Partial recovery and weak consistency in the non-uniform hypergraph stochastic block model

Abstract

We consider the community detection problem in sparse random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a $\gamma$ fraction of the vertices classified correctly, where $\gamma \in (0.5,1)$ depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in Ghoshdastidar and Dukkipati ((2017) Ann. Stat. 45(1) 289–315.) for non-uniform HSBMs.

Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularised adjacency matrix and obtain an approximate partition based on singular vectors; (3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularisation of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

1. Introduction

Clustering is one of the central problems in network analysis and machine learning [59, 60, 65]. Many clustering algorithms make use of graph models, which represent pairwise relationships among data. A well-studied probabilistic model is the stochastic block model (SBM), which was first introduced in [39] as a random graph model that generates community structure with given ground truth for clusters so that one can study algorithm accuracy. The past decades have brought many notable results in the analysis of different algorithms and fundamental limits for community detection in SBMs in different settings [20, 37, 54, 70]. A major breakthrough was the proof of phase transition behaviours of community detection algorithms in various connectivity regimes [2, 5, 12, 52, 55, 57, 58]. See the survey [1] for more references.

Hypergraphs can represent more complex relationships among data [10, 11], including recommendation systems [13, 49], computer vision [34, 73], and biological networks [53, 68], and they have been shown empirically to have advantages over graphs [79]. Besides community detection problems, sparse hypergraphs and their spectral theory have also found applications in data science [38, 40, 80], combinatorics [26, 29, 66], and statistical physics [14, 64].

With the motivation from a broad set of applications, many efforts have been made in recent years to study community detection on random hypergraphs. The hypergraph stochastic block model (HSBM), as a generalisation of graph SBM, was first introduced and studied in [31]. In this model, we observe a random uniform hypergraph where each hyperedge appears independently with some given probability depending on the community structure of the vertices in the hyperedge.

Succinctly put, the HSBM recovery problem is to find the ground truth clusters either approximately or exactly, given a sample hypergraph and estimates of model parameters. We may ask the following questions about the quality of the solutions (see [1] for further details in the graph case).

1. 1. Exact recovery (strong consistency): With high probability, find all clusters exactly (up to permutation).

2. 2. Almost exact recovery (weak consistency): With high probability, find a partition of the vertex set such that at most $o(n)$ vertices are misclassified.

3. 3. Partial recovery: Given a fixed $\gamma \in (0.5,1)$, with high probability, find a partition of the vertex set such that at least a fraction $\gamma$ of the vertices are clustered correctly.

4. 4. Weak recovery (detection): With high probability, find a partition correlated with the true partition.

For exact recovery of uniform HSBMs, it was shown that the phase transition occurs in the regime of logarithmic expected degrees in [16, 17, 50]. The thresholds are given for binary [30, 43] and multiple [77] community cases, by generalising the techniques in [2–4]. After our work appeared on arXiv, thresholds for exact recovery on non-uniform HSBMs were given by [25, 71]. Strong consistency on the degree-corrected non-uniform HSBM was studied in [24]. Spectral methods were considered in [6, 16, 21, 30, 75, 77], while semidefinite programming methods were analysed in [8, 43, 46]. Weak consistency for HSBMs was studied in [16, 17, 32, 33, 42].

For detection of the HSBM, the authors of [9] proposed a conjecture that the phase transition occurs in the regime of constant expected degrees. The positive part of the conjecture for the binary and multi-block case was solved in [62] and [67], respectively. Their algorithms can output a partition better than a random guess when above the Kesten-Stigum threshold, but can not guarantee the correctness ratio. [35, 36] proved that detection is impossible and the Kesten-Stigum threshold is tight for $m$-uniform hypergraphs with binary communities when $m =3, 4$, while KS threshold is not tight when $m\geq 7$, and some regimes remain unknown.

1.1. Non-uniform hypergraph stochastic block model

The non-uniform HSBM was first studied in [32], which removed the uniform hypergraph assumption in previous works, and it is a more realistic model to study higher-order interaction on networks [51, 73]. It can be seen as a superposition of several uniform HSBMs with different model parameters. We first define the uniform HSBM in our setting and extend it to non-uniform hypergraphs.

Definition 1.1 (Uniform HSBM). Let $V=\{V_1,\dots V_k\}$ be a partition of the set $[n]$ into $k$ blocks of size $\frac{n}{k}$ (assuming $n$ is divisible by $k$). Let $m \in \mathbb{N}$ be some fixed integer. For any set of $m$ distinct vertices $i_1,\dots i_m$, a hyperedge $\{i_1,\dots i_m\}$ is generated with probability $a_m/\binom{n}{m-1}$ if vertices $i_1,\dots i_m$ are in the same block; otherwise with probability $b_m/ \binom{n}{m-1}$. We denote this distribution on the set of $m$-uniform hypergraphs as

(1) \begin{equation} H_m\sim \mathrm{HSBM}_{m} \bigg ( \frac{n}{k}, \frac{a_m}{\binom{n}{m-1}}, \frac{b_m}{\binom{n}{m-1}} \bigg )\,. \end{equation}

Definition 1.2 (Non-uniform HSBM). Let $H = (V, E)$ be a non-uniform random hypergraph, which can be considered as a collection of $m$-uniform hypergraphs, i.e., $H = \bigcup _{m = 2}^{M} H_m$ with each $H_m$ sampled from (1).

Examples of $2$-uniform and $3$-uniform HSBM, and an example of non-uniform HSBM with ${\mathcal M} = \{2, 3\}$ and $k=3$ is displayed in Fig. 1a, b, c respectively.

Figure 1. An example of non-uniform HSBM sampled from model 1.2.

1.2. Main results

To illustrate our main results, we first introduce the concepts $\gamma$-correctness and sigal-to-noise ratio to measure the accuracy of the obtained partitions.

Definition 1.3 ($\gamma$-correctness). Suppose we have $k$ disjoint blocks $V_1,\dots, V_k$. A collection of subsets $\widehat{V}_1,\dots, \widehat{V}_k$ of $V$ is $\gamma$-correct if $|V_i\cap \widehat{V}_i|\geq \gamma |V_i|$ for all $i\in [k]$.

Definition 1.4. For model 1.2 under Assumption 1.5, we define the signal-to-noise ratio ($\mathrm{SNR}$) as

(2) \begin{align} \mathrm{SNR}_{{\mathcal M}}(k) \,:\!=\,\frac{ \left [\sum _{m\in{\mathcal M}} (m-1)\left (\frac{a_m - b_m}{k^{m-1}} \right ) \right ]^2 }{\sum _{m\in{\mathcal M}} (m-1)\left (\frac{a_m - b_m}{k^{m-1}} + b_m \right )} \,. \end{align}

Let ${\mathcal M}_{\max }$ denote the maximum element in the set $\mathcal M$. The following constant $C_{{\mathcal M}}(k)$ is used to characterise the accuracy of the clustering result,

(3) \begin{align} C_{{\mathcal M}}(k)\,:\!=\,\frac{[\nu ^{{\mathcal M}_{\max }-1} - (1-\nu )^{{\mathcal M}_{\max }-1}]^2 }{2^{3}\cdot ({\mathcal M}_{\max } - 1)^2} \cdot \bigg ( \unicode{x1D7D9}_{\{ k = 2\}} + \frac{1}{2^{2{\mathcal M}_{\max }}} \cdot \unicode{x1D7D9}_{\{ k\geq 3\}} \bigg ) \end{align}

Note that a non-uniform HSBM can be seen as a collection of noisy observations for the same underlying community structure through several uniform HSBMs of different orders. A possible issue is that some uniform hypergraphs with small SNR might not be informative (if we observe an $m$-uniform hypergraph with parameters $a_m=b_m$, including hyperedge information from it ultimately increases the noise). To improve our error rate guarantees, we start by adding a pre-processing step (Algorithm 3) for hyperedge selection according to SNR and then apply the algorithm on the sub-hypergraph with maximal SNR.

We state the following assumption that will be used in our analysis of Algorithms 1 ($k=2$) and 2 ($k\geq 3$).

Assumption 1.5. For each $m\in{\mathcal M}$, assume $a_m, b_m$ are constants independent of $n$, and $a_m \geq b_m$. Let ${\mathcal M}_{\max }$ denote the maximum element in the set $\mathcal M$. Given $\nu \in (1/k, 1)$, assume that there exists a universal constant $C$ and some $\nu$-dependent constant $C_{\nu } \gt 0$, such that

(4a) \begin{align} d\,:\!=\, \sum _{m\in{\mathcal{M}}} (m-1)a_m &\geq C\,, \end{align}

(4b) \begin{align} \sum _{m\in{\mathcal{M}}} (m-1)(a_m-b_m) &\geq C_{\nu } \sqrt{d}\cdot k^{{\mathcal{M}}_{\max }-1} \cdot \bigg ( 2^{3} \cdot \unicode{x1D7D9}_{\{ k = 2\}} + \sqrt{\log \Big (\frac{k}{1-\nu }\Big )} \cdot \unicode{x1D7D9}_{\{ k\geq 3\}} \bigg ) . \end{align}

One does not have to take too large a $C$ for (4a); for example, $C = (2^{1/{\mathcal M}_{\max }} - 1)^{-1/3}$ should suffice, but even smaller $C$ may work. Both of the two inequalities above constant prevent the hypergraph from being too sparse, while (4b) also requires that the difference between in-block and across-blocks densities is large enough. The choices of $C, C_{\nu }$ and their relationship will be discussed in Remark 5.15.

1.2.1. The 2-block case

We start with Algorithm 1, which outputs a $\gamma$-correct partition when the non-uniform HSBM $H$ is sampled from model 1.2 with only $2$ communities. Inspired by the innovative graph algorithm in [19], we generalise it to non-uniform hypergraphs while we provide a complete and detailed analysis at the same time.

Algorithm 1. Binary Partition

Theorem 1.6 ($k=2$). Let $\nu \in (0.5, 1)$ and $\rho = 2\exp ({-}C_{{\mathcal M}}(2) \cdot \mathrm{SNR}_{{\mathcal M}}(2))$ with $\mathrm{SNR}_{{\mathcal M}}(k)$, $C_{{\mathcal M}}(k)$ defined in (2), (3), and let $\gamma = \max \{\nu,\, 1 - 2\rho \}$. Then under Assumption 1.5, Algorithm 1 outputs a $\gamma$-correct partition for sufficiently large $n$ with probability at least $1 - O(n^{-2})$.

1.2.2. The $k$-block case

For the multi-community case ($k \geq 3$), another algorithm with more subroutines is developed in Algorithm 2, which outputs a $\gamma$-correct partition with high probability. We state the result as follows.

Algorithm 2. General Partition

Theorem 1.7 ($k\geq 3$). Let $\nu \in (1/k, 1)$ and $\rho = \exp ({-}C_{{\mathcal M}}(k) \cdot \mathrm{SNR}_{{\mathcal M}}(k))$ with $\mathrm{SNR}_{{\mathcal M}}(k), C_{{\mathcal M}}(k)$ defined in (2), (3), and let $\gamma = \max \{\nu,\, 1 - k\rho \}$. Then under Assumption 1.5, Algorithm 2 outputs a $\gamma$-correct partition for sufficiently large $n$ with probability at least $1 - O(n^{-2})$.

The time complexities of Algorithms 1 and 2 are $O(n^{3})$, with the bulk of time spent in Stage 1 by the spectral method.

To the best of our knowledge, Theorems 1.6 and 1.7 are the first results for partial recovery of non-uniform HSBMs. When the number of blocks is $2$, Algorithm 1 guarantees a better error rate for partial recovery as in Theorem 1.6. This happens because Algorithm 1 does not need the merging routine in Algorithm 2: if one of the communities is obtained, then the other one is also obtained via the complement.

Remark 1.8. Taking ${\mathcal M} = \{2 \}$, Theorem 1.7 can be reduced to [19, Lemma 9] for the graph case. The failure probability $O(n^{-2})$ can be decreased to $O(n^{-p})$ for any $p\gt 0$, as long as one is willing to pay the price by increasing the constants $C$, $C_v$ in (4a), (4b).

Our Algorithms 1 and 2 can be summarised in 3 steps:

1. 1. Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio (SNR) for the induced sub-hypergraph.

2. 2. Spectral partition: construct a regularised adjacency matrix and obtain an approximate partition based on singular vectors (first approximation).

3. 3. Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee (second, better approximation).

The algorithm requires the input of model parameters $a_m,b_m$, which can be estimated by counting cycles in hypergraphs as shown in [55, 74]. Estimation of the number of blocks can be done by counting the outliers in the spectrum of the non-backtracking operator, e.g., as shown (for different regimes and different parameters) in [9, 44, 63, 67].

1.2.3. Weak consistency

Throughout the proofs for Theorems 1.6 and 1.7, we make only one assumption on the growth or finiteness of $d$ and $\mathrm{SNR}_{{\mathcal M}}(k)$, and it happens in estimating the failure probability as noted in Remark 1.10. Consequently, the corollary below follows, which covers the case when $d$ and $\mathrm{SNR}_{{\mathcal M}}(k)$ grow with $n$.

Corollary 1.9 (Weak consistency). For fixed $M$ and $k$, if $\mathrm{SNR}_{{\mathcal M}}(k)$ defined in (2) goes to infinity as $n\to \infty$ and $\mathrm{SNR}_{{\mathcal M}}(k) = o(\log n)$, then with probability $1- O(n^{-2})$, Algorithms 1 and 2 output a partition with only $o(n)$ misclassified vertices.

The paper [32] also proves weak consistency for non-uniform HSBMs, but in a much denser regime than we do here ($d = \Omega (\log ^2(n))$, instead of $d=\omega (1)$, as in Corollary 1.9). In fact, we now know that strong consistency should be achievable in this denser regime, as [25] shows. When restricting to the uniform HSBM case, Corollary 1.9 achieves weak consistency under the same sparsity condition as in [7].

