From: ICLR.cc/2023/Conference/Program_Chairs
title: Paper Decision
decision: Reject
metareview:_summary,_strengths_and_weaknesses: The paper introduces a new clustering algorithm that works in two phases. In the first phase the input graph is processed using a density-based technique and the output of this phase is an order matrix with clear block structure. In the second phase the matrix is processed and a clustering is returned by the algorithm.

The paper contains some novel and interesting ideas and the experiments show that the method is effective in practice.

The main concern for the paper is its readability, most reviewers agree that the paper in the current form(after the rebuttal phase) is still very hard to read and the paper would benefit from a major rewriting before acceptance. In particular, section 3 is a bit problematic. The theoretical results seem over-complicated and the analyzed model is a bit unrealistic.

Overall, the paper is interesting but it does not seem ready for publication in the current state.

Minor comment:
The fact that “the density-based clustering method, which is designed without knowing the cluster number, we tune the parameter of the density-based clustering method, e.g., DBSCAN, until the number of output clusters is K” is a bit concerning. In fact, the density based method could be forced to run with suboptimal parameters in these experiments.
justification_for_why_not_higher_score: The paper contains some nice idea but the level of writing is not good enough for ICLR.
justification_for_why_not_lower_score: N/A

From: ICLR.cc/2023/Conference/Paper498/Reviewer_eHSt
title: Response
comment: I thank the authors for their attempts to improve the theoretical aspects of the paper. However, with respect, my concern is now that the result (the deterministic case) seems both unrealistic and potentially trivial.

I want to take a step back to my original review, where I described my opinion being along the lines that the approach which is being proposed seems sensible and that I believe there is a way to motivate the approach soundly and persuasively, but that the way it is being presented currently is not clear nor persuasive enough. As a point of advice, which the authors may choose to take, or not, I would not try to provide a theoretical motivation unless a very persuasive and clear result can be obtained. There is nothing wrong with a well argued heuristic method and many sensible approaches which have been very influential do not have deep theoretical underpinnings. I would suggest that the authors, rather than attempting to salvage the paper in its current form, which I unfortunately think is not ready for publication, focus on the intuitive justifications for their approach and try to portray those as persuasively as possible and revise their paper for submission to a future conference.

Having said this, I do still believe the approach which is being introduced has merit, and I have decided to increase my score somewhat. I will not recommend acceptance, however because I believe the approach does have potential, and because the other reviewers have shown a higher opinion of the work, I would not be unsatisfied to see the paper accepted.

From: ICLR.cc/2023/Conference/Paper498/Authors
title: Response to Reviewer eHSt
comment: **Case 3:** $K\geq3$. Consider $t\in[c_{k},c_{k+1}]$, $k=1,2,...,K-2$.

\begin{align*}
	F(t) 
	 =\frac{c_{k}^{2}b(1,k)+(t-c_{k})^{2}\mu_{k+1}}{c_{k}^{2}b(1,k)+(c_{k+1}-c_{k})(t-c_{k})\mu_{k+1}}+\frac{\left(N-c_{k+1}\right)^{2}b(k+2,K)+(c_{k+1}-t)^{2}\mu_{k+1}}{\left(N-c_{k+1}\right)^{2}b(k+2,K)+(c_{k+1}-t)(c_{k+1}-c_{k})\mu_{k+1}}
\end{align*}
where 
$b(1,k)=\frac{c_{1}^{2}\mu_{1}+...,+\left(c_{k}-c_{k-1}\right)^{2}\mu_{k}}{c_{k}^{2}}$
and $b(k+2,K)=\frac{\left(c_{k+2}-c_{k+1}\right)^{2}\mu_{k+2}+...,+\left(N-c_{k-1}\right)^{2}\mu_{K}}{\left(N-c_{k+1}\right)^{2}}.$
Denote $d=c_{k+1}-c_{k}$, and $x=t-\frac{c_{k+1}+c_{k}}{2}$. Since
$t\in(c_{k+1},c_{k})$, we have $x\in(-\frac{d}{2},\frac{d}{2})$.
Denote $B_{1}=\frac{\mu_{k+1}}{c_{k}^{2}b(1,k)}$, and $B_{2}=\frac{\mu_{k+1}}{(N-c_{k+1})^{2}b(k+2,K)}$.
So, $F(t)$ can be written as
\begin{align*}
F(x)=\frac{x^{2}(-d^{2}B_{1}B_{2}+B_{1}+B_{2})+2dx(B_{1}-B_{2})+\frac{3}{4}d^{2}(B_{1}+B_{2})+\frac{d^{4}}{4}B_{1}B_{2}+2}{-x^{2}d^{2}B_{1}B_{2}+xd(B_{1}-B_{2})+\frac{d^{4}}{4}B_{1}B_{2}+\frac{1}{2}d^{2}(B_{1}+B_{2})+1}
\end{align*}
$F(x)$ is monotonically decreasing for $x\leq x_0$, and monotonically increasing for $x\geq x_0$, where
$x_{0}=
	\frac{\left(\sqrt{B_{1}d^{2}+1}+\sqrt{B_{2}d^{2}+1}\right)^{2}}{2d(B_{2}-B_{1})}$ if $B_{1}\neq B_{2}$ and
	$0$ if $B_{1}=B_{2}$.

