Domain-specific hardware is becoming a promising topic in the backdrop of improvement slow down for general-purpose processors due to the foreseeable end of Moore’s Law. Machine learning, especially deep neural networks (DNNs), has become the most dazzling domain witnessing successful applications in a wide spectrum of artificial intelligence (AI) tasks. The incomparable accuracy of DNNs is achieved by paying the cost of hungry memory consumption and high computational complexity, which greatly impedes their deployment in embedded systems. Therefore, the DNN compression concept was naturally proposed and widely used for memory saving and compute acceleration. In the past few years, a tremendous number of compression techniques have sprung up to pursue a satisfactory tradeoff between processing efficiency and application accuracy. Recently, this wave has spread to the design of neural network accelerators for gaining extremely high performance. However, the amount of related works is incredibly huge and the reported approaches are quite divergent. This research chaos motivates us to provide a comprehensive survey on the recent advances toward the goal of efficient compression and execution of DNNs without significantly compromising accuracy, involving both the high-level algorithms and their applications in hardware design. In this article, we review the mainstream compression approaches such as compact model, tensor decomposition, data quantization, and network sparsification. We explain their compression principles, evaluation metrics, sensitivity analysis, and joint-way use. Then, we answer the question of how to leverage these methods in the design of neural network accelerators and present the state-of-the-art hardware architectures. In the end, we discuss several existing issues such as fair comparison, testing workloads, automatic compression, influence on security, and framework/hardware-level support, and give promising topics in this field and the possible challenges as well. This article attempts to enable readers to quickly build up a big picture of neural network compression and acceleration, clearly evaluate various methods, and confidently get started in the right way.

Model Compression and Hardware Acceleration for Neural Networks: A Comprehensive Survey

Tensor methods are among the most prominent tools for the numerical solution of high-dimensional problems where functions of multiple variables have to be approximated. These methods exploit the tensor structure of function spaces and apply to many problems in computational science which are formulated in tensor spaces, such as problems arising in stochastic calculus, uncertainty quantification or parametric analyses. Here, we present complexity reduction methods based on low-rank approximation methods. We analyze the problem of best approximation in subsets of low-rank tensors and discuss its connection with the problem of optimal model reduction in low-dimensional reduced spaces. We present different algorithms for computing approximations of a function in low-rank formats. In particular, we present constructive algorithms which are based either on a greedy construction of an approximation (with successive corrections in subsets of low-rank tensors) or on the greedy construction of tensor subspaces (for subspace-based low-rank formats). These algorithms can be applied for tensor compression, tensor completion or for the numerical solution of equations in low-rank tensor formats. A special emphasis is given to the solution of stochastic or parameter-dependent models. Different approaches are presented for the approximation of vector-valued or multivariate functions (identified with tensors), based on samples of the functions (black-box approaches) or on the models equations which are satisfied by the functions.

Low-rank methods for high-dimensional approximation and model order reduction

This paper proposes a novel approach to tensor completion, which recovers missing entries of data represented by tensors. The approach is based on the tensor train (TT) rank, which is able to capture hidden information from tensors thanks to its definition from a well-balanced matricization scheme. Accordingly, new optimization formulations for tensor completion are proposed as well as two new algorithms for their solution. The first one called simple low-rank tensor completion via TT (SiLRTC-TT) is intimately related to minimizing a nuclear norm based on TT rank. The second one is from a multilinear matrix factorization model to approximate the TT rank of a tensor, and is called tensor completion by parallel matrix factorization via TT (TMac-TT). A tensor augmentation scheme of transforming a low-order tensor to higher orders is also proposed to enhance the effectiveness of SiLRTC-TT and TMac-TT. Simulation results for color image and video recovery show the clear advantage of our method over all other methods.

/pdf/efficient-tensor-completion-for-color-image-and-video-1bp9yvh2g7.pdf

Efficient Tensor Completion for Color Image and Video Recovery: Low-Rank Tensor Train

Tensor completion is a problem of filling the missing or unobserved entries of partially observed tensors. Due to the multidimensional character of tensors in describing complex datasets, tensor completion algorithms and their applications have received wide attention and achievement in areas like data mining, computer vision, signal processing, and neuroscience. In this survey, we provide a modern overview of recent advances in tensor completion algorithms from the perspective of big data analytics characterized by diverse variety, large volume, and high velocity. We characterize these advances from the following four perspectives: general tensor completion algorithms, tensor completion with auxiliary information (variety), scalable tensor completion algorithms (volume), and dynamic tensor completion algorithms (velocity). Further, we identify several tensor completion applications on real-world data-driven problems and present some common experimental frameworks popularized in the literature along with several available software repositories. Our goal is to summarize these popular methods and introduce them to researchers and practitioners for promoting future research and applications. We conclude with a discussion of key challenges and promising research directions in this community for future exploration.

