Tensors Explained

Figure 1 depicts a series of vectors. A vector is a tensor of rank 1. Specific vectors have a fixed number of dimensions, one for each value. So the vector in Figure 1b represents a point in three dimensional space. The vector in 1c describes a way of labelling the dimensions of vectors, where the first element, Xi, is labelled with a subscript 1, the second with a subscript 2, and so on for how ever many elements there are.

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Figure 2 depicts a series of matrices. A matrix is a tensor of rank 2. The matrix in Figure 2c shows how you label the dimensions of a rank 2 tensor, by numbering the row (horizontal line) and column (vertical line) in which each dimension belongs. So the element in the first row and first column, Xi,j, is labelled with the subscript 1,1, while the element in the third row and fourth column is labelled with the subscript 3,4.

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Figure 3 depicts a tensor of rank 3. You can now generalise from the first three figures what a tensor of rank 4, and even rank n, would look like (although these are hard things to visualise). A tensor of rank zero is a scalar - a single value, eg. 0.37.

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Operations on tensors

There are three tensor operations of use in modelling memory processes: tensor addition, dot (or inner) products, and cross (or outer) products.

Tensor addition superimposes tensors of the same dimensions over each other to produce a new aggregate tensor. It is calculated simply by adding together the elements in a particular position in each tensor (see Figure 4). So Tij..n = Aij..n + Bij..n +..+ Mij..n. A tensor memory of multiple items is formed via this superimposition process.

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A dot product between two tensors is calculated by multiplying together the values which are in the same position within the two tensors, and then adding the results of these multiplications together to get a scalar (see Figure 5). The dot product of two tensors represents the level of similarity between them.

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A cross product between two tensors is calculated by multiplying each element in one tensor by each element in the other tensor (see Figure 6). The resulting values are arranged to form a new tensor which has a rank equivalent to the original two ?added? together. So the cross product of a rank 2 tensor with a rank three tensor produces a rank 5 tensor. Within memory modelling, the cross product of a cue tensor with an item tensor forms a memory trace.

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