Remark 1.10. To be precise, Algorithms 1 and 2 work optimally in the $\mathrm{SNR}_{{\mathcal M}} = o (\log n)$ regime. When $\mathrm{SNR}_{{\mathcal M}}(k) = \Omega (\log n)$, it implies that $\rho = n^{-\Omega (1)}$, and one may have $e^{-n\rho } = \Omega (1)$ in (31), which may not decrease to $0$ as $n\to \infty$. Therefore the theoretical guarantees of Algorithms 5 and 6 may not remain valid. This, however, should not matter: in the regime when $\mathrm{SNR}_{{\mathcal M}}(k) = \Omega (\log n)$, strong (rather than weak) consistency is expected, as per [25]. Therefore, the regime of interest for weak consistency is $\mathrm{SNR}_{{\mathcal M}} = o(\log n)$.

1.3. Comparison with existing results

Although many algorithms and theoretical results have been developed for hypergraph community detection, most of them are restricted to uniform hypergraphs, and few results are known for non-uniform ones. We will discuss the most relevant results.

In [42], the authors studied the degree-corrected HSBM with general connection probability parameters by using a tensor power iteration algorithm and Tucker decomposition. Their algorithm achieves weak consistency for uniform hypergraphs when the average degree is $\omega (\log ^2 n)$, which is the regime complementary to the regime we studied here. They discussed a way to generalise the algorithm to non-uniform hypergraphs, but the theoretical analysis remains open. The recent paper [78] analysed non-uniform hypergraph community detection by using hypergraph embedding and optimisation algorithms and obtained weak consistency when the expected degrees are of $\omega (\log n)$, again a complementary regime to ours. Results on spectral norm concentration of sparse random tensors were obtained in [23, 40, 47, 61, 80], but no provable tensor algorithm in the bounded expected degree case is known. Testing for the community structure for non-uniform hypergraphs was studied in [41, 74], which is a problem different from community detection.

In our approach, we relied on knowing the tensors for each uniform hypergraph. However, in computations, we only ran the spectral algorithm on the adjacency matrix of the entire hypergraph since the stability of tensor algorithms does not yet come with guarantees due to the lack of concentration, and for non-uniform hypergraphs, $M-1$ adjacency tensors would be needed. This approach presented the challenge that, unlike for graphs, the adjacency matrix of a random non-uniform hypergraph has dependent entries, and the concentration properties of such a random matrix were previously unknown. We overcame this issue and proved concentration bounds from scratch down to the bounded degree regime. Similar to [28, 45], we provided here a regularisation analysis by removing rows in the adjacency matrix with large row sums (suggestive of large degree vertices) and proving a concentration result for the regularised matrix down to the bounded expected degree regime (see Theorem 3.3).

In terms of partial recovery for hypergraphs, our results are new, even in the uniform case. In [7, Theorem 1], for uniform hypergraphs, the authors showed detection (not partial recovery) is possible when the average degree is $\Omega (1)$; in addition, the error rate is not exponential in the model parameters, but only polynomial. Here, we mention two more results for the graph case. In the arbitrarily slowly growing degrees regime, it was shown in [27, 76] that the error rate in (2) is optimal up to a constant in the exponent. In the bounded expected degrees regime, the authors in [18, 56] provided algorithms that can asymptotically recover the optimal fraction of vertices, when the signal-to-noise ratio is large enough. It’s an open problem to extend their analysis to obtain a minimax error rate for hypergraphs.

In [32], the authors considered weak consistency in a non-uniform HSBM model with a spectral algorithm based on the hypergraph Laplacian matrix, and showed that weak consistency is achievable if the expected degree is of $\Omega (\log ^2 n)$ with high probability [31, Theorem 4.2]. Their algorithm can’t be applied to sparse regimes straightforwardly since the normalised Laplacian is not well-defined due to the existence of isolated vertices in the bounded degree case. In addition, our weak consistency results obtained here are valid as long as the expected degree is $\omega (1)$ and $o(\log n)$, which is the entire set of problems on which weak consistency is expected. By contrast, in [32], weak consistency is shown only when the minimum expected degree is $\Omega (\log ^2(n))$, which is a regime complementary to ours and where exact recovery should (in principle) be possible: for example, this is known to be an exact recovery regime in the uniform case [17, 43, 46, 77].

In subsequent works [25, 71] we proposed algorithms to achieve weak consistency. However, their methods can not cover the regime when the expected degree is $\Omega (1)$ due to the lack of concentration. Additionally, [72] proposed Projected Tensor Power Method as the refinement stage to achieve strong consistency, as long as the first stage partition is partially correct, as ours.

1.4. Organization of the paper

In Section 2, we include the definitions of adjacency matrices of hypergraphs. The concentration results for the adjacency matrices are provided in Section 3. The algorithms for partial recovery are presented in Section 4. The proof for the correctness of our algorithms for Theorem 1.7 and Corollary 1.9 are given in Section 5. The proof of Theorem 1.6, as well as the proofs of many auxiliary lemmas and useful lemmas in the literature, are provided in the supplemental materials.

2. Preliminaries

Definition 2.1 (Adjacency tensor). Given an $m$-uniform hypergraph $H_m=([n], E_m)$, we can associate to it an order- $m$ adjacency tensor $\boldsymbol{\mathcal{A}}^{(m)}$. For any $m$-hyperedge $e = \{ i_1, \dots, i_m \}$, let $\boldsymbol{\mathcal{A}}^{(m)}_e$ denote the corresponding entry $\boldsymbol{\mathcal{A}}_{[i_{1},\dots,i_{m}]}^{(m)}$, such that

(5) \begin{equation} \boldsymbol{\mathcal{A}}_{e}^{(m)} \,:\!=\, \boldsymbol{\mathcal{A}}_{[i_1,\dots, i_m]}^{(m)} = \unicode{x1D7D9}_{\{ e \in E_m\} }\,. \end{equation}

Definition 2.2 (Adjacency matrix). For the non-uniform hypergraph $H$ sampled from model 1.2, let $\boldsymbol{\mathcal{A}}^{(m)}$ be the order- $m$ adjacency tensor corresponding to the underlying $m$-uniform hypergraph for each $m\in{\mathcal M}$. The adjacency matrix ${\mathbf{A}} \,:\!=\, [{\mathbf{A}}_{ij}]_{n \times n}$ of the non-uniform hypergraph $H$ is defined by

(6) \begin{equation} {\mathbf{A}}_{ij} = \unicode{x1D7D9}_{\{ i \neq j\} } \cdot \sum _{m\in{\mathcal M}}\,\,\sum _{\substack{e\in E_m\\ \{i,j\}\subset e} }\boldsymbol{\mathcal{A}}^{(m)}_{e}\,. \end{equation}

We compute the expectation of ${\mathbf{A}}$ first. In each $m$-uniform hypergraph $H_m$, two distinct vertices $i, j\in V$ with $i \neq j$ are picked arbitrarily since our model does not allow for loops. Assume for a moment $\frac{n}{k} \in \mathbb{N}$, then the expected number of $m$-hyperedge containing $i$ and $j$ can be computed as follows.

• • If $i$ and $j$ are from the same block, the $m$-hyperedge is sampled with probability $a_m/\binom{n}{m-1}$ when the other $m-2$ vertices are from the same block as $i$, $j$, otherwise with probability $b_m/\binom{n}{m-1}$. Then

  \begin{align*} \alpha _m\,:\!=\,{\mathbb{E}}{\mathbf{A}}_{ij} = \binom{\frac{n}{k} -2}{m-2} \frac{a_m}{\binom{n}{m-1}} + \left [ \binom{n-2}{m-2} - \binom{\frac{n}{k} - 2}{m-2} \right ]\frac{b_m}{\binom{n}{m-1}} \,. \end{align*}

• • If $i$ and $j$ are not from the same block, we sample the $m$-hyperedge with probability $b_m/\binom{n}{m-1}$, and

  \begin{align*} \beta _m \,:\!=\,{\mathbb{E}}{\mathbf{A}}_{ij} = \binom{n-2}{m-2} \frac{b_m}{\binom{n}{m-1}}\,. \end{align*}

By assumption $a_m \geq b_m$, then $\alpha _m \geq \beta _m$ for each $m\in{\mathcal M}$. Summing over $m$, the expected adjacency matrix under the $k$-block non-uniform $\mathrm{HSBM}$ can be written as

(7) \begin{align} {\mathbb{E}}{\mathbf{A}} = \begin{bmatrix} \alpha{\mathbf{J}}_{\frac{n}{k}} & \quad \beta{\mathbf{J}}_{\frac{n}{k}} & \quad \cdots & \quad \beta{\mathbf{J}}_{\frac{n}{k}} \\[4pt] \beta{\mathbf{J}}_{\frac{n}{k}} & \quad \alpha{\mathbf{J}}_{\frac{n}{k}} & \quad \cdots & \quad \beta{\mathbf{J}}_{\frac{n}{k}} \\[4pt] \vdots & \quad \vdots & \quad \ddots & \quad \vdots \\[4pt] \beta{\mathbf{J}}_{\frac{n}{k}} & \quad \beta{\mathbf{J}}_{\frac{n}{k}} & \quad \cdots & \quad \alpha{\mathbf{J}}_{\frac{n}{k}} \end{bmatrix} - \alpha{\mathbf{I}}_{n}\,, \end{align}

where ${\mathbf{J}}_{\frac{n}{k}}\in{\mathbb{R}}^{\frac{n}{k} \times \frac{n}{k}}$ denotes the all-one matrix and

(8) \begin{align} \alpha \,:\!=\, \sum _{m\in{\mathcal M}} \alpha _m\,, \quad \beta \,:\!=\, \sum _{m\in{\mathcal M}} \beta _m\,. \end{align}

Lemma 2.3. The eigenvalues of ${\mathbb{E}}{\mathbf{A}}$ are given below:

\begin{align*} \lambda _1({\mathbb{E}}{\mathbf{A}}) =&\, \frac{n}{k}(\alpha + (k-1)\beta ) - \alpha \,,\\[4pt] \lambda _i({\mathbb{E}}{\mathbf{A}}) =&\, \frac{n}{k}(\alpha - \beta ) -\alpha \,, \quad 2\leq i \leq k\,,\\[4pt] \lambda _i({\mathbb{E}}{\mathbf{A}}) =&\, - \alpha \,, \,\quad \quad \quad k+1\leq i \leq n\,. \end{align*}

Lemma 2.3 can be verified via direct computation. Lemma 2.4 is used for approximately equi-partitions, meaning that eigenvalues of $\widetilde{{\mathbb{E}}{\mathbf{A}}}$ can be approximated by eigenvalues of ${\mathbb{E}}{\mathbf{A}}$ when $n$ is sufficiently large.

Lemma 2.4. For any partition $(V_1, \dots, V_k)$ of $V$ where $n_i \,:\!=\, |V_i|$, consider the following matrix

\begin{align*} \widetilde{{\mathbb{E}}{\mathbf{A}}} = \begin{bmatrix} \alpha{\mathbf{J}}_{n_1} & \quad \beta{\mathbf{J}}_{n_1 \times n_2} & \quad \cdots & \quad \beta{\mathbf{J}}_{n_1 \times n_{k-1}} & \quad \beta{\mathbf{J}}_{n_1 \times n_k} \\[4pt] \beta{\mathbf{J}}_{n_2 \times n_1} & \quad \alpha{\mathbf{J}}_{n_2} & \quad \cdots & \quad \beta{\mathbf{J}}_{n_2 \times n_{k-1}} & \quad \beta{\mathbf{J}}_{n_2 \times n_k} \\[4pt] \vdots & \quad \vdots & \quad \ddots & \quad \vdots & \quad \vdots \\[4pt] \beta{\mathbf{J}}_{n_{k-1} \times n_1} & \quad \beta{\mathbf{J}}_{n_{k-1} \times n_2} & \quad \cdots & \quad \alpha{\mathbf{J}}_{n_{k-1}} & \quad \beta{\mathbf{J}}_{n_{k-1} \times n_k} \\[4pt] \beta{\mathbf{J}}_{n_k \times n_1} & \quad \beta{\mathbf{J}}_{n_k \times n_2} & \quad \cdots & \quad \beta{\mathbf{J}}_{n_k \times n_{k-1}} & \quad \alpha{\mathbf{J}}_{n_k} \end{bmatrix} - \alpha{\mathbf{I}}_{n}\,. \end{align*}

Assume that $n_i = \frac{n}{k} + O(\sqrt{n} \log n)$ for all $i\in [k]$. Then, for all $1\leq i \leq k$,

\begin{align*} \frac{|\lambda _i( \widetilde{{\mathbb{E}}{\mathbf{A}}} ) - \lambda _i({\mathbb{E}}{\mathbf{A}})|}{|\lambda _i({\mathbb{E}}{\mathbf{A}})| } = O\Big (n^{-\frac{1}{4}}\log ^{\frac{1}{2}}(n)\Big )\,. \end{align*}

Note that both $(\,\widetilde{{\mathbb{E}}{\mathbf{A}}} + \alpha{\mathbf{I}}_{n})$ and $(\,{\mathbb{E}}{\mathbf{A}} + \alpha{\mathbf{I}}_{n})$ are rank $k$ matrices, then $\lambda _i(\widetilde{{\mathbb{E}}{\mathbf{A}}}) = \lambda _i({\mathbb{E}}{\mathbf{A}}) = -\alpha$ for all $(k+1) \leq i \leq n$. At the same time, $\mathrm{SNR}$ in (2) is related to the following quantity

\begin{equation*} \begin{aligned} &\, \frac{[\lambda _2({\mathbb{E}}{\mathbf{A}})]^{2}}{\lambda _1({\mathbb{E}}{\mathbf{A}})} = \frac{[(n-k)\alpha - n \beta ]^2}{k[(n - k)\alpha + n(k-1)\beta ]} = \frac{ \left [\sum _{m\in{\mathcal M}} (m-1)\left (\frac{a_m - b_m}{k^{m-1}} \right ) \right ]^2 }{\sum _{m\in{\mathcal M}} (m-1)\left (\frac{a_m - b_m}{k^{m-1}} + b_m \right )}(1+o(1)). \end{aligned} \end{equation*}

When ${\mathcal M} = \{2\}$ and $k$ is fixed, $\mathrm{SNR}$ in (2) is equal to $\frac{(a - b)^2}{k[a + (k-1)b]}$, which corresponds to the $\mathrm{SNR}$ for the undirected graph in [19], see also [1, Section 6].