Recall the definition of $d$, and $x$, $F(t)$ is monotonically decreasing for $t\leq \hat{t}$, and monotonically increasing for $t\geq \hat{t}$, where
$\hat{t}=
	\frac{\left(\sqrt{B_{1}\left(c_{k+1}-c_{k}\right)^{2}+1}+\sqrt{B_{2}\left(c_{k+1}-c_{k}\right)^{2}+1}\right)^{2}}{2\left(c_{k+1}-c_{k}\right)(B_{2}-B_{1})}+\frac{1}{2}\left(c_{k+1}+c_{k}\right) $ if $B_{1}\neq B_{2}$ and
	$\frac{1}{2}\left(c_{k+1}+c_{k}\right)$ if $B_{1}=B_{2}$.

Thus, for the interval $[c_{k},c_{k+1}]$, $k=1,...,K-2$, $F(t)$
decreases for $t\in[c_{k},\hat{t}]$ and increases for $t\in[\hat{t},c_{k+1}]$
if $\hat{t}\in(c_{k},c_{k+1})$; $F(t)$ decreases for $t\in[c_{k},c_{k+1}]$
if $\hat{t}\geq c_{k+1}$; $F(t)$ increases for $t\in[c_{k},c_{k+1}]$
if $\hat{t}\leq c_{k}$. 


**Theorem 3.2** If the within-cluster weight of the $k$th cluster {$\{c_{k-1}+1,c_{k-1}+2,...,c_{k}\}$}  is $\mu_{k}>0$, $\forall k\in[1,K]$, and the between-cluster weight is zero. Then, our BDSL algorithm will converge to $\{c_k\}_{k=1}^{K-1}$ by maximizing the objective function of problem (2).

From: ICLR.cc/2023/Conference/Paper498/Authors
title: Response to Reviewer eHSt
comment: Dear Reviewer eHSt,

Thank you so much for your reply. We sincerely appreciate your questions. They are beneficial for us to improve the quality of our paper. In order to make the paper clear. We remove the usage of the expectation and suppose the weight is constant directly.

Suppose there exists a $K$-component graph matrix  $\mathbf{W}\in\mathbb{R}^{N\times N}$ which is ordered correctly, and the $k$th component corresponds to the $k$th cluster $[c_{k-1}+1,c_{k}]$, where $c_{k-1}<c_{k},\forall k\in[1,K]$, and the dummy indexes $c_{0}:=0$, $c_{K}:=N$. Suppose the within-cluster weight of the $k$th cluster, $w_{ij}=\mu_{k}$, $\mu_{k}>0$, $\forall k\in[1,K]$, and the between-cluster weight, $w_{ij}=0$, $i\in[c_{k-1}+1,c_{k}]$, $j\notin[c_{k-1}+1,c_{k}], \forall k\in[1,K]$. Suppose the max weight is $\alpha$. Obviously, the min weight is $0$. We apply min-max normalization on all of the weights to make sure $w_{ij}\in[0,1]$ for any $i,j\in[1,N]$, i.e., the weight $w_{ij}$ is updated by $\frac{w_{ij}}{\alpha}$. 

Suppose we partition a $K$-component graph matrix into a two-block-diagonal form by index $t$. Then, the function $F(t)$ satisfies Theorem 3.1.
**Proof of Theorem 3.1**
Consider
\begin{align*}
F(t) &=\frac{\sum_{i,j\in[1,t]}\frac{w_{ij}}{\alpha}}{\sum_{i\in[1,t]}\sum_{j\in[1,N]}\frac{w_{ij}}{\alpha}}+\frac{\sum_{i,j\in[t+1,N]}\frac{w_{ij}}{\alpha}}{\sum_{i\in[t+1,N]}\sum_{j\in[1,N]}\frac{w_{ij}}{\alpha}} \\
&=\frac{\sum_{i,j\in[1,t]}w_{ij}}{\sum_{i\in[1,t]}\sum_{j\in[1,N]}w_{ij}}+\frac{\sum_{i,j\in[t+1,N]}w_{ij}}{\sum_{i\in[t+1,N]}\sum_{j\in[1,N]}w_{ij}} 
\end{align*}


**Case 0:** $K=1$, we have $F(t)=\frac{t^{2}\mu_{1}}{tN\mu_{1}}+\frac{(N-t)^{2}\mu_{1}}{(N-t)N\mu_{1}}=1$.

**Case 1:** $K\geq2$. Consider $t\in(0,c_{1}]$, we have
\begin{align*}
	F(t) & =\frac{t^{2}\mu_{1}}{tc_{1}\mu_{1}}+\frac{(c_{1}-t)^{2}\mu_{1}+(c_{2}-c_{1})^{2}\mu_{2}+...+(N-c_{K-1})^{2}\mu_{K}}{(c_{1}-t)c_{1}\mu_{1}+(c_{2}-c_{1})^{2}\mu_{2}+...+(N-c_{K-1})^{2}\mu_{K}}\\
	& =\frac{t}{c_{1}}+\frac{1+(c_{1}-t)^{2}C_{1}}{1+(c_{1}-t)c_{1}C_{1}}
\end{align*}
where $C_{1}=\frac{\mu_{1}}{(N-c_{1})^{2}b(2,K)}$ and $b(2,K)=\frac{(c_{2}-c_{1})^{2}\mu_{2}+...+(N-c_{k-1})^{2}\mu_{K}}{(N-c_{1})^{2}}$. Then, we have
\begin{align*}
	F(t)
	& =\frac{t(1+(c_{1}-t)c_{1}C_{1})+c_{1}(1+(c_{1}-t)^{2}C_{1})}{c_{1}(1+(c_{1}-t)c_{1}C_{1})}\\
	& =\frac{t+c_{1}+(c_{1}-t)c_{1}C_{1}(t+c_{1}-t)}{c_{1}+(c_{1}-t)c_{1}^{2}C_{1}}\\
	& =\frac{t+c_{1}+(c_{1}-t)c_{1}^{2}C_{1}}{c_{1}+(c_{1}-t)c_{1}^{2}C_{1}}\\
	& =1+\frac{1}{(1+c_{1}^{2}C_{1})c_{1}t^{-1}-c_{1}^{2}C_{1}}
\end{align*}
$F(t)$ is monotonically increasing for $t\in(0,c_{1}]$.

