yan's advice

main.tex

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  % Created Date: Thursday, July 11th 2024
  % Author: Zihan
  % -----
- % Last Modified: Sunday, 4th August 2024 11:02:16 am
+ % Last Modified: Sunday, 4th August 2024 7:41:27 pm
  % Modified By: the developer formerly known as Zihan at <[email protected]>
  % -----
  % HISTORY:

sections/conclusion.tex

@@ -13,4 +13,4 @@
 
 \section{Conclusion}
 \label{sec:conclude}
-This paper introduces a novel, scalable co-clustering method for large matrices, addressing the computational challenges of high-dimensional data analysis. Our method first partitions large matrices into smaller, parallel-processed submatrices, significantly reducing processing time. Next, a hierarchical co-cluster merging algorithm integrates the submatrix results, ensuring accurate and consistent final co-clustering. Extensive evaluations demonstrate that our method outperforms existing solutions in handling large-scale datasets, proving its effectiveness, efficiency, and scalability. This work sets a new benchmark for future research in scalable data analysis technologies.
+This paper introduces a novel, scalable co-clustering method for large matrices, addressing the computational challenges of high-dimensional data analysis. Our method first partitions large matrices into smaller, parallel-processed submatrices, significantly reducing processing time. Next, a hierarchical co-cluster merging algorithm integrates the submatrix results, ensuring accurate and consistent final co-clustering. Extensive evaluations demonstrate that our method outperforms existing solutions in handling large-scale datasets, proving its effectiveness, efficiency, and scalability.

sections/experiment.tex

@@ -13,7 +13,7 @@
 
 \section{Experimental Evaluation}
 \label{sec:experiment}
-\subsection{Experimental Setup}
+\subsection{Experiment Setup}
 
 \textbf{Datasets.}
 The experiments were conducted using three distinct datasets to demonstrate the versatility and robustness of our method:

sections/introduction.tex

@@ -12,7 +12,7 @@
 %%%
 
 \section{Introduction}
-Artificial Intelligence is a rapidly advancing technology facilitating complex data analysis, pattern recognition, and decision-making processes. Clustering, a fundamental unsupervised learning technique, groups data points based on shared features, aiding in interpreting complex data structures. However, traditional clustering algorithms \cite{zhang2023AdaptiveGraphConvolution, wu2023EffectiveClusteringStructured} tend to treat all features of data uniformly and solely cluster either rows (samples) or columns (features),  as shown in Figure \ref{fig:cluster}. They oversimplified interpretations and overlooked critical context-specific relationships within the data, especially when dealing with large, high-dimensional datasets \cite{chen2023FastFlexibleBipartite, zhao2023MultiviewCoclusteringMultisimilarity, kumar2023CoclusteringBasedMethods}.
+Artificial Intelligence is a rapidly advancing technology facilitating complex data analysis, pattern recognition, and decision-making processes. Clustering, a fundamental unsupervised learning technique, groups data points based on shared features, aiding in interpreting complex data structures. However, traditional clustering algorithms \cite{zhang2023AdaptiveGraphConvolution, wu2023EffectiveClusteringStructured} treat all features of data uniformly and solely cluster either rows (samples) or columns (features),  as shown in Figure \ref{fig:cluster}. They oversimplified interpretations and overlooked critical context-specific relationships within the data, especially when dealing with large, high-dimensional datasets \cite{chen2023FastFlexibleBipartite, zhao2023MultiviewCoclusteringMultisimilarity, kumar2023CoclusteringBasedMethods}.
 
 \textit{Co-clustering} \cite{kluger2003SpectralBiclusteringMicroarray, yan2017CoclusteringMultidimensionalBig} is a technique that groups rows (samples) and columns (features) simultaneously, as shown in Figure \ref{fig:cocluster}. It can reveal complex correlations between two different data types and is transformative in scenarios where the relationships between rows and columns are as important as the individual entities themselves. For example, in bioinformatics, co-clustering could identify gene-related patterns leading to biological insights by concurrently analyzing genes and conditions \cite{higham2007SpectralClusteringIts, kluger2003SpectralBiclusteringMicroarray, zhao2012BiclusteringAnalysisPattern}. In recommendation systems, co-clustering can simultaneously discover more fine-grained relationships between users and projects \cite{dhillon2007WeightedGraphCuts, chen2023ParallelNonNegativeMatrix}. Co-clustering extends traditional clustering methods, enhancing accuracy in pattern detection and broadening the scope of analyses.
 
