Industry-grade ML models are carefully designed to meet rapidly evolving serving constraints, which requires significant resources for model development. In this paper, we propose MatTA, a framework for training multiple accurate Student models using a novel Teacher-TA-Student recipe. TA models are larger versions of the Student models with higher capacity, and thus allow Student models to better relate to the Teacher model and also bring in more domain-specific expertise. Furthermore, multiple accurate Student models can be extracted from the TA model. Therefore, despite only one training run, our methodology provides multiple servable options to trade off accuracy for lower serving cost. We demonstrate the proposed method, MatTA, on proprietary datasets and models. Its practical efficacy is underscored by live A/B tests within a production ML system, demonstrating 20% improvement on a key metric. We also demonstrate our method on GPT-2 Medium, a public model, and achieve relative improvements of over 24% on SAT Math and over 10% on the LAMBADA benchmark.

Approximate K-Nearest Neighbor Search (AKNNS) has now become ubiquitous in modern applications, such as a fast search procedure with two-tower deep learning models. Graph-based methods for AKNNS in particular have received great attention due to their superior performance. These methods rely on greedy graph search to traverse the data points as embedding vectors in a database. Under this greedy search scheme, we make a key observation: many distance computations do not influence search updates so that these computations can be approximated without hurting performance. As a result, we propose FINGER, a fast inference method for efficient graph search in AKNNS. FINGER approximates the distance function by estimating angles between neighboring residual vectors. The approximated distance can be used to bypass unnecessary computations for faster searches. Empirically, when it comes to speeding up the inference of HNSW, which is one of the most popular graph-based AKNNS methods, FINGER significantly outperforms existing acceleration approaches and conventional libraries by 20 to 60 across different benchmark datasets.

Conventional methods for query autocompletion aim to predict which completed query a user will select from a list. A shortcoming of this approach is that users often do not know which query will provide the best retrieval performance on the current information retrieval system, meaning that any query autocompletion methods trained to mimic user behavior can lead to suboptimal query suggestions. To overcome this limitation, we propose a new approach that explicitly optimizes the query suggestions for downstream retrieval performance. We formulate this as a problem of ranking a set of rankings, where each query suggestion is represented by the downstream item ranking it produces. We then present a learning method that ranks query suggestions by the quality of their item rankings. The algorithm is based on a counterfactual learning approach that is able to leverage feedback on the items (e.g., clicks, purchases) to evaluate query suggestions through an unbiased estimator, thus avoiding the assumption that users write or select optimal queries. We establish theoretical support for the proposed approach and provide learning-theoretic guarantees. We also present empirical results on publicly available datasets, and demonstrate real-world applicability using data from an online shopping store.

Tree-based models underpin many modern semantic search engines and recommender systems due to their sub-linear inference times. In industrial applications, these models operate at extreme scales, where every bit of performance is critical. Memory constraints at extreme scales also require that models be sparse, hence tree-based models are often back-ended by sparse matrix algebra routines. However, there are currently no sparse matrix techniques specifically designed for the sparsity structure one encounters in tree-based models for extreme multi-label ranking/classification (XMR/XMC) problems. To address this issue, we present the masked sparse chunk multiplication (MSCM) technique, a sparse matrix technique specifically tailored to XMR trees. MSCM is easy to implement, embarrassingly parallelizable, and offers a significant performance boost to any existing tree inference pipeline at no cost. We perform a comprehensive study of MSCM applied to several different sparse inference schemes and benchmark our methods on a general purpose extreme multi-label ranking framework. We observe that MSCM gives consistently dramatic speedups across both the online and batch inference settings, single- and multi-threaded settings, and on many different tree models and datasets. To demonstrate its utility in industrial applications, we apply MSCM to an enterprise-scale semantic product search problem with 100 million products and achieve sub-millisecond latency of 0.88 ms per query on a single thread — an 8x reduction in latency over vanilla inference techniques. The MSCM technique requires absolutely no sacrifices to model accuracy as it gives exactly the same results as standard sparse matrix techniques. Therefore, we believe that MSCM will enable users of XMR trees to save a substantial amount of compute resources in their inference pipelines at very little cost. Our code is publicly available at https://github.com/amzn/pecos , as well as our complete benchmarks and code for reproduction at https://github.com/UniqueUpToPermutation/pecos/tree/benchmark.

Extreme multi-label classification (XMC) aims to learn a model that can tag data points with a subset of relevant labels from an extremely large label set. Real world e-commerce applications like personalized recommendations and product advertising can be formulated as XMC problems, where the objective is to predict for a user a small subset of items from a catalog of several million products. For such applications, a common approach is to organize these labels into a tree, enabling training and inference times that are logarithmic in the number of labels [23]. While training a model once a label tree is available is well studied, designing the structure of the tree is a difficult task that is not yet well understood, and can dramatically impact both model latency and statistical performance. Existing approaches to tree construction either optimize exclusively for statistical performance or optimize exclusively for latency. We propose an efficient information theory inspired algorithm to construct intermediate operating points that trade off between the benefits of both, which was not previously possible. We corroborate our theoretical analysis with numerical results, showing that on the Wiki-500K [4] benchmark dataset our method can reduce a proxy for expected latency by up to 28% while maintaining the same accuracy as Parabel [23]. On several datasets derived from e-commerce customer logs, our modified label tree is able to improve this expected latency metric by up to 20% while maintaining the same accuracy. Finally, we discuss challenges in realizing these latency improvements in deployed models.

We consider the problem of semantic matching in product search: given a customer query, retrieve all semantically related products from a huge catalog of size 100 million, or more. Because of large catalog spaces and real-time latency constraints, semantic matching algorithms not only desire high recall but also need to have low latency. Conventional lexical matching approaches (e.g., Okapi-BM25) exploit inverted indices to achieve fast inference time, but fail to capture behavioral signals between queries and products. In contrast, embedding-based models learn semantic representations from customer behavior data, but the performance is often limited by shallow neural encoders due to latency constraints. Semantic product search can be viewed as an eXtreme Multi-label Classification (XMC) problem, where customer queries are input instances and products are output labels. In this paper, we aim to improve semantic product search by using tree-based XMC models where inference time complexity is logarithmic in the number of products. We consider hierarchical linear models with n-gram features for fast real-time inference. Quantitatively, our method maintains a low latency of 1.25 milliseconds per query and achieves a 65% improvement of [email protected] (60.9% v.s. 36.8%) over a competing embedding-based DSSM model. Our model is robust to weight pruning with varying thresholds, which can flexibly meet different system requirements for online deployments. Qualitatively, our method can retrieve products that are complementary to existing product search system and add diversity to the match set.