https://dl.acm.org/doi/pdf/10.1145/3278607

Tensor Completion Algorithms in Big Data Analytics

This paper proposes a novel approach to tensor completion, which recovers missing entries of data represented by tensors. The approach is based on the tensor train (TT) rank, which is able to capture hidden information from tensors thanks to its definition from a well-balanced matricization scheme. Accordingly, new optimization formulations for tensor completion are proposed as well as two new algorithms for their solution. The first one called simple low-rank tensor completion via tensor train (SiLRTC-TT) is intimately related to minimizing a nuclear norm based on TT rank. The second one is from a multilinear matrix factorization model to approximate the TT rank of a tensor, and is called tensor completion by parallel matrix factorization via tensor train (TMac-TT). A tensor augmentation scheme of transforming a low-order tensor to higher-orders is also proposed to enhance the effectiveness of SiLRTC-TT and TMac-TT. Simulation results for color image and video recovery show the clear advantage of our method over all other methods.

Efficient tensor completion for color image and video recovery: Low-rank tensor train

Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete information. Here we consider a further extension to the reconstruction of tensors of low multi-linear rank in recently introduced hierarchical tensor formats from a small number of measurements. Hierarchical tensors are a flexible generalization of the well-known Tucker representation, which have the advantage that the number of degrees of freedom of a low rank tensor does not scale exponentially with the order of the tensor. While corresponding tensor decompositions can be computed efficiently via successive applications of (matrix) singular value decompositions, some important properties of the singular value decomposition do not extend from the matrix to the tensor case. This results in major computational and theoretical difficulties in designing and analyzing algorithms for low rank tensor recovery. For instance, a canonical analogue of the tensor nuclear norm is NP-hard to compute in general, which is in stark contrast to the matrix case. In this book chapter we consider versions of iterative hard thresholding schemes adapted to hierarchical tensor formats. A variant builds on methods from Riemannian optimization and uses a retraction mapping from the tangent space of the manifold of low rank tensors back to this manifold. We provide first partial convergence results based on a tensor version of the restricted isometry property (TRIP) of the measurement map. Moreover, an estimate of the number of measurements is provided that ensures the TRIP of a given tensor rank with high probability for Gaussian measurement maps.

Tensor Completion in Hierarchical Tensor Representations

We study extensions of compressive sensing and low rank matrix recovery to the recovery of low rank tensors from incomplete linear information. While the reconstruction of low rank matrices via nuclear norm minimization is rather well-understand by now, almost no theory is available for the extension to higher order tensors due to various theoretical and computational difficulties arising for tensor decompositions. In fact, nuclear norm minimization for matrix recovery is a tractable convex relaxation approach, but the extension of the nuclear norm to tensors is in general NP-hard to compute. In this article, we introduce convex relaxations of the tensor nuclear norm which are computable in polynomial time via semidefinite programming. Our approach is based on theta bodies, a concept from computational algebraic geometry similar to the Lasserre relaxations. We introduce polynomial ideals which are generated by the second order minors corresponding to different matricizations of the tensor (where the tensor entries are treated as variables) such that the nuclear norm ball is the convex hull of the algebraic variety of the ideal. The $k$-th theta body for such an ideal generates a new norm which we call the $\theta_k$-norm. We show that in the matrix case, these norms reduce to the nuclear norm. For tensors of order $d \geq 3$ however, we obtain new norms. The sequence of the corresponding unit-$\theta_k$-norm balls converges asymptotically to the unit tensor nuclear norm ball. By providing the Grobner basis for the ideals, we explicitly give semidefinite programs for the computation of the $\theta_k$-norm and for the minimization of the $\theta_k$-norm under an affine constraint. Numerical experiments for order-3 tensor recovery via $\theta_1$-norm minimization suggest that our approach successfully reconstructs tensors of low rank from incomplete linear (random) measurements.

Željka Stojanac

Papers

Tensor Completion in Hierarchical Tensor Representations

Tensor theta norms and low rank recovery