Original: C LUSTERING S TRUCTURE I DENTIFICATION W ITH O RDERING G RAPH
Revised: C LUSTERING S TRUCTURE I DENTIFICATION W ITH O RDERING G RAPH

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: In machine learning, data is often presented in the form of a graph or similarity (or distance) values between samples.
Revised: In machine learning, data is often presented in the form of a graph or similarity (or distance) values between samples.

Original: Graph-based clustering methods such as spectral clustering are deﬁned for general weighted graphs to identify the clustering structure.

Original: Graph construction research has developed signiﬁcantly for decades, but the graph-based partition study still requires more attention because of its poor performance.
Revised: Graph construction research has developed signiﬁcantly for decades, but the graph-based partition study still requires more attention because of its poor performance.

Original: For example, spectral clustering needs a post-processing (e.g., KMeans) step to uncover the clustering indicators.
Revised: For example, spectral clustering needs a post-processing (e.g., K-Means) step to uncover the clustering indicators.

Original: Yet, K-Means is sensitive to the initial center setting and easily falls into a local optimum.
Revised: However, K-Means is sensitive to the initial center setting and easily falls into a local optimum.

Original: In this paper, we investigate a new type of graph-based clustering approach.
Revised: In this paper, we investigate an integrated density and graph clustering approach.

Original: Firstly, we introduce a new algorithm forthe purpose of cluster analysis which does not explicitly produce a clustering of a dataset but instead creates an augmented graph representing its density-based ordered clustering structure.
Revised: Firstly, we introduce a matrix ordering algorithm for cluster analysis that does not explicitly produce a clustering of a dataset but instead creates an augmented graph representing its density-based ordered clustering structure.

Original: This ordered graph contains information equivalent to density-based clustering corresponding to a broad range of parameter settings.

Original: Secondly, we found that the graph matrix is shown in a blockdiagonal form because of the nature of ordering.
Revised: Secondly, we ﬁnd that the graph matrix is shown in a block-diagonal form because of the nature of ordered clusters.

Original: We propose a partition method to learn thegraph matrix’s block-diagonal structure  and identify the clustering directly.
Revised: We proposed a partition-based graph-cut clustering model to learn the block-diagonal structure of the graph matrix and identify the clustering structure simultaneously.

Revised: If the between- and within-cluster weight of an ordered graph follow Gaussian distribution, then the true cluster segmentation will maximize the expectation of the objective function, and our algorithm can converge to the true cluster segmentation from the expectation view.

Original: Theglobal optimality is guaranteed theoretically.
Revised: The optimality has been guaranteed theoretically.

Original: We test the proposed method on synthetic datasets and ﬁve high-dimensional  datasets.
Revised: We test the proposed method on synthetic datasets and ﬁve high-dimensional real datasets.

Original: Experimental results show that the proposed method outperforms state-of-the-artgraph-based clustering methods and improves their performance  by roughly 10% ∼ 50%.
Revised: Experimental results show that the proposed method outperforms state-of-the-art clustering methods and improves the performance of subspace clustering approaches by roughly 10% ∼ 50%.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Larger and larger amounts of data are collected and stored in databases, increasing the need for efﬁcient and effective analysis methods to implicitly use the information in the data.
Revised: Larger and larger amounts of data are collected and stored in databases, increasing the need for efﬁcient and effective analysis methods to implicitly use the information in the data.

Original: One of the primary data analysis tasks is cluster analysis which is intended to help a user to understand the natural grouping or structure in a dataset.
Revised: One of the primary data analysis tasks is cluster analysis which is intended to help a user to understand the natural grouping or structure in a dataset.

Original: Therefore, improved clustering algorithms have received much attention in the last few years.
Revised: Therefore, improved clustering algorithms have received much attention in the last few years.

Original: The goal of a clustering algorithm is to group the objects of a database into a set of meaningful subclasses.
Revised: The goal of a clustering algorithm is to group the objects of a database into a set of meaningful subclasses.

Original: A clustering algorithm can be used either as a standalone tool to get insight into the distribution of a dataset, e.g., to focus further analysis and data processing, or as a preprocessing step for other algorithms that operate on the detected clusters.
Revised: A clustering algorithm can be used either as a standalone tool to get insight into the distribution of a dataset, e.g., to focus further analysis and data processing or as a preprocessing step for other algorithms that operate on the detected clusters.

Original: Most of the recent research related to the task of clustering has been directed towards efﬁciency. The more critical problem, however, is effectivity, i.e., the quality or usefulness of the  result. 
Revised: Many recent works pay attention to the effectivity of clustering, i.e., the quality or usefulness of the clustering result. However, their characteristic of effectiveness is still unsatisfactory.

Original: There are three interconnected reasons why the effectiveness of clustering algorithms is more important. First, almost all clustering algorithms require values for input parameters that are hard to determine, especially for real-world datasets containing high-dimensional objects.
Revised: There are three interconnected reasons: 1) Almost all clustering algorithms require values for input parameters that are hard to determine, especially for real-world datasets containing high-dimensional objects.

Original: Second, the algorithms are susceptible to these parameter values, often producing very different segmentations of the dataset even for slightly different parameter settings.
Revised: The algorithms are susceptible to these parameter values, often producing very different segmentations of the dataset, even for slightly different parameter settings.

Original: Third, high-dimensional real-datasets usually have avery skewed distribution that can not be revealed by a clustering algorithm using only one global parameter setting.
Revised: 3) High-dimensional real datasets usually have a skewed distribution that can not be revealed by a clustering algorithm using only one global parameter setting.

Original: State-of-the art clustering methods based on graphical representations of the relationships among data points have be widely used for decades.
Revised: State-of-the-art clustering methods based on graphical representations of the relationships among data points have been widely used for decades.

Original: For example, spectral clustering Von Luxburg (2007), normalized cut Shi & Malik (2000) and radio cut Hagen & Kahng (1992)
Revised: For example, spectral clustering Von Luxburg (2007), normalized cut Shi & Malik (2000), and radio cut Hagen & Kahng (1992)

Original: all transform the data into a weighted, undirected graph based on pairwise similarities.
Revised: all transform the data into a weighted, undirected graph based on pairwise similarities.

Original: Clustering is then accomplished by spectral or graph-theoretic optimization procedures.
Revised: Clustering is then accomplished by spectral or graph-theoretic optimization procedures.

Original: See Ding et al.
Revised: See Ding et al.

Original: (2005; 2006) for a discussion of the relations among these graph-based methods, and also the connections to nonnegative matrix factorization.
Revised: (2005; 2006) for a discussion of the relations among these graph-based methods and also the connections to nonnegative matrix factorization.

Original: All of these methods involve a two-stage process in which an data graph is formed from the data, and then various optimization procedures are invoked on this ﬁxed input data graph.
Revised: All of these methods involve a two-stage process in which a data graph is formed from the data, and then various optimization procedures are invoked on this ﬁxed input data graph.

Original: On the one hand, the clustering results are dependent on the quality of the input data graph (i.e., they are sensitive to the particular graph construction methods).
Revised: On the one hand, the clustering results depend on the quality of the input data graph (i.e., they are sensitive to the particular graph construction methods).

Original: Thus, almost all researchers focus on the graph construction task, including good neighbor pruning Yang et al.
Revised: Thus, almost all researchers focus on the graph construction task, including good neighbor pruning Yang et al.

Original: ; Cui et al.
Revised: ; Cui et al.

Original: (2021), multi-view afﬁnity learning Xiao et al.
Revised: (2021), multi-view afﬁnity learning Xiao et al.

Original: ; Tang et al.
Revised: ; Tang et al.

Original: ; Kang et al.
Revised: ; Kang et al.

Original: (2020; 2021), sparse representation learning Elhamifar & Vidal (2013b), low-rank representation learning Xu et al.
Revised: (2020; 2021), sparse representation learning Elhamifar & Vidal (2013b), low-rank representation learning Xu et al.

Original: (2021), self-expression training based on neural network Lv et al.
Revised: (2021), self-expression training based on neural network Lv et al.

Original: ; Cai et al.
Revised: ; Cai et al.

Original: ; Zhang et al.
Revised: ; Zhang et al.

Original: (2021a).
Revised: (2021a).

Original: However, the ﬁnal clustering structures are not represented explicitly in the data graph (e.g., graph-cut methods often use K-means algorithm to post-process the results to get the clustering indicators).
Revised: However, the ﬁnal clustering structures are not represented explicitly in the data graph (e.g., graph-cut methods often use the K-Means algorithm to post-process the results to get the clustering indicators).

Original: More and more researchers have discovered that it is not the graph construction but the graph partition that limits improving clustering accuracy.
Revised: More and more researchers have discovered that it is not the graph construction but the graph partition that limits improving clustering accuracy.

Original: Although a lot of graph partition approaches have been proposed recently, e.g., MST S¸aar & Topcu (2022), PrimZhu & Chen (2021), Kruskal Battaglia & Pensa (2021), SNN Liu et al.
Revised: Although a lot of graph partition approaches have been proposed recently, e.g., MST S¸aar & Topcu (2022), PrimZhu & Chen (2021), Kruskal Battaglia & Pensa (2021), SNN Liu et al.

Original: Jarvis-Patrick Wang et al.
Revised: Jarvis-Patrick Wang et al.

Original:  Extended-DBSCAN Chen et al.
Revised: (2022), Extended-DBSCAN Chen et al.

Original: (2021), Chameleon Zhang et al.
Revised: (2021), Chameleon Zhang et al.

Original: (2021b), and Extended-Spectral Sharma & Seal (2021), however, none of them have characteristics of effectiveness and robustness at the same time.
Revised: (2021b), and ExtendedSpectral Sharma & Seal (2021). However, their characteristics of effectiveness and robustness on real datasets are still unsatisfactory.

Original: A clustering technique is considered to be good if it satisﬁes the following requirements Fahim et al.
Revised: A clustering technique is considered to be good if it satisﬁes the following requirements Fahim et al.