3. Spectral norm concentration

The correctness of Algorithms 2 and 1 relies on the concentration of the adjacency matrix of $H$. The following two concentration results for general random hypergraphs are included, which are independent of HSBM model. The proofs are deferred to Section A.

Theorem 3.1. Let $H=\bigcup _{m =2}^{M} H_m$, where $H_m = ([n], E_m)$ is an Erdős-Rényi inhomogeneous hypergraph of order $m$ for each $m\in \{2, \cdots, M\}$. Let $\boldsymbol{\mathcal{T}}^{\,(m)}$ denote the probability tensor such that $\boldsymbol{\mathcal{T}}^{(m)} ={\mathbb{E}} \boldsymbol{\mathcal{A}}^{(m)}$ and $\boldsymbol{\mathcal{T}}^{(m)}_{[i_1,\dots, i_m]} = d_{[i_1,\dots,i_m]}/ \binom{n}{m-1}$, denoting $d_m=\max d_{[i_1,\dots, i_m]}$. Suppose for some constant $c\gt 0$,

(9) \begin{align} d\,:\!=\,\sum _{m =2}^{M} (m-1)\cdot d_m \geq c\log n\,. \end{align}

Then for any $K\gt 0$, there exists a constant $ C= 512M(M-1)(K+6)\left [ 2 + (M-1)(1+K)/c \right ]$ such that with probability at least $1-2n^{-K}-2e^{-n}$, the adjacency matrix ${\mathbf{A}}$ of $H$ satisfies

(10) \begin{align} \|{\mathbf{A}} -{\mathbb{E}}{\mathbf{A}}\|\leq C \sqrt{d}\,. \end{align}

The inequality (10) can be reduced to the result for graph case obtained in [28, 48] by taking ${\mathcal M} = \{2\}$. The result for a uniform hypergraph is obtained in [46]. Note that $d$ is a fixed constant in our community detection problem, thus the Assumption 3.1 does not hold and the inequality (9) cannot be directly applied. However, we can still prove a concentration bound for a regularised version of ${\mathbf{A}}$, following the same strategy of the proof for Theorem 3.1.

Definition 3.2 (Regularized matrix). Given any $n\times n$ matrix ${\mathbf{A}}$ and an index set $\mathcal{I}$, let ${\mathbf{A}}_{\mathcal{I}}$ be the $n\times n$ matrix obtained from ${\mathbf{A}}$ by zeroing out the rows and columns not indexed by $\mathcal{I}$. Namely,

(11) \begin{equation} ({\mathbf{A}}_{\mathcal{I}})_{ij} = \unicode{x1D7D9}_{ \{i, j\in \mathcal{I}\} } \cdot{\mathbf{A}}_{ij}\,. \end{equation}

Since every hyperedge of size $m$ containing $i$ is counted $(m-1)$ times in the $i$-th row sum of ${\mathbf{A}}$, the $i$-th row sum of ${\mathbf{A}}$ is given by

\begin{align*} \mathrm{row}(i) \,:\!=\,\sum _{j}{\mathbf{A}}_{ij} \,:\!=\, \sum _{j} \unicode{x1D7D9}_{\{ i \neq j\} } \sum _{m\in{\mathcal M}}\,\,\sum _{\substack{e\in E_m\\ \{i,j\}\subset e} }\boldsymbol{\mathcal{A}}^{(m)}_{e} = \sum _{m\in{\mathcal M}}\,(m-1) \sum _{e\in E_m:\, i\in e} \boldsymbol{\mathcal{A}}_e^{(m)}. \end{align*}

Theorem 3.3 is the concentration result for the regularised ${\mathbf{A}}_{\mathcal{I}}$, by zeroing out rows and columns corresponding to vertices with high row sums.

Theorem 3.3. Following all the notations in Theorem 3.1, for any constant $\tau \gt 1$, define

\begin{equation*} \mathcal {I}=\{i\in [n]\,:\, \mathrm {row}(i)\leq \tau d\}. \end{equation*}

Let ${\mathbf{A}}_{\mathcal{I}}$ be the regularised version of ${\mathbf{A}}$, as in Definition 3.2. Then for any $K\gt 0$, there exists a constant $C_{\tau }=2( (5M+1)(M-1)+\alpha _0\sqrt{\tau })$ with $\alpha _0=16+\frac{32}{\tau }(1+e^2)+128M(M-1)(K+4)\left (1+\frac{1}{e^2}\right )$, such that $ \| ({\mathbf{A}}-{\mathbb{E}}{\mathbf{A}})_{\mathcal{I}}\|\leq C_{\tau }\sqrt{d}$ with probability at least $1-2(e/2)^{-n} -n^{-K}$.

Algorithm 3. Pre-processing

Algorithm 4. Spectral Partition

4. Algorithms blocks

In this section, we are going to present the algorithmic blocks constructing our main partition method (Algorithm 2): pre-processing (Algorithm 3), initial result by spectral method (Algorithm 4), correction of blemishes via majority rule (Algorithm 5), and merging (Algorithm 6).

Algorithm 5. Correction

Algorithm 6. Merging

4.1. Three or more blocks ($k\geq 3$)

The proof of Theorem 1.7 is structured as follows.

Lemma 4.1. Under the assumptions of Theorem 1.7, Algorithm 4 outputs a $\nu$-correct partition $U_1^{\prime }, \cdots, U_k^{\prime }$ of $Z = (Z\cap V_{1}) \cup \cdots \cup (Z\cap V_{k})$ with probability at least $1 - O(n^{-2})$.

Lines $4$ and $6$ contribute most complexity in Algorithm 4, requiring $O(n^{3})$ and $O(n^{2}\log ^2(n))$ each (technically, one should be able to get away with $O(n^2 \log (1/\varepsilon ))$ in line 4, for some desired accuracy $\varepsilon$ to get the singular vectors). We will conservatively estimate the time complexity of Algorithm 4 as $O(n^3)$.

Lemma 4.2. Under the assumptions of Theorem 1.7, for any $\nu$-correct partition $U_1^{\prime }, \cdots, U_k^{\prime }$ of $Z = (Z\cap V_{1}) \cup \cdots \cup (Z\cap V_{k})$ and the red hypergraph over $Z$, Algorithm 5 computes a $\gamma _{\mathrm{C}}$-correct partition $\widehat{U}_1, \cdots, \widehat{U}_k$ with probability $1 -O(e^{-n\rho })$, while $\gamma _{\mathrm{C}} = \max \{\nu,\, 1 - k\rho \}$ with $ \rho \,:\!=\, k\exp\!\left ({-}C_{{\mathcal M}}(k) \cdot \mathrm{SNR}_{{\mathcal M}}(k) \right )$ where $\mathcal M$ is obtained from Algorithm 3, and $\mathrm{SNR}_{{\mathcal M}}(k)$ and $C_{{\mathcal M}}(k)$ are defined in (2), (3).

Lemma 4.3. Given any $\nu$-correct partition $\widehat{U}_1, \cdots, \widehat{U}_k$ of $Z = (Z\cap V_{1}) \cup \cdots \cup (Z\cap V_{k})$ and the blue hypergraph between $Y$ and $Z$, with probability $1 -O(e^{-n\rho })$, Algorithm 6 outputs a $\gamma$-correct partition $\widehat{V}_1, \cdots, \widehat{V}_{k}$ of $V_1 \cup V_2 \cup \cdots \cup V_k$, while $\gamma =\max \{\nu,\, 1 - k\rho \}$.

The time complexities of Algorithms 5 and 6 are $O(n)$, since each vertex is adjacent to only constant many hyperedges.

Algorithm 7. Spectral Partition

Algorithm 8. Correction

4.2. The binary case ($k = 2$)

The spectral partition step is given in Algorithm 7, and the correction step is given in Algorithm 8.

Lemma 4.4. Under the conditions of Theorem 1.6, the Algorithm 7 outputs a $\nu$-correct partition $V_1^{\prime }, V_2^{\prime }$ of $V = V_1 \cup V_2$ with probability at least $1 - O(n^{-2})$.

Lemma 4.5. Given any $\nu$-correct partition $V_1^{\prime }, V_2^{\prime }$ of $V = V_1 \cup V_2$, with probability at least $1 -O(e^{-n\rho })$, the Algorithm 8 computes a $\gamma$-correct partition $\widehat{V}_1, \widehat{V}_2$ with $\gamma = \{\nu,\, 1 - 2\rho \}$ and $\rho = 2\exp ({-}C_{{\mathcal M}}(2) \cdot \mathrm{SNR}_{{\mathcal M}}(2))$, where $\mathrm{SNR}_{{\mathcal M}}(2)$ and $C_{{\mathcal M}}(2)$ are defined in (2), (3).

5. Algorithm’s correctness

We are going to present the correctness of Algorithm 2 in this section. The correctness of Algorithm 1 is deferred to Section C. We first introduce some definitions.

Vertex set splitting and adjacency matrix

In Algorithm 2, we first randomly partition the vertex set $V$ into two disjoint subsets $Z$ and $Y$ by assigning $+1$ and $-1$ to each vertex independently with equal probability. Let ${\mathbf{B}} \in{\mathbb{R}}^{|Z|\times |Y|}$ denote the submatrix of ${\mathbf{A}}$, while ${\mathbf{A}}$ was defined in (6), where rows and columns of ${\mathbf{B}}$ correspond to vertices in $Z$ and $Y$ respectively. Let $n_i$ denote the number of vertices in $Z\cap V_i$, where $V_i$ denotes the true partition with $|V_i| = \frac{n}{k}$ for all $i\in [k]$, then $n_i$ can be written as a sum of independent Bernoulli random variables, i.e.,

(12) \begin{align} n_i = |Z \cap V_i| = \sum _{v\in V_i} \unicode{x1D7D9}_{ \{v\in Z\} }\,, \end{align}

and $|Y\cap V_i| = |V_i| - |Z\cap V_i| = \frac{n}{k} - n_i$ for each $i \in [k]$.

Definition 5.1. The splitting $V = Z \cup Y$ is perfect if $|Z\cap V_i| = |Y\cap V_i| = n/(2k)$ for all $i\in [k]$. And the splitting $Y = Y_1 \cup Y_2$ is perfect if $|Y_1\cap V_i| = |Y_2\cap V_i| = n/(4k)$ for all $i\in [k]$.

However, the splitting is imperfect in most cases since the size of $Z$ and $Y$ would not be exactly the same under the independence assumption. The random matrix ${\mathbf{B}}$ is parameterised by $\{\boldsymbol{\mathcal{A}}^{(m)}\}_{m \in{\mathcal M}}$ and $\{n_i\}_{i=1}^{k}$. If we take expectation over $\{\boldsymbol{\mathcal{A}}^{(m)}\}_{m \in{\mathcal M}}$ given the block size information $\{n_i\}_{i=1}^{k}$, then it gives rise to the expectation of the imperfect splitting, denoted by $\widetilde{{\mathbf{B}}}$,

\begin{equation*} \widetilde {{\mathbf{B}}} \,:\!=\, \begin {bmatrix} \alpha {\mathbf{J}}_{n_1 \times ( \frac {n}{k} - n_1)} & \quad \beta {\mathbf{J}}_{n_1 \times (\frac {n}{k} - n_2)} & \quad \dots & \quad \beta {\mathbf{J}}_{n_1 \times (\frac {n}{k}- n_k)} \\[4pt] \beta {\mathbf{J}}_{n_2 \times (\frac {n}{k}- n_1)} & \quad \alpha {\mathbf{J}}_{n_2 \times (\frac {n}{k}- n_2)} & \quad\dots & \quad \beta {\mathbf{J}}_{n_2 \times (\frac {n}{k}- n_k)} \\[4pt] \vdots & \quad \vdots & \quad \ddots & \quad \vdots \\[4pt] \beta {\mathbf{J}}_{n_k \times (\frac {n}{k}- n_1)} & \quad \beta {\mathbf{J}}_{n_k \times (\frac {n}{k}- n_2)} & \quad \dots & \quad \alpha {\mathbf{J}}_{n_k \times (\frac {n}{k}- n_k)} \end {bmatrix}\,, \end{equation*}

where $\alpha$, $\beta$ are defined in (8). In the perfect splitting case, the dimension of each block is $n/(2k)\times n/(2k)$ since ${\mathbb{E}} n_i = n/(2k)$ for all $i \in [k]$, and the expectation matrix $\overline{{\mathbf{B}}}$ can be written as

\begin{equation*} \overline {{\mathbf{B}}} \,:\!=\, \begin {bmatrix} \alpha {\mathbf{J}}_{\frac {n}{2k}} & \quad \beta {\mathbf{J}}_{\frac {n}{2k}} & \quad \dots & \quad \beta {\mathbf{J}}_{\frac {n}{2k}} \\[4pt] \beta {\mathbf{J}}_{\frac {n}{2k}} & \quad \alpha {\mathbf{J}}_{\frac {n}{2k}} & \quad \dots & \quad \beta {\mathbf{J}}_{\frac {n}{2k}} \\[4pt] \vdots & \quad \vdots & \quad \ddots & \quad \vdots \\[4pt] \beta {\mathbf{J}}_{\frac {n}{2k}} & \quad \beta {\mathbf{J}}_{\frac {n}{2k}} & \quad\dots & \quad \alpha {\mathbf{J}}_{\frac {n}{2k}} \end {bmatrix}\,. \end{equation*}

In Algorithm 4, $Y_1$ is a random subset of $Y$ obtained by selecting each element with probability $1/2$ independently, and $Y_2 = Y\setminus Y_1$. Let $n^{\prime }_{i}$ denote the number of vertices in $Y_1\cap V_i$, then $n^{\prime }_{i}$ can be written as a sum of independent Bernoulli random variables,

(13) \begin{align} n^{\prime }_{i} = |Y_1 \cap V_i| = \sum _{v\in V_i} \mathbf{1}_{ \{v\in Y_1\} }\,, \end{align}

and $|Y_2\cap V_i| = |V_i| - |Z\cap V_i| - |Y_1\cap V_i| = n/k - n_i - n^{\prime }_{i}$ for all $i \in [k]$.

