{j}$.

**Proof ** See Appendix B.  

Following the analysis in (Rohe et al., 2011), we define the set of misclustered vertices $\mathscr{M}$ as containing the vertices that do not satisfy the sufficient condition in Lemma 4.1.1.

[TABLE]

Now that we have the definition for misclustered vertices, we analyze the performance of $k$-means. Let the matrix $C\in\mathbb{R}^{n\times k}$ be the result of $k$-means clustering where the $i$th row, $c_{i}$ is the centroid corresponding to the $i$th vertex. $C\in\mathscr{C}_{n,k}$ where $\mathscr{C}_{n,k}$ represents the family of matrices with $n$ rows out of which only $k$ are unique. $C$ can be defined as

[TABLE]

The next theorem bounds the number of misclustered vertices, that is the size of the set $\mathscr{M}$.

Theorem 4.1.2** (Bound on the number of Misclustered Vertices)**

[TABLE]

**Proof ** See Appendix B.

4.2 Consistency in Special Cases

We consider a simplified SBM with four parameters $k$, $q$, $r$ and $s$ with $k$ blocks each of which contains $s$ nodes so that the total number of vertices in the graph, $n=ks$. The probability of an edge between two vertices of the same block is given by $q+r\in[0,1]$ and that of different blocks is given by $r\in[0,1]$. For the simplified SBM, the population of the largest block, $P_{n}=s$. The smallest non-zero eigenvalue of the sampled graph Laplacian $L$ is given by $\bar{\lambda}_{k_{n}}=\frac{1}{k(r/q)+1}$ and the parameter $\tau_{n}=q/k+r$ (Rohe et al., 2011). The proportion of the misclustered vertices is given by

[TABLE]

For weak consistency, we need $\lim_{n\to\infty}\frac{|\mathscr{M}|}{n}=0$. From (3), the condition on the number of clusters for weak consistency is $k=o(n^{1/3}/(\log n)^{2/3})$ and the worst case condition on the polynomial approximation error is $e_{n}=o(n^{-5/6}(\log n)^{1/3})$.

5 Experiments

We perform experiments on the simplified four parameter SBM presented in Section 4.2. For polynomial approximation of the ideal filter, we use Chebyshev polynomials with Jackson damping coefficients (Di Napoli et al., 2016).

In our first experiment, we analyze the error rate for Algorithm 2 for fixed number of clusters as the number of nodes is increased. As expected, the proportion of misclustered vertices, $\frac{|\mathscr{M}|}{n}$ tends to zero as $n$ grows large. However, for the case of high polynomial error ($p=5$) we see that the error rate diverges. This validates the presence of $e_{n}$ in (3).

In our second experiment, we analyze the effect of the polynomial error $e_{n}$ in finer detail, by fixing all the other variables, $n$, $k$, $q$ and $r$. From (3) the proportion of misclustered vertices should grow linearly with the squared polynomial error $e_{n}^{2}$. From Figure 2, this behavior is evident.

6 Conclusion

In this paper, we prove some basic theorems that provide the theoretical basis for spectral clustering done via graph filtering. By Theorem 3.1.1, we prove the fundamental conditions required for the consistency of the spectral clustering algorithm via graph filtering, namely separability and convergence. By Theorem 4.1.2, we have shown that the algorithm can retrieve the planted clusters in a stochastic block model consistently, and derive a bound on the number of misclustered vertices. Through Lemma 3.2.1 and Lemma 3.3.1, we quantify the maximum tolerable filtering error for the algorithm to succeed. We then validate our results by performing experiments on the simulated stochastic block model.

While the results we prove in this paper provide evidence for the weak consistency of Algorithm 2 under the stochastic block model under certain assumptions on sparsity and separability, several problems still remain open. First, the bound on the accuracy of the $\lambda_{k}$ estimate as given in Lemma 3.3.1 is derived in terms of the polynomial approximation error $e_{n}$. However, it is not trivial to estimate the polynomial order required to achieve a specific absolute error ($L_{1}$ norm) even in case of popular choices like the Jackson-Chebyshev polynomials Di Napoli et al. (2016). This results in complications in deriving explicit expressions for the algorithm’s computational complexity. It also remains to be seen if the algorithm remains consistent under a milder assumption on the graph sparsity ($\tau_{n}$) as is the case with the original spectral clustering algorithm Lei et al. (2015). While it is inevitable that the approximations involved in estimating $\lambda_{k}$ (Lemma 3.3.1) and in obtaining the approximate spectral embedding (Lemma 3.2.1) will result in a weaker bound on the performance, we do not know if the results we derived are optimal. With this work, we hope to see a renewed interest in graph filtering approaches to spectral algorithms which promise significant speed-ups in computation while (provably) maintaining almost the same performance.

A Proofs for Theorems in Section 3

A.1 Proof of Theorem 3.1.1

**Proof ** For the sake of compactness, we omit the superscript $(n)$ for the sequences of matrices, as the analysis is valid at every $n$.

