Clustering based tensor decomposition
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1 Clustering based tensor decomposition Huan He Shihua Wang Emory University November 29, 2017 (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
2 Outline 1 Introduction Recommendation System Common methods 2 K-means as an Clustering method K-means K-prototype 3 Tensor Decomposition as an Latent Factor model What s a tensor and its applications? Tensor Decomposition Model 4 Our proposed methods Tensor Decomposition Continued Problem define and Algorithms 5 Experiments (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
3 Recommendation System Traditional Definition: Estimate a utility function that automatically predicts how a user will like an item Based on 1 Past behavior 2 Relations to other users 3 Item Similarity 4... (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
4 Common methods Collaborative Filtering: It analyzes relationships between users and interdependencies among products to identify new user-item associations. Clustering: It looks for homogenous subgroups among observations. Latent factor methods: Dimensionality reduction (e.g., PCA and MF) looks for a low dimensional representation of the observations. (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
5 K-means K-means as a recommendation system method. The goal is to cluster users and compute per-cluster typical preferences What s the problem? Users receive recommendations computed at the cluster level In addition, It can not handle datasets with categorical features. Figure: Snapshot of the members data (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
6 K-mode Figure: Summation: the dissimilarity measure-the total mismatches of the corresponding attribute categories of the two objects K-mode: Suppose X is a subset of the original dataset, and there are n observations in X. For each observation, there are m categorical features. For attribute j, if observation X and Y has the same value, the measure will be one. A vector Q[q1, q2, qm]minimizes the dissimilarity measure of the mode of X [X 1, X 2..Xn]. (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
7 K-mode Figure: Cost function K-mode: Suppose X is a subset of the original dataset, and there are n observations in X. For each observation, there are m categorical features. For attribute j, if observation X and Y has the same value, the measure will be one. A vector Q[q1, q2, qm]minimizes the dissimilarity measure of the mode of X [X 1, X 2..Xn]. (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
8 K-prototype k-prototype Figure: Distance function with mixture types of features λ is the weighting factor to avoid favoring either type of attribute Figure: Cost function becomes: where there are m features in total and p of them are continuous (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
9 What s a tensor and its applications? A tensor is a multidimensional array. A first-order tensor is a vector, a second-order tensor is a matrix, and tensors of order three or higher are called higher-order tensors. Standardly, TD is a unsupervised dimension reduction method that can generate latent factors. It can be a clustering method as well! Stay with me one more slide. Figure: A third order tensor (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
10 CP Decomposition Basic Idea: The CP decomposition factorizes a tensor into a sum of component rank-one tensors. Optimization CP-ALS Figure: CP decomposition of three-way tensor as sum of R outer products (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
11 CP Decomposition Basic Idea: The CP decomposition factorizes a tensor into a sum of component rank-one tensors. Optimization CP-ALS Figure: CP decomposition of three-way tensor as sum of R outer products (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
12 CP Decomposition Basic Idea: The CP decomposition factorizes a tensor into a sum of component rank-one tensors. Optimization CP-ALS Figure: CP decomposition of three-way tensor as sum of R outer products (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
13 Two examples to convince you (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
14 Two examples to convince you (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
15 Tensor Decomposition Continued ALS has a computational issue and needs to modify to train on incomplete datasets. Recall two examples in the previous slide, how to determine the rank of a CP Decomposition is a NP-hard problem in real world. (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
16 Our Proposed Method Can we go a step further? Motivated by a GLMix model by Linkedin 1, can we add a Personalized effect model to a Global fixed effect model? We consider each cluster as a personalized effect model. A proportion of popular songs, active users as our Global fixed effect model. Motivated by a CP algorithm named Parcube 2, we can also correctly combine them nobreakspace epapalex/papers/parcube PKDD2012 (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
17 Algorithm: The algorithm, roughly, consists of the following steps: 1 Biased sampling: We use biased sampling to select indices from all three modes of the tensor. Given k clusters, we create k independent samples, each is indexed by a p proportion of Global fixed effect indexes and a full cluster indexes. 2 Parallel decomposition on the clustering based samples: It includes fitting of the CP decomposition to the sample tensors, obtained from the previous step. 3 Factor and Merge: The final step is merging the intermediate decomposition results into a full sized set of decomposition factors. (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
18 Algorithm: Biased Sampling: Figure: Example of rank-1 CP Decomposition. This is, given r clusters, each vector of a consists of the cluster members and songs plus a proportion of popular indexes. Then run CP-SGD in parallel and obtain r factor matrices. As a final step, combine those factors (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
19 Experiments Completed work: 1 K-mode 2 Biased Sampling 3 Parallel SGD using multiprocessing 4 Sparse format tensor storage, a hash table for implicit feedbacks and incomplete data. Future work: K-prototype and Factor-merge (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
20 Conclusion Advatages based on Guess 1 Don t need to determine the rank of a CP decomposition 2 This hybrid Recsys can improve the accuracy of only one model 3 Relative simplicity, speed, and parallelizable execution Bottleneck: Based on our completed work, the number of optimal k for the whole datasets could be at least 40. Figure: example caption (Huan)(Shihua) (Emory University) Clustering based tensor decomposition November 29, / 15
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