Original: (2008): a) Minimal requirements of the domain knowledge to determine the values of its input parameters, which is very important problem especially for large datasets.
Revised: (2008): a) Minimal requirements of the domain knowledge to determine the values of its input parameters, which is a signiﬁcant problem, especially for large datasets.

Original: b) Discovery of arbitrary shaped clusters.
Revised: b) Discovery of arbitrarily shaped clusters.

Original: c) Good efﬁciency on large datasets.
Revised: c) Good efﬁciency on large datasets.

Original: It is still challenging to design a clustering method that has the above three features simultaneously.
Revised: It is still challenging to design a clustering method that has the above three features simultaneously.

Original: In this paper, we introduce an effective and robust approach for the purpose of cluster analysis which does not produce a clustering explicitly; but instead permutes a given graph representing the density-based clustering structure of the data.
Revised: In this paper, we introduce an effective and robust approach for the purpose of cluster analysis which does not produce a clustering explicitly; but instead permutes a given graph representing the density-based clustering structure of the data.

Original: This ordered graph contains information equivalent to the density-based clusterings corresponding to a broad range of parameter settings.
Revised: This ordered graph contains information similar to the density-based clusterings corresponding to a broad range of parameter settings.

Original: It is a versatile basis for both automatic and interactive cluster analysis.
Revised: It is a versatile basis for both automatic and interactive cluster analysis.

Original: The ordering algorithm is designed for ﬁnding ordered density-based clusters in spatial data andit can also detect meaningful clusters in data of varying densities.
Revised: We proposed an ordering algorithm to ﬁnd ordered density-based clusters in spatial data and detect meaningful clusters in data of varying densities.

Original: The graph weight matrix is shown in block-diagonal form because of the nature of ordered graph.
Revised: The graph weight matrix is shown in block-diagonal form because of the nature of ordered clusters.

Original: We propose a novel graph-based partition model that identiﬁes the clustering structure from an ordered similarity matrix which is in a K -block-diagonal form ( K is the number of clusters).
Revised: Thus, we proposed a novel partition-based normalized cut clustering model that identiﬁes the clustering structure from the ordered similarity matrix with a K -block-diagonal form, where K is the number of clusters.

Original: In order to identify the clustering structure directly without requiring any post-processing, we impose a partition approach based on the ordered graph.
Revised: In order to identify the clustering structure directly without requiring any post-processing, we impose an efﬁcient binary segmentation algorithm to learn the proposed graph partition model.

Original: The ordered graph will be cut into a K -block-diagonal form by K − 1 partition indices.
Revised: The ordered graph will be cut into a K -block-diagonal form by K − 1 partition indexes.

Original: The true partition indices will be given by the proposed algorithm  theoretically.
Revised: If the between- and within-cluster weight follows Gaussian distribution, then the true cluster segmentation will maximize the expectation of the objective function, and our algorithm can converge to the true cluster partition from an expectation view theoretically.

Original: We conduct empirical studies on  simulateddatasets and seven real-world benchmark datasets to validate our proposed methods.
Revised: We conduct empirical studies on four simulated and ﬁve real-world benchmark datasets to validate our proposed methods.

Original: Our graphbased clustering method consistently outperforms other related methods in general clustering tasks and subspace clustering tasks.
Revised: Our integrated density and graph clustering method consistently outperforms other related methods in general low-dimensional clustering tasks.

Revised: For the high-dimensional clustering task, the performance of conventional subspace clustering methods can be improved by roughly 10% ∼ 50% by replacing spectral clustering with our proposed method.

Original:  R EVIEW OF G RAPH C ONSTRUCTION AND P ARTITIONING 2. G RAPH C ONSTRUCTION
Revised:  R EVIEW OF G RAPH C ONSTRUCTION AND P ARTITIONING 2. G RAPH C ONSTRUCTION

Original: The general graph is often given by the distance measurement between points.
Revised: The general graph is often given by the distance measurement between points.

Original: The usage of distance depends on the property of the data.
Revised: The usage of distance depends on the property of the data.

Original: The traditional distance measurements include Euclidean, Mahalanobis, Cityblock, Minkowski, Chebychev, Cosine, Correlation, Hamming, Jaccard, and Spearman Choi et al.
Revised: The traditional distance measurements include Euclidean, Mahalanobis, Cityblock, Minkowski, Chebychev, Cosine, Correlation, Hamming, Jaccard, and Spearman Choi et al.

Original: For example, the work Menon et al.
Revised: For example, the work Menon et al.

Original: (2020) supposes the data in each cluster are located in a high-dimensional subspace, and use the cosine angle between points to measure the similarity of the samples in the same or different clusters.
Revised: (2020) supposes the data in each cluster are located in a high-dimensional subspace and use the cosine angle between points to measure the similarity of the samples in the same or different clusters.

Original: Speciﬁcally, the point x i is normalized by ¯ x i = x i ||
Revised: Speciﬁcally, the point x i is normalized by ¯ x i = x i ||

Original: x
Revised: x

Original: i || 2 ﬁrstly, where || · || denotes the l 2 norm.
Revised: i || 2 ﬁrstly, where || · || denotes the l 2 norm.

Original: These points ¯ x i ∈ R D , ∀ i ∈ { 1 , 2 , ..., N } will lie in the high dimensional hypersphere S D − 1 .
Revised: These points ¯ x i ∈ R D , ∀ i ∈ { 1 , 2 , ..., N } will lie in the high dimensional hypersphere S D − 1 .

Original: Let θ ij denote the angle between two data points x i and x j , i.e., θ ij = cos − 1 (¯ x T i ¯ x j ) , θ ij ∈ [0 , π ] . Since the similarity measurement is linearly smaller as two points becoming nearer.
Revised: Let θ ij denote the angle between two data points x i and x j , i.e., θ ij = cos − 1 (¯ x T i ¯ x j ) , θ ij ∈ [0 , π ] . We convert the radian θ ij to its degree form ˜ θ ij , i.e., ˜ θ ij = θ ij 180 /π , ˜ θ ij ∈ [0 , 180] .

Original: To make similar features more prominent, people usually use the exponential function to normalize the original measurement,(The value will in [0 , 1] , and the closer the distance, the larger the value).
Revised: To make similar features more prominent, people usually perform the exponential normalization on the original measurement, and the closer the distance, the larger the value.

Original: Suppose the similarity between i th sample and j th sample is w ij , then the new similarity is given by˜ w ij = exp( − w ij /σ 2 ) , where σ  is a parameter to control the attenuation degree with the increasing of the difference.
Revised: The weight is given by w ij = exp( − ˜ θ 2 ij /σ 2 ) , where σ 2 = Var[˜ θ ij ] , which is a parameter to control the attenuation degree with the increasing of the difference, and we usually set it as the variance of the weight directly for convenience in this paper.

Original: Another type of method of constructing the similarity graph between points is mainly used for the data located in high-dimensional subspaces.
Revised: Another type of method of constructing the similarity graph between points is mainly used for the data located in high-dimensional subspaces.

Original: These works use the self-expression property to construct an optimization problem based on  sparse feature Elhamifar & Vidal (2013b), blockdiagonal constraint Lu et al.
Revised: These works use the self-expression property to construct an optimization problem based on the sparse feature Elhamifar & Vidal (2013b), blockdiagonal constraint Lu et al.

Original: (2018), or low-rank property Vidal & Favaro (2014).
Revised: (2018), or low-rank property Vidal & Favaro (2014).

Original: Based on these models, deep neural network Ji et al.
Revised: Based on these models, deep neural network Ji et al.

Original: (2017), multi-view combination Elhamifar & Vidal (2013b), and neighbor pruning Yang et al.
Revised: (2017), multi-view combination Elhamifar & Vidal (2013b), and neighbor pruning Yang et al.

Original: ; Cui et al.
Revised: ; Cui et al.

Original: (2021) are used to improve the selfrepresentation performance of the similarity matrix.
Revised: (2021) are used to improve the self-representation performance of the similarity matrix.

Original: For example, consider a matrix of column wise samples X
Revised: For example, consider a matrix of column-wise samples X

Original: = [ x 1 , x 2 , ..., x N ] ∈ R D × N .
Revised: = [ x 1 , x 2 , ..., x N ] ∈ R D × N .

Original: Each sample or datum can be represented by a linear combination of n atoms in a dictionary A =
Revised: Each sample or datum can be represented by a linear combination of n atoms in a dictionary A =

Original: [ a 1 , a 2 , ..., a n ] ∈ R D × n : X = AZ , where
Revised: [ a 1 , a 2 , ..., a n ] ∈ R D × n : X = AZ , where

Original: Z = [ z 1 , z 2 , ..., z N ] ∈ R n × N is a coefﬁcient matrix.
Revised: Z = [ z 1 , z 2 , ..., z N ] ∈ R n × N is a coefﬁcient matrix.

Original: Under this representation, each column z i of the coefﬁcient matrix Z can be interpreted as a new representation for each data sample x i under the dictionary.
Revised: Under this representation, each column z i of the coefﬁcient matrix Z can be interpreted as a new representation for each data sample x i under the dictionary.

Original: Suppose the column vectors of X are drawn from a union of K subspaces { Q k } Kk =1 of unknown dimensions { d k } Kk =1 .
Revised: Suppose the column vectors of X are drawn from a union of K subspaces { Q k } Kk =1 of unknown dimensions { d k } Kk =1 .

Original: Without knowing the dictionary A , one can use the self-expressiveness property of data to ﬁnd the subspaces: each data point in a union of subspaces can be efﬁciently reconstructed by a combination of other points in the data.
Revised: Without knowing the dictionary A , one can use the self-expressiveness property of data to ﬁnd the subspaces: each data point in a union of subspaces can be efﬁciently reconstructed by a combination of other points in the data.

Original: For instance, the work Elhamifar & Vidal (2013a) uses the data samples themselves as the dictionary, i.e., A = X .
Revised: For instance, the work Elhamifar & Vidal (2013a) uses the data samples themselves as the dictionary, i.e., A = X .