**Case 2:** $K\geq2$. Consider $t\in[c_{K-1},N)$, we have
\begin{align*}
	F(t) & =\frac{(N-t)^{2}\mu_{1}}{(N-t)(N-c_{K-1})\mu_{1}}
	 +\frac{c_{1}^{2}\mu_{1}+(c_{2}-c_{1})^{2}\mu_{2}+...+(c_{K-1}-c_{K-2})^{2}\mu_{K-1}+(t-c_{K-1})^{2}\mu_{K}}{c_{1}^{2}\mu_{1}+(c_{2}-c_{1})^{2}\mu_{2}+...+(c_{K-1}-c_{K-2})^{2}\mu_{K-1}+(t-c_{K-1})(N-c_{K-1})\mu_{K}}\\
	& =\frac{(N-t)^{2}\mu_{1}}{(N-t)(N-c_{K-1})\mu_{1}}+\frac{c_{K-1}^{2}b(1,K-1)+(t-c_{K-1})^{2}\mu_{K}}{c_{K-1}^{2}b(1,K-1)+(t-c_{K-1})(N-c_{K-1})\mu_{K}}\\
	& =\frac{N-t}{N-c_{K-1}}+\frac{1+(t-c_{K-1})^{2}C_{2}}{1+(t-c_{K-1})(N-c_{K-1})C_{2}}\\
\end{align*}
where $C_{2}=\frac{\mu_{K}}{c_{K-1}^{2}b(1,K-1)}$ and $b(1,K-1)=\frac{c_{1}^{2}\mu_{1}+(c_{2}-c_{1})^{2}\mu_{2}+...+(c_{K-1}-c_{K-2})^{2}\mu_{K-1}}{c_{K-1}^{2}}$. Then, we have
\begin{align*}
	F(t) 
	& =1+\frac{N-t}{N-c_{K-1}+(N-c_{K-1})^{2}C_{2}(t-c_{K-1})}\\
	& =1+\frac{1}{(N-c_{K-1})(1+(N-c_{K-1})^{2}C_{2})(N-t)^{-1}-(N-c_{K-1})^{2}C_{2}}
\end{align*}
$F(t)$ is monotonically decreasing for $t\in[c_{K-1},N)$.

From: ICLR.cc/2023/Conference/Paper498/Reviewer_eHSt
title: Thanks
comment: Thanks for your responses to my concerns.

I was just looking over the paper again; I cannot see where this operation of modifying the weights by min-max normalisation is done.

Furthermore, apologies if I am wrong, but in your proof of Thm 3.1 it looks as though you have assumed E[1/X] = 1/E[X], or something similar. Otherwise I don't know how you even get the first step. It is not trivial to me to evaluate the expected value of a sum of i.i.d. normal random variables divided by another sum of non-identical normal random variables, where the sums overlap and so are non-independent. This alone seems like a hard problem, but you state your result as though it is fact.

From: ICLR.cc/2023/Conference/Paper498/Authors
title: Response to Reviewer eHSt
comment: (Q3) We plus $\beta$ on all of the weight. This operation does not affect the property of Ncut problem. Specifically, recall the min-max normalization operation on the graph, i.e., $\frac{w_{ij}-w_{min}}{w_{max}-w_{min}}$. We set $\beta=-w_{min}$. The constant term $w_{max}-w_{min}$ can be offset for the Ncut problem. Thus, the operation of adding $\beta$ is the same as the min-max normalization operation. As we all know, min-max normalization does not affect the property of Ncut. So, we do not need to care about the value of $\beta$. We just use $\beta$ to avoid the same confusion as you.

Due to the effect of the noise on the weight, the maximizer of CNcut is possible to be $t_1,t_2,...,t_{K-1}$, which is not equal to the true clustering partition $c_1,c_2,...,c_{K-1}$. It is not meaningful to explore the value of $t_1,t_2,...,t_{K-1}$, despite the dynamic programming-based searching method with $O(KN^2)$ complexity can find $t_1,t_2,...,t_{K-1}$, as said in Q0. The expectation operation only works on removing partial effects of noise on the weight. It is reasonable and meaningful for our algorithm to converge to $c_1,c_2,...,c_{K-1}$ with low complexity from the expectation view. 

The proof of Theorem 3.2 has been explained while describing the proposed BDSL algorithm, so we do not prove it alone. In short, if the noise is negligible, we insert $K-1$ partition indexes one by one. Each partition should be one of the true clustering partitions $c_1,c_2,...,c_{K-1}$ according to Corollary 3.1.1.

From: ICLR.cc/2023/Conference/Paper498/Authors
title: Response to Reviewer eHSt
comment: Dear Reviewer eHSt,

Thank you so much for your reply. We sincerely hope that our answers fully address your concerns.

(Q0) Overall, we proposed a joint density and graph-based clustering method, which remains the advantages of density-based and graph-based clustering methods and overcomes their disadvantages. Since DBSCAN and Ncut are the most famous and influential methods of density-based and graph-based clustering methods separately, this paper chooses DBSCAN and Ncut to describe the proposed integrated density and graph clustering framework. We can also replace Ncut with Radiocut, and replace DBSCAN with OPTICS. The difference and similarities between our method and DBSCAN, Ncut should be clarified in the paper. Thank you very much for your reminder.