Query auto-completion (QAC) is a fundamental feature in search engines where the task is to suggest plausible completions of a prefix typed in the search bar. Previous queries in the user session can provide useful context for the user's intent and can be leveraged to suggest auto-completions that are more relevant while adhering to the user's prefix. Such session-aware QACs can be generated by recent sequence-to-sequence deep learning models; however, these generative approaches often do not meet the stringent latency requirements of responding to each user keystroke. Moreover, these generative approaches pose the risk of showing nonsensical queries. One can pre-compute a relatively small subset of relevant queries for common prefixes and rank them based on the context. However, such an approach fails when no relevant queries for the current context are present in the pre-computed set.

We consider the extreme multi-label text classification (XMC) problem: given an input text, return the most relevant labels from a large label collection. For example, the input text could be a product description on Amazon.com and the labels could be product categories. XMC is an important yet challenging problem in the NLP community. Recently, deep pretrained transformer models have achieved state-of-the-art performance on many NLP tasks including sentence classification, albeit with small label sets. However, naively applying deep transformer models to the XMC problem leads to sub-optimal performance due to the large output space and the label sparsity issue. In this paper, we propose X-Transformer, the first scalable approach to fine-tuning deep transformer models for the XMC problem. The proposed method achieves new state-of-the-art results on four XMC benchmark datasets. In particular, on a Wiki dataset with around 0.5 million labels, the [email protected] of X-Transformer is 77.28%, a substantial improvement over state-of-the-art XMC approaches Parabel (linear) and AttentionXML (neural), which achieve 68.70% and 76.95% [email protected], respectively. We further apply X-Transformer to a product2query dataset from Amazon and gained 10.7% relative improvement on [email protected] over Parabel.

The goal of co-clustering is to simultaneously identify a clustering of the rows as well as the columns of a two dimensional data matrix. Most existing co-clustering algorithms are designed to find pairwise disjoint and exhaustive co-clusters. However, many real-world datasets might contain not only a large overlap between co-clusters but also outliers which should not belong to any co-cluster. We formulate the problem of Non-Exhaustive, Overlapping Co-Clustering where both of the row and column clusters are allowed to overlap with each other and the outliers for each dimension of the data matrix are not assigned to any cluster. To solve this problem, we propose an intuitive objective function, and develop an efficient iterative algorithm which we call the NEO-CC algorithm. We theoretically show that the NEO-CC algorithm monotonically decreases the proposed objective function. Experimental results show that the NEO-CC algorithm is able to effectively capture the underlying co-clustering structure of real-world data, and thus outperforms state-of-the-art clustering and co-clustering methods.

Nonlinear kernel machines often yield superior predictive performance on various tasks; however, they suffer from severe computational challenges. In this paper, we show how to overcome the important challenge of speeding up kernel machines using multiple computers. In particular, we develop a parallel block minimization framework, and demonstrate its good scalability in solving nonlinear kernel SVM and logistic regression. Our framework proceeds by dividing the problem into smaller subproblems by forming a block-diagonal approximation of the Hessian matrix. The subproblems are then solved approximately in parallel. After that, a communication efficient line search procedure is developed to ensure sufficient reduction of the objective function value by exploiting the problem structure of kernel machines. We prove global linear convergence rate of the proposed method with a wide class of subproblem solvers, and our analysis covers strongly convex and some non-strongly convex functions. We apply our algorithm to solve large-scale kernel SVM problems on distributed systems, and show a significant improvement over existing parallel solvers. As an example, on the covtype dataset with half-a-million samples, our algorithm can obtain an approximate solution with 96% accuracy in 20 seconds using 32 machines, while all the other parallel kernel SVM solvers require more than 2000 seconds to achieve a solution with 95% accuracy. Moreover, our algorithm is the first distributed kernel SVM solver that can scale to massive data sets. On the KDDB dataset (20 million samples and 30 million features), our parallel solver can compute the kernel SVM solution within half an hour using 32 machines with 640 cores in total, while existing solvers can not scale to this dataset.

Extreme Classification comprises multi-class or multi-label prediction where there is a large number of classes, and is increasingly relevant to many real-world applications such as text and image tagging. In this setting, standard classification methods, with complexity linear in the number of classes, become intractable, while enforcing structural constraints among classes (such as low-rank or tree-structure) to reduce complexity often sacrifices accuracy for efficiency. The recent PD-Sparse method addresses this via an algorithm that is sub-linear in the number of variables, by exploiting primal-dual sparsity inherent in a specific loss function, namely the max-margin loss. In this work, we extend PD-Sparse to be efficiently parallelized in large-scale distributed settings. By introducing separable loss functions, we can scale out the training, with network communication and space efficiency comparable to those in one-versus-all approaches while maintaining an overall complexity sub-linear in the number of classes. On several large-scale benchmarks our proposed method achieves accuracy competitive to the state-of-the-art while reducing the training time from days to tens of minutes compared with existing parallel or sparse methods on a cluster of 100 cores.

Matrix completion (MC) with additional information has found wide applicability in several machine learning applications. Among algorithms for solving such problems, Inductive Matrix Completion(IMC) has drawn a considerable amount of attention, not only for its well established theoretical guarantees but also for its superior performance in various real-world applications. However, IMC based methods usually place very strong constraints on the quality of the features(side information) to ensure accurate recovery, which might not be met in practice. In this paper, we propose Goal-directed Inductive Matrix Completion(GIMC) to learn a nonlinear mapping of the features so that they satisfy the required properties. A key distinction between GIMC and IMC is that the feature mapping is learnt in a supervised manner, deviating from the traditional approach of unsupervised feature learning followed by model training. We establish the superiority of our method on several popular machine learning applications including multi-label learning, multi-class classification, and semi-supervised clustering.

Clustering is one of the most fundamental and important tasks in data mining. Traditional clustering algorithms, such as K-means, assign every data point to exactly one cluster. However, in real-world datasets, the clusters may overlap with each other. Furthermore, often, there are outliers that should not belong to any cluster. We recently proposed the NEO-K-Means (Non-Exhaustive, Overlapping K-Means) objective as a way to address both issues in an integrated fashion. Optimizing this discrete objective is NP-hard, and even though there is a convex relaxation of the objective, straightforward convex optimization approaches are too expensive for large datasets. A practical alternative is to use a low-rank factorization of the solution matrix in the convex formulation. The resulting optimization problem is non-convex, and we can locally optimize the objective function using an augmented Lagrangian method. In this paper, we consider two fast multiplier methods to accelerate the convergence of an augmented Lagrangian scheme: a proximal method of multipliers and an alternating direction method of multipliers (ADMM). For the proximal augmented Lagrangian or proximal method of multipliers, we show a convergence result for the non-convex case with bound-constrained subproblems. These methods are up to 13 times faster—with no change in quality—compared with a standard augmented Lagrangian method on problems with over 10,000 variables and bring runtimes down from over an hour to around 5 minutes.