Induced sub-hypergraph

Definition 5.2 (Induced Sub-hypergraph). Let $H = (V, E)$ be a non-uniform random hypergraph and $S\subset V$ be any subset of the vertices of $H$. Then the induced sub-hypergraph $H[S]$ is the hypergraph whose vertex set is $S$ and whose hyperedge set $E[S]$ consists of all of the edges in $E$ that have all endpoints located in $S$.

Let $H[Y_1 \cup Z]$(resp. $H[Y_2 \cup Z]$) denote the induced sub-hypergraph on vertex set $Y_1\cup Z$ (resp. $Y_2 \cup Z$), and ${\mathbf{B}}_1 \in{\mathbb{R}}^{|Z|\times |Y_1|}$ (resp. ${\mathbf{B}}_2 \in{\mathbb{R}}^{|Z|\times |Y_2|}$) denote the adjacency matrices corresponding to the sub-hypergraphs, where rows and columns of ${\mathbf{B}}_1$ (resp. ${\mathbf{B}}_2$) are corresponding to elements in $Z$ and $Y_1$ (resp., $Z$ and $Y_2$). Therefore, ${\mathbf{B}}_1$ and ${\mathbf{B}}_2$ are parameterised by $\{\boldsymbol{\mathcal{A}}^{(m)}\}_{m \in{\mathcal M}}$, $\{n_i\}_{i=1}^{k}$ and $\{n^{\prime }_{i}\}_{i=1}^{k}$, and the entries in ${\mathbf{B}}_1$ are independent of the entries in ${\mathbf{B}}_2$, due to the independence of hyperedges. If we take expectation over $\{\boldsymbol{\mathcal{A}}^{(m)}\}_{m \in{\mathcal M}}$ conditioning on $\{n_i\}_{i=1}^{k}$ and $\{n^{\prime }_{i}\}_{i=1}^{k}$, then it gives rise to the expectation of the imperfect splitting, denoted by $\widetilde{{\mathbf{B}}}_1$,

(14) \begin{equation} \widetilde{{\mathbf{B}}}_1 \,:\!=\, \begin{bmatrix} \widetilde{\alpha }_{11}{\mathbf{J}}_{n_1 \times n^{\prime }_{1}} & \quad \dots & \quad \widetilde{\beta }_{1k}{\mathbf{J}}_{n_1 \times n^{\prime }_{k}} \\[4pt] \vdots & \quad \ddots & \quad \vdots \\[4pt] \widetilde{\beta }_{k1}{\mathbf{J}}_{n_k \times n^{\prime }_{1}} & \quad \dots & \quad \widetilde{\alpha }_{kk}{\mathbf{J}}_{n_k \times n^{\prime }_{k}} \end{bmatrix}\,, \end{equation}

where

(15a) \begin{align} \widetilde{\alpha }_{ii} \,:\!=\,&\, \sum _{m \in{\mathcal{M}}}\left \{ \binom{n_i + n_i^{\prime } - 2}{m-2} \frac{a_m - b_m}{\binom{n}{m-1}} + \binom{\sum _{l=1}^{k}(n_l + n_l^{\prime }) - 2}{m-2} \frac{b_m}{\binom{n}{m-1}} \right \}\,, \end{align}

(15b) \begin{align} \widetilde{\beta }_{ij} \,:\!=\,&\, \sum _{m \in{\mathcal{M}}} \binom{\sum _{l=1}^{k}(n_l + n_l^{\prime }) - 2}{m-2}\frac{b_m}{\binom{n}{m-1}}\,,\quad i\neq j, \, i, j\in [k]\,. \end{align}

The expectation of the perfect splitting, denoted by $\overline{{\mathbf{B}}}_1$, can be written as

(16) \begin{equation} \overline{{\mathbf{B}}}_1 \,:\!=\, \begin{bmatrix} \overline{\alpha }{\mathbf{J}}_{\frac{n}{2k} \times \frac{n}{4k}} & \quad \overline{\beta }{\mathbf{J}}_{\frac{n}{2k} \times \frac{n}{4k}} & \quad \dots & \quad \overline{\beta }{\mathbf{J}}_{\frac{n}{2k} \times \frac{n}{4k}} \\[4pt] \overline{\beta }{\mathbf{J}}_{\frac{n}{2k} \times \frac{n}{4k}} & \quad \overline{\alpha }{\mathbf{J}}_{\frac{n}{2k} \times \frac{n}{4k}} & \quad \dots & \quad \overline{\beta }{\mathbf{J}}_{\frac{n}{2k} \times \frac{n}{4k}} \\[4pt] \vdots & \quad \vdots & \quad \ddots & \quad \vdots \\[4pt] \overline{\beta }{\mathbf{J}}_{\frac{n}{2k} \times \frac{n}{4k}} & \quad \overline{\beta }{\mathbf{J}}_{\frac{n}{2k} \times \frac{n}{4k}} & \quad\dots & \quad \overline{\alpha }{\mathbf{J}}_{\frac{n}{2k} \times \frac{n}{4k}} \end{bmatrix}\,, \end{equation}

where

(17) \begin{align} \overline{\alpha } \,:\!=\, \sum _{m \in{\mathcal M}}\left \{ \binom{ \frac{3n}{4k} - 2}{m-2} \frac{a_m - b_m}{\binom{n}{m-1}} + \binom{\frac{3n}{4} - 2}{m-2}\frac{b_m}{\binom{n}{m-1}} \right \}\,,\quad \overline{\beta } \,:\!=\, \sum _{m \in{\mathcal M}} \binom{\frac{3n}{4} - 2}{m-2}\frac{b_m}{\binom{n}{m-1}}\,. \end{align}

The matrices $\widetilde{{\mathbf{B}}}_2, \overline{{\mathbf{B}}}_2$ can be defined similarly, since dimensions of $|Y_2\cap V_i|$ are also determined by $n_i$ and $n^{\prime }_{i}$. Obviously, $\overline{{\mathbf{B}}}_2 = \overline{{\mathbf{B}}}_1$ since ${\mathbb{E}} n^{\prime }_{i} ={\mathbb{E}} (n/k- n_{i} - n_i^{\prime }) = n/(4k)$ for all $i\in [k]$.

Fixing Dimensions

The dimensions of $\widetilde{{\mathbf{B}}}_1$ and $\widetilde{{\mathbf{B}}}_2$, as well as blocks they consist of, are not deterministic – since $n_i$ and $n^{\prime }_i$, defined in (12) and (13) respectively, are sums of independent random variables. As such, we cannot directly compare them. In order to overcome this difficulty, we embed ${\mathbf{B}}_1$ and ${\mathbf{B}}_2$ into the following $n\times n$ matrices:

(18) \begin{equation} {\mathbf{A}}_1\,:\!=\, \begin{bmatrix} \mathbf{0}_{|Z|\times |Z|} & \quad{\mathbf{B}}_1 & \quad \mathbf{0}_{|Z|\times |Y_2|} \\[4pt] \mathbf{0}_{|Y|\times |Z|} & \quad \mathbf{0}_{|Y|\times |Y_1|} & \quad \mathbf{0}_{|Y|\times |Y_2|} \end{bmatrix}\,,\quad{\mathbf{A}}_2\,:\!=\, \begin{bmatrix} \mathbf{0}_{|Z|\times |Z|} & \quad \mathbf{0}_{|Z|\times |Y_1|} & \quad{\mathbf{B}}_2 \\[4pt] \mathbf{0}_{|Y|\times |Z|} & \quad \mathbf{0}_{|Y|\times |Y_1|} & \quad \mathbf{0}_{|Y|\times |Y_2|} \end{bmatrix}\,. \end{equation}

Note that ${\mathbf{A}}_1$ and ${\mathbf{A}}_2$ have the same size. Also by definition, the entries in ${\mathbf{A}}_1$ are independent of the entries in ${\mathbf{A}}_2$. If we take expectation over $\{\boldsymbol{\mathcal{A}}^{(m)}\}_{m\in{\mathcal M}}$ conditioning on $\{n_i\}_{i=1}^{k}$ and $\{n^{\prime }_{i}\}_{i=1}^{k}$, then we obtain the expectation matrices of the imperfect splitting, denoted by $\widetilde{{\mathbf{A}}}_1$(resp. $\widetilde{{\mathbf{A}}}_2$), written as

(19) \begin{equation} \widetilde{{\mathbf{A}}}_1\,:\!=\, \begin{bmatrix} \mathbf{0}_{|Z|\times |Z|} & \quad \widetilde{{\mathbf{B}}}_1 & \quad \mathbf{0}_{|Z|\times |Y_2|} \\[4pt] \mathbf{0}_{|Y|\times |Z|} & \quad \mathbf{0}_{|Y|\times |Y_1|} & \quad \mathbf{0}_{|Y|\times |Y_2|} \end{bmatrix}\,,\quad \widetilde{{\mathbf{A}}}_2\,:\!=\, \begin{bmatrix} \mathbf{0}_{|Z|\times |Z|} & \quad \mathbf{0}_{|Z|\times |Y_1|} & \quad \widetilde{{\mathbf{B}}}_2 \\[4pt] \mathbf{0}_{|Y|\times |Z|} & \quad \mathbf{0}_{|Y|\times |Y_1|} & \quad \mathbf{0}_{|Y|\times |Y_2|} \end{bmatrix}\,. \end{equation}

The expectation matrix of the perfect splitting, denoted by $\overline{{\mathbf{A}}}_1$(resp. $\overline{{\mathbf{A}}}_2$), can be written as

(20) \begin{equation} \overline{{\mathbf{A}}}_1\,:\!=\, \begin{bmatrix} \mathbf{0}_{ \frac{n}{2} \times \frac{n}{2}} & \quad \overline{{\mathbf{B}}}_1 & \quad \mathbf{0}_{\frac{n}{2}\times \frac{n}{4}} \\[4pt] \mathbf{0}_{\frac{n}{2}\times \frac{n}{2}} & \quad \mathbf{0}_{\frac{n}{2}\times \frac{n}{4}} & \quad \mathbf{0}_{\frac{n}{2}\times \frac{n}{4}} \end{bmatrix}\,,\quad \overline{{\mathbf{A}}}_2\,:\!=\, \begin{bmatrix} \mathbf{0}_{ \frac{n}{2} \times \frac{n}{2}} & \quad \mathbf{0}_{\frac{n}{2}\times \frac{n}{4}} & \quad \overline{{\mathbf{B}}}_2\\[4pt] \mathbf{0}_{\frac{n}{2}\times \frac{n}{2}} & \quad \mathbf{0}_{\frac{n}{2}\times \frac{n}{4}} & \quad \mathbf{0}_{\frac{n}{2}\times \frac{n}{4}} \end{bmatrix}\,. \end{equation}

Obviously, $\widetilde{{\mathbf{A}}}_i$ and $\widetilde{{\mathbf{B}}}_i$(resp. $\overline{{\mathbf{A}}}_i$ and $\overline{{\mathbf{B}}}_i$) have the same non-zero singular values for $i = 1, 2$. In the remaining of this section, we will deal with $\widetilde{{\mathbf{A}}}_i$ and $\overline{{\mathbf{A}}}_i$ instead of $\widetilde{{\mathbf{B}}}_i$ and $\overline{{\mathbf{B}}}_i$ for $i = 1, 2$.

5.1. Spectral partition: Proof of Lemma 4.1

5.1.1. Proof outline

Recall that ${\mathbf{A}}_1$ is defined as the adjacency matrix of the induced sub-hypergraph $H[Y_1\cup Z]$ in equation (5.2). Consequently, the index set should contain information only from $H[Y_1\cup Z]$. Define the index sets

\begin{align*} \mathcal{I} = \left \{i\in [n]\,:\, \mathrm{row}(i)\leq 20{\mathcal M}_{\max }d \right \}, \quad \mathcal{I}_{1} = \left \{i\in [n]\,:\, \mathrm{row}(i)\Big |_{Y_1 \cup Z} \leq 20{\mathcal M}_{\max }d \right \}\,, \end{align*}

where $d = \sum _{m \in{\mathcal M}}(m-1)a_m$, and $\mathrm{row}(i)\big |_{Y_1 \cup Z}$ is the row sum of $i$ on $H[Y_1\cup Z]$. We say $\mathrm{row}(i)\big |_{Y_1 \cup Z}=0$ if $i\not \in Y_1\cup Z$, and for vertex $i\in Y_1\cup Z$,

\begin{align*} \mathrm{row}(i)\Big |_{Y_1 \cup Z} \,:\!=\, \sum _{j=1}^{n} \sum _{m\in{\mathcal M}}\,\,\sum _{ \substack{e\in E_m[Y_1\cup Z]\\ \{i, j\}\subset e} } \,\, \boldsymbol{\mathcal{A}}_e^{(m)} = \sum _{m\in{\mathcal M}}(m-1)\sum _{ \substack{e\in E_m[Y_1\cup Z]\\ \{i, j\}\subset e} } \,\, \boldsymbol{\mathcal{A}}_e^{(m)}\,. \end{align*}

As a result, the matrix $({\mathbf{A}}_1)_{\mathcal{I}_1}$ is obtained by restricting ${\mathbf{A}}_1$ on index set $\mathcal{I}_1$. The next $4$ steps guarantee that Algorithm 4 outputs a $\nu$-correct partition.

1. (i) Find the singular subspace ${\mathbf{U}}$ spanned by the first $k$ left singular vectors of $({\mathbf{A}}_1)_{\mathcal{I}_1}$.

2. (ii) Randomly pick $s = 2k \log ^2 n$ vertices from $Y_2$ and denote the corresponding columns in ${\mathbf{A}}_2$ by $\boldsymbol{a}_{i_1},\dots,\boldsymbol{a}_{i_s}$. Project each vector $\boldsymbol{a}_i - \overline{\boldsymbol{a}}$ onto the singular subspace ${\mathbf{U}}$, with $\overline{\boldsymbol{a}}\in{\mathbb{R}}^{n}$ defined by $\overline{\boldsymbol{a}}(j) = \unicode{x1D7D9}_{j\in Z} \cdot (\overline{\alpha } + \overline{\beta })/2$, where $\overline{\alpha }$, $\overline{\beta }$ were defined in (17).

3. (iii) For each projected vector $P_{{\mathbf{U}}}(\boldsymbol{a}_i - \overline{\boldsymbol{a}})$, identify the top $n/(2k)$ coordinates in value and place the corresponding vertices into a set $U^{\prime }_i$. Discard half of the obtained $s$ subsets, those with the lowest blue edge densities.

