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 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., K-Means) 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: However, 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 an integrated density and graph clustering approach.
Revised: In this paper, we investigate an integrated density and graph clustering approach.

Original: 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.
Revised: Firstly, we introduce a matrix ordering algorithm, namely DBGO, 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: Secondly, we ﬁnd that the graph matrix is shown in a block-diagonal form because of the nature of ordered clusters.
Revised: Secondly, we ﬁnd that the graph matrix is shown in a block-diagonal form because of the nature of ordered clusters.

Original: 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: 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.

Original: 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. The optimality has been guaranteed theoretically.
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 graph-cut function, and our proposed BDSL algorithm can converge to the true cluster segmentation by maximizing the expectation of the graph-cut function, which has been guaranteed theoretically.

Original: We test the proposed method on synthetic datasets and ﬁve high-dimensional real 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-art clustering methods and improves the performance of subspace clustering approaches 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: Many recent works pay attention to the effectivity of clustering, i.e., the quality or usefulness of the clustering result.
Revised: Many recent works pay attention to the effectivity of clustering, i.e., the quality or usefulness of the clustering result.

Original: However, their characteristic of effectiveness is still unsatisfactory.
Revised: However, their characteristic of effectiveness is still unsatisfactory.

Original: 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.
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: 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: 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.
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 been 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 accom- plished 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 a 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 depend 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 the 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 &
Revised: Although a lot of graph partition approaches have been proposed recently, e.g., MST S¸aar & Topcu (2022), PrimZhu &

Original: Chen (2021), Kruskal Battaglia & Pensa (2021), SNN Liu et al.
Revised: Chen (2021), Kruskal Battaglia & Pensa (2021), SNN Liu et al.

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

Original: (2022), 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 ExtendedSpectral Sharma & Seal (2021).
Revised: (2021b), and ExtendedSpectral Sharma & Seal (2021).

Original: However, their characteristics of effectiveness and robustness on real datasets are still unsatisfactory.
Revised: 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 a signiﬁcant 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 arbitrarily 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 similar 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: We proposed an ordering algorithm to ﬁnd ordered density-based clusters in spatial data and 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 clusters.
Revised: The graph weight matrix is shown in block-diagonal form because of the nature of ordered clusters.

Original: 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
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

Original: K is the number of clusters.
Revised: K is the number of clusters.

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

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

Original: 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.
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 ARTITION

Revised: The graph-based method includes three phases: graph construction, graph reﬁnement (e.g., prune and generate), and graph partition.

Revised: Since the graph reﬁnement method is complicated and various, we only introduce the graph construction and the graph partition technique.

Revised: 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 ] .
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 ] .

Original: We convert the radian θ ij to its degree form ˜ θ ij , i.e., ˜ θ ij = θ ij 180 /π , ˜ θ ij ∈ [0 , 180] .
Revised: 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 perform the exponential normalization on the original measurement, 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: 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.
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 the 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 self-representation 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 main 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 is 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

Revised: [Equation]

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 ARTITION

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 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).
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.
Revised: Recall the symmetric and positive properties of the graph.

Original: We assume w ij = w ji ≥ 0 .
Revised: 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 paths 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 a 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 partition 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 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).
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 mini cut criterion, which has to be minimized: cut ( A 1 , A 2 , ..., A K )
Revised: The ﬁrst requirement for the partition 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 mini cut 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) [Equation] ¯ A k ) vol [Equation]
Revised: = (cid:80) Kk =1 cut ( A k , ¯ A k ) vol ( A k )

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

Original: 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.
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.

Revised: Thus, the density-based clustering method is a commonly used clustering strategy.

Original: DBSCAN Ester et al.
Revised: DBSCAN Ester et al.

Original: (1996) is a well-known density-based clustering method.
Revised: (1996) is a well-known density-based clustering method.

Original: It relies on choosing suitable parameters to order the sample and identify the clustering structure simultaneously. However, DBSCAN cannot cluster data setswell with large differences in densities since the minPts- (cid:15) combination cannot be chosen appropriately for all clusters. 
Revised: However, 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 or nested clusters. Although many methodologies have been proposed for solving this problem, e.g., HDBSCANCampello et al.