Original: In this case, the coefﬁcient matrix Z becomes a square matrix of size N × N , i.e., X = XZ .
Revised: In this case, the coefﬁcient matrix Z becomes a square matrix of size N × N , i.e., X = XZ .

Original: In general N > D , in this unrestricted case there are near inﬁnite possibilities for the coefﬁcient matrix Z .
Revised: In general, N > D , in this unrestricted case, there are near inﬁnite possibilities for the coefﬁcient matrix Z .

Original: The choice of Z is the mainpoint of difference among subspace clustering techniques.
Revised: The choice of Z is the main difference among subspace clustering techniques.

Original: The work Elhamifar & Vidal (2013a) aims to ﬁnd the sparsest representation using l 1 approximation.
Revised: The work Elhamifar & Vidal (2013a) aims to ﬁnd the sparsest representation using l 1 approximation.

Original: More speciﬁcally every point in the data as a set of sparse linear combinations of other points from the same subspace.
Revised: More speciﬁcally, every point in the data is a set of sparse linear combinations of other points from the same subspace.

Original: Mathematically the sparse formulation is written as
Revised: Mathematically the sparse formulation is written as

Original: [Equation]
Revised: [Equation]

Original: where E is the noise matrix.
Revised: where E is the noise matrix.

Original: From these sparse representations, an afﬁnity matrix Z is compiled.
Revised: From these sparse representations, an afﬁnity matrix Z is compiled.

Original: The problem can be solved by ADMM effectively Elhamifar & Vidal (2013a).
Revised: The problem can be solved by ADMM effectively Elhamifar & Vidal (2013a).

Original: In general, the graph is given by W = | Z | + | Z | Tbecause of the symmetric and positive properties of the graph.
Revised: In general, the graph is given by W = | Z | + | Z | Tbecause of the symmetric and positive properties of the graph.

Original: 2. G RAPH P ARTITIONING
Revised: 2. G RAPH P ARTITIONING

Original: Graph-based clustering approaches typically optimize their objectives based on a given data graph associated with a symmetric afﬁnity matrix W ∈ R N × N , where N is the number of nodes (data points) in the graph.
Revised: Graph-based clustering approaches typically optimize their objectives based on a given data graph associated with a symmetric afﬁnity matrix W ∈ R N × N , where N is the number of nodes (data points) in the graph.

Original: A weighted undirected complete graph G (i.e., fully connected graph) can be written as an ordered pair G :=
Revised: A weighted undirected complete graph G (i.e., fully connected graph) can be written as an ordered pair G :=

Original: ( V , E , W ) , where V is a set of vertices or nodes, E is a set of pairs (unordered) of distinct vertices, called edges or lines, and W is a N × N adjacency matrix (also similarity matrix or weight matrix) of the ﬁnite undirected graph G on N vertices, where the nondiagonal entry w ij is the number of edges from vertex i to vertex j , and the diagonal entry w ii is either twice the number of loops at vertex i or just the number of loops (usages differ, depending on the mathematical needs; this paper is not concerned with reﬂexive connections).
Revised: ( V , E , W ) , where V is a set of vertices or nodes, E is a set of pairs (unordered) of distinct vertices, called edges or lines, and W is an N × N adjacency matrix (also similarity matrix or weight matrix) of the ﬁnite undirected graph G on N vertices, where the nondiagonal entry w ij is the number of edges from vertex i to vertex j , and the diagonal entry w ii is either twice the number of loops at vertex i or just the number of loops (usages differ, depending on the mathematical needs; this paper is not concerned with reﬂexive connections).

Original: Recall the symmetric and positive properties of the graph, we assume w ij = w ji ≥ 0 .
Revised: Recall the symmetric and positive properties of the graph. We assume w ij = w ji ≥ 0 .

Original: Given A ⊂ V , its complement, V\A will be denoted as ¯ A .
Revised: Given A ⊂ V , its complement, V\A will be denoted as ¯ A .

Original: Intuitively, the subset A is connected if pathes between any two points in A need only points in A .
Revised: Intuitively, the subset A is connected if paths between any two points in A need only points in A .

Original: A is called  connected component with respect to ¯ A, if A is connected, and there are no edges between vertices in A and ¯ A .
Revised: A is called a connected component with respect to ¯ A if A is connected, and there are no edges between vertices in A and ¯ A .

Original: Subsets A 1 , A 2 , ...,
Revised: Subsets A 1 , A 2 , ...,

Original: A K represent a partitioning of V , if, for all i, j ∈ [1 , 2 , ..., K ] , A i ∩ A j = ∅ , and ∪ Kk =

Revised: A K represent a partitioning of V , if, for all i, j ∈ [1 , 2 , ..., K ] , A i ∩ A j = ∅ , and ∪ Kk =

Original: A k = V .
Revised: A k = V .

Original: Generally speaking, clustering means partitioningof a graph, so that the edges between different groups have low similarity (low distance) and the edges within a group have high weight (low distance).
Revised: Generally speaking, clustering means partitioning a graph so that the edges between different groups have low similarity (low distance) and the edges within a group have high weight (low distance).

Original: The ﬁrst requirement for the partitioning can be stated as the minicut criterion, which has to be minimized: cut ( A 1 , A 2 , ..., A K )
Revised: The ﬁrst requirement for the partitioning can be stated as the mini cut criterion, which has to be minimized: cut ( A 1 , A 2 , ..., A K )

Original: = (cid:80) Kk =1 cut ( A k , ¯ A k ) where cut ( A k , ¯ A k ) =
Revised: = (cid:80) Kk =1 cut ( A k , ¯ A k ) where cut ( A k , ¯ A k ) =

Original: (cid:80) i ∈A k ,
Revised: (cid:80) i ∈A k ,

Original: j ∈ ¯
Revised: j ∈ ¯

Original: A k w ij .
Revised: A k w ij .

Original: Following the minicut requirement, often only a little group of points is isolated.
Revised: Following the mini cut requirement, often only a little group of points is isolated.

Original: For this reason, the famous cutting methods, which are widely used in spectral clustering, are introduced: Ncut ( A 1 , A 2 , ..., A K )
Revised: For this reason, the famous cutting methods, which are widely used in spectral clustering, are introduced: Ncut ( A 1 , A 2 , ..., A K )

Original: = (cid:80) Kk =1 cut ( A k , ¯ A k ) vol ( A k )
Revised: = (cid:80) [Equation] ¯ A k ) vol [Equation]

Original:  Shi & Malik (2000) where vol ( A ) measures the size of A by the weights of its edges, i.e. vol ( A k )
Revised: A k ) Shi & Malik (2000) where vol ( A ) measures the size of A by the weights of its edges, i.e. vol ( A k )

Original: = (cid:80) i ∈A k (cid:80)
Revised: = (cid:80) i ∈A k (cid:80)

Original: j ∈V w ij .
Revised: j ∈V w ij .

Original: Within-cluster similarity is maximized if cut ( A k , ¯ A k ) is small and vol ( A k ) is big.
Revised: Within-cluster similarity is maximized if cut ( A k , ¯ A k ) is small and vol ( A k ) is large.

Original: Therefore, Ncut can be interpreted as cutting through edges rarely transitions by a random walk.
Revised: Therefore, Ncut can be interpreted as cutting through edges rarely transitions by a random walk.

Original: Denote the converse
Revised: Denote the converse

Original: Ncut as CNcut ( A 1 , A 2 , ..., A K ) =
Revised: Ncut as CNcut ( A 1 , A 2 , ..., A K ) =

Original: (cid:80) Kk =1 sasso ( A k ) vol ( A k ) where sasso ( A k ) =
Revised: (cid:80) Kk =1 sasso ( A k ) vol ( A k ) where sasso ( A k ) =

Original: vol ( A k ) − cut ( A k , ¯ A k ) .
Revised: vol ( A k ) − cut ( A k , ¯ A k ) .

Original: Since Ncut ( A 1 , A 2 , ..., A K )
Revised: Since Ncut ( A 1 , A 2 , ..., A K )

Original: =
Revised: =

Original: K − CNcut ( A 1 , A 2 , ..., A K ) , minimizing Ncut
Revised: K − CNcut ( A 1 , A 2 , ..., A K ) , minimizing Ncut

Original: ( A 1 , A 2 , ..., A K ) is equivalent to maximize CNcut ( A 1 , A 2 , ..., A K ) with ﬁxed K .
Revised: ( A 1 , A 2 , ..., A K ) is equivalent to maximize CNcut ( A 1 , A 2 , ..., A K ) with ﬁxed K .

Original:  C LUSTERING A LGORITHM3. D ENSITY -B ASED G RAPH O RDERING
Revised:  C LUSTERING A LGORITHM

Original: 𝐴 𝐵 𝐶 𝐴 ! 𝐴 " An important property of many real-datasets is that their intrinsic cluster structure cannot be characterized by global density parameters. Different local densities may be needed to reveal clusters in different regions of the data space.
Revised: An essential property of many real datasets is that different local densities may be needed to reveal clusters in different regions of the data space.

Original: For example, in the dataset depicted in Figure 1, it is not possible to detect the clusters A 1 , A 2 , B , and C simultaneously using one global density parameter. A global density-based decomposition would consist only of the clusters A , B , and C , or A 1 , A 2 , and C . In the second case, the objects from B are noise.
Revised: DBSCAN Ester et al.

Original: Two types of methods are considered. The ﬁrst alternative to recognize and analyze such clustering structures isto use a hierarchical clusteringalgorithm, for example, the single-link method. 
Revised: (1996) is a well-known density-based clustering method. It relies on choosing suitable parameters to order the sample and identify the clustering structure simultaneously.

Original: Yet, this alternative has two disadvantages. First, it suffers signiﬁcantly from the fact that clusters connected by a line of few points having a small inter-object distance are not separated.
Revised: However, DBSCAN cannot cluster data sets well with large differences in densities since the minPts- (cid:15) combination cannot be chosen appropriately for all clusters.