**The disadvantage of DBSCAN:** It can only provide flat labeling of the data objects based on a global density threshold. Using a single density threshold can often not properly characterize common data sets with clusters of very different densities and/or nested clusters. 

**The disadvantage of Ncut:** Ncut is an NP-hard problem, and the famous spectral clustering algorithm can only give us a local clustering by applying Kmeans on the eigenvector.

We propose a two-stage clustering method. It remains the advantages of DBSCAN and Ncut, i.e., the usage of the density-based relationship and the graph construction, refinement, and partition technique. In addition, **it overcomes the disadvantage of DBSCAN**: CNcut is used to identify the clustering structure instead of using a single density threshold as DBSCAN, and Ncut does not face the uneven density clustering problem of DBSCAN. **It also overcomes the disadvantage of Ncut**: Due to the density-based ordering step, the Ncut becomes a non-NP-hard problem from the partition view. Our new dynamic programming-based searching method with $O(KN^2)$ complexity can find the optimal solution to the proposed partition-based CNcut problem with any graph. However, in this paper, we proposed the BDSL algorithm with $O(NlogN)$ complexity to solve the partition-based CNcut problem. Although the proposed BDSL can not guarantee global optimality with any graph (the dynamic programming-based searching can do it), it can converge to the true clustering partition intuitively.

(Q1) HDBSCAN is a well-known density-based method that is proposed for solving the uneven density clustering problem of DBSCAN. There are two main contributions of HDBSCAN:

1. HDBSCAN uses mutual reachability distance to measure the similarity between samples. This setting can also be used in our clustering framework. Specifically, the graph-based method includes three phases: graph construction, graph refinement (e.g., prune and generate), and graph partitioning. Our paper focus on the third phase, i.e., graph partitioning. The usage of mutual reachability distance can be seen as the second phase, i.e., graph refinement. In our paper, we can replace the common weight with the mutual reachability distance. However, there are also many graph refinement method which is more complicated and effective than the mutual reachability distance. We will not mention them here. This is one of the advantages of combining density-based clustering with graph-based clustering.

2. HDBSCAN uses the minimum spanning tree and the proposed density splitting metric to identify the clustering structure. This strategy is proposed to solve the uneven density clustering problem of DBSCAN. We have added HDBSCAN as one of the comparisons in our experimental section, and its performance is unsatisfactory. The reason is 
   **a)** HDBSCAN replies on the performance of the minimum spanning tree. The tree only remains $N-1$ edges among the $N^2$ edges in the graph. Thus, the $N-1$ edges must contain enough information on sample relationships to identify the clustering structure. Even a small fraction of errors in the tree can lead to a failed clustering.
   **b)** Hierarchical clustering is an unpopular clustering method because of its poor performance. It is not advisable for HDBSCAN to identify the clustering structure based on the hierarchy.

(Q2) As you said, any dissimilarity matrices can be applied to DBSCAN. However, almost all works use graph-based clustering and density-based clustering alone. Our paper reminds the researcher that density-based clustering can also be used in the subspace clustering task and enjoy the superiority of graph construction and refinement technique. In addition, our main contribution is not only combining density-based clustering and graph-based clustering but also developing a new clustering strategy that can overcome the disadvantage of DBSCAN and Ncut, as illustrated in Q0.

From: ICLR.cc/2023/Conference/Paper498/Reviewer_eHSt
title: Responses
comment: Dear authors,

thank you for your responses to my comments. To begin, I understand the differences between your approach and DBSCAN, and, to reiterate, I believe your proposal may have advantages in many situations. However, my concern is not that it is similar to DBSCAN (in my opinion) but rather that this fact was not addressed/discussed in the paper. Regarding specific comments:

- There are variants of DBSCAN which use varying bandwidths (\epsilon in DBSCAN notation) which can allow for the detection of clusters of different density. See, e.g.:
Campello, Ricardo JGB, Davoud Moulavi, and Jörg Sander. “Density-based clustering based on hierarchical density estimates.” Pacific-Asia conference on knowledge discovery and data mining. Springer, Berlin, Heidelberg, 2013.

- DBSCAN can easily, as far as I am aware, operate on distance/dissimilarity matrices directly, rather than the original points. The duality between dissimilarity and similarity matrices means this difference between yours and DBSCAN's approach may be overstated?

- Regarding Theorem 3.2: First, there is no \beta which guarantees positivity, although fairly a large enough value can be set a priori to ensure positivity with very high probability. Further, perhaps I am misunderstanding the theorem, but maximising the expected value of an objective is very different from maximising the actual objective. Can you elaborate? Also, there is no proof for Theorem 3.2. I'd be interested to see how the authors have approached the proof, and also to fully understand what the theorem is stating.

From: ICLR.cc/2023/Conference/Paper498/Authors
title: Feedback to reviewer Fmso
comment: Dear Reviewer Fmso:

Thank you so much for your careful reading and lovely review comments. In the revised version, we will carefully add the following limitations of our work.

- Our proposed BDSL needs to know the number of clusters, and it can only guarantee to converge to the optimal global solution under the Gaussian distribution assumption. In our newest research, we have developed an alternative algorithm of BDSL which can identify the optimal clustering structure without knowing the number of clusters. 
- Our clustering method requires that the cluster is ordered correctly by the proposed DBGO according to the graph matrix. However, there are two parameters in the proposed DBGO. Although DBGO is not sensitive to these two parameters, we still need to pay more attention to them.