Original: In addition, if the data and scale are poorly understood, choosing a meaningful distance threshold (cid:15) can be difﬁcult.
Revised: , their performance is still unsatisfactory. NcutShi & Malik (2000) is the most famous graph-based clustering method. However, 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.

Original: In this paper, we proposed an integrated density and graph clustering method by combining the knowledge of  density-based clustering and graph-based clustering.
Revised: 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 and Ncut.

Original: The proposed clustering method has the potential to overcome some drawbacks of  theconventional graph-based method and density-based method. Firstly, we consider beginning with a graph because of the superiority of the existing state-of-the-art graph construction method.
Revised: Speciﬁcally, 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. Due to the density-based ordering step, the Ncut becomes a non-NP-hard problem from the partition view.

Original: Secondly, we consider separating the  clustering task into two parts: ordering and partitioning.
Revised: We proposed the BDSL algorithm with O ( NlogN ) complexity to solve the partition-based CNcut problem, and the proposed BDSL can converge to the true clustering partition from the expectation view. Note that we can replace Ncut with RadiocutHagen & Kahng (1992) and replace DBSCAN with OPTICSAnkerst et al. (1999). This paper only chooses DBSCAN and Ncut to describe the proposed integrated density and graph clustering framework.

Original: 3. D ENSITY -B ASED G RAPH O RDERING
Revised: 3. D ENSITY -B ASED G RAPH O RDERING

Original: 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.
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.

Original: 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.
Revised: 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: 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.
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: Then, the graph matrix can be permuted according to the iteration order.
Revised: Then, the graph matrix can be permuted according to the iteration order.

Original: Since the nature of ordered clusters, the ordering graph should be in a block-diagonal form.
Revised: Since the nature of ordered clusters, the ordering graph should be in a block-diagonal form.

Original: We use the same greedy ordering strategy as DBSCAN Ester et al.
Revised: We use the same greedy ordering strategy as DBSCAN Ester et al.

Original: (1996) to iterate through each sample based on the similarity given by the weight matrix.
Revised: (1996) to iterate through each sample based on the similarity given by the weight matrix.

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

Original: The element w ij 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 more similar the i th sample and the j th sample are, the greater the weight w ij .
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.

Original: 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.
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: 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
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) .

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

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

Original: In our ordering step, we only care about the order.
Revised: Our method can use the ratio threshold rule because we only need to consider the order.

Original: 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.
Revised: However, DBSCAN can only use the constant threshold rule because it still relies on this constant threshold rule to identify the clustering structure. 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 .

Original: Deﬁnition 3.1 ( Density-reachable ) .
Revised: 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. 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 ( i ) . Then, the reachability score is deﬁned as follows.
Revised: Algorithm

Original: Deﬁnition 3.2 ( Reachability-score ) .

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

Original: Let ε be a natural number.

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

Original: [Equation]

Original: Undefined, otherwise

Original: Input: the graph W , and the parameter δ , ε .
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 i in C
Revised: for each unprocessed element i in C

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

Original: [ O ; i ] .
Revised: [ O ; i ] .

Original: [ Q , s ]=
Revised: [ Q , s ]=

Original: QueueUpdate( i , Q , s ) repeat
Revised: QueueUpdate( i , Q , s ) repeat

Original: Pop j from Q .
Revised: Pop j from Q .

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

Original: [ O ; j ] .
Revised: [ O ; j ] .

Original: if | N δ ( j ) | ≥ ε
Revised: if | N δ ( j ) | ≥ ε

Original: [ Q , s ]=QueueUpdate( j , Q , s ) end if until Q is empty end for
Revised: [ Q , s ]=QueueUpdate( j , Q , s ) end if until Q is empty end for

Original: Permute W based on O .
Revised: Permute W based on O .

Original: Output: W
Revised: 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. 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.
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. 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 ( i ) .

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 theweight between i th sample and j th sample directly.
Revised: Then, the reachability score is deﬁned as follows. Deﬁnition 3.2 ( Reachability-score ) . Suppose i, j ∈ [1 , N ] , where N is the sample number.

Original: The reachability-score of j thobject depends on the core sample with respect to which it is calculated.
Revised: Let ε be a natural number. Then, the reachability-score of j th sample with respect to i th sample is deﬁned as s ε ( j, i ) = (cid:26) min ( c ε ( i ) , w ij ) , if | N δ ( i ) | ≥ ε Undefined, otherwise

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. 1 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.