Community detection is an important task in network analysis. A community (also referred to as a cluster) is a set of cohesive vertices that have more connections inside the set than outside. In many social and information networks, these communities naturally overlap. For instance, in a social network, each vertex in a graph corresponds to an individual who usually participates in multiple communities. In this paper, we propose an efficient overlapping community detection algorithm using a seed expansion approach. The key idea of our algorithm is to find good seeds, and then greedily expand these seeds based on a community metric. Within this seed expansion method, we investigate the problem of how to determine good seed nodes in a graph. In particular, we develop new seeding strategies for a personalized PageRank clustering scheme that optimizes the conductance community score. An important step in our method is the neighborhood inflation step where seeds are modified to represent their entire vertex neighborhood. Experimental results show that our seed expansion algorithm outperforms other state-of-the-art overlapping community detection methods in terms of producing cohesive clusters and identifying ground-truth communities. We also show that our new seeding strategies are better than existing strategies, and are thus effective in finding good overlapping communities in real-world networks.

Popular microblogging sites such as Tumblr have attracted hundreds of millions of users as a content sharing platform, where users can create rich content in the form of posts that are shared with other users who follow them. Due to the sheer amount of posts created on such services, an important task is to make quality recommendations of blogs for users to follow. Apart from traditional recommender system settings where the follower graph is the main data source, additional side-information of users and blogs such as user activity (e.g., like and reblog) and rich content (e.g., text and images) are also available to be exploited for enhanced recommendation performance. In this paper, we propose a novel boosted inductive matrix completion method (BIMC) for blog recommendation. BIMC is an additive low-rank model for user-blog preferences consisting of two components; one component captures the low-rank structure of follow relationships and the other captures the latent structure using side-information. Our model formulation combines the power of the recently proposed inductive matrix completion (IMC) model (for side-information) together with a standard matrix completion (MC) model (for low-rank structure). Furthermore, we utilize recently developed deep learning techniques to obtain semantically rich feature representations of text and images that are incorporated in BIMC. Experiments on a large-scale real-world dataset from Tumblr illustrate the effectiveness of the proposed BIMC method.

Forecasting accurately is essential to successful inventory planning in retail. Unfortunately, there is not always enough historical data to forecast items individually- this is particularly true in e-commerce where there is a long tail of low selling items, and items are introduced and phased out quite frequently, unlike physical stores. In such scenarios, it is preferable to forecast items in well-designed groups of similar items, so that data for different items can be pooled together to fit a single model. In this paper, we first discuss the desiderata for such a grouping and how it differs from the traditional clustering problem. We then describe our approach which is a scalable local search heuristic that can naturally handle the constraints required in this setting, besides being capable of producing solutions competitive with well-known clustering algorithms. We also address the complementary problem of estimating similarity, particularly in the case of new items which have no past sales. Our solution is to regress the sales profile of items against their semantic features, so that given just the semantic features of a new item we can predict its relation to other items, in terms of as yet unobserved sales. Our experiments demonstrate both the scalability of our approach and implications for forecast accuracy.

Clustering is one of the most fundamental tasks in data mining. To analyze complex real-world data emerging in many data-centric applications, the problem of non-exhaustive, overlapping clustering has been studied where the goal is to find overlapping clusters and also detect outliers simultaneously. We propose a novel convex semidefinite program (SDP) as a relaxation of the non-exhaustive, overlapping clustering problem. Although the SDP formulation enjoys attractive theoretical properties with respect to global optimization, it is computationally intractable for large problem sizes. As an alternative, we optimize a low-rank factorization of the solution. The resulting problem is non-convex, but has a smaller number of solution variables. We construct an optimization solver using an augmented Lagrangian methodology that enables us to deal with problems with tens of thousands of data points. The new solver provides more accurate and reliable answers than other approaches. By exploiting the connection between graph clustering objective functions and a kernel k-means objective, our new low-rank solver can also compute overlapping communities of social networks with state-of-the-art accuracy.

Traditional clustering algorithms, such as k-means, output a clustering that is disjoint and exhaustive, that is, every single data point is assigned to exactly one cluster. However, in real datasets, clusters can overlap and there are often outliers that do not belong to any cluster. This is a well recognized problem that has received much attention in the past, and several algorithms, such as fuzzy k-means have been proposed for overlapping clustering. However, most existing algorithms address either overlap or outlier detection and do not tackle the problem in a unified way. In this paper, we propose a simple and intuitive objective function that captures the issues of overlap and non-exhaustiveness in a unified manner. Our objective function can be viewed as a reformulation of the traditional k-means objective, with easy-to-understand parameters that capture the degrees of overlap and non-exhaustiveness. By studying the objective, we are able to obtain a simple iterative algorithm which we call NEO-K-Means (Non-Exhaustive Overlapping K-Means). Furthermore, by considering an extension to weighted kernel k-means, we can tackle the case of non-exhaustive and overlapping graph clustering. This extension allows us to apply our NEO-K-Means algorithm to the community detection problem, which is an important task in network analysis. Our experimental results show that the new objective and algorithm are effective in finding ground-truth clusterings that have varied overlap and non-exhaustiveness; for the case of graphs, we show that our algorithm outperforms state-of-the-art overlapping community detection methods.

Learning meaningful topic models with massive document collections which contain millions of documents and billions of tokens is challenging because of two reasons. First, one needs to deal with a large number of topics (typically on the order of thousands). Second, one needs a scalable and efficient way of distributing the computation across multiple machines. In this paper, we present a novel algorithm F+Nomad LDA which simultaneously tackles both these problems. In order to handle large number of topics we use an appropriately modified Fenwick tree. This data structure allows us to sample from a multinomial distribution over T items in O(log T) time. Moreover, when topic counts change the data structure can be updated in O(log T) time. In order to distribute the computation across multiple processors, we present a novel asynchronous framework inspired by the Nomad algorithm of Yun et al, 2014. We show that F+Nomad LDA significantly outperforms recent state-of-the-art topic modeling approaches on massive problems which involve millions of documents, billions of words, and thousands of topics.