From: ICLR.cc/2023/Conference/Paper498/Authors
title: Feedback to reviewer eHSt
comment: Dear eHSt:

Thank you so much for your careful reading and constructive review comments. We have carefully addressed all your questions below and revised the manuscript accordingly. We sincerely hope that our answers fully address your concerns.

- (Q1) Overriding any potential technical issues

  In this paper, we proposed a two-stage clustering scheme, including density-based graph ordering (DBGO) and partition-based graph cut (BDSL). The differences and similarities between DBSCAN and the proposed DBGO are depicted as follows:

  The **differences** between the proposed DBGO and the conventional DBSCAN:

  ​	**1)** **The defination of neighborhood is different**. For the sample $x_i$, in DBSCAN, if $w_{ij}<\epsilon$, then $x_j$ is seen as the sample in the neighborhood of $x_i$. In proposed DBGO, we first sort $w_{ij}$, $j=1,2,...,N$ in descending order. Then, the largest several samples satisfying the ratio requirement of $\delta$ are seen as the sample in the neighborhood of $\mathbf{x}_i$. The proposed ratio threshold is more suitable for the dataset containing clusters with uneven weight density. For example, if the average within-cluster density of cluster one is 1, and the average within-cluster density of cluster two is 0.3, then the weight to the sample in cluster one can be assumed as 1 1 1 1... 1 0.4 0.4 ... 0.4, and the weight to the sample in cluster two can be assumed as 0.3 0.3 ... 0.3 0.1 0.1 ... 0.1. For this case, the ratio threshold is better than the constant threshold to identify the neighborhood of samples.

  ​	 **2)** **The task is different**. The input of DBSCAN is all of the samples, and the output of DBSCAN is the predicted label. However, the input of DBGO is a square graph matrix, and the output of DBGO is the matrix after ordering. DBSCAN relies on cautious parameter choices to identify the clustering structure. It requires a clear, low-density boundary between two clusters. However, our DBGO is not sensitive to the parameter. We only need to iterate over all the points one cluster by one cluster according to the given graph weight instead of identifying the clustering structure. Our non-parameter BDSL method can identify the clusters even if there is no clear boundary between clusters. In our experiments, we usually set $\delta=0.5$ and $\varepsilon=6$ for DBGO. However, we should carefully tune the parameter of DBSCAN for each dataset because there is a huge difference in the within-cluster density of different datasets.

  

  The **similarity** between the proposed DBGO and the conventional DBSCAN: 

  ​	1) They use a similar greedy strategy to search the samples one by one.

    **We have clarified the difference between DBSCAN and DBGO in the revised version as said above.**

- (Q2) Thank you for your question about the unreality of the theoretical assumption. According to your advice, we make all of the weight plus a large enough $\beta>0$ to ensure the weight satisfies the property of the non-negativeness. We added Theorem 3.2 to illustrate it in the revised version.

  Suppose there exists a $K$-component graph matrix  $\mathbf{W}\in\mathbb{R}^{N\times N}$ which is ordered correctly, and the $k$th component corresponds to the $k$th cluster $[c_{k-1}+1,c_{k}]$, where $c_{k-1}<c_{k},\forall k\in[1,K]$, and the dummy indexes $c_{0}:=0$, $c_{K}:=N$.

  **Theorem 3.2** 	If the within-cluster weight of the $k$th cluster  follows the Gaussian distribution $\mathcal{N}(\mu_{k}+\beta,\sigma^{2})$, $\mu_{k}>0$, $\forall k\in[1,K]$, and the between-cluster weight follows the Gaussian distribution $\mathcal{N}(\beta,\sigma^{2})$, where $\beta>0$ is a large enough number to make sure all of the weight are non-negative. Then, our BDSL algorithm will converge to $c_k$, $k=1,2,...,K-1$ by maximizing the expectation of the objective function of Problem (2). 

  **Explanation**: Since the Gaussian assumption of the weight, there may exist negative weights, which is inconsistent with the non-negative property of the graph, we make all of the weight plus a large enough $\beta>0$ to ensure the weight satisfies the property of the non-negativeness. 

  

  In order to make Theorem 3.1 clear, we rewrite it to the following form

  **Theorem 3.1** The function $F(t)$ has the following property:

  1) If $K=1$, then $F(t)=1$ for $t\in(0,N)$.

  2) If $K\geq2$, then $F(t)$ increase for $t\in(0,c_{1}]$ and decrease for $t\in[c_{K-1},N)$.

  3) If $K\geq3$, then, for the interval $[c_{k},c_{k+1}]$, $\forall k=1,...,K-2$, $F(t)$ decreases for $t\in[c_{k},\hat{t}]$ and increases for $t\in[\hat{t},c_{k+1}]$ if $\hat{t}\in(c_k,c_{k+1})$; $F(t)$ decreases for $t\in[c_{k},c_{k+1}]$ if $\hat{t}\geq c_{k+1}$; $F(t)$ increases for $t\in[c_{k},c_{k+1}]$ if $\hat{t}\leq c_{k}$.

- (Q3) The paper has been proofread carefully. We try to make the revised version clear and avoid typos.

From: ICLR.cc/2023/Conference/Paper498/Authors
title: Feedback to reviewer vLzH
comment: Dear vLzH:

Thank you so much for your careful reading and constructive review comments. We have carefully revised the manuscript accordingly. We sincerely hope that our answers fully address your concerns.

- (Q1) We will follow your advice to generate the synthetic data in a much more systematic manner. For example, we generate some clusters following different Gaussian distributions with different specific parameters. We also plan to generate a synthetic dataset similar to the conventional density-based or graph-based clustering article. 
- (Q2) The revised version has been edited carefully. Thank you very much for pointing out the specific inappropriate language. The sweeping statement and radical sentence have been replaced by an impartial expression. In addition, the paper has been proofread carefully. We try to make the revised version clear and avoid typos.