Original: Second, the results produced by hierarchical algorithms are difﬁcult to understand or analyze for more than a few hundred objects.
Revised: In addition, if the data and scale are poorly understood, choosing a meaningful distance threshold (cid:15) can be difﬁcult.

Original:  The second alternative is to use a density-based partitioning algorithm with different parameter settings. However, there are an inﬁnite number of possible parameter values.
Revised: In this paper, we proposed an integrated density and graph clustering method by combining the knowledge of density-based clustering and graph-based clustering. The proposed clustering method has the potential to overcome some drawbacks of the conventional graph-based method and density-based method.

Original: Even if we store a vast number of similarities ofdifferent cluster memberships for each point, it is not apparent how to analyze the results, and we may still miss theinteresting clustering levels.
Revised: Firstly, we consider beginning with a graph because of the superiority of the existing state-of-the-art graph construction method. Secondly, we consider separating the clustering task into two parts: ordering and partitioning.

Original: The basic idea to overcome these problems is to run an algorithm which produces a particular order of the database with respect to its density-based clustering structure containing information about every clustering level of the dataset, and is very easy to analyze.
Revised: Due to the superiority of graph construction methodology, we start from a graph matrix, which can be a distance measurement or sparse representation, as said in Section 2.1. In this subsection, we consider combining the knowledge of density and graph to permute the graph into a block diagonal form, which implies the clustering structure.

Original: The key idea of density-based graph ordering is that each object of a cluster in the neighborhood of a given unspeciﬁed radius must contain at least a minimal number of objects, i.e., the cardinality of the neighborhood must exceed a threshold.
Revised: Speciﬁcally, since the similarity between each sample has been given by the entry of the graph matrix, the density-based greedy searching strategy has the potential ability to iterate through each sample one cluster by one cluster.

Original: The formal deﬁnitions for this notion of clustering are brieﬂy presented below.
Revised: Then, the graph matrix can be permuted according to the iteration order. Since the nature of ordered clusters, the ordering graph should be in a block-diagonal form. We use the same greedy ordering strategy as DBSCAN Ester et al. (1996) to iterate through each sample based on the similarity given by the weight matrix.

Original: Given an undirected complete  graph G with a weight matrix W .
Revised: Speciﬁcally, given a complete undirected graph G with a weight matrix W .

Original: The weight w ijin W measures the similarity between the i th sample and the j th sample.
Revised: The element w ij measures the similarity between the i th sample and the j th sample.

Original: The weight, and it becomes larger with the nearing of the i th sample and the j th sample.
Revised: The more similar the i th sample and the j th sample are, the greater the weight w ij .

Original: Denote N δ
Revised: Denote N δ

Original: ( i ) as the δ -neighborhood index set of the i th object.
Revised: ( i ) as the δ -neighborhood index set of the i th object.

Original: We sort the weight w ij , j = 1 , 2 , ..., N , connected to the i th sample in descending order.
Revised: We sort the weight w ij , j = 1 , 2 , ..., N , connected to the i th sample in descending order.

Revised: Since DBSCAN cannot cluster data sets well with large differences in densities, a ratio threshold is proposed instead of using the constant threshold used in DBSCAN.

Original: The index of the largest values will be put into the set N δ ( i ) one-by-one until the sum of the weights in N δ
Revised: Speciﬁcally, the index of the largest values will be put into the set N δ ( i ) one by one until the sum of the weights in N δ

Original: ( i ) over the total weight is larger than δ , i.e., (cid:80) l ∈
Revised: ( i ) over the total weight is larger than δ , i.e., (cid:80) l ∈

Original: Nδ ( i ) w il (cid:80) j ∈ [1 ,N ] w ij > δ , δ ∈ (0 , 1) .
Revised: Nδ ( i ) w il (cid:80) j ∈ [1 ,N ] w ij > δ , δ ∈ (0 , 1) .

Revised: The proposed ratio threshold rule is more suitable for the dataset containing clusters with uneven within-cluster weight density than the constant threshold rule in DBSCAN.

Original: The i th object satisfying | N δ ( i ) | ≥ ε is called a ε -core sample , where | X | denotes the number of the element in the set X .
Revised: Based on the deﬁnition of δ -neighborhood index set, the i th object satisfying | N δ ( i ) | ≥ ε is called a ε -core sample , where | X | denotes the number of the element in the set X .

Revised: However, the proposed ratio threshold also has a potential drawback.

Revised: If a cluster contains a small number of samples, the low δ and high ε will make the sample in this cluster unable to be a core sample.

Original:  Deﬁnition 3.1 ( Density-reachable ) .
Revised: In our ordering step, we only care about the order. Thus, overcoming the uneven density problem is more important, and we usually set a lower δ to avoid the small cluster missing case. However, DBSCAN cares more about the clustering structure, and it prefers to choose a more suitable constant threshold to control the output cluster number. Deﬁnition 3.1 ( Density-reachable ) .

Original: The j th sample is directly density-reachable from the i th sample w.r.t.
Revised: The j th sample is directly density-reachable from the i th sample w.r.t.

Original: δ and ε in a set of samples X if 1) j ∈ N δ
Revised: δ and ε in a set of samples X if 1) j ∈ N δ

Original: ( i ) ; 2) | N δ
Revised: ( i ) ; 2) | N δ

Original: ( i ) | ≥ ε .
Revised: ( i ) | ≥ ε .

Original: Instead of ordering the graph based on density-reachable directly, we denote the core-weight c ε ( i ) as the weight between i th sample and its ε th-largest-weight neighbor.
Revised: Instead of ordering the graph based on density-reachable directly, we denote the core-weight c ε ( i ) as the weight between i th sample and its ε th-largest-weight neighbor.

Original: The core-weight of the i th object is simply the largest weight between i th sample and an object in its δ -neighborhood index set such that i th sample would be a core sample with respect to ε if this neighbor is contained in N δ
Revised: The core-weight of the i th object is simply the largest weight between i th sample and an object in its δ -neighborhood index set such that i th sample would be a core sample with respect to ε if this neighbor is contained in N δ

Original: ( i ) .
Revised: ( i ) .

Original: Then, the reachability-score is deﬁned as follows.
Revised: Then, the reachability score is deﬁned as follows.

Original: Deﬁnition 3.2 ( Reachability-score ) .
Revised: Deﬁnition 3.2 ( Reachability-score ) .

Original: Suppose i, j ∈ [1 , N ] , where N is the sample number.
Revised: Suppose i, j ∈ [1 , N ] , where N is the sample number.

Original: Let ε be a natural number.
Revised: Let ε be a natural number.

Original: Then, the reachability-score of j th sample with respect to i th sample is deﬁned as
Revised: Then, the reachability-score of j th sample with respect to i th sample is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: Undefined, otherwise
Revised: Undefined, otherwise

Revised: Input: the graph W , and the parameter δ , ε .

Original: Initialize an empty set C , and put the index of all εcore samples into C .
Revised: Initialize an empty set C , and put the index of all εcore samples into C .

Original: Calculate core-weight c
Revised: Calculate core-weight c

Original: ε ( i ) , i ∈ [1 , N ] .
Revised: ε ( i ) , i ∈ [1 , N ] .

Original: Initialize an empty order list O , and a null reachability-score recording vector s ∈ R N .
Revised: Initialize an empty order list O , and a null reachability-score recording vector s ∈ R N .

Original: for each unprocessed element 
Revised: for each unprocessed element i in C

Original:  i 
Revised: Mark i as processed, and O = [ O ; i ] . [ Q , s ]= QueueUpdate( i , Q , s ) repeat

Original: C
Revised: Pop j from Q .

Original: Mark i processed, 
Revised: Mark j as processed, and O =

Original:  O i ε
Revised: [ O ; j ] .

Original: =
Revised: if | N δ ( j ) | ≥ ε [ Q , s ]=QueueUpdate( j , Q , s ) end if until Q is empty end for Permute W based on O . Output: W

Original: Intuitively, the reachability-score of the j th object with respect to another i th object is the smallest weight such that j th sample is density-reachable from i th sample if i th sample is a core sample.
Revised: Intuitively, the reachability-score of the j th object with respect to another i th object is the smallest weight such that j th sample is density-reachable from i th sample if i th sample is a core sample.

Original: In this case, the reachability-score cannot be smaller than the core-weight c ε ( i ) of i th sample because for smaller weights no object is densityreachable from i th sample.
Revised: In this case, the reachability-score cannot be smaller than the core-weight c ε ( i ) of i th sample because, for smaller weights, no object is density-reachable from i th sample.

Original: Otherwise, if i th sample is not a ε -core sample, the reachability-score of j th sample with respect to i th sample is the weight between i th sample and j th sample directly.
Revised: Otherwise, if i th sample is not a ε -core sample, the reachability-score of j th sample with respect to i th sample is the weight between i th sample and j th sample directly.

Original: The reachability-score of j th object depends on the core sample with respect to which it is calculated.
Revised: The reachability-score of j th object depends on the core sample with respect to which it is calculated.

Original: Based on the above deﬁnitions, we propose a density-based graph ordering approach, namely DBGO, which permutes a random graph into the block-diagonal form.
Revised: Based on the above deﬁnitions, we propose a density-based graph ordering approach, namely DBGO, which permutes a random graph into the block-diagonal form.

Original: Alg.illustrates the main loop of the algorithm DBGO, and Alg.
Revised: Alg.illustrates the main loop of the algorithm DBGO, and Alg.

Original: 2 is a subfunction used in Alg. 1.
Revised: 2 is a subfunction used in Alg. 1.

Revised: Algorithm

Revised: Input: the processing sample i , and the queue Q , the recording vector s .