Original: Algorithm 
Revised: Algorithm 2 Queue updating.

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

Original: for each unprocessed j in N δ
Revised: for each unprocessed j in N δ

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

Original: s
Revised: 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 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 .
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 operation of the queue Q will provide us with the element in
Revised: The pop operation of the queue Q will provide us with the element in

Original: Q , which has the largest existing-reachability-score.
Revised: Q , which has the largest existing-reachability-score.

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

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 ARTITION

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

Original: 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.

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

Original: 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.

Original: 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 in Section 2.2: 
Revised: Recall the proposed converse Ncut in Section 2.2: CNcut

Original: [Equation]
Revised: [Equation]

Original: Since the nature of ordered clusters, the graph matrix is shown in a blockdiagonal form.
Revised: Since the nature of ordered clusters, the graph matrix is shown in a block-diagonal form as illustrated in Fig.1.

Original: Thus, the CNut problem can be seen as partitioning 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: 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 } .
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 indexes 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.

Original: We have the constraint that t k − 1 < t k for any k ∈ [1 , K ] .
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 sets 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 K parts.
Revised: ( V , E , W ) can be partitioned into K disjoint subgraph with vertex sets 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 K parts.

Original: We propose the following partition-based CNut clustering model:
Revised: We propose the following partition-based CNut clustering model:

Original: [Equation]
Revised: [Equation]

Original: 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.
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 − 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.
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 solve (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 Fig.1) into a two-block-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 as
Revised: Problem (2) can be written as

Original: [Equation]
Revised: [Equation]

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

Original: If K ≥ 2 , then F ( t ) increase for t ∈ (0 , c 1 ] and decrease for t ∈ [ c K − 1 , N ) .
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 , c k +1 ] , ∀ k
Revised: If K ≥ 3 , then, for the interval [ c k , c k +1 ] , ∀ k

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

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

Original: +1 ] if ˆ t ≤ c k , where
Revised: if ˆ t ≤ c k , where

Original: [Equation]
Revised: [Equation]

Original: 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 ).
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: We denote the bounded binary-segmentation function as
Revised: We denote the bounded binary-segmentation function as

Original: F B ( c i , t, c j ) =
Revised: F B ( c i , t, c j ) =

Original: E [ f ( c i , t ) + f ( t, c j )] , where c 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: and c i < c j .
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, we have the following corollary.
Revised: Then, we have the following corollary.

Original: Corollary 3.1.1.
Revised: Corollary 3.1.1.

Original: For the interval ( c i , c j ) ⊆ (0 , N ) , where c i , c
Revised: For the interval ( c i , c j ) ⊆ (0 , N ) , where c i , c

Original: 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 ) :
Revised: 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: 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
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

Original: +1 , ..., c q will maximize F B (
Revised: +1 , ..., c q will maximize F B (

Original: c i , t, c j ) .
Revised: c i , t, c j ) .

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

Original: [Equation]
Revised: [Equation]

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

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

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

Original: [Equation]
Revised: [Equation]

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

Original: 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: 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 ) .

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

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

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

Original: 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: 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.

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

Original: c 1 , τ 2 = N .
Revised: c 1 , τ 2 = N .

Original: 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: 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.

Original: Then, we have the following problem.
Revised: Then, we have the following problem.

Original: [Equation]
Revised: [Equation]

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

Original: [Equation]

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

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

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

Original: − 
Revised: − 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 }

Original: As said before, Corollary 3.1.1 also holds for E ( h ( c i , t, c j )) .
Revised: As said before, Corollary 3.1.1 also holds for E ( h ( c i , t, c j )) .

Original: 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 ) .
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: If c 2 maximizes (5), then c 2 also maximizes (4).
Revised: If c 2 maximizes (5), then c 2 also maximizes (4).

Original: Suppose the indexes τ 0 = 0 , τ 1 = c 1 , τ 2 = c 2 τ 3 = N .
Revised: Suppose the indexes τ 0 = 0 , τ 1 = c 1 , τ 2 = c 2 τ 3 = N .

