Tensor Computation: A New Framework for High-Dimensional Problems in EDA

Many critical EDA problems suffer from the curse of dimensionality, i.e. the very fast-scaling computational burden produced by large number of parameters and/or unknown variables. This phenomenon may be caused by multiple spatial or temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit simulation), nonlinearity of devices and circuits, large number of design or optimization parameters (e.g. full-chip routing/placement and circuit sizing), or extensive process variations (e.g. variability/reliability analysis and design for manufacturability). The computational challenges generated by such high dimensional problems are generally hard to handle efficiently with traditional EDA core algorithms that are based on matrix and vector computation. This paper presents "tensor computation" as an alternative general framework for the development of efficient EDA algorithms and tools. A tensor is a high-dimensional generalization of a matrix and a vector, and is a natural choice for both storing and solving efficiently high-dimensional EDA problems. This paper gives a basic tutorial on tensors, demonstrates some recent examples of EDA applications (e.g., nonlinear circuit modeling and high-dimensional uncertainty quantification), and suggests further open EDA problems where the use of tensor computation could be of advantage.Comment: 14 figures. Accepted by IEEE Trans. CAD of Integrated Circuits and System

Tensor Analysis and Fusion of Multimodal Brain Images

Current high-throughput data acquisition technologies probe dynamical systems with different imaging modalities, generating massive data sets at different spatial and temporal resolutions posing challenging problems in multimodal data fusion. A case in point is the attempt to parse out the brain structures and networks that underpin human cognitive processes by analysis of different neuroimaging modalities (functional MRI, EEG, NIRS etc.). We emphasize that the multimodal, multi-scale nature of neuroimaging data is well reflected by a multi-way (tensor) structure where the underlying processes can be summarized by a relatively small number of components or "atoms". We introduce Markov-Penrose diagrams - an integration of Bayesian DAG and tensor network notation in order to analyze these models. These diagrams not only clarify matrix and tensor EEG and fMRI time/frequency analysis and inverse problems, but also help understand multimodal fusion via Multiway Partial Least Squares and Coupled Matrix-Tensor Factorization. We show here, for the first time, that Granger causal analysis of brain networks is a tensor regression problem, thus allowing the atomic decomposition of brain networks. Analysis of EEG and fMRI recordings shows the potential of the methods and suggests their use in other scientific domains.Comment: 23 pages, 15 figures, submitted to Proceedings of the IEE

Enhancing representation learning with tensor decompositions for knowledge graphs and high dimensional sequence modeling

The capability of processing and digesting raw data is one of the key features of a human-like artificial intelligence system. For instance, real-time machine translation should be able to process and understand spoken natural language, and autonomous driving relies on the comprehension of visual inputs. Representation learning is a class of machine learning techniques that autonomously learn to derive latent features from raw data. These new features are expected to represent the data instances in a vector space that facilitates the machine learning task. This thesis studies two specific data situations that require efficient representation learning: knowledge graph data and high dimensional sequences. In the first part of this thesis, we first review multiple relational learning models based on tensor decomposition for knowledge graphs. We point out that relational learning is in fact a means of learning representations through one-hot mapping of entities. Furthermore, we generalize this mapping function to consume a feature vector that encodes all known facts about each entity. It enables the relational model to derive the latent representation instantly for a new entity, without having to re-train the tensor decomposition. In the second part, we focus on learning representations from high dimensional sequential data. Sequential data often pose the challenge that they are of variable lengths. Electronic health records, for instance, could consist of clinical event data that have been collected at subsequent time steps. But each patient may have a medical history of variable length. We apply recurrent neural networks to produce fixed-size latent representations from the raw feature sequences of various lengths. By exposing a prediction model to these learned representations instead of the raw features, we can predict the therapy prescriptions more accurately as a means of clinical decision support. We further propose Tensor-Train recurrent neural networks. We give a detailed introduction to the technique of tensorizing and decomposing large weight matrices into a few smaller tensors. We demonstrate the specific algorithms to perform the forward-pass and the back-propagation in this setting. Then we apply this approach to the input-to-hidden weight matrix in recurrent neural networks. This novel architecture can process extremely high dimensional sequential features such as video data. The model also provides a promising solution to processing sequential features with high sparsity. This is, for instance, the case with electronic health records, since they are often of categorical nature and have to be binary-coded. We incorporate a statistical survival model with this representation learning model, which shows superior prediction quality