4. (iv) Sort the remaining sets according to blue hyperedge density and identify $k$ distinct subsets $U^{\prime }_1, \cdots, U^{\prime }_k$ such that $|U^{\prime }_i \cap U^{\prime }_j| \lt \lceil (1 - \nu )n/k\rceil$ if $i\neq j$.

Based on the $4$ steps above in Algorithm 4, the proof of Lemma 4.1 is structured in $4$ parts.

1. (i) Let $\widetilde{{\mathbf{U}}}$ denote the subspace spanned by first $k$ left singular vectors of $\widetilde{{\mathbf{A}}}_1$ defined in (19). Subsection 5.1.2 shows that the subspace angle between ${\mathbf{U}}$ and $\widetilde{{\mathbf{U}}}$ is smaller than any $c\in (0, 1)$ as long as $a_m, b_m$ satisfy certain conditions depending on $c$.

2. (ii) The vector $\widetilde{{\boldsymbol \delta }}_i$, defined in (22), reflects the underlying true partition $Z\cap V_{k(i)}$ for each $i\in [s]$, where $k(i)$ denotes the membership of vertex $i$. Subsection 5.1.3 shows that $\overline{{\boldsymbol \delta }}_i$, an approximation of $\widetilde{{\boldsymbol \delta }}_i$ defined in (23), can be recovered by the projected vector $P_{{\mathbf{U}}}(\boldsymbol{a}_{i} - \overline{\boldsymbol{a}})$, since projection error $\|P_{{\mathbf{U}}}(\boldsymbol{a}_{i} - \overline{\boldsymbol{a}}) - \overline{{\boldsymbol \delta }}_{i}\|_2 \lt c\|\overline{{\boldsymbol \delta }}_{i}\|_2$ for any $c\in (0, 1)$ if $a_m, b_m$ satisfy the desired property in part (i).

3. (iii) Subsection 5.1.4 indicates that the coincidence ratio between the remaining sets and the true partition is at least $\nu$, after discarding half of the sets with the lowest blue edge densities.

4. (iv) Lemma 5.13 proves that we can find $k$ distinct subsets $U^{\prime }_i$ within $k\log ^2 n$ trials with high probability.

5.1.2. Bounding the angle between ${\mathbf{U}}$ and $\widetilde{{\mathbf{U}}}$

The angle between subspaces ${\mathbf{U}}$ and $\widetilde{{\mathbf{U}}}$ is defined as

\begin{equation*}\sin \angle ({\mathbf{U}}, \widetilde {{\mathbf{U}}}) \,:\!=\, \|P_{{\mathbf{U}}} - P_{\widetilde {{\mathbf{U}}}}\|.\end{equation*}

A natural idea is to apply Wedin’s $\sin \Theta$ Theorem (Lemma D.7). Lemma 5.3 indicates that the difference of $\sigma _i(\widetilde{{\mathbf{A}}}_1)$ and $\sigma _i(\overline{{\mathbf{A}}}_1)$ is relatively small, compared to $\sigma _i(\overline{{\mathbf{A}}}_1)$.

Lemma 5.3. Let $\sigma _i(\overline{{\mathbf{A}}}_1)$(resp. $\sigma _i(\widetilde{{\mathbf{A}}}_1)$) denote the singular values of $\overline{{\mathbf{A}}}_1$ (resp. $\widetilde{{\mathbf{A}}}_1$) for all $i\in [k]$, where the matrices $\overline{{\mathbf{A}}}_1$ and $\widetilde{{\mathbf{A}}}_1$ are defined in (20) and (19) respectively. Then

\begin{align*} \sigma _1(\overline{{\mathbf{A}}}_1) =&\,\frac{n\left [ \overline{\alpha } + (k-1)\overline{\beta } \right ] }{2\sqrt{2}k}= \frac{n}{2\sqrt{2}k}\sum _{m\in{\mathcal M}} \left [ \binom{\frac{3n}{4k} - 2}{m-2} \frac{a_m - b_m}{ \binom{n}{m-1}} + k\binom{\frac{3n}{4} - 2}{m-2}\frac{b_m}{\binom{n}{m-1}} \right ] \,,&\\[5pt] \sigma _i(\overline{{\mathbf{A}}}_1) =&\, \frac{n( \overline{\alpha } - \overline{\beta })}{2\sqrt{2}k} = \frac{n}{2\sqrt{2}k}\sum _{m\in{\mathcal M}} \binom{\frac{3n}{4k} - 2}{m-2} \frac{a_m - b_m}{ \binom{n}{m-1}}\,, 2 \leq i \leq k\,,&\\[5pt] \sigma _i(\overline{{\mathbf{A}}}_1) =&\, 0\,, k+1 \leq i \leq n\,.& \end{align*}

with $\overline{\alpha }$, $\overline{\beta }$ defined in (17). Moreover, with probability at least $1 - 2k\exp ({-}k\log ^2(n))$,

\begin{align*} \frac{|\sigma _i(\overline{{\mathbf{A}}}_1) - \sigma _i(\widetilde{{\mathbf{A}}}_1)|}{\sigma _i(\overline{{\mathbf{A}}}_1)} = O\left (n^{-\frac{1}{4}}\log ^{\frac{1}{2} }(n)\right ). \end{align*}

Therefore, with Lemma 5.3, we can write $ \sigma _i(\widetilde{{\mathbf{A}}}_1) = \sigma _i(\overline{{\mathbf{A}}}_1)(1 + o(1)).$ Define ${\mathbf{E}}_1 \,:\!=\,{\mathbf{A}}_1 - \widetilde{{\mathbf{A}}}_1$ and its restriction on $\mathcal{I}_1$ as

(21) \begin{equation} ({\mathbf{E}}_1)_{\mathcal{I}_1} \,:\!=\, ({\mathbf{A}}_1 - \widetilde{{\mathbf{A}}}_1)_{\mathcal{I}_1} = ({\mathbf{A}}_1)_{\mathcal{I}_1} - (\widetilde{{\mathbf{A}}}_1)_{\mathcal{I}_1}\,, \end{equation}

as well as ${\boldsymbol \Delta }_{1}\,:\!=\, (\widetilde{{\mathbf{A}}}_1)_{\mathcal{I}_1} - \widetilde{{\mathbf{A}}}_1$. Then $({\mathbf{A}}_1)_{\mathcal{I}_1} - \widetilde{{\mathbf{A}}}_1$ is decomposed as

\begin{align*} ({\mathbf{A}}_1)_{\mathcal{I}_1} - \widetilde{{\mathbf{A}}}_1 = [({\mathbf{A}}_1)_{\mathcal{I}_1} - (\widetilde{{\mathbf{A}}}_1)_{\mathcal{I}_1}] + [(\widetilde{{\mathbf{A}}}_1)_{\mathcal{I}_1} - \widetilde{{\mathbf{A}}}_1 ] = ({\mathbf{E}}_1)_{\mathcal{I}_1} +{\boldsymbol \Delta }_{1}\,. \end{align*}

Lemma 5.4. Let $d = \sum _{m\in{\mathcal M}} (m-1)a_m$, where $\mathcal M$ is obtained from Algorithm 3. There exists a constant $C_1\geq (2^{1/{\mathcal M}_{\max }} - 1)^{-1/3}$ such that if $d \geq C_1$, then with probability at least $1 - \exp\!\left ({-} d^{-2}n/{\mathcal M}_{\max } \right )$, no more than $d^{-3}n$ vertices have row sums greater than $20{\mathcal M}_{\max }d$.

Lemma 5.4 shows that the number of high-degree vertices is relatively small. Consequently, Corollary 5.5 indicates $\|{\boldsymbol \Delta }_1\| \leq \sqrt{d}$ with high probability.

Corollary 5.5. Assume $d \geq \max \{C_1, \sqrt{2}\}$, where $C_1$ is the constant in Lemma 5.4, then $\|{\boldsymbol \Delta }_1\| \leq \sqrt{d}$ with probability at least $1 - \exp\!\left ({-} d^{-2}n/{\mathcal M}_{\max } \right )$.

Proof of Corollary 5.5. Note that $n - |\mathcal{I}| \leq d^{-3}n$ and $\mathcal{I} \subset \mathcal{I}_1$, then $n - |\mathcal{I}_1| \leq d^{-3}n$. From Lemma 5.4, there are at most $d^{-3} n$ vertices with row sum greater than $20{\mathcal M}_{\max }d$ in the adjacency matrix ${\mathbf{A}}_1$, then the matrix ${\boldsymbol \Delta }_1 = (\widetilde{{\mathbf{A}}}_1)_{\mathcal{I}_1} - \widetilde{{\mathbf{A}}}_1$ has at most $2d^{-3} n^{2}$ non-zero entries. Every entry of $\widetilde{{\mathbf{A}}}_1$ in (19) is bounded by $\alpha$, then,

\begin{align*} \|{\boldsymbol \Delta }_1\| \leq &\, \|{\boldsymbol \Delta }_1 \|_{{\mathrm{F}}} = \|(\widetilde{{\mathbf{A}}}_1)_{\mathcal{I}_1} - \widetilde{{\mathbf{A}}}_1\|_{{\mathrm{F}}} \\ \leq &\, \sqrt{2 d^{-3}n^{2}} \, \alpha = \sqrt{2 d^{-3}} n \sum _{m\in{\mathcal M}} \left [ \binom{ \frac{n}{k}-2}{m-2} \frac{a_m - b_m}{\binom{n}{m-1}} +\left({n - 2 \atop m-2}\right) \frac{b_m}{\binom{n}{m-1}} \right ]\\ \leq &\, \sqrt{2d^{-3} } \sum _{m\in{\mathcal M}} (m-1)a_m \leq \sqrt{2d^{-1}} \leq \sqrt{d}\,. \end{align*}

Moreover, taking $\tau =20{\mathcal M}_{\max }, K=3$ in Theorem 3.3, with probability at least $1 - n^{-2}$

(22) \begin{align} \|({\mathbf{E}}_1)_{\mathcal{I}_1}\|\leq C_3\sqrt{d}\,, \end{align}

where constant $C_3$ depends on ${\mathcal M}_{\max }$. Together with upper bounds for $\|({\mathbf{E}}_1)_{\mathcal{I}_1}\|$ and $\|{\boldsymbol \Delta }_1\|$, Lemma 5.6 shows that the angle between ${\mathbf{U}}$ and $\widetilde{{\mathbf{U}}}$ is relatively small with high probability.

Lemma 5.6. For any $c\in (0,1)$, there exists some constant $C_2$ such that, if

\begin{equation*} \sum _{m\in {\mathcal M}}(m-1)(a_m-b_m)\geq C_2k^{{\mathcal M}_{\max } - 1}\sqrt {d}\,, \end{equation*}

then $\sin \angle ({\mathbf{U}}, \widetilde{{\mathbf{U}}}) \leq c$ with probability $1 - n^{-2}$. Here $\angle ({\mathbf{U}}, \widetilde{{\mathbf{U}}})$ is the angle between ${\mathbf{U}}$ and $\widetilde{{\mathbf{U}}}$.

Proof of Lemma 5.6. From (22) and Corollary 5.5, with probability at least $1 - n^{-2}$,

\begin{align*} \|({\mathbf{A}}_1)_{\mathcal{I}_1} - \widetilde{{\mathbf{A}}}_1\| \leq \|({\mathbf{E}}_1)_{\mathcal{I}_1}\| + \|{\boldsymbol \Delta }_1 \|\leq (C_3 + 1)\sqrt{d}. \end{align*}

Since $\sigma _{k+1}(\widetilde{{\mathbf{A}}}_1)=0$, using Lemma 5.3 to approximate $\sigma _k(\widetilde{{\mathbf{A}}}_1)$, we obtain

\begin{align*} &\,\sigma _k(\widetilde{{\mathbf{A}}}_1) - \sigma _{k+1}(\widetilde{{\mathbf{A}}}_1) =\sigma _k(\widetilde{{\mathbf{A}}}_1)=(1+o(1))\sigma _k(\overline{{\mathbf{A}}}_1) \geq \frac{1}{2}\sigma _k(\overline{{\mathbf{A}}}_1)\\[6pt] \geq &\, \frac{n}{4\sqrt{2}k }\sum _{m \in{\mathcal M}}\left({ \frac{3n}{4k} -2 \atop m-2}\right) \frac{a_m - b_m}{\left({n \atop m-1}\right)} \geq \frac{1}{8k}\sum _{m\in{\mathcal M}} \bigg (\frac{3}{4k} \bigg )^{m-2}(m-1)(a_m-b_m)\\[6pt] \geq &\, \frac{1}{8k}\left (\frac{1}{2k}\right )^{{\mathcal M}_{\max } - 2}\sum _{m\in{\mathcal M}} (m-1)(a_m-b_m)\geq \frac{C_2 \sqrt{d}}{2^{{\mathcal M}_{\max } + 1}}\,. \end{align*}

Then for any $c\in (0, 1)$, we can find $C_2 = [2^{{\mathcal M}_{\max } + 2}(C_3 + 1)/c]$ such that $\|({\mathbf{A}}_1)_{\mathcal{I}_1} - \widetilde{{\mathbf{A}}}_1\|\leq (1-1/\sqrt{2}) \sigma _k(\widetilde{{\mathbf{A}}}_1)$. By Wedin’s Theorem (Lemma D.7), the angle $\angle ({\mathbf{U}}, \widetilde{{\mathbf{U}}})$ is bounded by

\begin{align*} \sin \angle ({\mathbf{U}}, \widetilde{{\mathbf{U}}}) \,:\!=\, \|P_{{\mathbf{U}}} - P_{\widetilde{{\mathbf{U}}}}\| \leq &\, \frac{\sqrt{2}\|({\mathbf{A}}_1)_{\mathcal{I}_1} - \widetilde{{\mathbf{A}}}_1\|}{ \sigma _k(\widetilde{{\mathbf{A}}}_1)} \leq \frac{\sqrt{2}(C_3+1)\sqrt{d}}{C_2\sqrt{d}/ 2^{{\mathcal M}_{\max } + 1}} = \frac{\sqrt{2}}{2}c\lt c\,. \end{align*}