Revised: for each unprocessed j in N δ

Original:  = min ( c ε ( i ) , w ij ) if s j is unassigned s j = ˆ s
Revised: ( i ) ˆ s = min ( c ε ( i ) , w ij ) if s j is unassigned s j = ˆ s

Original: Push j into Q else if ˆ s > s j s j
Revised: Push j into Q else if ˆ s > s j s j

Original: =
Revised: =

Original: ˆ s end if end if end for
Revised: ˆ s end if end if end for

Original: Output: Q , s
Revised: Output: Q , s

Original: Each ε -core sample is handed over to a procedure of expanding cluster order if the core-sample is not yet processed.
Revised: Each ε -core sample is handed over to a procedure of expanding cluster order if the core-sample is not yet processed.

Original: In the beginning, one of the unprocessed core samples is marked as processed and appended to the order list
Revised: In the beginning, one of the unprocessed core samples is marked as processed and appended to the order list

Original: O .
Revised: O .

Original: Then, an empty priority queue Q is initialized, and the samples that may be in the same cluster as the picked core sample will be put in the queue.
Revised: Then, an empty priority queue Q is initialized, and the samples that may be in the same cluster as the picked core sample will be put in the queue.

Original: This operadion is accomplished by the QueueUpdate function in Alg.2, i.e., put the index of all of the δ -neighborhood of the i sample into the queue Q .
Revised: This operation is accomplished by the QueueUpdate function in Alg.2, i.e., put the index of all of the δ -neighborhood of the i sample into the queue Q .

Original: The queue Q stores the index and sorts the index in descending order of their existing-reachability-score.
Revised: The queue Q stores the index and sorts the index in descending order of their existing-reachability-score.

Original: We use s j to save the existing-reachability-score of the index j in Q .
Revised: We use s j to save the existing-reachability-score of the index j in Q .

Original: The existing-reachability-score s j of the index j will be updated if the reachability-score of j to the processing sample m is larger than the current value of s j .
Revised: The existing-reachability-score s j of the index j will be updated if the reachability-score of j to the processing sample m is larger than the current value of s j .

Original: The pop operadion of the queue Q will provide us with the element in Q , which has the largest existing-reachability-score.
Revised: The pop operation of the queue Q will provide us with the element in Q , which has the largest existing-reachability-score.

Original: The push operadion of the queue Q will append an element to Q , and the elements in Q will be sorted according to their existing-reachability-score automatically.
Revised: The push operation of the queue Q will append an element to Q , and the elements in Q will be sorted according to their existing-reachability-score automatically.

Original: We pop one of the element m from Q , and push all of the δ -neighborhood of the element m to Q if m th sample is a core sample.
Revised: We pop one of the element m from Q , and push all of the δ -neighborhood of the element m to Q if m th sample is a core sample.

Original: These two steps will repeat until Q is empty.
Revised: These two steps will repeat until Q is empty.

Original: The processing order of the sample will be recorded in the list O .
Revised: The processing order of the sample will be recorded in the list O .

Original: Finally, the graph  W will be adjusted according to O .
Revised: Finally, the graph matrix W is adjusted according to O .

Original: The run-time for DBGO algorithm is O ( N 2 ) .

Original: If a tree-based spatial index is used, e.g., Grid-R-tree

Original: Goyal et al.

Original: (2020),the run-time is reduced to O ( N log N ) .

Original: Furthermore, if we have direct access to the δ -neighborhood, e.g., if the objects are organized in a grid.

Original: The run-time is further reduced to

Original: O ( N ) because in a grid, the complexity of a single neighborhood query is O (1) .

Original: 3. B LOCK -D IAGONAL S TRUCTURE L EARNING B ASED ON G RAPH P ARTITIONING
Revised: 3. B LOCK -D IAGONAL S TRUCTURE L EARNING B ASED ON G RAPH P ARTITIONING

Original: We found that the ordered graph should be in a block-diagonal form.
Revised: We found that the ordered graph is shown in a block-diagonal form.

Original: Figure 2 shows an example of three clusters with the average within-cluster weight: 0.6, 0.9, and 0.7, which indicate the withincluster density.
Revised: Fig.1 shows an example of three clusters with the average within-cluster weight: 0.6, 0.9, and 0.7, which indicate the within-cluster density.

Revised: We suppose the between-cluster weight is zero.

Original: The weightsare disturbed by a Gaussian noise N (0 , 1) to simulate the randomness of the similarity between any two samples.
Revised: We assume all of the weights disturbed by a Gaussian noise N (0 , 1) to simulate the randomness of the similarity between any two samples.

Revised: We make all weights subtract by the minimum of the weight to make sure all of the weights are non-negative.

Original: Recall the proposed converse Ncut CNcut ( {A [Equation] [Equation] ) =
Revised: Recall the proposed converse Ncut in Section 2.2:

Original: [Equation]
Revised: [Equation]

Original: sasso [Equation] vol
Revised: Since the nature of ordered clusters, the graph matrix is shown in a blockdiagonal form.

Original: [Equation] k ) . For the ordered graph, CNcut actually partitions an ordered graph matrix into a K -block-diagonal form.
Revised: Thus, the CNut problem can be seen as partitioning an ordered graph matrix into a K -block-diagonal form.

Original: Thus, in this section, we will learn the block-diagonal structure of the self-representation, and identify the clustering structure simultaneously.

Original: Since the graph is ordered, the graph can be clustered by a set of K − 1 indices, which is denoted by a calligraphic letter: T = { t 1 , t 2 , ..., t K − 1 } ⊂ { 1 , 2 , ..., N }, t k − 1 < t k for any k ∈ [2 , K − 1] , and its cardinal is |T | .
Revised: Speciﬁcally, since the graph is ordered, the graph can be clustered by a set of K − 1 indexes, which is denoted by a calligraphic letter: T = { t 1 , t 2 , ..., t K − 1 } ⊂ { 1 , 2 , ..., N } .

Original: The dummy indices t 0 :
Revised: The dummy indexes t 0 :

Original: = 0 and t K := N are implicitly available.
Revised: = 0 and t K := N are implicitly available.

Revised: We have the constraint that t k − 1 < t k for any k ∈ [1 , K ] .

Original: Thus, a complete graph G =
Revised: Thus, a complete graph G =

Original: ( V , E , W ) can be partitioned into K disjoint subgraph with vertex setsV t
Revised: ( V , E , W ) can be partitioned into K disjoint subgraph with vertex sets

Original: [Equation] 1 , t k − 1 +2 , ..., t k } , k = 1 , 2 , ..., K , by simply removing edges connecting theK parts.
Revised: V t k − 1 ,t k = { t k − 1 + 1 , t k − 1 + 2 , ..., t k } , k = 1 , 2 , ..., K , by simply removing edges connecting the

Original: In other words, the graph matrix is partitioned into a K -block-diagonal form.
Revised: K parts.

Original: The converse Ncut problem for ordered graph is formulated as
Revised: We propose the following partition-based CNut clustering model:

Original: [Equation]
Revised: [Equation]

Revised: We learn the block-diagonal structure of the graph matrix by solving (2), and the clustering structure will be identiﬁed by the K − 1 partition indexes simultaneously.

Original: Since there are N K − 1 combinations of K − 1unknown variables { t k } K − 1 k =1 , the global optimal solution ofProblem (2) can be searched exhaustively with a O ( N K − 1 ) complexity.
Revised: Since there are N K − 1 combinations of K − 1 variables { t k } K − 1 k =1 , the global optimal solution of (2) can be searched exhaustively with a O ( N K − 1 ) complexity.

Original: In order to explore a low-complexity approach to solvethe problem (2), we give the following theoretical analysis.
Revised: In order to explore a low-complexity approach to solve (2), we give the following theoretical analysis.

Original: Consider partitioning a K -component graph matrix (e.g., the matrix shown in Figure 2) into a twoblock-diagonal form.
Revised: Consider partitioning a K -component graph matrix (e.g., the matrix shown in Fig.1) into a two-block-diagonal form.

Original: Problem (2) can be written asmaximize
Revised: Problem (2) can be written as

Original: [Equation]
Revised: [Equation]

Original: Given a k -component graph  which is ordered correctly, and separated by K − 1 indices { c k} K − 1  k = perfectly.
Revised: Suppose there exists a K -component graph matrix W ∈ R N × 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 , ∀ k ∈ [1 , K ] , and the dummy indexes c 0 := 0 , c K := N .

Original: Suppose the within-cluster weight  w ij , i, j ∈ [ c k − 1 + 1 , c k ] follow Gaussian distribution N ( µ k , σ 2 ) ,where µ k is the average within-cluster weight of the k th cluster measuring the k th within-cluster density, andσ 2 is the variance of the noise simulating the uncertainty of the similarity between points.
Revised: Suppose the within-cluster weight of the k th cluster, w ij , i, j ∈ [ c k − 1 + 1 , c k ] , follows the Gaussian distribution N ( µ k + β, σ 2 ) , µ k > 0 , ∀ k ∈ [1 , K ] , and the between-cluster weight, w ij , i ∈

Original: In addition, suppose the without-cluster weight w ij , i ∈ [ c k − 1 +1 , c k ] , j / ∈ [ c k − 1 + 1 , c k ] follow zero means Gaussian distribution N (0 , σ 2 ) .
Revised: [ c k − 1 + 1 , c k ] , j / ∈ [ c k − 1 + 1 , c k ] , ∀ k ∈ [1 , K ] , follows the Gaussian distribution N ( β, σ 2 ) , where β > 0 is a large enough number to make sure all of the weights are non-negative.

Original: Denote the expectation of the objective function ofthe problem (3) as F ( t ) .
Revised: Denote the expectation of the objective function of (3) as F ( t ) .

Original: We have the following observation:
Revised: We have the following observation:

Original: Theorem 3.1.
Revised: Theorem 3.1.

Original: The function F ( t ) has the following property:
Revised: The function F ( t ) has the following property:

Original: 1) If K = 1 , then F ( t ) =2)
Revised: 1) If K = 1 , then F ( t ) = 1 for t ∈ (0 , N ) .

