ACCAMS: Additive Co-Clustering to Approximate Matrices Succinctly

Matrix completion and approximation are popular tools to capture a user's preferences for recommendation and to approximate missing data. Instead of using low-rank factorization we take a drastically different approach, based on the simple insight that an additive model of co-clusterings allows one to approximate matrices efficiently. This allows us to build a concise model that, per bit of model learned, significantly beats all factorization approaches to matrix approximation. Even more surprisingly, we find that summing over small co-clusterings is more effective in modeling matrices than classic co-clustering, which uses just one large partitioning of the matrix. Following Occam's razor principle suggests that the simple structure induced by our model better captures the latent preferences and decision making processes present in the real world than classic co-clustering or matrix factorization. We provide an iterative minimization algorithm, a collapsed Gibbs sampler, theoretical guarantees for matrix approximation, and excellent empirical evidence for the efficacy of our approach. We achieve state-of-the-art results on the Netflix problem with a fraction of the model complexity.Comment: 22 pages, under review for conference publicatio

Prediction of Treatment Response to Neoadjuvant Chemotherapy for Breast Cancer via Early Changes in Tumor Heterogeneity Captured by DCE-MRI Registration.

We analyzed DCE-MR images from 132 women with locally advanced breast cancer from the I-SPY1 trial to evaluate changes of intra-tumor heterogeneity for augmenting early prediction of pathologic complete response (pCR) and recurrence-free survival (RFS) after neoadjuvant chemotherapy (NAC). Utilizing image registration, voxel-wise changes including tumor deformations and changes in DCE-MRI kinetic features were computed to characterize heterogeneous changes within the tumor. Using five-fold cross-validation, logistic regression and Cox regression were performed to model pCR and RFS, respectively. The extracted imaging features were evaluated in augmenting established predictors, including functional tumor volume (FTV) and histopathologic and demographic factors, using the area under the curve (AUC) and the C-statistic as performance measures. The extracted voxel-wise features were also compared to analogous conventional aggregated features to evaluate the potential advantage of voxel-wise analysis. Voxel-wise features improved prediction of pCR (AUC = 0.78 (±0.03) vs 0.71 (±0.04), p < 0.05 and RFS (C-statistic = 0.76 ( ± 0.05), vs 0.63 ( ± 0.01)), p < 0.05, while models based on analogous aggregate imaging features did not show appreciable performance changes (p > 0.05). Furthermore, all selected voxel-wise features demonstrated significant association with outcome (p < 0.05). Thus, precise measures of voxel-wise changes in tumor heterogeneity extracted from registered DCE-MRI scans can improve early prediction of neoadjuvant treatment outcomes in locally advanced breast cancer

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page