By Theorem 2.3.1, there are at most $k$ unique rows out of the $n$ rows of the matrix $\mathcal{X}$, while the $n$ rows of the matrix $X$, can potentially be unique. The same inference can be made for the matrices $\mathcal{X}OX^{T}$ and $XX^{T}$, where $O$ is the orthonormal matrix from Theorem 2.2.1.

Treating the combined $n+k$ unique rows of the two matrices as data points in $\mathbb{R}^{k}$, we can use the Johnson-Lindenstrauss Lemma to approximately preserve the pairwise Euclidian distances between any two rows up to a factor of $\epsilon_{1}$. Applying Theorem 1.1 from Achlioptas (2003), if $d_{n}$ is larger than

[TABLE]

then with probability at least $1-n^{-\beta}$, we have

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for any $Z_{i*}\neq Z_{j*}$,

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and

[TABLE]

where $i,j\in\{1,\cdots,n\}$.

Combining the inequality on the let side of (4) with Statement 3 of Theorem 2.3.1, we get

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for any $Z_{i*}\neq Z_{j*}$. Since $X^{T}X$ is an identity matrix, the rows of $OX^{T}$ are orthogonal. Hence, multiplication of a vector by $OX^{T}$ from the right does not change the norm. By a similar procedure, combining the inequality on the right side of (5) with Theorem 2.2.1, we get

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A.2 Proof of Lemma 3.2.1

**Proof ** Firstly, we note that $\lVert R^{(n)}\rVert_{F}^{2}$ is a chi-squared random variable with $nd_{n}$ degrees of freedom and mean $n$. Using the Chernoff bound on $\lVert R^{(n)}\rVert_{F}^{2}$, we have

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Now to bound the difference between the ideal and polynomial filters,

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Using the result from (6) and (A.2), we can bound the difference between the ideal and approximate spectral embedding as follows.

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where the last step follows with a probability of at least $1-e^{-nd_{n}(\epsilon_{2}^{2}-\epsilon_{2}^{3})/4}$.

A.3 Proof of Lemma 3.3.1

**Proof ** For the sake of compactness, we omit the superscript $(n)$ for the sequences of matrices, as the analysis is valid at every $n$.

From Lemma 3.2.1, we have a bound on the term $\lVert\widetilde{X}_{R}-X_{R}\rVert_{F}^{2}$. So, we proceed to prove Lemma 3.3.1 by bounding the term $\lVert X_{R}\rVert_{F}^{2}$. For this, we make use of the fact that the $k_{n}$ columns of $X^{(n)}$ are orthonormal.

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Combining (8) with (6), we have the following with probability exceeding $1-e^{-nd_{n}(\epsilon_{2}^{2}-\epsilon_{2}^{3})/4}$.

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Now, we prove the upper bound on $\widetilde{X}_{R}$.

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where the last statement follows from the fact that the matrices $\widetilde{X}_{R}^{T}\widetilde{X}_{R}$, $(\widetilde{h}_{\lambda_{k}}(\Lambda))^{2}$ and $U^{T}RR^{T}U$ are non-negative semi-definite.

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The last statement follows from (6) with a probability of at least $1-e^{-nd_{n}(\epsilon_{2}^{2}-\epsilon_{2}^{3})/4}$.

Using the definition of the maximum filter error $e_{n}$, we get

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Combining (10) and (11) with (A.3), we get

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Now we proceed to proving the lower bound.

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[TABLE]

Here, $I_{n}\in\mathbb{R}^{n\times n}$ is the Identity matrix. By the definition of $e_{n}$, the diagonal entries of $(\widetilde{h}_{\lambda_{k}}(\Lambda)-h_{\lambda_{k}}(\Lambda)+e_{n}I_{n})$ are non-negative. Hence, the first term in (A.3) is non-negative. For the second term we have,

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In addition, the term $\lVert\widetilde{X}_{R}-X_{R}\rVert_{F}^{2}$ in (13) is non-negative. Combining (15) with (13), we get

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Putting together (12) and (16), we prove Lemma 3.3.1.

B Proofs for Theorems in Section 4

B.1 Proof of Lemma 4.1.1

**Proof ** We follow a similar technique as that of Lemma 3.2 in Rohe et al. (2011). Suppose that $\lVert c_{i}^{(n)}-\mathcal{X}_{R_{i*}}^{(n)}\rVert_{2}<(1-\epsilon_{1})\frac{1}{\sqrt{2P_{n}}}$ for some $i$. For any $z_{j}\neq z_{i}$, we have

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Here we have used the result of Theorem 3.1.1 on the separability of the rows of $\mathcal{X}_{R}^{(n)}$.

B.2 Proof of Theorem 4.1.2

**Proof ** From the output of the $k$-means we have

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From Theorem 2.3.1 we know that only $k$ rows of the matrix $\mathcal{X}$ are unique out of its $n$ rows. The same inference can be made about $\mathcal{X}_{R}$. Hence, $\mathcal{X}_{R}\in\mathscr{C}_{n,k}$. By the optimality of $k$-means we have

[TABLE]

Hence

[TABLE]

From the definition of the misclustered vertices,

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The last statement follows from Theorem 3.1.1 and Lemma 3.2.1.