We develop an efficient parallel distributed algorithm for matrix completion, named NOMAD (Non-locking, stOchastic Multi-machine algorithm for Asynchronous and Decentralized matrix completion). NOMAD is a decentralized algorithm with non-blocking communication between processors. One of the key features of NOMAD is that the ownership of a variable is asynchronously transferred between processors in a decentralized fashion. As a consequence it is a lock-free parallel algorithm. In spite of being asynchronous, the variable updates of NOMAD are serializable, that is, there is an equivalent update ordering in a serial implementation. NOMAD outperforms synchronous algorithms which require explicit bulk synchronization after every iteration: our extensive empirical evaluation shows that not only does our algorithm perform well in distributed setting on commodity hardware, but also outperforms state-of-the-art algorithms on a HPC cluster both in multi-core and distributed memory settings.

Community detection is an important task in network analysis. A community (also referred to as a cluster) is a set of cohesive vertices that have more connections inside the set than outside. In many social and information networks, these communities naturally overlap. For instance, in a social network, each vertex in a graph corresponds to an individual who usually participates in multiple communities. One of the most successful techniques for finding overlapping communities is based on local optimization and expansion of a community metric around a seed set of vertices. In this paper, we propose an efficient overlapping community detection algorithm using a seed set expansion approach. In particular, we develop new seeding strategies for a personalized PageRank scheme that optimizes the conductance community score. The key idea of our algorithm is to find good seeds, and then expand these seed sets using the personalized PageRank clustering procedure. Experimental results show that this seed set expansion approach outperforms other state-of-the-art overlapping community detection methods. We also show that our new seeding strategies are better than previous strategies, and are thus effective in finding good overlapping clusters in a graph.

Stochastic block models provide a rich, probabilistic framework for modeling relational data by expressing the objects being modeled in terms of a latent vector representation. This representation can be a latent indicator vector denoting the cluster membership (hard clustering), a vector of cluster membership probabilities (soft clustering), or more generally a real-valued vector (latent space representation). Recently, a new class of overlapping stochastic block models has been proposed where the idea is to allow the objects to have hard memberships in multiple clusters (in form of a latent binary vector). This aspect captures the properties of many real-world networks in domains such as biology and social networks where objects can simultaneously have memberships in multiple clusters owing to the multiple roles they may have. In this paper, we improve upon this model in three key ways: (1) we extend the overlapping stochastic block model to the bipartite graph case which enables us to simultaneously learn the overlapping clustering of two different sets of objects in the graph, the unipartite graph is just a special case of our model, (2) we allow objects (in either set) to not have membership in any cluster by using a relevant object selection mechanism, and (3) we make use of additionally available object features (or a kernel matrix of pair wise object similarities) to further improve the overlapping clustering performance. We do this by explicitly encouraging similar objects to have similar cluster membership vectors. Moreover, using nonparametric Bayesian prior distributions on the key model parameters, we side-step the model selection issues such as selecting the number of clusters a priori. Our model is quite general and can be applied for both overlapping clustering and link prediction tasks in unipartite and bipartite networks (directed/undirected), or for overlapping co-clustering of general binary-valued data. Experiments on synthetic and real-world datasets from biology and social networks demonstrate that our model outperforms several state-of-the-art methods.

The application a smart phone user will launch next intuitively depends on the sequence of apps used recently. More generally, when users interact with systems such as shopping websites or online radio, they click on items that are of interest in the current context. We call the sequence of clicks made in the current session interactional context. It is desirable for a recommender system to use the context set by the user to update recommendations. Most current context-aware recommender systems focus on a relatively less dynamic representational context defined by attributes such as season, location and tastes. In this paper, we study the problem of collaborative filtering with interactional context, where the goal is to make personalized and dynamic recommendations to a user engaged in a session. To this end, we propose the methodname algorithm that works in two stages. First, users are clustered by their transition behavior (one-step Markov transition probabilities between items), and cluster-level Markov models are computed. Then personalized PageRank is computed for a given user on the corresponding cluster Markov graph, with a personalization vector derived from the current context. We give an interpretation of the second stage of the algorithm as adding an appropriate context bias, in addition to click bias (or rating bias), to a classical neighborhood-based collaborative filtering model, where the neighborhood is determined from a Markov graph. Experimental results on two real-life datasets demonstrate the superior performance of our algorithm, where we achieve at least 20% (up to 37%) improvement over competitive methods in the recall level at top-20.

We consider the general $k$-way clustering problem in signed social networks where relationships between entities can be either positive or negative. Motivated by social balance theory, the clustering problem in signed networks aims to find mutually antagonistic groups such that entities within the same group are friends with each other. A recent method proposed in [13] extended the spectral clustering algorithm to the signed network setting by considering the signed graph Laplacian. This has been shown to be equivalent to finding clusters that minimize the 2-way signed ratio cut. In this paper, we show that there is a fundamental weakness when we directly extend the signed Laplacian to the k-way clustering problem. To overcome this weakness, we formulate new k-way objectives for signed networks. In particular, we propose a criterion that is analogous to the normalized cut, called balance normalized cut, which is not only theoretically sound but also experimentally effective in k-way clustering. In addition, we prove that these objectives are equivalent to weighted kernel k-means objectives by choosing an appropriate kernel matrix. Employing this equivalence, we develop a multilevel clustering framework for signed networks. In this framework, we coarsen the graph level by level and refine the clustering results at each level via a k-means based algorithm so that the signed clustering objectives are optimized. This approach gives good quality clustering results, and is also highly efficient and scalable. In experiments, we see that our multilevel approach is competitive to other state-of-the-art methods, while it is much faster and more scalable. In particular, the largest graph we have considered in our experiments contains 1 million nodes and 100 million edges --- this graph can be clustered in less than four hundred seconds using our algorithm.

The automated analysis of social networks has become an important problem due to the proliferation of social networks, such as LiveJournal, Flickr and Facebook. The scale of these social networks is massive and continues to grow rapidly. An important problem in social network analysis is proximity estimation that infers the closeness of different users. Link prediction, in turn, is an important application of proximity estimation. However, many methods for computing proximity measures have high computational complexity and are thus prohibitive for large-scale link prediction problems. One way to address this problem is to estimate proximity measures via low-rank approximation. However, a single low-rank approximation may not be sufficient to represent the behavior of the entire network. In this paper, we propose Multi-Scale Link Prediction (MSLP), a framework for link prediction, which can handle massive networks. The basic idea of MSLP is to construct low-rank approximations of the network at multiple scales in an efficient manner. To achieve this, we propose a fast tree-structured approximation algorithm.