Original: If K ≥ 2 , then F ( t ) increase for t ≤ c 1  and decrease for t > c K −3)
Revised: If K ≥ 2 , then F ( t ) increase for t ∈ (0 , c 1 ] and decrease for t ∈ [ c K − 1 , N ) .

Original: If K ≥ 3 , then, for the interval [ c k− 1 + 1 , c k  ] ,  k = 2 , ..., K − 1 , F ( t ) decrease for t ∈ ( c k , ˆ t ) and increase for t ∈ (ˆ t, c k+1 ) , where
Revised: If K ≥ 3 , then, for the interval [ c k , c k +1 ] , ∀ k = 1 , ..., K − 2 , F ( t ) decreases for t ∈ [ c k

Original: [Equation]
Revised: ˆ t ] and increases

Original: √ B 2 ( c k +1 − c k ) 2 +1 (cid:17)2( c k +1− c k )  ( B 2 − B 1 ) + 1 2 ( c k +1 + c k ) B 1 (cid:54) = B12 ( c k +1 + c k  ) B 1 = Bwhere B 1 = µ k +1 (cid:80) ki
Revised: for t ∈ [ˆ t, c k +1 ] if ˆ t ∈ ( c k , c k +1 ) ; F ( t ) decreases for t ∈ [ c k , c k +1 ] if ˆ t ≥ c k +1 ; F ( t ) increases for t ∈ [ c k , c

Original: =1 ( c i − c i − 1 ) [Equation]
Revised: k

Original: µ i , and B 2 = µ k +1 (cid:80)
Revised: +1 ] if ˆ t ≤ c k , where

Original: [Equation]
Revised: [Equation]

Original: +2 ( c i − c i − 1 ) 2 µ i . The proof of Theorem 3.1 is shown in Appendix A. Theorem 3.1 illustrate that  one ofthe true partition { c k } K − 1 k =1 will minimize F ( t ) .
Revised: The proof of Theorem 3.1 is shown in Appendix A. Theorem 3.1 illustrate that F ( t ) will be constant if none c k in (0 , N ) ( K = 1 ), and one of { c k } K − 1 k =1 will maximize F ( t ) if there exists partition indexes { c k } K − 1 k =1 in (0 , N ) ( K ≥ 2 ).

Original: Denote the bounded  function as F B ( t i , t, t j ) =
Revised: We denote the bounded binary-segmentation function as F B ( c i , t, c j ) =

Original: E [ f ( t i , t ) + f ( t, t j )] , 0 < t i <{+, c j are two of the true partition { c k } Kk =
+}
Revised: E [ f ( c i , t ) + f ( t, c j )] , where c i {+, c j are two of the true partition { c k } Kk =
+}

Original: t j < N .
Revised: and c i < c j .

Original: Obviously, F B (0 , t, N ) =
Revised: Obviously, F B (0 , t, N ) =

Original: F ( t ) .
Revised: F ( t ) .

Original: Then, wealso have the following corollary.
Revised: Then, we have the following corollary.

Original: Corollary 3.1.1.
Revised: Corollary 3.1.1.

Original: For any interval ( t i , t j ] ⊆ (0 , N ] , the function F B ( t i , t, t j ) has the same property as the function F ( t ) :
Revised: For the interval ( c i , c j ) ⊆ (0 , N ) , where c i , c j are two of the partition { c k } Kk =0 , and c i < c j , the function F B ( c i , t, c j ) has the same property as the function F ( t ) :

Original:  If none  c k  in ( t i+ 1 , t j ) , then F B ( t i , t, t j ) =2) If there existing partition indices c p , c p +1 , ..., c q  , 1 ≤ p ≤ q ≤ K − 1  in ( t i+ 1 , t j ) , then one of them will maximize F B ( t i , t, t j ) .
Revised: 1) If none of { c k } K − 1 k =1 in ( c i , c j ) , then F B ( c i , t, c j ) =2) If there exists partition indexes c p , c p +1 , ..., c q ∈ { c k } K − 1 k =1 , 1 ≤ p ≤ q ≤ K − 1 , in ( c i , c j ) , then one of c p , c p +1 , ..., c q will maximize F B ( c i , t, c j ) .

Original: Denote the binary paritition loss function as
Revised: Denote the binary-segmentation reward function as

Original: [Equation]
Revised: [Equation]

Revised: The function h ( c i , t, c j ) represents the reward when we replace one segment f

Revised: ( c i , c j ) with two segments f ( c i , t ) and f ( t, c j ) .

Revised: The expectation of h ( c i , t, c j ) is

Original: [Figure]
Revised: [Equation]

Revised: Since E { f ( c i , c j ) } is constant with given c

Revised: i and c j , max t ∈ ( c i ,c j ) E ( h ( c i , t, c j )) is equivalent to max t ∈ ( c i ,c j ) F B ( c i , t, c j ) .

Revised: Thus, the expectation function E ( h ( c i , t, c j )) has the same property as

Revised: F B ( c i , t, c j ) , i.e., Corollary 3.1.1 also holds for E ( h ( c i , t, c j )) .

Revised: The property of function F B ( · ) implies the character of the objective function of the problem (2).

Revised: If we partition the graph matrix into two-block-diagonal matrix by solving (3), Theorem 3.1 points out that one of the true partition { c k } K − 1 k =1 will maximize (3) from the expectation view.

Revised: Suppose the ﬁrst estimated partition is c 1 , and the indexes τ 0 = 0 , τ 1 =

Revised: c 1 , τ 2 = N .

Revised: If we want to obtain the second partition by inserting a new partition among { τ i } 2 i =1 , in other words, we want to choose one of the two intervals ( τ 0 , τ 1 ) , and ( τ 1 , τ 2 ) to insert a new partition.

Revised: Then, we have the following problem.

Revised: [Equation]

Revised: Since (cid:80) 2 k =1 f ( τ k − 1 , τ k ) is constant, Problem (4) is equivalent to

Revised: [Equation]

Revised: [Equation]

Original: Input: an ordered graph W , and the cluster number K
Revised: Input: an ordered graph W , and the cluster number K

Original: Initialize: T = {} repeat
Revised: Initialize: T = {} repeat

Original: Initialize L = |T
Revised: Initialize L = |T

Original: | + 1 , t 0 = 0 , t L = N ,
Revised: | + 1 , t 0 = 0 , t L = N ,

Original: T = { t 1 , ..., t L − 1 } , and two empty vector v, g ∈ R L .
Revised: T = { t 1 , ..., t L − 1 } , and two empty vector v, g ∈ R L .

Original: for k = 0 : ( L − 1) do v k = argmax t ∈ ( t k +1 ,t k +1 )
Revised: for k = 0 : ( L − 1) do v k = argmax t ∈ ( t k +1 ,t k +1 )

Original: f ( t k , t ) + f ( t, t k +1 )g k = f
Revised: f ( t k , t ) + f ( t, t k +1 )

Original: ( t k, v k )
Revised: ( t k v k )

Original: + f ( v k, t k +1 )
Revised: + f ( v k t −

Original: −f ( t k , t k +1 ) end for ˆ k = argmax k g k , and ˆ t = v ˆ k T = T ∪ { ˆ t } until L = K − Output: {T }
Revised: −

Original:  Corollary 3.1. indicates that Φ( τ 1 , τ 2 ) > Φ( τ 3 , τ 4 ) if there existing c k in ( τ 1 + 1 , τ 2 ) , and there are none c k in ( τ 3 + 1 , τ 4 ) .
Revised: As said before, Corollary 3.1.1 also holds for E ( h ( c i , t, c j )) .

Original: In addition, one of { c K } K − 1 k =1 will maximize Φ( τ 1 , τ 2 ) for any interval ( τ 1 + 1 ,  τ 2 ) .
Revised: Corollary 3.1.1 indicates that one of { c k } K − 1 k =1 will maximize the expectation of h ( τ j − 1 , t, τ j ) .

Original: The property of functions F ( · ) and F B ( · ) implies the characteristic of the objective function of the problem (2). Ifwe partition the graph into two-block-diagonal matrix, i.e., the problem (3), one of the true partition c k will maximize the problem (3) will a high probability. Based on this observation, we propose a block-diagonal structure learning method, namely BDSL, to solve the problem (2).
Revised: If c 2 maximizes (5), then c 2 also maximizes (4).

Original: Firstly, we partition the interval ( t 0 , t K ] , t 0 = 0 , t K = N into two segments, Corollary 3.1.1 indicates that one of the true partition { c k } K − 1 k =1 will maximize the objective function, which is denoted as ˆ c l .
Revised: Suppose the indexes τ 0 = 0 , τ 1 = c 1 , τ 2 = c 2 τ 3 = N . We choose one of the three intervals to insert the third partition.

Original: Then, we insert the third partition index by searching in ( t 0 , ˆ c l ] , and (ˆ c l , t K ] . We can obtain the second optimal partition index ˆ c j in ( t 0 , ˆ c l ] , or (ˆ c l , t K ] .
Revised: The problem will be shown in a similar form as (5).

Original: We repeat this progress until we obtain K −optimal partition index {ˆ c l } K − 1 l =1 for the problem (2).
Revised: We repeat the updating process until we obtain K − 1 optimal partition index { c k } K − 1 k =1 for the problem (2).

Original: The pseudo-code for the procedure of graph partition is depicted in Alg. 3.
Revised: The pseudo-code for the procedure of graph partition is depicted in Alg. 3, namely BDSL, which is of the order of O ( N log N ) . Theorem 3.2.

Original: Even if the data are extracted from K clusters partitioned by theindices { c k } K − 1 k =1 , the indices { c k } K − 1 k =1 will not maximize the objective function of the problem (2) with severe noises. The only way to obtain the global maximum is the exhaustive searching method, which has a O ( N K − 1 ) complexity.
Revised: If the within-cluster weight of the k th cluster follows the Gaussian distribution N ( µ k + β, σ 2 ) , µ k > 0 , ∀ k ∈ [1 , K ] , and the between-cluster weight follows the Gaussian distribution N ( β, σ 2 ) , where β > 0 is a large enough number to make sure all of the weight are non-negative.