Original: We choose one of the three intervals to insert the third partition.
Revised: We choose one of the three intervals to insert the third partition.

Original: The problem will be shown in a similar form as (5).
Revised: The problem will be shown in a similar form as (5).

Original: We repeat the updating process until we obtain K − 1 optimal partition index { c k } K − 1 k =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, namely BDSL, which is of the order of O ( N log N ) .
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 ) .

Original: Theorem 3.2.
Revised: Theorem 3.2.

Original: If the within-cluster weight of the k th cluster  follows theGaussian distribution
Revised: If the within-cluster weight of the k th cluster { c k − 1 +1 , c k − 1 +2 , ..., c k } follows the

Original:  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.
Revised: 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: Then, our BDSL algorithm will converge to { c k } K − 1 k
Revised: Then, our BDSL algorithm will converge to { c k } K − 1 k

Original: =1 by maximizing the expectation of the objective function of Problem (2).
Revised: =1 by maximizing the expectation of the objective function of Problem (2).

Original: [Figure]
Revised: [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 effectiveness 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. First

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: [Table] a synthetic dataset is generated to show the performance of the proposed method on the dataset with different densities, distributions [Table] and connectivity.

Original: Then the widely used image 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

Revised: [Table]

Original: 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.
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: Dataset A contains four clusters with different densities and shapes.
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 boundaries.
Revised: Dataset C contains four oddly shaped clusters with clear boundaries.

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

Original: As said in Section 2.1, different distance measurements can be used to show the similarity between samples.
Revised: As said in Section 2.1, different distance measurements can be used to show the similarity between samples.

Original: We pick Euclidean distance after exponential normalization as the weight.
Revised: We pick Euclidean distance after exponential normalization as the weight.

Original: We compare the proposed approach with six types of clustering methods, including K-means MacQueen (1967), STREAM O’callaghan 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: PAM Kaufman & Rousseeuw (2009), BIRCH Zhang et al.
Revised: BIRCH Zhang et al.

Original: DBSCAN Ester et al.
Revised: DBSCAN Ester et al.

Original: OPTICS Ankerst et al.
Revised: OPTICS Ankerst et al.

Revised: HDBSCAN Campello et al.

Original: (1999), ST-DBSCAN Birant & Kut (2007), STING Wang et al.
Revised: (2013), ST-DBSCAN Birant & Kut (2007), STING Wang et al.

Original: , GMM Rasmussen (1999),COBWEB Fisher (1987), spectral clustering Von Luxburg (2007), CLICK Sharan & Shamir (2000), and Chinese Whisper (CW) Biemann (2006).
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 ten 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 Fig.4.
Revised: The experimental results are shown in Tab.1.

Original: We provide a visual comparison of graph weight matrices in Fig.3.
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 that 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 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. 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]
Revised: LSR means the original version of the conventional subspace clustering technique, and LSR-DB means we apply our DBGO-BDSL on the afﬁnity matrix generated by LSR.

Original: [Figure]
Revised: [Table]

Original: 95.57 ORGEN 25. 61.50 ORGEN-DB 80. 91.32 iPursuit 65. 75.40 iPursuit-DB 82.
Revised: For good performance in dealing with complex local correlation and highdimensional structure of the image data, representation-based methods have become one of the hot topics for high-dimensional data clustering, in which spectral-based subspace clustering is a representative tool.

Original: 95.99 FGNSC 68. 52.40 FGNSC-DB 73. 92.69 BDRZ 51. 71.80 BDRZ-DB 88.
Revised: However, all of these state-of-theart spectral-based subspace clustering methods use the spectral clustering method to solve the Ncut problem and identify the clustering structure by K-Means.

Original:  spectral clustering method shows poor performance on the non-uniform density and distribution dataset, which is shown in Section 4.1.
Revised: The 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 clustering with our proposed DBGO-BDSLalgorithm in the conventional spectral-based subspace clustering methods to show the effectiveness of our DBGO-BDSL.
Revised: Thus, we replace the spectral clustering with our proposed DBGO-BDSL in the conventional spectral-based subspace clustering methods to show the effectiveness of our DBGO-BDSL.