Clustering of social networks is an important task for their analysis, however, most existing algorithms do not scale to the massive size of todayâs social networks. A popular class of graph clustering algorithms for large-scale networks, such as PMetis, KMetis and Graclus, is based on a multilevel framework. Generally, these multilevel algorithms work reasonably well on networks with a few million vertices. However, when the network size increases to the scale of 10 million vertices or greater, the performance of these algorithms rapidly degrades. Furthermore, an inherent property of social networks, the power law degree distribution, makes these algorithms infeasible to apply to large-scale social networks. In this paper, we propose a scalable and memory-efficient clustering algorithm for large-scale social networks. We name our algorithm GEM, by mixing two key concepts of the algorithm, Graph Extraction and weighted kernel k-Means. GEM efficiently extracts a good skeleton graph from the original graph, and propagates the clustering result of the extracted graph to the rest of the network. Experimental results show that GEM produces clusters of quality comparable to or better than existing state-of-the-art graph clustering algorithms, while it is much faster and consumes much less memory. Furthermore, the parallel implementation of GEM, called PGEM, not only produces higher quality of clusters but also achieves much better scalability than most current parallel graph clustering algorithms.

Matrix factorization, when the matrix has missing values, has become one of the leading techniques for recommender systems. To handle web-scale datasets with millions of users and billions of ratings, scalability becomes an important issue. Alternating Least Squares (ALS) and Stochastic Gradient Descent (SGD) are two popular approaches to compute matrix factorization. There has been a recent flurry of activity to parallelize these algorithms. However, due to the cubic time complexity in the target rank, ALS is not scalable to large-scale datasets. On the other hand, SGD conducts efficient updates but usually suffers from slow convergence that is sensitive to the parameters. Coordinate descent, a classical optimization approach, has been used for many other large-scale problems, but its application to matrix factorization for recommender systems has not been explored thoroughly. In this paper, we show that coordinate descent based methods have a more efficient update rule compared to ALS, and are faster and have more stable convergence than SGD. We study different update sequences and propose the CCD++ algorithm, which updatesrank-one factors one by one. In addition, CCD++ can be easily parallelized on both multi-core and distributed systems. We empirically show that CCD++ is much faster than ALS and SGD in both settings. As an example, on a synthetic dataset with 2 billion ratings, CCD++ is 4 times faster than both SGD and ALS using a distributed system with 20 machines.

Trust networks, where people leave trust and distrust feedback, are becoming increasingly common. These networks may be regarded as signed graphs, where a positive edge weight captures the degree of trust while a negative edge weight captures the degree of distrust. Analysis of such signed networks has become an increasingly important research topic. One important analysis task is that of sign inference, i.e., infer unknown (or future) trust or distrust relationships given a partially observed signed network. Most state-of-the-art approaches consider the notion of structural balance in signed networks, building inference algorithms based on information about links, triads, and cycles in the network. In this paper, we first show that the notion of weak structural balance in signed networks naturally leads to a global low-rank model for the network. Under such a model, the sign inference problem can be formulated as a low-rank matrix completion problem. We show that we can perfectly recover missing relationships, under certain conditions, using state-of-the-art matrix completion algorithms. We also propose the use of a low-rank matrix factorization approach with generalized loss functions as a practical method for sign inference - this approach yields high accuracy while being scalable to large signed networks, for instance, we show that this analysis can be performed on a synthetic graph with 1.1 million nodes and 120 million edges in 10 minutes. We further show that the low-rank model can be used for other analysis tasks on signed networks, such as user segmentation through signed graph clustering, with theoretical guarantees. Experiments on synthetic as well as real data show that our low rank model substantially improves accuracy of sign inference as well as clustering. As an example, on the largest real dataset available to us (Epinions data with 130K nodes and 840K edges), our matrix factorization approach yields 94.6% accuracy on the sign inference task as compared to 90.8% accuracy using a state-of-the-art cycle-based method - moreover, our method runs in 40 seconds as compared to 10,000 seconds for the cycle-based method.

We consider the problem of link prediction in signed networks. Such networks arise on the web in a variety of ways when users can implicitly or explicitly tag their relationship with other users as positive or negative. The signed links thus created reflect social attitudes of the users towards each other in terms of friendship or trust. Our first contribution is to show how any quantitative measure of social imbalance in a network can be used to derive a link prediction algorithm. Our framework allows us to reinterpret some existing algorithms as well as derive new ones. Second, we extend the approach of Leskovec et al. (2010) by presenting a supervised machine learning based link prediction method that uses features derived from longer cycles in the network. The supervised method outperforms all previous approaches on 3 networks drawn from sources such as Epinions, Slashdot and Wikipedia. The supervised approach easily scales to these networks, the largest of which has 132k nodes and 841k edges. Most real-world networks have an overwhelmingly large proportion of positive edges and it is therefore easy to get a high overall accuracy at the cost of a high false positive rate. We see that our supervised method not only achieves good accuracy for sign prediction but is also especially effective in lowering the false positive rate.

Nonnegative Matrix Factorization (NMF) is an effective dimension reduction method for non-negative dyadic data, and has proven to be useful in many areas, such as text mining, bioinformatics and image processing. NMF is usually formulated as a constrained non-convex optimization problem, and many algorithms have been developed for solving it. Recently, a coordinate descent method, called FastHals, has been proposed to solve least squares NMF and is regarded as one of the state-of-the-art techniques for the problem. In this paper, we first show that FastHals has an inefficiency in that it uses a cyclic coordinate descent scheme and thus, performs unneeded descent steps on unimportant variables. We then present a variable selection scheme that uses the gradient of the objective function to arrive at a new coordinate descent method. Our new method is considerably faster in practice and we show that it has theoretical convergence guarantees. Moreover when the solution is sparse, as is often the case in real applications, our new method benefits by selecting important variables to update more often, thus resulting in higher speed. As an example, on a text dataset RCV1, our method is 7 times faster than FastHals, and more than 15 times faster when the sparsity is increased by adding an L1 penalty. We also develop new coordinate descent methods when error in NMF is measured by KL-divergence by applying the Newton method to solve the one-variable sub-problems. Experiments indicate that our algorithm for minimizing the KL-divergence is faster than the Lee & Seung multiplicative rule by a factor of 10 on the CBCL image dataset.

In this paper we present a fast and accurate procedure called clustered low rank matrix approximation for massive graphs. The procedure involves a fast clustering of the graph and then approximates each cluster separately using existing methods, e.g. the singular value decomposition, or stochastic algorithms. The cluster-wise approximations are then extended to approximate the entire graph. This approach has several benefits: (1) important community structure of the graph is preserved due to the clustering; (2) highly accurate low rank approximations are achieved; (3) the procedure is efficient both in terms of computational speed and memory usage; (4) better performance in problems from various applications compared to standard low rank approximation. Further, we generalize stochastic algorithms to the clustered low rank approximation framework and present theoretical bounds for the approximation error. Finally, a set of experiments, using large scale and real-world graphs, show that our methods outperform standard low rank matrix approximation algorithms.