From: ICLR.cc/2023/Conference/Paper498/Authors
title: Feedback to reviewer ops2
comment: - (Q3.1) The ordered graph contains information that is equivalent to density-based clustering, however it seems to beat density-based clustering in some scenarios

  I have amended this misleading sentence in the revised version. Our method is DBGO+BDSL. Since DBGO and DBSCAN use a similar greedy searching strategy, the result of DBGO, i.e., ordered graph, contains information that is equivalent to density-based clustering, i.e., DBSCAN. DBSCAN outputs the clustering result directly, which is sensitive to the parameter. However, we proposed non-parameter BDSL to identify the clustering structure. Thus, our performance is better than DBSCAN.

  (Q3.2) In our experiment, for the comparison, e.g., the density-based clustering method, which is designed without knowing the cluster number, we tune the parameter of the density-based clustering method, e.g., DBSCAN, until the number of output clusters is K. 

  (Q3.3) If a data point has many points that are very very close to it (so similarity score is very big) so that when the ratio reached $\delta$ there are only a few points in the neighborhood, it won't be considered as a core sample.

  I think you mean the case when one of the clusters only contains a few samples (e.g., six samples) and the ratio is low. Then, for any sample $\mathbf{x}$ in this cluster, the weight to $\mathbf{x}$ maybe 1 1 1 1 1 1 0 0 0 0 0 .... 0. If the ratio $\delta=0.5$, only three weights are counted in the neighborhood. If the threshold number $\varepsilon<3$, then all of the samples in this cluster will not be seen as a core sample. This case can be avoided by increasing $\delta$ or decreasing $\varepsilon$. Since the proposed DBGO can only be required to order the graph matrix, it is robust to the parameter setting. In our experiments, we usually set $\delta=0.5$ and $\varepsilon=6$ for DBGO. The case you said never happened. Actually, the case you mentioned can be avoided by replacing the ratio threshold with the constant threshold, i.e., all of the weights which are larger than 0.8 are chosen to be the neighbor, then all of the six weights will be counted in the neighborhood. However, the threshold rule will fail in uneven-density cases. Specifically, if the average within-cluster density of cluster one is 1, and the average within-cluster density of cluster two is 0.3, then the weight to the sample in cluster one can be assumed as 1 1 1 1... 1 0.4 0.4 ... 0.4, and the weight to the sample in cluster two can be assumed as 0.3 0.3 ... 0.3 0 0 ... 0. In this case, our ratio threshold rule shows better performance than the constant threshold rule. In reality, the number of samples in each cluster is often large enough. However, the density of samples in each cluster is usually very different. 

  

- (Q4) The paper has been proofread carefully. We try to make the revised version clear and avoid typos.

From: ICLR.cc/2023/Conference/Paper498/Authors
title: Feedback to reviewer ops2
comment: Dear reviewer ops2:

Thank you so much for your careful reading and constructive review comments. We have carefully addressed all your questions below and revised the manuscript accordingly. We sincerely hope that our answers fully address your concerns.

- (Q1) the theoretical guarantees of this approach

  Thank you for your question about the condition of global optimality gaurantee. We add Theorem 3.2 to illustrate it in the revised verson.

  Suppose there exists a $K$-component graph matrix  $\mathbf{W}\in\mathbb{R}^{N\times N}$ which is ordered correctly, and the $k$th component corresponds to the $k$th cluster $[c_{k-1}+1,c_{k}]$, where $c_{k-1}<c_{k},\forall k\in[1,K]$, and the dummy indexes $c_{0}:=0$, $c_{K}:=N$.

  **Theorem 3.2** 	If the within-cluster weight of the $k$th cluster  follows the Gaussian distribution $\mathcal{N}(\mu_{k}+\beta,\sigma^{2})$, $\mu_{k}>0$, $\forall k\in[1,K]$, and the between-cluster weight follows the Gaussian distribution $\mathcal{N}(\beta,\sigma^{2})$, where $\beta>0$ is a large enough number to make sure all of the weight are non-negative. Then, our BDSL algorithm will converge to $c_k$, $k=1,2,...,K-1$ by maximizing the expectation of the objective function of Problem (2). 

  **Explanation**: Since the Gaussian assumption of the weight, there may exist negative weights, which is inconsistent with the non-negative property of the graph, we make all of the weight plus a large enough $\beta>0$ to ensure the weight satisfies the property of the non-negativeness. In addition, since the random nature of noise, the indices $c_k$, $k=1,2,...,K-1$ may not be able to maximizeze the objective function of Problem (2). However, we found that the indices $c_k$, $k=1,2,...,K-1$ can maximize the expectation of the objective function of Problem (2), and our BDSL algorithm will converge to $c_k$, $k=1,2,...,K-1$ by maximizing the expectation of the objective function of Problem (2). 



- (Q2)  the contrast between our approach and existing density-based clustering approaches: 

  In this paper, We proposed an integrated density and graph clustering method, which has the potential to overcome some drawbacks of the conventional graph-based method and density-based method. Specifically, DBSCAN is one of the representative density-based methods and performs well with arbitrarily shaped clusters. **However, DBSCAN cannot cluster data sets with significant differences in densities well and is not entirely deterministic.** Spectral clustering is one of the representative graph-based methods and needs a post-processing (e.g., K-Means) step to uncover the clustering indicators. **However, K-Means is sensitive to the initial center setting and easily falls into a local optimum.** The proposed two-stage clustering scheme includes density-based graph ordering (DBGO) and partition-based graph cut (BDSL). The superiority is depicted as follows:

  (1) We proposed a ratio threshold to adapt to the cluster with uneven densities. DBSCAN can not use this technique because they rely on a constant density threshold to separate the cluster. The samples which have a high density will be seen as a cluster. However, our DBGO aims to iterate the sample one cluster by one cluster, and we do not need to distinguish the cluster.

