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In machine learning, the term tensor informally refers to two different concepts (i) a way of organizing data and (ii) a multilinear (tensor) transformation. Data may be organized in a multidimensional array (M-way array), 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 ...
The transform has a null-space, assuming the components are smooth and decay at infinity any =, the symmetrized derivative of a rank m-1 tensor field , satisfies =. [1] More generally the Saint-Venant tensor W f {\displaystyle Wf} can be recovered uniquely by an explicit formula.
A metric tensor is a (symmetric) (0, 2)-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index.
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T • (V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product.It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property ...
the tensor product of two objects A 1, ..., A n and B 1, ..., B m is the concatenation A 1, ..., A n, B 1, ..., B m of the two lists, and, similarly, the tensor product of two morphisms is given by the concatenation of lists. The identity object is the empty list. This operation Σ mapping category C to Σ(C) can be extended to a strict 2-monad ...
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor [1]) is a tensor that can be written as a product of tensors of the form = where a, b, ..., d are nonzero and in V or V ∗ – that is, if the tensor is nonzero and completely factorizable. Every tensor can be expressed as a sum of simple tensors.
Tensor sketches can be used to decrease the number of variables needed when implementing Bilinear Pooling in a neural network. Bilinear pooling is the technique of taking two input vectors, x , y {\displaystyle x,y} from different sources, and using the tensor product x ⊗ y {\displaystyle x\otimes y} as the input layer to a neural network.