Social network analysis has attracted increasing attention in recent years. In many social networks, besides friendship links among users, the phenomenon of users associating themselves with groups or communities is common. Thus, two networks exist simultaneously: the friendship network among users, and the affiliation network between users and groups. In this article, we tackle the affiliation recommendation problem, where the task is to predict or suggest new affiliations between users and communities, given the current state of the friendship and affiliation networks. More generally, affiliations need not be community affiliations---they can be a user’s taste, so affiliation recommendation algorithms have applications beyond community recommendation. In this article, we show that information from the friendship network can indeed be fruitfully exploited in making affiliation recommendations. Using a simple way of combining these networks, we suggest two models of user-community affinity for the purpose of making affiliation recommendations: one based on graph proximity, and another using latent factors to model users and communities. We explore the affiliation recommendation algorithms suggested by these models and evaluate these algorithms on two real-world networks, Orkut and Youtube. In doing so, we motivate and propose a way of evaluating recommenders, by measuring how good the top 50 recommendations are for the average user, and demonstrate the importance of choosing the right evaluation strategy. The algorithms suggested by the graph proximity model turn out to be the most effective. We also introduce scalable versions of these algorithms, and demonstrate their effectiveness. This use of link prediction techniques for the purpose of affiliation recommendation is, to our knowledge, novel.

Link prediction is a fundamental problem in social network analysis and modern-day commercial applications such as Face book and My space. Most existing research approaches this problem by exploring the topological structure of a social network using only one source of information. However, in many application domains, in addition to the social network of interest, there are a number of auxiliary social networks and/or derived proximity networks available. The contribution of the paper is twofold: (1) a supervised learning framework that can effectively and efficiently learn the dynamics of social networks in the presence of auxiliary networks, (2) a feature design scheme for constructing a rich variety of path-based features using multiple sources, and an effective feature selection strategy based on structured sparsity. Extensive experiments on three real-world collaboration networks show that our model can effectively learn to predict new links using multiple sources, yielding higher prediction accuracy than unsupervised and single-source supervised models.

Social network analysis has attracted increasing attention in recent years. In many social networks, besides friendship links amongst users, the phenomenon of users associating themselves with groups or communities is common. Thus, two networks exist simultaneously: the friendship network among users, and the affiliation network between users and groups. In this paper, we tackle the affiliation recommendation problem, where the task is to predict or suggest new affiliations between users and communities, given the current state of the friendship and affiliation networks. More generally, affiliations need not be community affiliations - they can be a user's taste, so affiliation recommendation algorithms have applications beyond community recommendation. In this paper, we show that information from the friendship network can indeed be fruitfully exploited in making affiliation recommendations. Using a simple way of combining these networks, we suggest two models of user-community affinity for the purpose of making affiliation recommendations: one based on graph proximity, and another using latent factors to model users and communities. We explore the two classes of affiliation recommendation algorithms suggested by these models. We evaluate these algorithms on two real world networks - Orkut and Youtube. In doing so, we motivate and propose a way of evaluating recommenders, by measuring how good the top 50 recommendations are for the average user, and demonstrate the importance of choosing the right evaluation strategy. The algorithms suggested by the graph proximity model turn out to be the most effective and efficient. This use of link prediction techniques for the purpose of affiliation recommendation is, to our knowledge, novel.

In graph-based learning models, entities are often represented as vertices in an undirected graph with weighted edges describing the relationships between entities. In many real-world applications, however, entities are often associated with relations of different types and/or from different sources, which can be well captured by multiple undirected graphs over the same set of vertices. How to exploit such multiple sources of information to make better inferences on entities remains an interesting open problem. In this paper, we focus on the problem of clustering the vertices based on multiple graphs in both unsupervised and semi-supervised settings. As one of our contributions, we propose Linked Matrix Factorization (LMF) as a novel way of fusing information from multiple graph sources. In LMF, each graph is approximated by matrix factorization with a graph-specific factor and a factor common to all graphs, where the common factor provides features for all vertices. Experiments on SIAM journal data show that (1) we can improve the clustering accuracy through fusing multiple sources of information with several models, and (2) LMF yields superior or competitive results compared to other graph-based clustering methods.

In this paper, we propose a novel spatio-temporal model for collaborative filtering applications. Our model is based on low-rank matrix factorization that uses a spatio-temporal filtering approach to estimate user and item factors. The spatial component regularizes the factors by exploiting correlation across users and/or items, modeled as a function of some implicit feedback (e.g., who rated what) and/or some side information (e.g., user demographics, browsing history). In particular, we incorporate correlation in factors through a Markov random field prior in a probabilistic framework, whereby the neighborhood weights are functions of user and item covariates. The temporal component ensures that the user/item factors adapt to process changes that occur through time and is implemented in a state space framework with fast estimation through Kalman filtering. Our spatio-temporal filtering (ST-KF hereafter) approach provides a single joint model to simultaneously incorporate both spatial and temporal structure in ratings and therefore provides an accurate method to predict future ratings. To ensure scalability of ST-KF, we employ a mean-field approximation for inference. Incorporating user/item covariates in estimating neighborhood weights also helps in dealing with both cold-start and warm-start problems seamlessly in a single unified modeling framework; covariates predict factors for new users and items through the neighborhood. We illustrate our method on simulated data, benchmark data and data obtained from a relatively new recommender system application arising in the context of Yahoo! Front Page.

The success of popular algorithms such as k-means clustering or nearest neighbor searches depend on the assumption that the underlying distance functions reflect domain-specific notions of similarity for the problem at hand. The distance metric learning problem seeks to optimize a distance function subject to constraints that arise from fully-supervised or semisupervised information. Several recent algorithms have been proposed to learn such distance functions in low dimensional settings. One major shortcoming of these methods is their failure to scale to high dimensional problems that are becoming increasingly ubiquitous in modern data mining applications. In this paper, we present metric learning algorithms that scale linearly with dimensionality, permitting efficient optimization, storage, and evaluation of the learned metric. This is achieved through our main technical contribution which provides a framework based on the log-determinant matrix divergence which enables efficient optimization of structured, low-parameter Mahalanobis distances. Experimentally, we evaluate our methods across a variety of high dimensional domains, including text, statistical software analysis, and collaborative filtering, showing that our methods scale to data sets with tens of thousands or more features. We show that our learned metric can achieve excellent quality with respect to various criteria. For example, in the context of metric learning for nearest neighbor classification, we show that our methods achieve 24% higher accuracy over the baseline distance. Additionally, our methods yield very good precision while providing recall measures up to 20% higher than other baseline methods such as latent semantic analysis.