5.1.3. Bound the projection error

Randomly pick $s=2k\log ^2 n$ vertices from $Y_2$. Let $\boldsymbol{a}_{i_1},\dots,\boldsymbol{a}_{i_s}$, $\widetilde{\boldsymbol{a}}_{i_1}$,$\dots$, $\widetilde{\boldsymbol{a}}_{i_s}$, $\overline{\boldsymbol{a}}_{i_1},\dots, \overline{\boldsymbol{a}}_{i_s}$ and $\boldsymbol{e}_{i_1}, \dots,\boldsymbol{e}_{i_s}$ be the corresponding columns of ${\mathbf{A}}_2$, $\widetilde{{\mathbf{A}}}_2$, $\overline{{\mathbf{A}}}_2$ and ${\mathbf{E}}_2\,:\!=\,{\mathbf{A}}_2 - \widetilde{{\mathbf{A}}}_2$ respectively, where ${\mathbf{A}}_2$, $\widetilde{{\mathbf{A}}}_2$ and $\overline{{\mathbf{A}}}_2$ were defined in (18), (19) and (20). Let $k(i)$ denote the membership of vertex $i$. Note that entries of vector $\widetilde{\boldsymbol{a}}_{i}$ are $\widetilde{\alpha }_{ii}$, $\widetilde{\beta }_{ij}$ or $0$, according to the membership of vertices in $Z$, where $\widetilde{\alpha }_{ii}$, $\widetilde{\beta }_{ij}$ were defined in (15a), (15b). Then the corresponding vector $\widetilde{{\boldsymbol \delta }}_i \in{\mathbb{R}}^{n}$ with the entries given by

(23) \begin{equation} \widetilde{\boldsymbol{a}}_{i}(j) = \begin{cases} \widetilde{\alpha }_{ii}, & \mathrm{if } j\in Z\cap V_{k(i)}\\[6pt] \widetilde{\beta }_{ij}, & \mathrm{if } j\in Z\setminus V_{k(i)} \\[6pt] 0\,, & \mathrm{if } j\in Y \end{cases}\,,\quad \widetilde{{\boldsymbol \delta }}_i(j) = \begin{cases} (\widetilde{\alpha }_{ii} - \widetilde{\beta }_{ij})/2 \gt 0, & \mathrm{if } j\in Z\cap V_{k(i)} \\[6pt] (\widetilde{\beta }_{ij} - \widetilde{\alpha }_{ii})/2 \lt 0, & \mathrm{if } j\in Z\setminus V_{k(i)} \\[6pt] 0, & \mathrm{if } j\in Y \end{cases}, \end{equation}

can be used to recover the vertex set $Z\cap V_{k(i)}$ based on the sign of elements in $\widetilde{{\boldsymbol \delta }}_i$. However, it is hard to handle with $\widetilde{{\boldsymbol \delta }}_i$ due to the randomness of $\widetilde{\alpha }_{ii}$, $\widetilde{\beta }_{ij}$ originated from $n_i$ and $n^{\prime }_i$. Note that $n_i$ and $n^{\prime }_i$ concentrate around $n/(2k)$ and $n/(4k)$ respectively as shown in Lemma 5.3. Thus a good approximation of $\widetilde{{\boldsymbol \delta }}_i$, which rules out randomness of $n_i$ and $n^{\prime }_i$, was given by $\overline{{\boldsymbol \delta }}_i \,:\!=\, \overline{\boldsymbol{a}}_{i} - \overline{\boldsymbol{a}}$, with entries given by $\overline{\boldsymbol{a}}(j)\,:\!=\, \unicode{x1D7D9}_{j\in Z}\cdot (\overline{\alpha } + \overline{\beta })/2$, where $\overline{\alpha }$ and $\overline{\beta }$ were defined in (17), and

(24) \begin{equation} \overline{\boldsymbol{a}}_i (j) = \begin{cases} \overline{\alpha }, \quad &\mathrm{if } j\in Z\cap V_{k(i)} \\[6pt] \overline{\beta }, \quad &\mathrm{if } j\in Z\setminus V_{k(i)} \\[6pt] 0, \quad &\mathrm{if } j\in Y \end{cases} \,,\quad \overline{{\boldsymbol \delta }}_i(j) = \begin{cases} (\overline{\alpha } - \overline{\beta })/2 \gt 0, \quad & \mathrm{if } j\in Z\cap V_{k(i)} \\[6pt] (\overline{\beta } - \overline{\alpha })/2 \lt 0, \quad &\mathrm{if } j\in Z\setminus V_{k(i)} \\[6pt] 0, \quad &\mathrm{if } j\in Y \end{cases} \,. \end{equation}

By construction, $\overline{{\boldsymbol \delta }}_i$ identifies vertex set $Z \cap V_{k(i)}$ in the case of perfect splitting for any $i\in \{i_1, \cdots, i_s\}\cap Y_2\cap V_{k(i)}$. However, the access to $\overline{{\boldsymbol \delta }}_i$ is limited in practice, thus the projection $P_{{\mathbf{U}}}(\boldsymbol{a}_i - \overline{\boldsymbol{a}})$ is used instead as an approximation of $\overline{{\boldsymbol \delta }}_i$. Lemma 5.7 proves that at least half of the projected vectors have small projection errors.

Lemma 5.7. For any $c\in (0, 1)$, there exist constants $C_1$ and $C_2$ such that if $d\gt C_1$ and

\begin{equation*}\sum _{m \in {\mathcal M}}(m-1)(a_m -b_m) \gt C_2 k^{{\mathcal M}_{\max }}\sqrt {d},\end{equation*}

then among all projected vectors $P_{{\mathbf{U}}}(\boldsymbol{a}_{i} - \overline{\boldsymbol{a}})$ for $i\in \{i_1, \cdots, i_s\}\cap Y_2$, with probability $1 - O(n^{-k})$, at least half of them satisfy

(25) \begin{align} \|P_{{\mathbf{U}}}(\boldsymbol{a}_{i} - \overline{\boldsymbol{a}}) - \overline{{\boldsymbol \delta }}_{i}\|_2 \lt c \, \|\overline{{\boldsymbol \delta }}_{i}\|_2. \end{align}

Proof Lemma 5.7. Note that $\overline{{\boldsymbol \delta }}_i = P_{\overline{{\mathbf{U}}}}\overline{{\boldsymbol \delta }}_i$, where $\overline{{\mathbf{U}}}$ is spanned by the first $k$ left singular vectors of $\overline{{\mathbf{A}}}_1$ with $\mathrm{rank}(\overline{{\mathbf{A}}}_1) = k$, and $\overline{{\mathbf{A}}}_1$, $\overline{{\mathbf{A}}}_2$ preserve the same eigen-subspace. The approximation error between $ P_{{\mathbf{U}}}(\boldsymbol{a}_i - \overline{\boldsymbol{a}})$ and $\overline{{\boldsymbol \delta }}_i$ can be decomposed as

\begin{align*} P_{{\mathbf{U}}}(\boldsymbol{a}_i - \overline{\boldsymbol{a}}) - \overline{{\boldsymbol \delta }}_i =\,& P_{{\mathbf{U}}}\big [ (\boldsymbol{a}_i - \widetilde{\boldsymbol{a}}_i) + (\widetilde{\boldsymbol{a}}_i - \overline{\boldsymbol{a}}_{i}) + (\overline{\boldsymbol{a}}_i - \overline{\boldsymbol{a}}) \big ] - P_{\overline{{\mathbf{U}}}}\overline{{\boldsymbol \delta }}_i\\ =\,& P_{{\mathbf{U}}}\boldsymbol{e}_{i} + P_{{\mathbf{U}}}(\widetilde{\boldsymbol{a}}_i - \overline{\boldsymbol{a}}_{i}) + (P_{{\mathbf{U}}} - P_{\overline{{\mathbf{U}}}})\overline{{\boldsymbol \delta }}_i\,. \end{align*}

Then by triangle inequality,

\begin{align*} \| P_{{\mathbf{U}}}(\boldsymbol{a}_i - \overline{\boldsymbol{a}}) - \overline{{\boldsymbol \delta }}_i\|_2 \leq \|P_{{\mathbf{U}}}\boldsymbol{e}_{i}\|_2 + \|P_{{\mathbf{U}}}(\widetilde{\boldsymbol{a}}_i - \overline{\boldsymbol{a}}_{i})\|_2 + \|P_{{\mathbf{U}}} - P_{\overline{{\mathbf{U}}}}\|\cdot \|\overline{{\boldsymbol \delta }}_i\|_{2}\,. \end{align*}

Note that $\|\overline{{\boldsymbol \delta }}_i\| = O(n^{-\frac{1}{2} })$ and $n^{\prime }_{i}$ concentrates around $n/(4k)$ for each $i\in [k]$ with deviation at most $\sqrt{n}\log (n)$, then by definitions of $\overline{\alpha }$ and $\overline{\beta }$ in (17),

\begin{align*} \|P_{{\mathbf{U}}}(\widetilde{\boldsymbol{a}}_i - \overline{\boldsymbol{a}}_{i})\|_2 \leq \|\widetilde{\boldsymbol{a}}_i - \overline{\boldsymbol{a}}_{i}\|_2 = O\Big (\big [ k\sqrt{n}\log (n)(\overline{\alpha } -\overline{\beta })^2 \big ]^{\frac{1}{2}} \Big ) = O[ n^{-\frac{3}{4}}\log ^{\frac{1}{2}}(n)]=o(\|\overline{{\boldsymbol \delta }}_i\|_2). \end{align*}

Meanwhile, by an argument similar to Lemma 5.6, it can be proved that $\sin \angle ({\mathbf{U}}, \overline{{\mathbf{U}}}) \lt c/2$ for any $c\in (0, 1)$, if constants $C_1, C_2$ are chosen properly, hence $\|P_{{\mathbf{U}}} - P_{\overline{{\mathbf{U}}}}\|\cdot \|\overline{{\boldsymbol \delta }}_i\|_{2}\lt \frac{c}{2} \|\overline{{\boldsymbol \delta }}_i\|_{2}$. Lemma 5.8 shows that at least half of the indices from $\{i_1, \cdots, i_s\}\cap Y_2$ satisfy $\|P_{{\mathbf{U}}}\boldsymbol{e}_i \|_2 \lt \frac{c}{2}\|\overline{{\boldsymbol \delta }}_{i}\|_2$, which completes the proof.

Lemma 5.8. Let $d = \sum _{m\in{\mathcal M}_{\max }} (m-1)a_m$. For any $c\in (0, 1)$, with probability $1 - O(n^{-k\log n})$, at least $\frac{s}{2}$ of the vectors $\boldsymbol{e}_{i_1}, \dots,\boldsymbol{e}_{i_s}$ satisfy

\begin{align*} \|P_{{\mathbf{U}}}\boldsymbol{e}_i \|_2 \leq 2\sqrt{kd({\mathcal M}_{\max } + 2)/n} \lt \frac{c}{2}\|\overline{{\boldsymbol \delta }}_{i}\|_2 \,, \quad i\in \{ i_1, \cdots, i_s \} \subset Y_2\,. \end{align*}

Definition 5.9. The vector $\boldsymbol{a}_i$ satisfying (25) is referred as good vector. The index of the good vector is hence referred to as a good vertex.

To avoid introducing extra notations, let $\boldsymbol{a}_{i_1}, \dots,\boldsymbol{a}_{i_{s_1}}$ denote good vectors with $i_1, \cdots, i_{s_1}$ denoting good indices. Lemma 5.8 indicates that the number of good vectors is at least $s_1\geq \frac{s}{2} = k\log ^2 n$.

5.1.4. Accuracy

We are going to prove the accuracy of the initial partition obtained from Algorithm 4. Lemmas 5.10, 5.11 and 5.13 are crucial in proving our results. We present the proof logic first and defer the Lemma statements later.

For each projected vector $P_{{\mathbf{U}}}(\boldsymbol{a}_{i} - \overline{\boldsymbol{a}})$, let $U^{\prime }_{i}$ denote the set of its largest $\frac{n}{2k}$ coordinates, where $i\in \{i_1, \cdots, i_s\}$ and $s = 2k \log ^2 n$. Note that vector $\overline{{\boldsymbol \delta }}_{i_j}$ in (23) only identifies blocks $V_{k(i_j)}$ and $V\setminus V_{k(i_j)}$, which can be regarded as clustering two blocks with different sizes. By Lemma 5.7, good vectors satisfy $\|P_{{\mathbf{U}}}(\boldsymbol{a}_{i_j} - \overline{\boldsymbol{a}}) - \overline{{\boldsymbol \delta }}_{i_j}\|_2 \lt c \, \|\overline{{\boldsymbol \delta }}_{i_j}\|_2$ for any $c\in (0, 1)$. Then by Lemma 5.10 (after proper normalisation), for a good index $i_j$, the number of vertices in $U^{\prime }_{i_j}$ clustered correctly is at least $(1 - \frac{4}{3}kc^{2})\frac{n}{k}$. By choosing $c = \sqrt{3(1-\nu )/(8k)}$, the condition $|U^{\prime }_{i_j} \cap V_i| \gt (1+\nu )/2 |U^{\prime }_{i_j}|$ in part (ii) of Lemma 5.11 is satisfied. In Lemma 5.6, we choose

(26) \begin{align} C_2 = 2^{{\mathcal M}_{\max } + 2}(C_3 + 1)/c = 2^{{\mathcal M}_{\max } + 2}(C_3 + 1)\sqrt{8k/(3-3\nu )}\,\,, \end{align}

where $C_3$ defined in (22). Hence, with high probability, all good vectors have at least $\mu _{\mathrm{T}}$ blue hyperedges (we call this “high blue hyperedge density”). From Lemma 5.8, at least half of the selected vectors are good. Then, in Algorithm 4, throwing out half of the obtained sets $U^{\prime }_{i}$ (those with the lowest blue hyperedge density) guarantees that the remaining sets are good.