Original: The complexity of the proposed BDSL is of the order of O ( N log N ) .
Revised: Then, our BDSL algorithm will converge to { c k } K − 1 k

Original: This low complexity comes at the expense of optimality. Although BDSL can only give us an approximation of the optimal partition, rather than obtaining the optimal partition, we are more concerned with identifying the clustering structure based on a low-complexity algorithm, which is theoretically guaranteed to be optimal. In addition, the solution given by BDSL can approach the optimal partition with a high probability if the clusters are far away from each other, the density ofeach cluster is uniform, and the noise is negligible, which will be proved by the experiment (see Section 4.1).
Revised: =1 by maximizing the expectation of the objective function of Problem (2). [Figure]

Original:  E XPERIMENTAL R ESULTS AND A PPLICATIONS
Revised:  E XPERIMENTAL R ESULTS AND A PPLICATIONS

Original: In this section, extensive experiments are performed on different types of datasets to demonstrate the effectivenessand superiority of the proposed method in comparison with the state-of-the-art clustering methods.
Revised: In this section, extensive experiments are performed on different types of datasets to demonstrate the effectiveness of the proposed method in comparison with the state-of-the-art clustering methods.

Original: First, a synthetic dataset is generated to show the performance of the proposed method on the dataset with different densities, distributions, and connectivity.
Revised: First, a synthetic dataset is generated to show the performance of the proposed method on the dataset with different densities, distributions, and connectivity.

Original: Then the widely used high-dimensional datasets are used to show the effectiveness of the proposed method on the subspace clustering task.
Revised: Then the widely used image datasets are used to show the effectiveness of the proposed method on the subspace clustering task.

Original: 4. S YNTHETIC E XPERIMENTS
Revised: 4. S YNTHETIC E XPERIMENTS

Original: To test the effectiveness and robustness of the proposed methods on the general clustering task, we generate four basic datasets. As shown in Figure 3, we make an analysis of different datasets as [Figure] [Figure]
Revised: We generate four basic datasets to test the effectiveness and robustness of the proposed methods on the general clustering task, which is shown in Figure 2.

Original: 57.80 LSR-DB 82.

Original: 96.29 NSN 42.

Original: 55.50 NSN-DB 80.

Original: 95.66 SMR 68.

Original: 68.87 SMR-DB 79.

Original: 94.74 SSC 24.

Original: 40.16 SSC-DB 75.

Original: 90.26 QOSC 63.

Original: 65.30 QOSC-DB 79.

Original: 95.57 ORGEN 25.

Original: 61.50 ORGEN-DB 80.

Original: 91.32 iPursuit 65.

Original: 75.40 iPursuit-DB 82.

Original: 95.99 FGNSC 68.

Original: 52.40 FGNSC-DB 73. 92.69 BDRZ 51. 71.80 BDRZ-DB 88. follows. dataset A contains four clusters with different density and shape.
Revised: Dataset A contains four clusters with different densities and shapes.

Original: dataset B contains four clusters following different Gaussian distributions.
Revised: Dataset B contains four clusters following different Gaussian distributions.

Original: dataset C contains four oddly shaped clusters with clear boundary.
Revised: Dataset C contains four oddly shaped clusters with clear boundaries.

Original: dataset D contains two coupled clusters without clear boundary.
Revised: Dataset D contains two coupled clusters without clear boundaries.

Original: As said in Section 2.1, different distance measurement can be used to show the similarity between samples, we pick Euclidean distance after expernational normalizition as the weight.
Revised: As said in Section 2.1, different distance measurements can be used to show the similarity between samples. We pick Euclidean distance after exponential normalization as the weight.

Original: We compare the proposed approach with  K-means MacQueen (1967), BIRCH Zhang et al. , FCM Bezdek et al.
Revised: We compare the proposed approach with six types of clustering methods, including K-means MacQueen (1967), STREAM O’callaghan et al.

Original: GMM Rasmussen (1999), DBSCAN Ester et al. , CLICK Sharan & Shamir (2000), STING Wang et al., FC Barbar ´ a & Chen (2000), COBWEB Fisher (1987), kernel
Revised: PAM Kaufman & Rousseeuw (2009), BIRCH Zhang et al. DBSCAN Ester et al.

Original: Km Sch¨olkopf et al.
Revised: OPTICS Ankerst et al.

Original: LF Lumer & Faieta (1994), QC Horn & Gottlieb (2001), NCut Shi & Malik (2000), DD Shang et al.
Revised: (1999), ST-DBSCAN Birant & Kut (2007), STING Wang et al.

Original: STREAM O’callaghan et al. (2002), OPTICS Ankerst et al. (1999), PAM Kaufman & Rousseeuw (2009), and spectral clustering Von Luxburg (2007).
Revised: , GMM Rasmussen (1999),COBWEB Fisher (1987), spectral clustering Von Luxburg (2007), CLICK Sharan & Shamir (2000), and Chinese Whisper (CW) Biemann (2006).

Original: We tune the parameters for these methods to achieve the best results.
Revised: We tune the parameters for these methods to achieve the best results.

Original: The experiments are repeated 10 times.
Revised: The experiments are repeated ten times.

Original: The mean clustering accuracy is computed to show the performance.
Revised: The mean clustering accuracy is computed to show the performance.

Original: The experimental results are shown in Figure.
Revised: The experimental results are shown in Fig.4.

Original: We provide a visual comparison of graph weight matrices in Figure.
Revised: We provide a visual comparison of graph weight matrices in Fig.3.

Original: It can be observed that our method DBGO provides weight matrices which are in a block-diagonal form roughly.
Revised: It can be observed that our method DBGO provides weight matrices that are in a block-diagonal form roughly.

Original: The red dotted boxes show the graph partition result provided by BDSL.
Revised: The red dotted boxes show the graph partition result provided by BDSL.

Original: As anticipated, our proposed DBGO-BDSL outperforms all of the comparisons.
Revised: As anticipated, our proposed DBGO-BDSL outperforms all of the comparisons.

Original: This experiment highlights the robustness of our methods to noises and densities.
Revised: This experiment highlights the robustness of our methods to noises and densities.

Original: 4. E XPERIMENTS ON H IGH - DIMENSIONAL D ATASETS
Revised: 4. E XPERIMENTS ON H IGH - DIMENSIONAL D ATASETS

Original: For good performance in dealing with complex local correlation and high-dimensional structure of sequential data, representation-based methods have become one of the hot topics for highdimensional data clustering, in which  subspace clustering is a representative tool.
Revised: For good performance in dealing with complex local correlation and high-dimensional structure of the image data, representation-based methods have become one of the hot topics for highdimensional data clustering, in which spectral-based subspace clustering is a representative tool.

Original: However, all of these state-of-the-arts use  spectral clustering method to identify the clustering structure, and  thespectral clustering method shows poor performance on the non-uniform density and distribution dataset, which is shown in Section 4.1.
Revised: However, all of these state-of-the-art spectral-based subspace clustering methods use the spectral clustering method to solve the Ncut problem and identify the clustering structure by K-Means. The [Figure] [Figure]

Revised: 95.57 ORGEN 25.

Revised: 61.50 ORGEN-DB 80.

Revised: 91.32 iPursuit 65.

Revised: 75.40 iPursuit-DB 82.

Revised: 95.99 FGNSC 68.

Revised: 52.40 FGNSC-DB 73.

Revised: 92.69 BDRZ 51.

Revised: 71.80 BDRZ-DB 88.

Revised: spectral clustering method shows poor performance on the non-uniform density and distribution dataset, which is shown in Section 4.1.

Original: Thus, we replace the spectral clusteringoperadion of the famous state-of-the-art subspace clustering methods with our proposed DBGO-BDSL algorithm  to show the effectiveness of  DBGO-BDSL. 
Revised: Thus, we replace the spectral clustering with our proposed DBGO-BDSL algorithm in the conventional spectral-based subspace clustering methods to show the effectiveness of our DBGO-BDSL. [Figure]

Original: We test our method on ﬁve benchmark datasets  for famous tough high-dimensional clustering tasks, including face recognition (E-Yale B Georghiades et al.
Revised: We test our method on ﬁve benchmark datasets (as shown in Fig.6) for famous tough highdimensional clustering tasks, including face recognition (E-Yale B Georghiades et al.

Original:  AR Martinez & Benavente (1998)), object detection (COIL-20 Nene et al.
Revised: (2001), AR Martinez & Benavente (1998)), object detection (COIL-20 Nene et al.

Original: and handwritten digits identiﬁcation (MNIST LeCun et al.
Revised: and handwritten digits identiﬁcation (MNIST LeCun et al.

Original: USPS Hastie et al.
Revised: USPS Hastie et al.

Original: The detail of these datasets is introduced as follows: 1)

Original: The AR dataset consists of 1400 frontal face images of 100 subjects.

Original: For each subject, there areimages.

Original: Each cropped face image consists of 165 × 120 pixels.

Original: The dataset COIL-20 consists of 1440 object images of 20 subjects.

Original: For each subject, there are 72 images.

Original: Each cropped face image consists of 64 × 64 pixels.

Original: The USPS dataset contains grey scale images of handwritten digits 0 ∼ 9 .

Original: There are 10 subjects of digits.

Original: The Extended Yale B dataset consists of 2,414 frontal face images of 38 subjects under 9 poses and 64 illumination conditions.

Original: For each subject, there areimages.

Original: Each cropped face image consists of 192168 pixels.

Original: The MNIST dataset contains grey scale images of handwritten digits 0 ∼ 9 .

Original: There are 10 subjects of digits.

