On the case that the number of dangling nodes is large, PageRank computation can be proceeded with a much smaller matrix through lumping all dangling nodes of a web graph into a single node. Thus, it saves many computational cost and operations. There are also some theoretical contributions on Jordan canonical form of the Google matrix. Motivated by these theoretical contributions, in this note, we provide alternative proofs for some results of Google matrix through the lumping method due to Ipsen and Selee. Specifically we find that the result is also suitable for some subsequent work based on lumping dangling nodes into a node. Besides, an entirely new proof from the matrix decomposition viewpoint is also proposed.
相關內容
We employ a toolset -- dubbed Dr. Frankenstein -- to analyse the similarity of representations in deep neural networks. With this toolset, we aim to match the activations on given layers of two trained neural networks by joining them with a stitching layer. We demonstrate that the inner representations emerging in deep convolutional neural networks with the same architecture but different initializations can be matched with a surprisingly high degree of accuracy even with a single, affine stitching layer. We choose the stitching layer from several possible classes of linear transformations and investigate their performance and properties. The task of matching representations is closely related to notions of similarity. Using this toolset, we also provide a novel viewpoint on the current line of research regarding similarity indices of neural network representations: the perspective of the performance on a task.
Click-through rate (CTR) prediction plays a critical role in recommender systems and online advertising. The data used in these applications are multi-field categorical data, where each feature belongs to one field. Field information is proved to be important and there are several works considering fields in their models. In this paper, we proposed a novel approach to model the field information effectively and efficiently. The proposed approach is a direct improvement of FwFM, and is named as Field-matrixed Factorization Machines (FmFM, or $FM^2$). We also proposed a new explanation of FM and FwFM within the FmFM framework, and compared it with the FFM. Besides pruning the cross terms, our model supports field-specific variable dimensions of embedding vectors, which acts as soft pruning. We also proposed an efficient way to minimize the dimension while keeping the model performance. The FmFM model can also be optimized further by caching the intermediate vectors, and it only takes thousands of floating-point operations (FLOPs) to make a prediction. Our experiment results show that it can out-perform the FFM, which is more complex. The FmFM model's performance is also comparable to DNN models which require much more FLOPs in runtime.
Network embedding aims to learn a latent, low-dimensional vector representations of network nodes, effective in supporting various network analytic tasks. While prior arts on network embedding focus primarily on preserving network topology structure to learn node representations, recently proposed attributed network embedding algorithms attempt to integrate rich node content information with network topological structure for enhancing the quality of network embedding. In reality, networks often have sparse content, incomplete node attributes, as well as the discrepancy between node attribute feature space and network structure space, which severely deteriorates the performance of existing methods. In this paper, we propose a unified framework for attributed network embedding-attri2vec-that learns node embeddings by discovering a latent node attribute subspace via a network structure guided transformation performed on the original attribute space. The resultant latent subspace can respect network structure in a more consistent way towards learning high-quality node representations. We formulate an optimization problem which is solved by an efficient stochastic gradient descent algorithm, with linear time complexity to the number of nodes. We investigate a series of linear and non-linear transformations performed on node attributes and empirically validate their effectiveness on various types of networks. Another advantage of attri2vec is its ability to solve out-of-sample problems, where embeddings of new coming nodes can be inferred from their node attributes through the learned mapping function. Experiments on various types of networks confirm that attri2vec is superior to state-of-the-art baselines for node classification, node clustering, as well as out-of-sample link prediction tasks. The source code of this paper is available at //github.com/daokunzhang/attri2vec.
We show that for the problem of testing if a matrix $A \in F^{n \times n}$ has rank at most $d$, or requires changing an $\epsilon$-fraction of entries to have rank at most $d$, there is a non-adaptive query algorithm making $\widetilde{O}(d^2/\epsilon)$ queries. Our algorithm works for any field $F$. This improves upon the previous $O(d^2/\epsilon^2)$ bound (SODA'03), and bypasses an $\Omega(d^2/\epsilon^2)$ lower bound of (KDD'14) which holds if the algorithm is required to read a submatrix. Our algorithm is the first such algorithm which does not read a submatrix, and instead reads a carefully selected non-adaptive pattern of entries in rows and columns of $A$. We complement our algorithm with a matching query complexity lower bound for non-adaptive testers over any field. We also give tight bounds of $\widetilde{\Theta}(d^2)$ queries in the sensing model for which query access comes in the form of $\langle X_i, A\rangle:=tr(X_i^\top A)$; perhaps surprisingly these bounds do not depend on $\epsilon$. We next develop a novel property testing framework for testing numerical properties of a real-valued matrix $A$ more generally, which includes the stable rank, Schatten-$p$ norms, and SVD entropy. Specifically, we propose a bounded entry model, where $A$ is required to have entries bounded by $1$ in absolute value. We give upper and lower bounds for a wide range of problems in this model, and discuss connections to the sensing model above.
