ترغب بنشر مسار تعليمي؟ اضغط هنا

Reducing Computational Complexity of Tensor Contractions via Tensor-Train Networks

125   0   0.0 ( 0 )
 نشر من قبل Yao Lei Xu
 تاريخ النشر 2021
  مجال البحث الهندسة المعلوماتية
والبحث باللغة English




اسأل ChatGPT حول البحث

There is a significant expansion in both volume and range of applications along with the concomitant increase in the variety of data sources. These ever-expanding trends have highlighted the necessity for more versatile analysis tools that offer greater opportunities for algorithmic developments and computationally faster operations than the standard flat-view matrix approach. Tensors, or multi-way arrays, provide such an algebraic framework which is naturally suited to data of such large volume, diversity, and veracity. Indeed, the associated tensor decompositions have demonstrated their potential in breaking the Curse of Dimensionality associated with traditional matrix methods, where a necessary exponential increase in data volume leads to adverse or even intractable consequences on computational complexity. A key tool underpinning multi-linear manipulation of tensors and tensor networks is the standard Tensor Contraction Product (TCP). However, depending on the dimensionality of the underlying tensors, the TCP also comes at the price of high computational complexity in tensor manipulation. In this work, we resort to diagrammatic tensor network manipulation to calculate such products in an efficient and computationally tractable manner, by making use of Tensor Train decomposition (TTD). This has rendered the underlying concepts easy to perceive, thereby enhancing intuition of the associated underlying operations, while preserving mathematical rigour. In addition to bypassing the cumbersome mathematical multi-linear expressions, the proposed Tensor Train Contraction Product model is shown to accelerate significantly the underlying computational operations, as it is independent of tensor order and linear in the tensor dimension, as opposed to performing the full computations through the standard approach (exponential in tensor order).



قيم البحث

اقرأ أيضاً

The accurate approximation of high-dimensional functions is an essential task in uncertainty quantification and many other fields. We propose a new function approximation scheme based on a spectral extension of the tensor-train (TT) decomposition. We first define a functional version of the TT decomposition and analyze its properties. We obtain results on the convergence of the decomposition, revealing links between the regularity of the function, the dimension of the input space, and the TT ranks. We also show that the regularity of the target function is preserved by the univariate functions (i.e., the cores) comprising the functional TT decomposition. This result motivates an approximation scheme employing polynomial approximations of the cores. For functions with appropriate regularity, the resulting textit{spectral tensor-train decomposition} combines the favorable dimension-scaling of the TT decomposition with the spectral convergence rate of polynomial approximations, yielding efficient and accurate surrogates for high-dimensional functions. To construct these decompositions, we use the sampling algorithm texttt{TT-DMRG-cross} to obtain the TT decomposition of tensors resulting from suitable discretizations of the target function. We assess the performance of the method on a range of numerical examples: a modifed set of Genz functions with dimension up to $100$, and functions with mixed Fourier modes or with local features. We observe significant improvements in performance over an anisotropic adaptive Smolyak approach. The method is also used to approximate the solution of an elliptic PDE with random input data. The open source software and examples presented in this work are available online.
105 - Yongming Zheng , An-Bao Xu 2020
In this paper, we consider the tensor completion problem, which has many researchers in the machine learning particularly concerned. Our fast and precise method is built on extending the $L_{2,1}$-norm minimization and Qatar Riyal decomposition (LNM- QR) method for matrix completions to tensor completions, and is different from the popular tensor completion methods using the tensor singular value decomposition (t-SVD). In terms of shortening the computing time, t-SVD is replaced with the method computing an approximate t-SVD based on Qatar Riyal decomposition (CTSVD-QR), which can be used to compute the largest $r left(r>0 right)$ singular values (tubes) and their associated singular vectors (of tubes) iteratively. We, in addition, use the tensor $L_{2,1}$-norm instead of the tensor nuclear norm to minimize our model on account of it is easy to optimize. Then in terms of improving accuracy, ADMM, a gradient-search-based method, plays a crucial part in our method. Numerical experimental results show that our method is faster than those state-of-the-art algorithms and have excellent accuracy.
123 - Chao Zeng 2021
The orthogonal decomposition factorizes a tensor into a sum of an orthogonal list of rankone tensors. We present several properties of orthogonal rank. We find that a subtensor may have a larger orthogonal rank than the whole tensor and prove the low er semicontinuity of orthogonal rank. The lower semicontinuity guarantees the existence of low orthogonal rank approximation. To fit the orthogonal decomposition, we propose an algorithm based on the augmented Lagrangian method and guarantee the orthogonality by a novel orthogonalization procedure. Numerical experiments show that the proposed method has a great advantage over the existing methods for strongly orthogonal decompositions in terms of the approximation error.
120 - Changxin Mo , Weiyang Ding 2021
Perturbation analysis has been primarily considered to be one of the main issues in many fields and considerable progress, especially getting involved with matrices, has been made from then to now. In this paper, we pay our attention to the perturbat ion analysis subject on tensor eigenvalues under tensor-tensor multiplication sense; and also {epsilon}-pseudospectra theory for third-order tensors. The definition of the generalized T-eigenvalue of third-order tensors is given. Several classical results, such as the Bauer-Fike theorem and its general case, Gershgorin circle theorem and Kahan theorem, are extended from matrix to tensor case. The study on {epsilon}-pseudospectra of tensors is presented, together with various pseudospectra properties and numerical examples which show the boundaries of the {epsilon}-pseudospectra of certain tensors under different levels.
Tensor Train decomposition is used across many branches of machine learning. We present T3F -- a library for Tensor Train decomposition based on TensorFlow. T3F supports GPU execution, batch processing, automatic differentiation, and versatile functi onality for the Riemannian optimization framework, which takes into account the underlying manifold structure to construct efficient optimization methods. The library makes it easier to implement machine learning papers that rely on the Tensor Train decomposition. T3F includes documentation, examples and 94% test coverage.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا