Learning Efficient Tensor Representations with Ring Structure Networks

Tensor train (TT) decomposition is a powerful representation for high-order tensors, which has been successfully applied to various machine learning tasks in recent years. However, since the tensor product is not commutative, permutation of data dimensions makes solutions and TT-ranks of TT decomposition inconsistent. To alleviate this problem, we propose a permutation symmetric network structure by employing circular multilinear products over a sequence of low-order core tensors. This network structure can be graphically interpreted as a cyclic interconnection of tensors, and thus we call it tensor ring (TR) representation. We develop several efficient algorithms to learn TR representation with adaptive TR-ranks by employing low-rank approximations. Furthermore, mathematical properties are investigated, which enables us to perform basic operations in a computationally efficiently way by using TR representations. Experimental results on synthetic signals and real-world datasets demonstrate that the proposed TR network is more expressive and consistently informative than existing TT networks.Comment: arXiv admin note: substantial text overlap with arXiv:1606.0553

Towards Efficient Large-Scale Graph Neural Network Computing

Recent deep learning models have moved beyond low-dimensional regular grids such as image, video, and speech, to high-dimensional graph-structured data, such as social networks, brain connections, and knowledge graphs. This evolution has led to large graph-based irregular and sparse models that go beyond what existing deep learning frameworks are designed for. Further, these models are not easily amenable to efficient, at scale, acceleration on parallel hardwares (e.g. GPUs). We introduce NGra, the first parallel processing framework for graph-based deep neural networks (GNNs). NGra presents a new SAGA-NN model for expressing deep neural networks as vertex programs with each layer in well-defined (Scatter, ApplyEdge, Gather, ApplyVertex) graph operation stages. This model not only allows GNNs to be expressed intuitively, but also facilitates the mapping to an efficient dataflow representation. NGra addresses the scalability challenge transparently through automatic graph partitioning and chunk-based stream processing out of GPU core or over multiple GPUs, which carefully considers data locality, data movement, and overlapping of parallel processing and data movement. NGra further achieves efficiency through highly optimized Scatter/Gather operators on GPUs despite its sparsity. Our evaluation shows that NGra scales to large real graphs that none of the existing frameworks can handle directly, while achieving up to about 4 times speedup even at small scales over the multiple-baseline design on TensorFlow

Tensor Ring Decomposition

Tensor networks have in recent years emerged as the powerful tools for solving the large-scale optimization problems. One of the most popular tensor network is tensor train (TT) decomposition that acts as the building blocks for the complicated tensor networks. However, the TT decomposition highly depends on permutations of tensor dimensions, due to its strictly sequential multilinear products over latent cores, which leads to difficulties in finding the optimal TT representation. In this paper, we introduce a fundamental tensor decomposition model to represent a large dimensional tensor by a circular multilinear products over a sequence of low dimensional cores, which can be graphically interpreted as a cyclic interconnection of 3rd-order tensors, and thus termed as tensor ring (TR) decomposition. The key advantage of TR model is the circular dimensional permutation invariance which is gained by employing the trace operation and treating the latent cores equivalently. TR model can be viewed as a linear combination of TT decompositions, thus obtaining the powerful and generalized representation abilities. For optimization of latent cores, we present four different algorithms based on the sequential SVDs, ALS scheme, and block-wise ALS techniques. Furthermore, the mathematical properties of TR model are investigated, which shows that the basic multilinear algebra can be performed efficiently by using TR representaions and the classical tensor decompositions can be conveniently transformed into the TR representation. Finally, the experiments on both synthetic signals and real-world datasets were conducted to evaluate the performance of different algorithms

Higher-dimension Tensor Completion via Low-rank Tensor Ring Decomposition

The problem of incomplete data is common in signal processing and machine learning. Tensor completion algorithms aim to recover the incomplete data from its partially observed entries. In this paper, taking advantages of high compressibility and flexibility of recently proposed tensor ring (TR) decomposition, we propose a new tensor completion approach named tensor ring weighted optimization (TR-WOPT). It finds the latent factors of the incomplete tensor by gradient descent algorithm, then the latent factors are employed to predict the missing entries of the tensor. We conduct various tensor completion experiments on synthetic data and real-world data. The simulation results show that TR-WOPT performs well in various high-dimension tensors. Furthermore, image completion results show that our proposed algorithm outperforms the state-of-the-art algorithms in many situations. Especially when the missing rate of the test images is high (e.g., over 0.9), the performance of our TR-WOPT is significantly better than the compared algorithms.Comment: APSIPA2018 conference paper. arXiv admin note: substantial text overlap with arXiv:1805.0846

Tensor Ring Decomposition with Rank Minimization on Latent Space: An Efficient Approach for Tensor Completion

In tensor completion tasks, the traditional low-rank tensor decomposition models suffer from the laborious model selection problem due to their high model sensitivity. In particular, for tensor ring (TR) decomposition, the number of model possibilities grows exponentially with the tensor order, which makes it rather challenging to find the optimal TR decomposition. In this paper, by exploiting the low-rank structure of the TR latent space, we propose a novel tensor completion method which is robust to model selection. In contrast to imposing the low-rank constraint on the data space, we introduce nuclear norm regularization on the latent TR factors, resulting in the optimization step using singular value decomposition (SVD) being performed at a much smaller scale. By leveraging the alternating direction method of multipliers (ADMM) scheme, the latent TR factors with optimal rank and the recovered tensor can be obtained simultaneously. Our proposed algorithm is shown to effectively alleviate the burden of TR-rank selection, thereby greatly reducing the computational cost. The extensive experimental results on both synthetic and real-world data demonstrate the superior performance and efficiency of the proposed approach against the state-of-the-art algorithms