Original: Moreover, We use the state-of-the-arts to generate the afﬁnity matrix, and then, apply the proposed DBGO-BDSL on the afﬁnity matrix.In another word, we replace the spectral clustering in the state-of-the-arts with proposed DBGO-BDSL.
Revised: Moreover, We use state-of-the-art spectral-based subspace clustering methods to generate the afﬁnity matrix and then apply the proposed DBGO-BDSL to the afﬁnity matrix.

Original: We compare the traditional version of these state-of-the-arts with the improved version.
Revised: We compare the traditional version of these state-of-the-art methods with the improved version.

Original: The comparison includes representation-based methods (such as SRC Xiao et al.
Revised: The comparison includes representation-based methods (SRC Xiao et al.

Original: LLC Tang et al.
Revised: LLC Tang et al.

Original:  CRC Wang et al. (2022),and LRC Zhang et al.
Revised: (2018), CRC Wang et al. (2022), LRC Zhang et al.

Original: (2021b)), and low-rank criterion-based methods (such as LSRLu et al.
Revised: (2021b)), and lowrank criterion-based methods (LSRLu et al.

Original: (2012), NSNPark et al.
Revised: (2012), NSNPark et al.

Original: (2014), SMRHu et al.
Revised: , SMRHu et al.

Original: (2014), SSCXu et al. , QOSCLu et 
Revised: SSCXu et al. (2015), QOSCLu et al.

Original: , ORGENYou et 
Revised: ORGENYou et al.

Original:  iPursuit Rahmani & Atia (
Revised: (2016), iPursuit Rahmani & Atia (2017), FGNSCYang et al.

Original: 2019), and BDRZLu 
Revised: (2019), and BDRZLu et al.

Original: We tune the parameters for the comparison methods to achieve the best results.
Revised: We tune the parameters for the comparison methods to achieve the best results.

Original: The experiments are repeated ten times, and the mean clustering accuracy is computed to show the performance.
Revised: The experiments are repeated ten times, and the mean clustering accuracy is computed to show the performance.

Original: The experimental result is shown in Figure 6.
Revised: The experimental result is shown in Fig.5.

Original: We compare the original version with our improved version, e.g., LSR is the original version, and LSR-DB is the improved version which integrates LSR with the proposed DBGO-BDSL algorithm.
Revised: We compare the original version with our improved version, e.g., LSR is the original version, and LSR-DB is the improved version which integrates LSR with the proposed DBGO-BDSL algorithm.

Original: Each traditional method’s performance can be improved by our DBGO-BDSL, and the clustering accuracy is improved by roughly 10% ∼ 50%.
Revised: Each conventional spectral-based subspace clustering method’s performance can be improved by our DBGO-BDSL, and the clustering accuracy is improved by roughly 10% ∼ 50%.

Original:  C ONCLUSIONS
Revised:  C ONCLUSIONS

Original: In this work, we proposed a new two-stage graph-based clustering method for general clustering task or subspace clustering task.
Revised: In this paper, we proposed a new two-stage integrated density and graph clustering method for lowor high-dimensional clustering tasks.

Original: Comprehensive experiments were conducted on synthetic datasets, realistic datasets.
Revised: Comprehensive experiments were conducted on synthetic datasets and real datasets.

Original: The proposed method not only shows signiﬁcantly better performance than current state-of-the-artgraph-based clustering methods  but also improves the performance of the traditional subspace clustering method by roughly 10% ∼ 50%.
Revised: The proposed method not only shows signiﬁcantly better performance than current state-of-the-art clustering methods on the synthetic low-dimensional dataset but also improves the performance of the conventional spectral-based subspace clustering method by roughly 10% ∼ 50% on the real high-dimensional dataset.

Original: A A PPENDIX A. P ROOF OF T HEOREM 3. 
Revised: A A PPENDIX A. P ROOF OF T HEOREM 3.

Original: Case 0: K = 1 , we have F ( t ) =
Revised: Case 0: K = 1 , we have F ( t ) =

Original: t 21  µ 1  t 1 Nµ 1 +  ( N − t [Equation] [Equation] N 
Revised: t 21 ( µ 1 + β ) t 1 N ( µ 1 + β ) + ( N − t 1 ) 2 ( µ 1 + β ) ( N − t 1 ) N ( µ 1 + β ) = 1 .

Original: [Equation]

Original: [Equation]

Original: 1 = 1 .

Original: Case 1: K ≥ 2 .
Revised: Case 1: K ≥ 2 .

Original: Consider t ∈ [1 , c 1 ] , the F ( t ) is given by
Revised: Consider t ∈ (0 , c 1 ] , the F ( t ) is given by

Original: [Equation]
Revised: [Equation]

Original: where the mean weight of the i th, i ∈ { 2 , 3 , ..., K } cluster with zero covariance is b (2 , K )
Revised: where the mean weight of the i th, i ∈ { 2 , 3 , ..., K } cluster with zero covariance is b (2 , K )

Original: = ( c 2 − c 1 ) 2  µ 2 +  ...
Revised: = ( c 2 − c 1 ) 2 ( µ 2 + β ) + ...

Original: +( N − c k − 1 ) 2  µ K  ( N − c 1 ), and C 1 = µ 1 ( N − c 1 ) 2 b (2 ,K ) .
Revised: +( N − c k − 1 ) 2 ( µ K + β ) ( N − c 1 ), and C 1 = µ 1 + β N − c 1 ) 2 b (2 ,K ) .

Original: F ( t ) will increase with the increase of t ∈ (0 , c 1 ] .
Revised: F ( t ) will increase with the increase of t ∈ (0 , c 1 ] .

Original: Case 2: K ≥ 2 .
Revised: Case 2: K ≥ 2 .

Original: Consider t ∈ [ c k − 1+ 1 , N ] , the F ( t ) is given by
Revised: Consider t ∈ [ c K − 1 , N ) , the F ( t ) is given by

Original: [Equation]
Revised: F

Original: [Equation]
Revised: [Equation]

Original: N
Revised: where b (1 , K − 1) = c 21 ( µ 1 +

Original: [Equation]
Revised: [Equation]

Original: C
Revised: c 2 −

Original: [Equation]
Revised: [Equation]

Original: where b (1 , K − 1) = c 21 µ 1 +( c 2 − c 1 ) 2 µ 2 + ...
Revised: [Equation] c K

Original: +( c k − 1 − c 
Revised: [Equation] c [Equation] µ

Original:  K − 2 ) 2 µ K − 1 ck −, and C 2 = µ K  c 2 k − 1 b (1 ,K −1) .
Revised: [Equation] K [Equation] , and C 2 = µ K + β c 2 K −

Original:  F ( t ) will decrease with the increase of t ∈ ( c k − 1 , N ] .
Revised: [Equation] (1 ,K − [Equation] F ( t ) will decrease with the increase of t ∈ [ c K − 1 , N ) .

Original: Case 3: K ≥ 3 .
Revised: Case 3: K ≥ 3 .

Original: Consider t ∈ ( c k , c k +1 ] , l = 1 , 2 , ..., K − 2 . F ( t ) =
Revised: Consider t ∈ [ c k , c k +1 ] , k =

Original: [Equation]
Revised: [Equation]

Original: where[-b (1 , k ) = c 21 µ 1 + ..., +( c k − c k − 1 ) 2 µ l ck , and b ( k + 2 , K ) = ( c l +2 − c k +1 ) 2 µ l +
-]
Revised: where[-b (1 , k ) = c 21 µ 1 + ..., +( c k − c k − 1 ) 2 µ l ck , and b ( k + 2 , K ) = ( c l +2 − c k +1 ) 2 µ l +
-]

Original: + ..., +( N − c
Revised: [Equation] and

Original: k − 1 ) 2 µ K ( N − c k +1 ).
Revised: [Equation]

Original: Denote d = c k +1 − c k , and x = t 1 − c k +1 + c k. Since t ∈ ( c k +1 , c k ) , we have x ∈ ( − d 2 , d 2 ) .
Revised: Denote d = c k +1 − c k , and x = t 1 − c k +1 + c k. Since t ∈ ( c k +1 , c k ) , we have x ∈ ( − d 2 , d 2 ) .

Original: Denote
Revised: Denote

Original: B 1 = µ k +1  c 2 k b (1 ,k ) , and B 2 =
Revised: B 1 = µ k +1 + β c 2 k b (1 ,k ) , and B 2 =

Original: [Equation]
Revised: [Equation]

Original: 2 b ( k
Revised: N − [Equation] k

Original: [Equation]
Revised: [Equation]

Original:  2  ,K ) .
Revised: 1 ) 2 b ( k +2 ,K ) .

Original: So, F ( t ) can be written as
Revised: So, F ( t ) can be written as

Original: [Equation]
Revised: [Equation]

Original: F ( x ) is convex for x ∈ ( − d 2 , d 2 ) .
Revised: F ( x ) is convex for x ∈ ( − d 2 , d 2 ) .

Original: If we set the differentiate of F ( x ) w.r.t.
Revised: If we set the differentiate of F ( x ) w.r.t.

Original: x as zero, we can obtain x 

Revised: x as zero, we can obtain x

Original: and we can say F ( x ) is minimized at x 0 , where
Revised: and we can say F ( x ) is minimized at x 0 , where

Original: [Equation]
Revised: [Equation]

Original: =
Revised: Recall the deﬁnition of d , and x , F ( t ) is minimized at ˆ t

Original: [Equation]
Revised: [Equation]

Original: Recall the deﬁnition of d , and x , F ( t ) is maximized at
Revised: B

Original: [Equation]
Revised: [Equation]

Revised: Thus, for the interval [ c k , c k +1 ] , k = 1 , ..., K − 2 , F ( t ) decreases for t ∈ [ c k , ˆ t ] and increases for t ∈ [ˆ t,

Revised: c k +1 ] if ˆ t ∈ ( c k , c k +1 ) ; F ( t ) decreases for t ∈ [ c k , c k +1 ] if ˆ t ≥ c k +1 ; F ( t ) increases for t ∈ [ c k , c k

Original: {++
+}
Revised: {++
+}

Revised: if ˆ t ≤ c k .