Original: [Figure]
Revised: [Figure]

Original: 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.
Revised: We test our method on ﬁve benchmark datasets (as shown in Fig.4) for famous tough high-dimensional clustering tasks, including face recognition (E-Yale B Georghiades et al.

Original: (2001), AR Martinez & Benavente (1998)), object detection (COIL-20 Nene et al.
Revised: 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: 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.
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-art methods with the improved version.
Revised: We compare the traditional version of these state-of-theart methods with the improved version.

Original: The comparison includes representation-based methods (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: (2018), CRC Wang et al.
Revised: CRC Wang et al.

Original: (2022), LRC Zhang et al.
Revised: LRC Zhang et al.

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

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

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

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

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

Original: ORGENYou et al.
Revised: ORGENYou et al.

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

Original: (2019), and BDRZLu et al.
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 Fig.5.
Revised: The experimental result is shown in Tab.2.

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. 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%. 
Revised: We compare the original version of the conventional subspace clustering technique with our improved version, Each state-of-the-art spectral-based subspace clustering method’s performance can be improved by our DBGO-BDSL, and their clustering accuracy is improved by roughly 10% ∼ 50%. The performance of the proposed DBGO-BDSL depends on the representation quality of the afﬁnity matrix given by the state-of-the-art method.

Original:  C ONCLUSIONS
Revised:  C ONCLUSIONS

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

Revised: The proposed clustering framework can overcome the drawback of the traditional density- and graph-based clustering strategies, and remains their advantages.

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

Original: The proposed method not only shows signiﬁcantly better performancethan 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. 
Revised: The proposed method not only shows the best performance on the synthetic low-dimensional dataset but also improves the clustering accuracy of the conventional spectral-based subspace clustering method by roughly 10% ∼ 50% on the real high-dimensional dataset. The proposed clustering structure can be improved in many aspects in the future.

Revised: For instance.

Revised: we can add a graph reﬁnement step to adjust the ordered matrix, reﬁne the DBGO step using a more reasonable density-based search strategy, and replace the BDSL step with a more precise partition method.

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 + β ) t 1 N ( µ 1 + β ) + ( N − t 1 ) 2 ( µ 1 + β ) ( N − t 1 ) N ( µ 1 + β ) = 1 .
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: Case 1: K ≥ 2 .
Revised: Case 1: K ≥ 2 .

Original: Consider t ∈ (0 , 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 + β )
Revised: = ( c 2 − c 1 ) 2 ( µ 2 + β )

Original: + ...
Revised: + ...

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

Original: N − c 1 ) 2 b (2 ,K ) .
Revised: 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 , N ) , the F ( t ) is given by
Revised: Consider t ∈ [ c K − 1 , N ) , the F ( t ) is given by

Original: F
Revised: F

Original: [Equation]
Revised: [Equation]

Original: where b (1 , K − 1)
Revised: where b (1 , K − 1)

Original: =
Revised: =

Original: c 21 ( µ 1 +
Revised: c 21 ( µ 1 +

Original: [Equation]
Revised: [Equation]

Original: c 2 −
Revised: c 2 −

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: c K
Revised: c K

Original: [Equation]
Revised: [Equation]

Original: c
Revised: c

Original: [Equation]
Revised: [Equation]

Original: µ
Revised: µ

Original: [Equation]
Revised: [Equation]

Original: K
Revised: K

Original: [Equation]
Revised: [Equation]

Original: , and C 2 = µ K + β c 2 K −
Revised: , and C 2 = µ K + β c 2 K −

Original: [Equation]
Revised: [Equation]

Original: (1 ,K −
Revised: (1 ,K −

Original: [Equation]
Revised: [Equation]

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

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
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 + β 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: N −
Revised: N −

Original: [Equation]
Revised: [Equation]

Original: k
Revised: k

Original: [Equation]
Revised: [Equation]

Original: 1 ) 2 b ( k +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: Recall the deﬁnition of d , and x , F ( t ) is minimized at ˆ t
Revised: Recall the deﬁnition of d , and x , F ( t ) is minimized at ˆ t

Original: [Equation]
Revised: [Equation]

Original: B
Revised: B

Original: [Equation]
Revised: [Equation]

Original: 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: 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,

Original: 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
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: +

Original: if ˆ t ≤ c k .
Revised: if ˆ t ≤ c k .