Most clustering algorithms produce a single clustering for a given data set even when the data can be clustered naturally in multiple ways. In this paper, we address the difficult problem of uncovering disparate clusterings from the data in a totally unsupervised manner. We propose two new approaches for this problem. In the first approach we aim to find good clusterings of the data that are also decorrelated with one another. To this end, we give a new and tractable characterization of decorrelation between clusterings, and present an objective function to capture it. We provide an iterative “decorrelated” k-means type algorithm to minimize this objective function. In the second approach, we model the data as a sum of mixtures and associate each mixture with a clustering. This approach leads us to the problem of learning a convolution of mixture distributions. Though the latter problem can be formulated as one of factorial learning [8, 13, 16], the existing formulations and methods do not perform well on many real high-dimensional data sets. We propose a new regularized factorial learning framework that is more suitable for capturing the notion of disparate clusterings in modern, high-dimensional data sets. The resulting algorithm does well in uncovering multiple clusterings, and is much improved over existing methods. We evaluate our methods on two real-world data sets - a music data set from the text mining domain, and a portrait data set from the computer vision domain. Our methods achieve a substantially higher accuracy than existing factorial learning as well as traditional clustering algorithms.

Nonnegative Matrix Approximation is an effective matrix decomposition technique that has proven to be useful for a wide variety of applications ranging from document analysis and image processing to bioinformatics. There exist a few algorithms for nonnegative matrix approximation (NNMA), for example, Lee & Seung's multiplicative updates, alternating least squares, and certain gradient descent based procedures. All of these procedures suffer from either slow convergence, numerical instabilities, or at worst, theoretical un-soundness. In this paper we present new and improved algorithms for the least-squares NNMA problem, which are not only theoretically well-founded, but also overcome many of the deficiencies of other methods. In particular, we use non-diagonal gradient scaling to obtain rapid convergence. Our methods provide numerical results superior to both Lee & Seung's method as well to the alternating least squares (ALS) heuristic, which is known to work well in some situations but has no theoretical guarantees (Berry et al. 2006). Our approach extends naturally to include regularization and box-constraints, without sacrificing convergence guarantees. We present experimental results on both synthetic and real-world datasets to demonstrate the superiority of our methods, in terms of better approximations as well as efficiency.

Localized search engines are small-scale systems that index a particular community on the web. They offer several benefits over their large-scale counterparts in that they are relatively inexpensive to build, and can provide more precise and complete search capability over their relevant domains. One disadvantage such systems have over large-scale search engines is the lack of global PageRank values. Such information is needed to assess the value of pages in the localized search domain within the context of the web as a whole. In this paper, we present well-motivated algorithms to estimate the global PageRank values of a local domain. The algorithms are all highly scalable in that, given a local domain of size n, they use O(n) resources that include computation time, bandwidth, and storage. We test our methods across a variety of localized domains, including site-specific domains and topic-specific domains. We demonstrate that by crawling as few as n or 2n additional pages, our methods can give excellent global PageRank estimates.

Graph clustering (also called graph partitioning) --- clustering the nodes of a graph --- is an important problem in diverse data mining applications. Traditional approaches involve optimization of graph clustering objectives such as normalized cut or ratio association; spectral methods are widely used for these objectives, but they require eigenvector computation which can be slow. Recently, graph clustering with a general cut objective has been shown to be mathematically equivalent to an appropriate weighted kernel k-means objective function. In this paper, we exploit this equivalence to develop a very fast multilevel algorithm for graph clustering. Multilevel approaches involve coarsening, initial partitioning and refinement phases, all of which may be specialized to different graph clustering objectives. Unlike existing multilevel clustering approaches, such as METIS, our algorithm does not constrain the cluster sizes to be nearly equal. Our approach gives a theoretical guarantee that the refinement step decreases the graph cut objective under consideration. Experiments show that we achieve better final objective function values as compared to a state-of-the-art spectral clustering algorithm: on a series of benchmark test graphs with up to thirty thousand nodes and one million edges, our algorithm achieves lower normalized cut values in 67% of our experiments and higher ratio association values in 100% of our experiments. Furthermore, on large graphs, our algorithm is significantly faster than spectral methods. Finally, our algorithm requires far less memory than spectral methods; we cluster a 1.2 million node movie network into 5000 clusters, which due to memory requirements cannot be done directly with spectral methods.

Co-clustering is a powerful data mining technique with varied applications such as text clustering, microarray analysis and recommender systems. Recently, an information-theoretic co-clustering approach applicable to empirical joint probability distributions was proposed. In many situations, co-clustering of more general matrices is desired. In this paper, we present a substantially generalized co-clustering framework wherein any Bregman divergence can be used in the objective function, and various conditional expectation based constraints can be considered based on the statistics that need to be preserved. Analysis of the co-clustering problem leads to the minimum Bregman information principle, which generalizes the maximum entropy principle, and yields an elegant meta algorithm that is guaranteed to achieve local optimality. Our methodology yields new algorithms and also encompasses several previously known clustering and co-clustering algorithms based on alternate minimization.

Kernel k-means and spectral clustering have both been used to identify clusters that are non-linearly separable in input space. Despite significant research, these methods have remained only loosely related. In this paper, we give an explicit theoretical connection between them. We show the generality of the weighted kernel k-means objective function, and derive the spectral clustering objective of normalized cut as a special case. Given a positive definite similarity matrix, our results lead to a novel weighted kernel k-means algorithm that monotonically decreases the normalized cut. This has important implications: a) eigenvector-based algorithms, which can be computationally prohibitive, are not essential for minimizing normalized cuts, b) various techniques, such as local search and acceleration schemes, may be used to improve the quality as well as speed of kernel k-means. Finally, we present results on several interesting data sets, including diametrical clustering of large gene-expression matrices and a handwriting recognition data set.