Network embedding aims to learn low-dimensional representations of nodes in a network, while the network structure and inherent properties are preserved. It has attracted tremendous attention recently due to significant progress in downstream network learning tasks, such as node classification, link prediction, and visualization. However, most existing network embedding methods suffer from the expensive computations due to the large volume of networks. In this paper, we propose a $10\times \sim 100\times$ faster network embedding method, called Progle, by elegantly utilizing the sparsity property of online networks and spectral analysis. In Progle, we first construct a \textit{sparse} proximity matrix and train the network embedding efficiently via sparse matrix decomposition. Then we introduce a network propagation pattern via spectral analysis to incorporate local and global structure information into the embedding. Besides, this model can be generalized to integrate network information into other insufficiently trained embeddings at speed. Benefiting from sparse spectral network embedding, our experiment on four different datasets shows that Progle outperforms or is comparable to state-of-the-art unsupervised comparison approaches---DeepWalk, LINE, node2vec, GraRep, and HOPE, regarding accuracy, while is $10\times$ faster than the fastest word2vec-based method. Finally, we validate the scalability of Progle both in real large-scale networks and multiple scales of synthetic networks.
Although Recommender Systems have been comprehensively studied in the past decade both in industry and academia, most of current recommender systems suffer from the fol- lowing issues: 1) The data sparsity of the user-item matrix seriously affect the recommender system quality. As a result, most of traditional recommender system approaches are not able to deal with the users who have rated few items, which is known as cold start problem in recommender system. 2) Traditional recommender systems assume that users are in- dependently and identically distributed and ignore the social relation between users. However, in real life scenario, due to the exponential growth of social networking service, such as facebook and Twitter, social connections between different users play an significant role for recommender system task. In this work, aiming at providing a better recommender sys- tems by incorporating user social network information, we propose a matrix factorization framework with user social connection constraints. Experimental results on the real-life dataset shows that the proposed method performs signifi- cantly better than the state-of-the-art approaches in terms of MAE and RMSE, especially for the cold start users.
Enhancing Network Embedding with Auxiliary Information: An Explicit Matrix Factorization Perspective
Recent advances in the field of network embedding have shown the low-dimensional network representation is playing a critical role in network analysis. However, most of the existing principles of network embedding do not incorporate auxiliary information such as content and labels of nodes flexibly. In this paper, we take a matrix factorization perspective of network embedding, and incorporate structure, content and label information of the network simultaneously. For structure, we validate that the matrix we construct preserves high-order proximities of the network. Label information can be further integrated into the matrix via the process of random walk sampling to enhance the quality of embedding in an unsupervised manner, i.e., without leveraging downstream classifiers. In addition, we generalize the Skip-Gram Negative Sampling model to integrate the content of the network in a matrix factorization framework. As a consequence, network embedding can be learned in a unified framework integrating network structure and node content as well as label information simultaneously. We demonstrate the efficacy of the proposed model with the tasks of semi-supervised node classification and link prediction on a variety of real-world benchmark network datasets.
We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.
Review-based recommender systems have gained noticeable ground in recent years. In addition to the rating scores, those systems are enriched with textual evaluations of items by the users. Neural language processing models, on the other hand, have already found application in recommender systems, mainly as a means of encoding user preference data, with the actual textual description of items serving only as side information. In this paper, a novel approach to incorporating the aforementioned models into the recommendation process is presented. Initially, a neural language processing model and more specifically the paragraph vector model is used to encode textual user reviews of variable length into feature vectors of fixed length. Subsequently this information is fused along with the rating scores in a probabilistic matrix factorization algorithm, based on maximum a-posteriori estimation. The resulting system, ParVecMF, is compared to a ratings' matrix factorization approach on a reference dataset. The obtained preliminary results on a set of two metrics are encouraging and may stimulate further research in this area.
Since the invention of word2vec, the skip-gram model has significantly advanced the research of network embedding, such as the recent emergence of the DeepWalk, LINE, PTE, and node2vec approaches. In this work, we show that all of the aforementioned models with negative sampling can be unified into the matrix factorization framework with closed forms. Our analysis and proofs reveal that: (1) DeepWalk empirically produces a low-rank transformation of a network's normalized Laplacian matrix; (2) LINE, in theory, is a special case of DeepWalk when the size of vertices' context is set to one; (3) As an extension of LINE, PTE can be viewed as the joint factorization of multiple networks' Laplacians; (4) node2vec is factorizing a matrix related to the stationary distribution and transition probability tensor of a 2nd-order random walk. We further provide the theoretical connections between skip-gram based network embedding algorithms and the theory of graph Laplacian. Finally, we present the NetMF method as well as its approximation algorithm for computing network embedding. Our method offers significant improvements over DeepWalk and LINE for conventional network mining tasks. This work lays the theoretical foundation for skip-gram based network embedding methods, leading to a better understanding of latent network representation learning.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191