@@ -39,7 +39,7 @@ \section{Introduction}
 \begin{itemize}
     \item{\textbf{High Computational Complexity.}} Co-clustering analyzes relationships both within and across the rows and columns of a dataset simultaneously. This dual-focus analysis requires evaluating a vast number of potential relationships, particularly as the dimensions of the data increase. The complexity can grow exponentially with the size of the data because the algorithm must process every possible combination of rows and columns to identify meaningful clusters \cite{hansen2011NonparametricCoclusteringLarge}.
     \item{\textbf{Significant Communication Overhead.}} Even when methods such as data partitioning are used to handle large-scale data, each partition may independently analyze a subset of the data. However, to optimize the clustering results globally, these partitions need to exchange intermediate results frequently. This requirement is inherent to iterative optimization techniques used in co-clustering, where each iteration aims to refine the clusters based on new data insights, necessitating continuous updates across the network. Such extensive communication can become a bottleneck, significantly slowing down the overall processing speed.
-    \item{\textbf{Dependency on Sparse Matrices.}} Many traditional co-clustering algorithms are designed to perform best with sparse matrices \cite{pan2008CRDFastCoclusteringa}. However, in many real-world applications, data matrices are often dense, meaning most elements are non-zero. Such scenarios present a significant challenge for standard co-clustering algorithms, as they must handle a larger volume of data without the computational shortcuts available with sparse matrices.
+    \item{\textbf{Dependency on Sparse Matrices.}} Several traditional co-clustering algorithms are designed to perform best with sparse matrices \cite{pan2008CRDFastCoclusteringa}. However, in many real-world applications, data matrices are often dense, meaning most elements are non-zero. Such scenarios present a significant challenge for standard co-clustering algorithms, as they must handle a larger volume of data without the computational shortcuts available with sparse matrices.
 \end{itemize}
 
 To address the inherent challenges associated with existing co-clustering methods, we propose a novel and scalable Adaptive Hierarchical Partitioning and Merging for Scalable Co-Clustering (\textbf{AHPM}) framework designed for large-scale datasets. First,  we propose a large matrix partitioning algorithm that divides the original data matrix into smaller submatrices. This partitioning facilitates parallel processing of co-clustering tasks across submatrices, significantly reducing both processing time and computational and storage demands for each processing unit. We also design a probabilistic model to determine the optimal number and configuration of these submatrices to ensure comprehensive data coverage.
@@ -51,7 +51,7 @@ \section{Introduction}
           We propose a novel matrix partitioning algorithm that enables parallel co-clustering by dividing a large matrix into optimally configured submatrices. This design is supported by a probabilistic model that calculates the optimal number and order of submatrices, balancing computational efficiency with the detection of relevant co-clusters.
     \item \textbf{Hierarchical Co-cluster Merging Algorithm:}
           We design a hierarchical co-cluster merging algorithm that combines co-clusters from submatrices, ensuring the completion of the co-clustering process within a pre-fixed number of iterations. This algorithm significantly enhances the robustness and reliability of the co-clustering process, effectively addressing model uncertainty.
-    \item \textbf{Experimental valuation:}
+    \item \textbf{Experimental Valuation:}
           We evaluate the effectiveness and efficiency of our method across a wide range of scenarios with large, complex data. Experimental results show an approximate 83\% decrease for dense matrices and up to 30\% for sparse matrices.
 \end{enumerate}
 

sections/method.tex

@@ -7,7 +7,7 @@
 \section{Mathematical Formulation and Problem Statement}\label{sec:formula}
 
 \subsection{Mathematical Formulation of Co-clustering}
-Co-clustering groups rows and columns of a data matrix $\mathbf{A} \in \mathbb{R}^{M \times N}$, where $M$ is the number of features and $N$ is the number of samples. Each element $a_{ij}$ represents the relationship between the $i$-th feature and the $j$-th sample. The goal is to partition $\mathbf{A}$ into $k$ row clusters and $d$ column clusters, creating $k \times d$ homogeneous submatrices $\mathbf{A}_{I, J}$.
+Co-clustering groups rows and columns of a data matrix $\mathbf{A} \in \mathbb{R}^{M \times N}$, where $M$ is the number of features and $N$ is the number of samples. Each element $a_{ij}$ represents the $i$-th feature of the $j$-th sample. The goal is to partition $\mathbf{A}$ into $k$ row clusters and $d$ column clusters, creating $k \times d$ homogeneous submatrices $\mathbf{A}_{I, J}$.
 
 When optimally reordered, $\mathbf{A}$ forms a block-diagonal structure where each block is a co-cluster with high internal similarity. Row and column labels are \( u \in \{1,\dots,k\}^M \) and \( v \in \{1,\dots,d\}^N \). Indicator matrices \( R \in \mathbb{R}^{M \times k} \) and \( C \in \mathbb{R}^{N \times d} \) assign rows and columns to clusters, ensuring unique assignments.
 