A wide variety of distortion functions are used for clustering, e.g., squared Euclidean distance, Mahalanobis distance and relative entropy. In this paper, we propose and analyze parametric hard and soft clustering algorithms based on a large class of distortion functions known as Bregman divergences. The proposed algorithms unify centroid-based parametric clustering approaches, such as classical kmeans and information-theoretic clustering, which arise by special choices of the Bregman divergence. The algorithms maintain the simplicity and scalability of the classical kmeans algorithm, while generalizing the basic idea to a very large class of clustering loss functions. There are two main contributions in this paper. First, we pose the hard clustering problem in terms of minimizing the loss in Bregman information, a quantity motivated by rate-distortion theory, and present an algorithm to minimize this loss. Secondly, we show an explicit bijection between Bregman divergences and exponential families. The bijection enables the development of an alternative interpretation of an efficient EM scheme for learning models involving mixtures of exponential distributions. This leads to a simple soft clustering algorithm for all Bregman divergences.

Microarray experiments have been extensively used for simultaneously measuring DNA expression levels of thousands of genes in genome research. A key step in the analysis of gene expression data is the clustering of genes into groups that show similar expression values over a range of conditions. Since only a small subset of the genes participate in any cellular process of interest, by focusing on subsets of genes and conditions, we can lower the noise induced by other genes and conditions — a co-cluster characterizes such a subset of interest. Cheng and Church [3] introduced an effective measure of co-cluster quality based on mean squared residue. In this paper, we use two similar squared residue measures and propose two fast k-means like co-clustering algorithms corresponding to the two residue measures. Our algorithms discover k row clusters and l column clusters simultaneously while monotonically decreasing the respective squared residues. Our co-clustering algorithms inherit the simplicity, efficiency and wide applicability of the k-means algorithm. Minimizing the residues may also be formulated as trace optimization problems that allow us to obtain a spectral relaxation that we use for a principled initialization for our iterative algorithms. We further enhance our algorithms by an incremental local search strategy that helps avoid empty clusters and escape poor local minima. We illustrate co-clustering results on a yeast cell cycle dataset and a human B-cell lymphoma dataset. Our experiments show that our co-clustering algorithms are efficient and are able to discover coherent co-clusters.

A novel approach to clustering cooccurrence data poses it as an optimization problem in information theory which minimizes the resulting loss in mutual information. A divisive clustering algorithm that monotonically reduces this loss function was recently proposed. We show that sparse high-dimensional data presents special challenges which can result in the algorithm getting stuck at poor local minima. We propose two solutions to this problem: (a) a "prior" to overcome infinite relative entropy values as in the supervised Naive Bayes algorithm, and (b) local search to escape local minima. Finally, we combine these solutions to get a robust algorithm that is computationally efficient. We present experimental results to show that the proposed method is effective in clustering document collections and outperform previous information-theoretic clustering approaches.

High dimensional directional data is becoming increasingly important in contemporary applications such as analysis of text and gene-expression data. A natural model for multi-variate directional data is provided by the von Mises-Fisher (vMF) distribution on the unit hypersphere that is analogous to the multi-variate Gaussian distribution in Rd. In this paper, we propose modeling complex directional data as a mixture of vMF distributions. We derive and analyze two variants of the Expectation Maximization (EM) framework for estimating the parameters of this mixture. We also propose two clustering algorithms corresponding to these variants. An interesting aspect of our methodology is that the spherical kmeans algorithm (kmeans with cosine similarity) can be shown to be a special case of both our algorithms. Thus, modeling text data by vMF distributions lends theoretical validity to the use of cosine similarity which has been widely used by the information retrieval community. As part of experimental validation, we present results on modeling high-dimensional text and gene-expression data as a mixture of vMF distributions. The results indicate that our approach yields superior clusterings especially for difficult clustering tasks in high-dimensional spaces.

Two-dimensional contingency or co-occurrence tables arise frequently in important applications such as text, web-log and market-basket data analysis. A basic problem in contingency table analysis is co-clustering: simultaneous clustering of the rows and columns. A novel theoretical formulation views the contingency table as an empirical joint probability distribution of two discrete random variables and poses the co-clustering problem as an optimization problem in information theory---the optimal co-clustering maximizes the mutual information between the clustered random variables subject to constraints on the number of row and column clusters. We present an innovative co-clustering algorithm that monotonically increases the preserved mutual information by intertwining both the row and column clusterings at all stages. Using the practical example of simultaneous word-document clustering, we demonstrate that our algorithm works well in practice, especially in the presence of sparsity and high-dimensionality.

The k-means algorithm with cosine similarity, also known as the spherical k-means algorithm, is a popular method for clustering document collections. However spherical k-means can often yield qualitatively poor results, especially when cluster sizes are small, say 25-30 documents per cluster, where it tends to get stuck at a local maximum far away from the optimal solution. In this paper, we present a local search procedure, which we call 'first-variation" that refines a given clustering by incrementally moving data points between clusters, thus achieving a higher objective function value. An enhancement of first variation allows a chain of such moves in a Kernighan-Lin fashion and leads to a better local maximum. Combining the enhanced first-variation with spherical k-means yields a powerful "ping-pong" strategy that often qualitatively improves k-means clustering and is computationally efficient. We present several experimental results to highlight the improvement achieved by our proposed algorithm in clustering high-dimensional and sparse text data.

In this paper we propose a new information-theoretic divisive algorithm for word clustering applied to text classification. In previous work, such "distributional clustering" of features has been found to achieve improvements over feature selection in terms of classification accuracy, especially at lower number of features [2, 28]. However the existing clustering techniques are agglomerative in nature and result in (i) sub-optimal word clusters and (ii) high computational cost. In order to explicitly capture the optimality of word clusters in an information theoretic framework, we first derive a global criterion for feature clustering. We then present a fast, divisive algorithm that monotonically decreases this objective function value, thus converging to a local minimum. We show that our algorithm minimizes the "within-cluster Jensen-Shannon divergence" while simultaneously maximizing the "between-cluster Jensen-Shannon divergence". In comparison to the previously proposed agglomerative strategies our divisive algorithm achieves higher classification accuracy especially at lower number of features. We further show that feature clustering is an effective technique for building smaller class models in hierarchical classification. We present detailed experimental results using Naive Bayes and Support Vector Machines on the 20 Newsgroups data set and a 3-level hierarchy of HTML documents collected from Dmoz Open Directory.

Both document clustering and word clustering are well studied problems. Most existing algorithms cluster documents and words separately but not simultaneously. In this paper we present the novel idea of modeling the document collection as a bipartite graph between documents and words, using which the simultaneous clustering problem can be posed as a bipartite graph partitioning problem. To solve the partitioning problem, we use a new spectral co-clustering algorithm that uses the second left and right singular vectors of an appropriately scaled word-document matrix to yield good bipartitionings. The spectral algorithm enjoys some optimality properties; it can be shown that the singular vectors solve a real relaxation to the NP-complete graph bipartitioning problem. We present experimental results to verify that the resulting co-clustering algorithm works well in practice.