![]() ![]() Tensor (multilinear) factor analysis disentangles and reduces the influence of different causal factors with multilinear subspace learning. Multilinear algebra, the algebra of higher-order tensors, is a suitable and transparent framework for analyzing the multifactor structure of an ensemble of obsevations and for addressing the difficult problem of disentangling the causal factors based on second order or higher order statistics associated with each causal factor. In the early 2000s, multilinear tensor methods crossed over into computer vision, computer graphics and machine learning with papers by Vasilescu or in collaboration with Terzopoulos, such as Human Motion Signatures, TensorFaces TensorTexures and Multilinear Projection. He noted several early limitations in determining the tensor rank and efficient tensor rank decomposition. Linear tensor rank methods (such as, Parafac/CANDECOMP) analyzed M-way arrays ("data tensors") composed of higher order statistics that were employed in blind source separation problems to compute a linear model of the data. Pierre Comon surveys the early adoption of tensor methods in the fields of telecommunications, radio surveillance, chemometrics and sensor processing. In 2001, the field of signal processing and statistics were making use of tensor methods. In machine learning, the exact use of tensors depends on the statistical approach being used. In physics, tensor fields, considered as tensors at each point in space, are useful in expressing mechanics such as stress or elasticity. In mathematics, this may express a multilinear relationship between sets of algebraic objects. These developments have greatly accelerated neural network architectures and increased the size and complexity of models that can be trained.Ī tensor is by definition a multilinear map. Ĭomputations are often performed on graphics processing units (GPUs) using CUDA and on dedicated hardware such as Google's Tensor Processing Unit or Nvidia's Tensor core. The computation of gradients, an important aspect of the backpropagation algorithm, can be performed using PyTorch and TensorFlow. Operations on data tensors can be expressed in terms of matrix multiplication and the Kronecker product. Tensor methods can factorize data tensors into smaller tensors. Observations, such as images, movies, volumes, sounds, and relationships among words and concepts, stored in an M-way array ("data tensor") may be analyzed either by artificial neural networks or tensor methods. Data may be organized in a multidimensional array ( M-way array) that is informally referred to as a "data tensor" however in the strict mathematical sense, a tensor is a multilinear mapping over a set of domain vector spaces to a range vector space. In machine learning, the word tensor informally refers to two different concepts that organize and represent data. ![]()
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