@@ -24,7 +24,6 @@ \subsection{Problem Statement}
         $a_{ij}$               & Element at the $i$-th row and $j$-th column of matrix $\mathbf{A}$.                                                            \\
         $I, J$                 & Indices of rows and columns selected for co-clustering.                                                                        \\
         $\mathbf{A}_{I, J}$    & Submatrix containing the rows indexed by $I$ and columns by $J$.                                                               \\
-        $k, d$                 & Number of row clusters and column clusters, respectively.                                                                      \\
         $R, C$                 & Indicator matrices for row and column cluster assignments.                                                                     \\
         $\phi_i, \psi_j$       & Block sizes in rows and columns, respectively.                                                                                 \\
         $s_i^{(k)}, t_j^{(k)}$ & Minimum row and column sizes of co-cluster $C_k$ in block $B_{(i,j)}$.                                                         \\
@@ -58,7 +57,7 @@ \subsection{Overview}
 
 \subsection{Large Matrix Partitioning}
 % Description of the matrix partitioning process and criteria for partitioning.
-The primary challenge in co-clustering large matrices is the risk of losing meaningful co-cluster relationships when the matrix is partitioned into smaller submatrices. To address this, we introduce an optimal partitioning algorithm underpinned by a probabilistic model. This model is meticulously designed to navigate the complexities of partitioning, ensuring that the integrity of co-clusters is maintained even as the matrix is divided. The objective of this algorithm is twofold: to determine the optimal partitioning strategy that minimizes the risk of fragmenting significant co-clusters and to define the appropriate number of repartitioning iterations needed to achieve a desired success rate of co-cluster identification.
+The primary challenge in co-clustering large matrices is the risk of losing co-clusters when the matrix is partitioned into smaller submatrices. To address this, we introduce an optimal partitioning algorithm underpinned by a probabilistic model. This model is meticulously designed to navigate the complexities of partitioning, ensuring that the integrity of co-clusters is maintained even as the matrix is divided. The objective of this algorithm is twofold: to determine the optimal partitioning strategy that minimizes the risk of fragmenting significant co-clusters and to define the appropriate number of repartitioning iterations needed to achieve a desired success rate of co-cluster identification.
 
 \subsubsection{Partitioning and Repartitioning Strategy based on the Probabilistic Model}
 Our probabilistic model serves as the cornerstone of the partitioning algorithm. It evaluates potential partitioning schemes based on their ability to preserve meaningful co-cluster structures within smaller submatrices. The model operates under the premise that each atom-co-cluster (the smallest identifiable co-cluster within a submatrix) can be identified with a probability $p$. This probabilistic model allows us to estimate the likelihood of successfully identifying all relevant co-clusters across the partitioned submatrices.

sections/related_work.tex

@@ -14,7 +14,7 @@
 \section{Related work}
 \label{sec:related_work}
 \subsection{Co-clustering Methods}
-Co-clustering methods, broadly categorized into graph-based and matrix factorization-based approaches, have limitations in handling large datasets. Graph-based methods like Flexible Bipartite Graph Co-clustering (FBGPC) \cite{chen2023FastFlexibleBipartitea} directly apply flexible bipartite graph models. Matrix factorization-based methods, such as Non-negative Matrix Tri-Factorization (NMTF) \cite{long2005CoclusteringBlockValue}, decompose data to cluster samples and features separately. Deep Co-Clustering (DeepCC) \cite{dongkuanxu2019DeepCoClustering}, which integrates deep autoencoders with Gaussian Mixture Models, also faces efficiency challenges with diverse data types and large datasets.
+Co-clustering methods, broadly categorized into graph-based and matrix factorization-based approaches, have limitations in handling large datasets. Graph-based methods like Flexible Bipartite Graph Co-clustering (FBGPC) \cite{chen2023FastFlexibleBipartite} directly apply flexible bipartite graph models. Matrix factorization-based methods, such as Non-negative Matrix Tri-Factorization (NMTF) \cite{long2005CoclusteringBlockValue}, decompose data to cluster samples and features separately. Deep Co-Clustering (DeepCC) \cite{dongkuanxu2019DeepCoClustering}, which integrates deep autoencoders with Gaussian Mixture Models, also faces efficiency challenges with diverse data types and large datasets.
 
 \subsection{Parallelizing Co-clustering}
 Parallel methods are crucial for big data processing. The CoClusterD framework \cite{cheng2015CoClusterDDistributedFramework} uses Alternating Minimization Co-clustering (AMCC) in a distributed environment but struggles with guaranteed convergence. Chen \textit{et al.} \cite{chen2023ParallelNonNegativeMatrix} introduced a parallel non-negative matrix tri-factorization method to accelerate computations but still faces difficulties with very large datasets.