  (2) DBSCAN can not identify the clustering structure by considering the whole similarity between samples, and it needs a fixed density threshold to confirm the clustering structure. Thus, the clustering result of DBSCAN is sensitive to the parameter. However, our graph partitioning method can identify the clustering structure by solving a partition-based graph cut problem, and the clustering result can be given by a non-parameter algorithm and is deterministic with a fixed-ordered graph.

  (3) Spectral clustering essentially converts the Ncut problem to the K-Means problem. The performance of K-Means is usually unsatisfactory. If the graph matrix is shown in a block-diagonal form by ordering, the Ncut problem can be solved efficiently from a partitioning view. Thus, we proposed a partition-based graph cut problem. The problem can be solved by the proposed BDSL, which is deterministic and can converge to the optimal global partition under some assumptions. Compared with spectral clustering, the performance of our algorithm depends on the ordering quality given by DBGO. However, the performance of spectral clustering depends on the performance of K-Means, which is not robust and deterministic.

From: ICLR.cc/2023/Conference/Paper498/Reviewer_ops2
confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: In Machine Learning, graph-based clustering methods such as spectral clustering are commonly used and well-studied. There are two parts of this problem: graph construction based on the data samples, and partition of the samples into clusters based on the constructed graph, the former being relatively more well-studied than the latter. This paper develops and investigates a new method that includes some innovations in both parts. The paper proposes a graph-construction algorithm which outputs an augmented graph representing the ordered clustering structure in a classic density-based clustering approach. The graph-matrix is then shown in a block-diagonal form because of the nature of ordering. Then a partition method is performed to learn the graph matrix’s block-diagonal structure and identify the clusters. The paper also provides both theoretical and empirical guarantees. Especially, they show on several synthetic and real datasets that the approach can be broadly combined with other existing clustering methods and improve their performance.
strength_and_weaknesses: The paper introduces a novel algorithm. Usually density-based clustering is directly used to output a clustering solution, but this paper constructs an ordered graph by creating an ordering using density-based approach and later permutes the matrix according to the ordering. The permuted matrix is now block diagonal and can easily induce a partition.

The graph construction and the algorithm proposed are relatively simple, compared with density-based clustering. I wish that the authors could discuss more about the following aspects: 1) the contrast between this approach and existing density-based clustering approaches; 2) the theoretical guarantees of this approach, maybe on some ground-truth datasets. I found it strange that the paper said in the abstract that global optimality is guaranteed, but there is no main theorem in section 3.2 that specifies under which conditions optimality is guaranteed.

I have several questions about the main points in this paper:

- In both the abstract and the introduction, the paper claimed that the ordered graph contains information that is equivalent to density-based clustering. However it seems to beat density-based clustering in some scenarios such as the example given at the beginning of section 3.1 and the synthesized datasets. I find it interesting that the results of the two algorithms can differ so much. Is it because DBGO does not use a global distance hyper-parameter to decide reachability? If so, the claim that the graph is "equivalent" to density-based clustering seems a bit misleading and the authors should emphasize this in Section 3.1.
- So DBGO requires an input of fixed K value, whereas other density-based clustering does not require that? What happens if the K value differs a bit from the "correct" one?
- Further, I find the definition of \epsilon-core samples non-intuitive: the weights measure similarity between pairs of samples. we put nodes into \delta-neighborhood index set in descending order of similarity, **until** the sum of weights goes over a certain ratio (\delta) of the total weights. Then the \epsilon-core sample condition requires that the cardinality of this index set is at least \epsilon. According to this definition, if a data point has many points that are very very close to it (so similarity score is very big) so that when the ratio reached \delta there are only a few points in the neighborhood, it won't be considered as a core sample?
clarity,_quality,_novelty_and_reproducibility: The paper is original. The proposed method of using an ordered graph to reformat the information contained by density-based clustering and identify the clusters based on that seemed to be novel to me.

The writing quality of the paper can be improved. After reading the paper I'm getting the sense that the paper attempts to make a few important points, but the message remains unclear. Some of the questions I have are listed in the "Strength and Weakness" section. I think the significance of the algorithm and the results are partially impaired by the clarity of this paper. Nevertheless, I got the main ideas of this paper and think they might be good to know for the community.

Further, there are more minor issues in the paper such as broken, unfinished or repetitive sentences here and there. For example: in the third paragraph of section 3.1, "The weight wij in W measures the similarity between the ith sample and the jth sample. The weight, and it becomes larger with the nearing of the ith sample and the jth sample." It will be helpful if the authors could proof-read the paper.
summary_of_the_review: Overall, I think this paper has some interesting ideas. The presentation style of these ideas can be improved to emphasize the novelty of the design. It would be more convincing if more can be said about the approach from a theoretical perspective, but it has shown great potentials in the experiments and can be coupled with many existing approaches to improve the clustering quality, which is impressive.
correctness: 4: All of the claims and statements are well-supported and correct.
technical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
empirical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
flag_for_ethics_review: NO.
recommendation: 6: marginally above the acceptance threshold

From: ICLR.cc/2023/Conference/Paper498/Reviewer_vLzH
confidence: 2: You are willing to defend your assessment, but it is quite likely that you did not understand the central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: This paper presents a novel approach to graph based clustering (or cluster structure identification)
strength_and_weaknesses: Strengths:
Fair novelty
Exhaustive experimental comparison with alternative methods
Weaknesses:
Poorly written.
Poor experiments with synthetic data
clarity,_quality,_novelty_and_reproducibility: The theoretical novelty (within my limited grasp of the specific subject) seems sound, although, the way the paper is written, I would say that with a limited clarity.
Experimental results seem reproducible at least to some extent.
summary_of_the_review: This paper presents a novel approach to graph based clustering (or cluster structure identification)

The study falls well within the scope of the conference. The use of English should be improved through a thorough editorial revision.
 