Recall that, by choosing constant appropriately, we can make the subspace angle $\sin \angle ({\mathbf{U}}, \overline{{\mathbf{U}}}) \lt c$ for any $c\in (0, 1)$ ($\overline{{\mathbf{U}}}$ is spanned by the first $k$ left singular vectors of $\overline{{\mathbf{A}}}_1$). Then for each vector $\overline{{\boldsymbol \delta }}_{i_1}, \cdots, \overline{{\boldsymbol \delta }}_{i_k}$ with each $i_j$ selected from different vertex set $V_j$, there is a vector $P_{{\mathbf{U}}}(\boldsymbol{a}_{i_j} - \overline{\boldsymbol{a}})$ in ${\mathbf{U}}$ arbitrarily close to $\overline{{\boldsymbol \delta }}_{i_j}$, which was proved by Lemma 5.7. From (i) of Lemma 5.11, so obtained $U_{i_j}^{\prime }$ must satisfy $|U_{i_j}^{\prime }\cap V_j|\geq \nu |U_{i_j}^{\prime }|$ for each $j\in [k]$. The remaining thing is to select $k$ different $U_{i_j}^{\prime }$ with each of them concentrating around distinct $V_j$ for each $j\in [k]$. This problem is equivalent to finding $k$ vertices in $Y_2$, each from a different partition class, which can be done with $k\log ^2(n)$ samplings as shown in Lemma 5.13.

To summarise, this section is a more precise and quantitative version of the following argument: with high probability,

\begin{align*} \left \{ i_j \,:\, \exists i\,\mbox{s.t.}\,|U^{\prime}_{i_j} \cap V_i| \geq \frac{1+\nu }{2} \frac{n}{k}\right \} & \subset \left \{ i_j \,:\, U^{\prime}_{i_j} \text{ has $\geq \mu _T$ blue hyperedges}\right \}\\ & \subset \left \{ i_j\,:\, \exists i\,\mbox{s.t.}\,|U^{\prime}_{i_j} \cap V_i| \geq \nu \frac{n}{k}\right \} \,. \end{align*}

Lemma 5.10 (Adapted from Lemma 23 in [19]). Suppose $n, k$ are such that $\frac{n}{k} \in \mathbb{N}$. Let $\boldsymbol{v},\bar{\boldsymbol{v}} \in{\mathbb{R}}^n$ be two unit vectors, and let $\bar{\boldsymbol{v}}$ be such that $\frac{n}{k}$ of its entries are equal to $\frac{1}{\sqrt{n}}$ and the rest are equal to $-\frac{1}{\sqrt{n}}$. If $\sin \angle (\bar{\boldsymbol{v}},\boldsymbol{v}) \lt c \leq 0.5$, then $\boldsymbol{v}$ contains at least $(1 - \frac{4}{3}kc^{2}) \frac{n}{k}$ positive entries $\boldsymbol{v}_i$ such that $\bar{\boldsymbol{v}}_i$ is also positive.

Lemma 5.11. Suppose that we are given a set $X \subset Z$ with size $|X| = n/(2k)$. Define

\begin{align*} \mu _1\,:\!=\,&\,\frac{1}{2} \sum _{m\in{\mathcal M}}m(m-1) \left \{ \left [ \binom{\frac{\nu n}{2k}}{m} + \binom{\frac{(1 - \nu )n}{2k}}{m} \right ] \frac{a_m - b_m }{ \binom{n}{m-1} } + \binom{\frac{n}{2k}}{m} \frac{b_{m}}{ \binom{n}{m-1} } \right \}\,,\\ \mu _{2} \,:\!=\,&\, \frac{1}{2} \sum _{m\in{\mathcal M}} m(m-1)\left \{ \left [ \binom{\frac{(1 + \nu ) n}{4k} }{m} + (k-1)\binom{ \frac{(1 - \nu )n}{4k(k-1)}}{m} \right ] \frac{a_m - b_m }{ \binom{n}{m-1} } + \binom{\frac{n}{2k}}{m} \frac{b_{m}}{ \binom{n}{m-1} } \right \}\,, \end{align*}

and $\mu _{\mathrm{T}} \,:\!=\, (\mu _1 + \mu _2)/2 \in [\mu _1, \mu _2]$. There is a constant $c\gt 0$ depending on $k, a_m, \nu$ such that for sufficiently large $n$,

1. (i) If $|X\cap V_i| \leq \nu |X|$ for each $i\in [k]$, then with probability $1 - e^{-cn}$, the number of blue hyperedges in the hypergraph induced by $X$ is at most $\mu _{\mathrm{T}}$.

2. (ii) Conversely, if $|X\cap V_i| \geq \frac{1+ \nu }{2}|X|$ for some $i\in \{1, \dots, k\}$, then with probability $1 - e^{-cn}$, the number of blue hyperedges in the hypergraph induced by $X$ is at least $\mu _{\mathrm{T}}$.

Remark 5.12. Lemma 5.11 is reduced to [19, Lemma 31] when ${\mathcal M} = \{2\}$.

Lemma 5.13. Through random sampling without replacement in Step $6$ of Algorithm 4, we can find at least $k$ indices $i_1,\dots, i_k$ in $Y_2$ among $k\log ^2 n$ samples such that with probability $1 -n^{-\Omega (\log n)}$,

\begin{equation*} |U^{\prime}_{i_j} \cap U^{\prime}_{i_l}| \leq (1-\nu )\frac {n}{k}, \,\text { for any $j,l\in [k]$} \text { with } j\not =l. \end{equation*}

5.2. Local correction: Proof of Lemma 4.2

For notation convenience, let $U_i\,:\!=\, Z \cap V_i$ denote the intersection of $Z$ and true partition $V_i$ for all $i\in [k]$. In Algorithm 2, we first colour the hyperedges with red and blue with equal probability. By running Algorithm 4 on the red hypergraph, we obtain a $\nu$-correct partition $U_1^{\prime }, \dots, U_k^{\prime }$, i.e.,

(27) \begin{equation} |U_{i}\setminus U_{i}^{\prime }| \leq (1 - \nu )\cdot |U_{i}^{\prime }| = (1 - \nu )\cdot \frac{n}{2k}\,, \quad \forall i\in [k]\,. \end{equation}

In the rest of this subsection, we condition on the event that (27) holds true.

Consider a hyperedge $e = \{i_{1}, \cdots, i_{m}\}$ in the underlying $m$-uniform hypergraph. If vertices $i_{1}, \cdots, i_{m}$ are from the same block, then $e$ is a red hyperedge with probability $a_{m}/2\binom{n}{m-1}$; if vertices $i_{1}, \cdots, i_{m}$ are not from the same block, then $e$ is a red hyperedge with probability $b_{m}/2\binom{n}{m-1}$. The presence of those two types of hyperedges can be denoted by

\begin{equation*} T_{e}^{(a_m)}\sim \mathrm {Bernoulli}\left ( \frac {a_{m}}{ 2\left({n \atop m-1}\right) }\right )\,, \quad T_{e}^{(b_m)}\sim \mathrm {Bernoulli}\left ( \frac {b_{m}}{ 2\left({n \atop m-1}\right) }\right )\,, \end{equation*}

respectively. For any finite set $S$, let $[S]^{l}$ denote the family of $l$-subsets of $S$, i.e., $[S]^{l} = \{Z| Z\subseteq S, |Z| = l \}$. Consider a vertex $u\in U_1\,:\!=\, Z \cap V_1$. The weighted number of red hyperedges, which contains $u\in U_1$ with the remaining vertices in $U_j^{\prime }$, can be written as

(28) \begin{align} S_{1j}^{\prime }(u)\,:\!=\, \sum _{m\in{\mathcal M}}(m-1)\cdot \left \{ \sum _{e\in \, \mathcal{E}^{(a_m)}_{1, j}} T_e^{(a_m)} + \sum _{e\in \mathcal{E}^{(b_m)}_{1, j} } T_e^{(b_m)}\right \}\,,\quad u\in U_1\,, \end{align}

where $\mathcal{E}^{(a_m)}_{1, j}\,:\!=\, E_m([U_1]^{1}, [ U_1 \cap U_j^{\prime }]^{m-1} )$ denotes the set of $m$-hyperedges with one vertex from $[U_1]^{1}$ and the other $m-1$ from $[U_1 \cap U^{\prime }_{j}]^{m-1}$, while $ \mathcal{E}^{(b_m)}_{1, j} \,:\!=\, E_m \Big ([U_1]^{1}, \,\, [U_j^{\prime }]^{m-1} \setminus [U_1\cap U_j^{\prime }]^{m-1} \Big )$ denotes the set of $m$-hyperedges with one vertex in $[U_1]^{1}$ while the remaining $m-1$ vertices in $[U_j^{\prime }]^{m-1}\setminus [U_1 \cap U_j^{\prime }]^{m-1}$(not all $m$ vertices are from $V_1$) with their cardinalities

\begin{align*} \big|\mathcal{E}^{(a_m)}_{1, j}\big| = \binom{|U_1 \cap U^{\prime }_{j}|}{m-1}\,,\quad |\mathcal{E}^{(b_m)}_{1, j}| = \left [ \binom{|U^{\prime }_{j}|}{m-1} - \binom{|U_1 \cap U^{\prime }_{j}|}{m-1} \right ]\,. \end{align*}

We multiply $(m-1)$ in (28) as weight since the rest $m-1$ vertices are all located in $U^{\prime }_{j}$, which can be regarded as $u$’s neighbours in $U^{\prime }_{j}$. According to the fact $|U^{\prime }_j \cap U_j| \geq (\nu n/2k)$ in (27) and $|U_{j}^{\prime }| = n/(2k)$ for $j\in [k]$,

\begin{align*} \big|\mathcal{E}^{(a_m)}_{1, 1}\big| \geq \binom{\frac{\nu n}{2k}}{m-1}\,,\quad \big|\mathcal{E}^{(a_m)}_{1, j}\big| \leq \binom{\frac{(1 - \nu )n}{2k}}{m-1}\,,\,\, j\neq 1\,. \end{align*}

To simplify the calculation, we take the lower and upper bound of $\big|\mathcal{E}^{(a_m)}_{1, 1}\big|$ and $\big|\mathcal{E}^{(a_m)}_{1, j}\big|\,(j\neq 1)$ respectively. By taking expectation with respect to $T_{e}^{(a_m)}$ and $T_{e}^{(b_m)}$, then for any $u \in U_1$, we have

\begin{align*}{\mathbb{E}} S_{11}^{\prime }(u) &= \sum _{m\in{\mathcal M}} (m-1)\cdot \left [ \binom{\frac{\nu n}{2k}}{m-1} \frac{a_m - b_m}{2\binom{n}{m-1}} + \binom{\frac{n}{2k}}{m-1} \frac{b_m}{2\binom{n}{m-1}}\right ]\,,\\ {\mathbb{E}} S_{1j}^{\prime }(u) &= \sum _{m\in{\mathcal M}} (m-1)\cdot \left [ \binom{\frac{(1-\nu )n}{2k}}{m-1} \frac{a_m - b_m}{2\binom{n}{m-1}} + \binom{\frac{n}{2k}}{m-1} \frac{b_m}{2\binom{n}{m-1}}\right ]\,, \quad j\neq 1\,. \end{align*}

By assumptions in Theorem 1.7, ${\mathbb{E}} S_{11}^{\prime }(u) -{\mathbb{E}} S_{1j}^{\prime }(u) = \Omega (1)$. Define

(29) \begin{align} \mu _{\mathrm{C}} \,:\!=\, \frac{1}{2}\sum _{m\in{\mathcal M}} (m-1)\cdot \left \{ \left [ \binom{\frac{\nu n}{2k}}{m-1} + \binom{\frac{(1 - \nu )n}{2k}}{m-1} \right ] \frac{a_m - b_m}{2\binom{n}{m-1}} + 2\cdot \binom{\frac{n}{2k}}{m-1}\frac{b_m}{2\binom{n}{m-1}}\right \}\,. \end{align}

In Algorithm 5, vertex $u$ is assigned to $\widehat{U}_{i}$ if it has the maximal number of neighbours in $U_i^{\prime }$. If $u\in U_1$ is mislabelled, then one of the following events must happen:

• • $S_{11}^{\prime }(u) \leq \mu _{\mathrm{C}}$, meaning that $u$ was mislabelled by Algorithm 5.

• • $S_{1j}^{\prime }(u) \geq \mu _{\mathrm{C}}$ for some $j\neq 1$, meaning that $u$ survived Algorithm 5 without being corrected.

Lemma 5.14 shows that the probabilities of those two events can be bounded in terms of the SNR.

Lemma 5.14. For sufficiently large $n$ and any $u\in U_1 = Z \cap V_1$, we have

(30) \begin{align} \rho _1^{\prime } \,:\!=\,{\mathbb{P}} \left ( S_{11}^{\prime }(u) \leq \mu _{\mathrm{C}}\right ) \leq \rho \,,\quad \rho _j^{\prime } \,:\!=\,{\mathbb{P}} \left ( S_{1j}^{\prime }(u) \geq \mu _{\mathrm{C}}\right ) \leq \rho \,,\,(j\neq 1), \end{align}

where $\rho \,:\!=\, \exp\!\left ({-}C_{{\mathcal M}}\cdot \mathrm{SNR}_{{\mathcal M}} \right )$ with $\mathrm{SNR}_{{\mathcal M}}$ and $C_{{\mathcal M}}$ defined in (2).

As a result, the probability that either of those events happened is bounded by $\rho$. The number of mislabelled vertices in $U_1$ after Algorithm 5 is at most

\begin{align*} R_1 = \sum _{t=1}^{|U_1|}\Gamma _{t}\, + \sum _{j=2}^{k}\sum _{t=1}^{|U_1\cap U_j^{\prime }|}\Lambda _{t}\,, \end{align*}

where $\Gamma _{t}$ (resp. $\Lambda _{t}$) are i.i.d indicator random variables with mean $\rho _1^{\prime }$ (resp. $\rho _j^{\prime }$, $j\neq 1$). Then

\begin{align*}{\mathbb{E}} R_1 \leq \frac{n}{2k} \rho _1^{\prime } + \sum _{j=2}^{k}\frac{(1 - \nu )n}{2k} \rho _j^{\prime } \leq \frac{n}{2k} \cdot k\rho = \frac{n\rho }{2}\,. \end{align*}

Let $t_1\,:\!=\, n \rho/2$, where $\nu$ denotes the correctness after Algorithm 4, then by Chernoff bound (Lemma D.1),

(31) \begin{align} {\mathbb{P}} \left ( R_1 \geq n\rho \right ) ={\mathbb{P}} \left ( R_1 - n \rho/2 \geq t_1 \right ) \leq{\mathbb{P}} \left ( R_1 -{\mathbb{E}} R_1 \geq t_1 \right ) \leq e^{-c t_1} = O(e^{-n\rho })\,. \end{align}

Then with probability $1 - O(e^{-n\rho })$, the fraction of mislabelled vertices in $U_1$ is smaller than $k\rho$, i.e., the correctness of $U_1$ is at least $\gamma _{\mathrm{C}} \,:\!=\, \max \{\nu \,, 1 - k\rho \}$. Therefore, Algorithm 5 outputs a $\gamma _{\mathrm{C}}$-correct partition $\widehat{U}_1, \cdots, \widehat{U}_{k}$ with probability $1- O(e^{-n\rho })$.