Provably Powerful Graph Networks

Recently, the Weisfeiler-Lehman (WL) graph isomorphism test was used to measure the expressive power of graph neural networks (GNN). It was shown that the popular message passing GNN cannot distinguish between graphs that are indistinguishable by the 1-WL test (Morris et al. 2018; Xu et al. 2019). Unfortunately, many simple instances of graphs are indistinguishable by the 1-WL test. In search for more expressive graph learning models we build upon the recent k-order invariant and equivariant graph neural networks (Maron et al. 2019a,b) and present two results: First, we show that such k-order networks can distinguish between non-isomorphic graphs as good as the k-WL tests, which are provably stronger than the 1-WL test for k>2. This makes these models strictly stronger than message passing models. Unfortunately, the higher expressiveness of these models comes with a computational cost of processing high order tensors. Second, setting our goal at building a provably stronger, simple and scalable model we show that a reduced 2-order network containing just scaled identity operator, augmented with a single quadratic operation (matrix multiplication) has a provable 3-WL expressive power. Differently put, we suggest a simple model that interleaves applications of standard Multilayer-Perceptron (MLP) applied to the feature dimension and matrix multiplication. We validate this model by presenting state of the art results on popular graph classification and regression tasks. To the best of our knowledge, this is the first practical invariant/equivariant model with guaranteed 3-WL expressiveness, strictly stronger than message passing models

Compressing Recurrent Neural Networks with Tensor Ring for Action Recognition

Recurrent Neural Networks (RNNs) and their variants, such as Long-Short Term Memory (LSTM) networks, and Gated Recurrent Unit (GRU) networks, have achieved promising performance in sequential data modeling. The hidden layers in RNNs can be regarded as the memory units, which are helpful in storing information in sequential contexts. However, when dealing with high dimensional input data, such as video and text, the input-to-hidden linear transformation in RNNs brings high memory usage and huge computational cost. This makes the training of RNNs unscalable and difficult. To address this challenge, we propose a novel compact LSTM model, named as TR-LSTM, by utilizing the low-rank tensor ring decomposition (TRD) to reformulate the input-to-hidden transformation. Compared with other tensor decomposition methods, TR-LSTM is more stable. In addition, TR-LSTM can complete an end-to-end training and also provide a fundamental building block for RNNs in handling large input data. Experiments on real-world action recognition datasets have demonstrated the promising performance of the proposed TR-LSTM compared with the tensor train LSTM and other state-of-the-art competitors.Comment: 9 page

From probabilistic graphical models to generalized tensor networks for supervised learning

Tensor networks have found a wide use in a variety of applications in physics and computer science, recently leading to both theoretical insights as well as practical algorithms in machine learning. In this work we explore the connection between tensor networks and probabilistic graphical models, and show that it motivates the definition of generalized tensor networks where information from a tensor can be copied and reused in other parts of the network. We discuss the relationship between generalized tensor network architectures used in quantum physics, such as string-bond states, and architectures commonly used in machine learning. We provide an algorithm to train these networks in a supervised-learning context and show that they overcome the limitations of regular tensor networks in higher dimensions, while keeping the computation efficient. A method to combine neural networks and tensor networks as part of a common deep learning architecture is also introduced. We benchmark our algorithm for several generalized tensor network architectures on the task of classifying images and sounds, and show that they outperform previously introduced tensor-network algorithms. The models we consider also have a natural implementation on a quantum computer and may guide the development of near-term quantum machine learning architectures.Comment: 15 pages, 18 figures, improved version with additional explanation

Tensor Decompositions for Modeling Inverse Dynamics

Modeling inverse dynamics is crucial for accurate feedforward robot control. The model computes the necessary joint torques, to perform a desired movement. The highly non-linear inverse function of the dynamical system can be approximated using regression techniques. We propose as regression method a tensor decomposition model that exploits the inherent three-way interaction of positions x velocities x accelerations. Most work in tensor factorization has addressed the decomposition of dense tensors. In this paper, we build upon the decomposition of sparse tensors, with only small amounts of nonzero entries. The decomposition of sparse tensors has successfully been used in relational learning, e.g., the modeling of large knowledge graphs. Recently, the approach has been extended to multi-class classification with discrete input variables. Representing the data in high dimensional sparse tensors enables the approximation of complex highly non-linear functions. In this paper we show how the decomposition of sparse tensors can be applied to regression problems. Furthermore, we extend the method to continuous inputs, by learning a mapping from the continuous inputs to the latent representations of the tensor decomposition, using basis functions. We evaluate our proposed model on a dataset with trajectories from a seven degrees of freedom SARCOS robot arm. Our experimental results show superior performance of the proposed functional tensor model, compared to challenging state-of-the art methods

Multi-Branch Tensor Network Structure for Tensor-Train Discriminant Analysis

Higher-order data with high dimensionality arise in a diverse set of application areas such as computer vision, video analytics and medical imaging. Tensors provide a natural tool for representing these types of data. Although there has been a lot of work in the area of tensor decomposition and low-rank tensor approximation, extensions to supervised learning, feature extraction and classification are still limited. Moreover, most of the existing supervised tensor learning approaches are based on the orthogonal Tucker model. However, this model has some limitations for large tensors including high memory and computational costs. In this paper, we introduce a supervised learning approach for tensor classification based on the tensor-train model. In particular, we introduce a multi-branch tensor network structure for efficient implementation of tensor-train discriminant analysis (TTDA). The proposed approach takes advantage of the flexibility of the tensor train structure to implement various computationally efficient versions of TTDA. This approach is then evaluated on image and video classification tasks with respect to computation time, storage cost and classification accuracy and is compared to both vector and tensor based discriminant analysis methods