In the introduction, authors state that "Most of the recent research related to the task of clustering has been directed towards efficiency.
The more critical problem, however, is effectivity ...". This is quite a sweeping statement that I do not quite agree about. If authors are confident about it, they should substantiate it further.

Then, in the review of the literature, several state-of-the-art graph partition methods are introduced, but they are summarily dismissed with the statement that "none of them have characteristics of effectiveness and robustness at the same time". This is, again, quite a radical categorization that would merit further clarification.

The theoretical development seems sound, and the results with high-dimensional datasets are convincing. I am not so happy about the synthetic data experiments, which are nothing more than indicative, but which should have been conducted in a much more systematic manner.
correctness: 3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.
technical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
empirical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
flag_for_ethics_review: NO.
recommendation: 6: marginally above the acceptance threshold

From: ICLR.cc/2023/Conference/Paper498/Reviewer_eHSt
confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
summary_of_the_paper: The paper introduces an algorithm for ordering the nodes in a weighted graph for the purpose of graph partitioning. The ordering is based on creating chains of high density nodes which are "reachable" from one another, which tends to lead to a close-to block-diagonal affinity matrix after permutation. An algorithm for extracting these blocks (corresponding to highly connected subgraphs, or clusters) based on the normalised cut objective (in a complementary form) is then proposed. The paper concludes with a fairly extensive set of experiments in which the proposed method is compared with other popular clustering methods.
strength_and_weaknesses: Strengths:
- The proposed method is sensible and has a potentially convincing justification
- The experiments suggest there may be potential for very good performance in practice

Weaknesses:
- Overriding any potential technical issues, the lack of clarity in the paper is very problematic. Strangely, the first two sections are well written and provide a nice introduction to the problem. However, I unfortunately found the remainder, mainly section 3, to be almost unintelligible. Although I believe I grasped the main concepts well enough, I am confident some of the textual descriptions of the definitions are incorrect, or at least inconsistent.
- The proposed method is remarkably similar to DBSCAN, and yet this point is not mentioned by the authors. Although DBSCAN is referenced, the very clear similarity between the methods is not mentioned.
- The assumptions given in the theoretical results seem at best unrealistic, and at worst totally inconsistent with the assumptions for the method. Specifically, it is assumed for the method that the affinity matrix is non-negative, and yet the weights between different clusters are assumed in the theory to be zero mean Gaussian random variables.
clarity,_quality,_novelty_and_reproducibility: To reiterate, the clarity in the paper needs to be addressed with urgency as it detracts significantly from the quality of the paper and to the extent that it is hard to assess the degree of novelty and technical correctness of the discussions given. Furthermore, the frequency of instances of poor clarity is so high that it is difficult as a reviewer to suggest where the authors should start. I believe the proposed method is novel, and that it has potential, but I have a lot of uncertainty because of the poor clarity. Furthermore, the extent of the novelty is potentially exaggerated due to the authors not commenting on the close connections to DBSCAN. Let me clarify that I do not mean the authors are reinventing DBSCAN, and I believe their proposal has some advantages over DBSCAN in some cases, however the general approach, even the terminology used, is very similar.
summary_of_the_review: The proposed approach appears to have some merit, and I would not be surprised if, after revision, the paper could be a good addition to the literature on graph clustering. However, in its current form the lack of clarity is too significant to recommend acceptance in a top tier conference.

UPDATE:
After communications with the authors, and accounting for their revisions and seeing the opinions of other reviewers, I have decided to increase my score slightly. I would prefer to see the authors revise their paper and take a different approach to motivating their proposed methodology, but would not be unhappy to see the paper accepted either, should the other reviewers see it as high enough in quality in its current form.
correctness: 2: Several of the paper’s claims are incorrect or not well-supported.
technical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
empirical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
flag_for_ethics_review: NO.
recommendation: 5: marginally below the acceptance threshold

From: ICLR.cc/2023/Conference/Paper498/Reviewer_Fmso
confidence: 2: You are willing to defend your assessment, but it is quite likely that you did not understand the central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: The authors present a two-step approach for graph-based clustering. In the first step, an ordered graph is created that represents the density-based cluster structure. In the 2nd step, the ordered graph is partitioned to give the clusters. The running time of the graph ordering in the worst case is O(n^2). Experimental results indicate that the clustering method performs better than other clustering methods
strength_and_weaknesses: + The idea is simple but well motivated through an example
+ The problem and algorithms are clearly formulated
+ Experiments are conducted for comparison with a number of well know algorithms on synthetic and real benchmark datasets  

- It would be nice if the authors describe limitations of their work 


clarity,_quality,_novelty_and_reproducibility: The paper is clearly written, seems to be correct and novel. Although experiments are conducted on benchmark datasets, the implementations may be provided for reproducibility
summary_of_the_review: The paper presents the idea clearly, algorithms are defined, and experiments are conducted to support the claim
correctness: 4: All of the claims and statements are well-supported and correct.
technical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
empirical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
flag_for_ethics_review: NO.
recommendation: 8: accept, good paper