5.3. Merging: Proof of Lemma 4.3

By running Algorithm 5 on the red hypergraph, we obtain a $\gamma _{\mathrm{C}}$-correct partition $\widehat{U}_1, \cdots, \widehat{U}_{k}$ where $\gamma _{\mathrm{C}}\,:\!=\,\max \{\nu \,, 1 - k\rho \} \geq \nu$, i.e.,

(32) \begin{equation} |U_{j}\cap \widehat{U}_j| \geq \nu \cdot |\widehat{U}_j| = \frac{\nu n}{2k}\,, \quad \forall j\in [k]\,. \end{equation}

In the rest of this subsection, we shall condition on this event and abbreviate $Y\cap V_l$ by $W_l\,:\!=\, Y\cap V_l$. The failure probability of Algorithm 6 is estimated by the presence of hyperedges between vertex sets $Y$ and $Z$.

Consider a hyperedge $e = \{i_{1}, \cdots, i_{m}\}$ in the underlying $m$-uniform hypergraph. If vertices $i_{1}, \cdots, i_{m}$ are all from the same cluster $V_{l}$, then the probability that $e$ is an existing blue edge conditioning on the event that $e$ is not a red edge is

(33) \begin{align} \psi _m\,:\!=\,{\mathbb{P}}\left [e \text{ is a blue edge } \Big | e \text{ is not a red edge } \right ] = \frac{\frac{a_{m}}{2\left({n \atop m-1}\right)}}{1 - \frac{a_{m}}{2\left({n \atop m-1}\right)}} = \frac{a_{m}}{2\left({n \atop m-1}\right)}(1+o(1))\,, \end{align}

and the presence of $e$ can be represented by an indicator random variable $\zeta _e^{(a_m)} \sim \mathrm{Bernoulli}\left (\psi _m\right )$. Similarly, if vertices $i_{1}, \cdots, i_{m}$ are not all from the same cluster $V_l$, the probability that $e$ is an existing blue edge conditioning on the event that $e$ is not red

(34) \begin{align} \phi _m\,:\!=\,{\mathbb{P}}\left [e \text{ is a blue edge } \Big | e \text{ is not a red edge } \right ] = \frac{\frac{b_{m}}{2\left({n \atop m-1}\right)}}{1 - \frac{b_{m}}{2\left({n \atop m-1}\right)}} =\frac{b_{m}}{2\left({n \atop m-1}\right)}(1+o(1))\,, \end{align}

and the presence of $e$ can be represented by an indicator variable $\xi _e^{(b_m)} \sim \mathrm{Bernoulli}\left (\phi _m\right )$.

For any vertex $w\in W_l \,:\!=\, Y\cap V_l$ with fixed $l \in [k]$, we want to compute the number of hyperedges containing $w$ with all remaining vertices located in vertex set $\widehat{U}_{j}$ for some fixed $j\in [k]$. Following a similar argument given in Subsection 5.2, this number can be written as

(35) \begin{align} \widehat{S}_{lj}(w)\,:\!=\, \sum _{m\in{\mathcal M}}(m-1)\cdot \left \{ \sum _{e\in \, \widehat{\mathcal{E}}^{(a_m)}_{l, j}} \zeta _e^{(a_m)} + \sum _{e\in \widehat{\mathcal{E}}^{(b_m)}_{l, j} } \xi _e^{(b_m)}\right \}\,,\quad w\in W_l\,, \end{align}

where $\widehat{\mathcal{E}}^{(a_m)}_{l, j}\,:\!=\, E_m([W_l]^{1}, [U_l\cap \widehat{U}_j]^{m-1})$ denotes the set of $m$-hyperedges with $1$ vertex from $[W_l]^{1}$ and the other $m-1$ vertices from $[U_l\cap \widehat{U}_j]^{m-1}$, while $ \widehat{\mathcal{E}}^{(b_m)}_{l, j} \,:\!=\, E_m ([W_l]^{1}, \,\, [\widehat{U}_j]^{m-1} \setminus [U_l\cap \widehat{U}_j]^{m-1})$ denotes the set of $m$-hyperedges with $1$ vertex in $[W_l]^{1}$ while the remaining $m-1$ vertices are in $[\widehat{U}_j]^{m-1}\setminus [U_l\cap \widehat{U}_j]^{m-1}$, with their cardinalities

\begin{align*} |\widehat{\mathcal{E}}^{(a_m)}_{l, j}| = \binom{|U_l \cap \widehat{U}_{j}|}{m-1}\,,\quad |\widehat{\mathcal{E}}^{(b_m)}_{l, j}| = \left [ \binom{|\widehat{U}_{j}|}{m-1} - \binom{|U_l \cap \widehat{U}_{j}|}{m-1} \right ]\,. \end{align*}

Similarly, we multiply $(m-1)$ in (35) as weight since the rest $m-1$ vertices can be regarded as $u$’s neighbours in $\widehat{U}_{j}$. By accuracy of Algorithm 5 in (32), $|\widehat{U}_{j} \cap U_j| \geq \nu n/(2k)$, then

\begin{align*} |\widehat{\mathcal{E}}^{(a_m)}_{l, l}| \geq \binom{\frac{\nu n}{2k}}{m-1}\,,\quad |\widehat{\mathcal{E}}^{(a_m)}_{l, j}| \leq \binom{\frac{(1 - \nu )n}{2k}}{m-1}\,,\,\, j\neq l\,. \end{align*}

Taking expectation with respect to $\zeta _e^{(a_m)}$ and $\xi _e^{(b_m)}$, for any $w \in W_l$, we have

\begin{align*}{\mathbb{E}} \widehat{S}_{ll}(w) &= \sum _{m\in{\mathcal M}} (m-1)\cdot \left [ \binom{\frac{\nu n}{2k}}{m-1} (\psi _{m} - \phi _{m}) + \binom{\frac{n}{2k}}{m-1} \phi _{m}\right ],\\[7pt] {\mathbb{E}} \widehat{S}_{lj}(w) &= \sum _{m\in{\mathcal M}} (m-1)\cdot \left [ \binom{\frac{(1 - \nu ) n}{2k}}{m-1} (\psi _{m} - \phi _{m}) + \binom{\frac{n}{2k}}{m-1} \phi _{m} \right ],\,\, j\neq l\,. \end{align*}

By assumptions in Theorem 1.7, ${\mathbb{E}} \widehat{S}_{ll}(w) -{\mathbb{E}} \widehat{S}_{lj}(w) = \Omega (1)$. We define

(36) \begin{align} \mu _{\mathrm{M}} \,:\!=\, \frac{1}{2}\sum _{m\in{\mathcal M}} (m-1)\cdot \left \{ \left [ \binom{\frac{\nu n}{2k}}{m-1} + \binom{\frac{(1 - \nu )n}{2k}}{m-1} \right ](\psi _{m} - \phi _{m}) + 2\binom{\frac{n}{2k}}{m-1} \phi _{m}\right \}\,. \end{align}

After Algorithm 6, if a vertex $w\in W_l$ is mislabelled, one of the following events must happen

• • $\widehat{S}_{ll}(w) \leq \mu _{\mathrm{M}}$, which implies that $u$ was mislabelled by Algorithm 6.

• • $\widehat{S}_{lj}(w) \geq \mu _{\mathrm{M}}$ for some $j\neq l$, which implies that $u$ survived Algorithm 6 without being corrected.

By an argument similar to Lemma 5.14, we can prove that for any $w\in W_l$,

\begin{align*} \hat{\rho }_{l} \,:\!=\,{\mathbb{P}}( \widehat{S}_{ll}(w)\leq \mu _{\mathrm{M}} ) \leq \rho \,,\quad \hat{\rho }_{j} \,:\!=\,{\mathbb{P}} ( \widehat{S}_{lj}(w) \geq \mu _{\mathrm{M}} ) \leq \rho \,, \,(j\neq l), \end{align*}

where $\rho \,:\!=\, \exp\!\left ({-}C_{{\mathcal M}}\cdot \mathrm{SNR}_{{\mathcal M}} \right )$. The misclassified probability for $w\in W_l$ is upper bounded by $\sum _{j=1}^{k}\hat{\rho }_{j} \leq k\rho$. The number of mislabelled vertices in $W_l$ is at most $ R_l = \sum _{t=1}^{|W_l|}\Gamma _{t} \,,$ where $\Gamma _{t}$ are i.i.d indicator random variables with mean $k \rho$ and ${\mathbb{E}} R_l \leq n/(2k)\cdot k\rho = n\rho/2$. Let $t_l\,:\!=\, n \rho/2$, by Chernoff bound (Lemma D.1),

\begin{align*}{\mathbb{P}} \left ( R_l \geq n\rho \right ) ={\mathbb{P}} \left ( R_l - n\rho/2\geq t_l \right ) \leq{\mathbb{P}} \left ( R_l -{\mathbb{E}} R_l \geq t_l \right ) \leq e^{-c t_l} = O(e^{-n\rho })\,. \end{align*}

Hence with probability $1 - O(e^{-n\rho })$, the fraction of mislabelled vertices in $W_l$ is smaller than $k\rho$, i.e., the correctness in $W_l$ is at least $\gamma _{\mathrm{M}} \,:\!=\,\max \{ \nu,\, 1 - k\rho \}$.

5.4. Proof of Theorem 1.7

Now we are ready to prove Theorem 1.7. The correctness of Algorithms 5 and 6 are denoted by $\gamma _{\mathrm{C}}$ and $\gamma _{\mathrm{M}}$ respectively, then with probability at least $1 - O(e^{-n\rho })$, the correctness $\gamma$ of Algorithm 2 is $ \gamma \,:\!=\, \min \{\gamma _{\mathrm{C}}, \gamma _{\mathrm{M}}\} = \max \{\nu,\, 1 - k\rho \}.$ We will have $\gamma = 1 - k\rho$ if $\nu \leq 1 - k\rho$, equivalently,

(37) \begin{align} \text{SNR}_{{\mathcal M}}(k) \geq \frac{1}{C_{{\mathcal M}}} \log \Big ( \frac{k}{1 - \nu }\Big )\,, \end{align}

otherwise $\gamma = \nu$. The inequality (37) holds since

\begin{align*} \text{SNR}_{{\mathcal M}}(k) =& \frac{ \left [\sum _{m\in{\mathcal M}} (m-1)\left (\frac{a_m - b_m}{k^{m-1}} \right ) \right ]^2 }{\sum _{m\in{\mathcal M}} (m-1)\left (\frac{a_m - b_m}{k^{m-1}} + b_m \right )}\\[7pt] \geq & \frac{[\sum _{m\in{\mathcal M}} (a_m-b_m)]^2}{k^{2{\mathcal M}_{\max }-2}({\mathcal M}_{\max }-1)d}\geq \frac{(C_{\nu })^2}{{\mathcal M}_{\max }-1} \log \Big (\frac{k}{1 - \nu } \Big ) \geq \frac{1}{C_{{\mathcal M}}}\log \Big (\frac{k}{1-\nu } \Big )\,. \end{align*}

where the first two inequalities hold since $d \,:\!=\, \sum _{m\in{\mathcal M}}(m-1)a_m$ and Condition (4b), while the last inequality holds by taking $C_{\nu }\geq \max \{ \sqrt{({\mathcal M}_{\max }-1)/C_{{\mathcal M}} }, C_2 \}$ with $C_2$ defined in (26).

Remark 5.15. The lower bound $C$ in (4a) comes from the requirement in Lemma 5.4 that only a few high-degree vertices be deleted. The constant $C_{\nu }$ in (4b) comes from the requirement in Lemma 5.6 that the subspace angle is small. When $C$ is not so large (or the hypergraph is too sparse), one could still achieve good accuracy $\gamma$ if $C_{\nu }$ is large enough (the difference between $a_m$ and $b_m$ is large enough).

Remark 5.16. Condition (37) indicates that the improvement of accuracy from local refinement (Algorithms 5 and 6) will be guaranteed when $\text{SNR}_{{\mathcal M}}(k)$ is large enough. If $\text{SNR}_{{\mathcal M}}(k)$ is small, we use correctness of Algorithm 4 instead, i.e., $\gamma = \nu$, to represent the correctness of Algorithm 2.

5.5. Proof of Corollary 1.9

For any fixed $\nu \in (1/k, 1)$, $\mathrm{SNR}_{{\mathcal M}}(k) \to \infty$ implies $\rho \to 0$ and

\begin{equation*}d =\sum _{m\in {\mathcal M}}(m-1)a_m \to \infty .\end{equation*}

Since

\begin{align*} \frac{\sum _{m\in{\mathcal M}} (m-1)(a_m-b_m)^2}{\sum _{m\in{\mathcal M}}(m-1)a_m} \geq \frac{\sum _{m\in{\mathcal M}} (m-1)(a_m-b_m)^2}{\sum _{m \in{\mathcal M}}(m-1)k^{m-1}(a_m+ k^{m-1}b_m)} = \mathrm{SNR}_{{\mathcal M}}(k)\,, \end{align*}

Condition (4b) is satisfied. Applying Theorem 1.7, we find $\gamma = 1 - o(1)$, which implies weak consistency. The constraint of $\mathrm{SNR}_{{\mathcal M}}(k) = o(\log n)$ is used in the proof of Lemma 4.2, see Remark 1.10.