Tensor (machine learning)

Neural network with tensors as node values

Tensors may be used as the unit values of neural networks which extend the concept of scalar, vector and matrix values to multiple dimensions.

The output value of single layer unit ${\displaystyle y_{m}}$ is the sum-product of its input units and the connection weights filtered through the activation function ${\displaystyle f}$:

${\displaystyle y_{m}=f\left(\sum _{n}x_{n}u_{m,n}\right),}$

where

${\displaystyle y_{m}\in \mathbb {R} .}$

Convolutional neural network with each unit as a 2D image represented by a tensor

If each output element of ${\displaystyle y_{m}}$ is a scalar, then we have the classical definition of an artificial neural network. By replacing each unit component with a tensor, the network is able to express higher dimensional data such as images or videos:

${\displaystyle y_{m}\in \mathbb {R} ^{I_{0}\times I_{1}\times ..\times I_{C}}.}$

This use of tensors to replace unit values is common in convolutional neural networks where each unit might be an image processed through multiple layers. By embedding the data in tensors such network structures enable learning of complex data types.

Two layers of a fully connected neural network and its expression as a tensor Kronecker product

Tensors may also be used to compute the layers of a fully connected neural network, where the tensor is applied to the entire layer instead of individual unit values.

The output value of single layer unit ${\displaystyle y_{m}}$ is the sum-product of its input units and the connection weights filtered through the activation function ${\displaystyle f}$:

${\displaystyle y_{m}=f\left(\sum _{n}x_{n}u_{m,n}\right).}$

The vectors ${\displaystyle x}$ and ${\displaystyle y}$ of output values can be expressed as a mode-1 tensors, while the hidden weights can be expressed as a mode-2 tensor. In this example the unit values are scalars while the tensor takes on the dimensions of the network layers:

${\displaystyle x_{n}\mapsto {\mathcal {X}}\in \mathbb {R} ^{1\times N},}$

${\displaystyle y_{n}\mapsto {\mathcal {Y}}\in \mathbb {R} ^{M\times 1},}$

${\displaystyle u_{n}\mapsto {\mathcal {U}}\in \mathbb {R} ^{N\times M}.}$

In this notation, the output values can be computed as a tensor product of the input and weight tensors:

${\displaystyle {\mathcal {Y}}=f({\mathcal {X}}{\mathcal {U}}).}$

which computes the sum-product as a tensor multiplication (similar to matrix multiplication).

This formulation of tensors enables the entire layer of a fully connected network to be efficiently computed by mapping the units and weights to tensors.

A different reformulation of neural networks allows tensors to express the convolution layers of a neural network. A convolutional layer has multiple inputs, each of which is a spatial structure such as an image or volume. The inputs are convolved by filtering before being passed to the next layer. A typical use is to perform feature detection or isolation in image recognition.

Convolution is often computed as the multiplication of an input signal ${\displaystyle g}$ with a filter kernel ${\displaystyle f}$. In two dimensions the discrete, finite form is:

${\displaystyle (f*g)_{x,y}=\sum _{j=-w}^{w}\sum _{k=-w}^{w}f_{j,k}g_{x+j,y+k},}$

where ${\displaystyle w}$ is the width of the kernel.

This definition can be rephrased as a matrix-vector product in terms of tensors that express the kernel, data and inverse transform of the kernel.[29]

${\displaystyle {\mathcal {Y}}={\mathcal {A}}[(Cg)\odot (Bd)],}$

where ${\displaystyle {\mathcal {A}},{\mathcal {B}}}$ and ${\displaystyle {\mathcal {C}}}$ are the inverse transform, data and kernel. The derivation is more complex when the filtering kernel also includes a non-linear activation function such as sigmoid or ReLU.

The hidden weights of the convolution layer are the parameters to the filter. These can be reduced with a pooling layer which reduces the resolution (size) of the data, and can also be expressed as a tensor operation.

An important contribution of tensors in machine learning is the ability to factorize tensors to decompose data into constituent factors or reduce the learned parameters. Data tensor modeling techniques stem from the linear tensor decomposition (CANDECOMP/Parafac decomposition) and the multilinear tensor decompositions (Tucker).

Tucker decomposition of a tensor

Tucker decomposition, for example, takes a 3-way array ${\displaystyle {\mathcal {X}}\in \mathbb {R} ^{I\times J\times K}}$ and decomposes the tensor into three matrices ${\displaystyle {\mathcal {A,B,C}}}$ and a smaller tensor ${\displaystyle {\mathcal {G}}}$. The shape of the matrices and new tensor are such that the total number of elements is reduced. The new tensors have shapes

${\displaystyle {\mathcal {A}}\in \mathbb {R} ^{I\times P},}$

${\displaystyle {\mathcal {B}}\in \mathbb {R} ^{J\times Q},}$

${\displaystyle {\mathcal {C}}\in \mathbb {R} ^{K\times R},}$

${\displaystyle {\mathcal {G}}\in \mathbb {R} ^{P\times Q\times R}.}$

Then the original tensor can be expressed as the tensor product of these four tensors:

${\displaystyle {\mathcal {X}}={\mathcal {G}}\times {\mathcal {A}}\times {\mathcal {B}}\times {\mathcal {C}}.}$

In the example shown in the figure, the dimensions of the tensors are

${\displaystyle {\mathcal {X}}}$: I=8, J=6, K=3, ${\displaystyle {\mathcal {A}}}$: I=8, P=5, ${\displaystyle {\mathcal {B}}}$: J=6, Q=4, ${\displaystyle {\mathcal {C}}}$: K=3, R=2, ${\displaystyle {\mathcal {G}}}$: P=5, Q=4, R=2.

The total number of elements in the Tucker factorization is

${\displaystyle |{\mathcal {A}}|+|{\mathcal {B}}|+|{\mathcal {C}}|+|{\mathcal {G}}|=}$

${\displaystyle (I\times P)+(J\times Q)+(K\times R)+(P\times Q\times R)=8\times 5+6\times 4+3\times 2+5\times 4\times 2=110.}$

The number of elements in the original ${\displaystyle {\mathcal {X}}}$ is 144, resulting in a data reduction from 144 down to 110 elements, a reduction of 23% in parameters or data size. For much larger initial tensors, and depending on the rank (redundancy) of the tensor, the gains can be more significant.

The work of Rabanser et al. provides an introduction to tensors with more details on the extension of Tucker decomposition to N-dimensions beyond the mode-3 example given here.[5]

Another technique for decomposing tensors rewrites the initial tensor as a sequence (train) of smaller sized tensors. A tensor-train (TT) is a sequence of tensors of reduced rank, called canonical factors. The original tensor can be expressed as the sum-product of the sequence.

${\displaystyle {\mathcal {X}}={\mathcal {G_{1}}}{\mathcal {G_{2}}}{\mathcal {G_{3}}}..{\mathcal {G_{d}}}}$

Developed in 2011 by Ivan Oseledts, the author observes that Tucker decomposition is "suitable for small dimensions, especially for the three-dimensional case. For large d it is not suitable."[30] Thus tensor-trains can be used to factorize larger tensors in higher dimensions.

The unified data architecture and automatic differentiation of tensors has enabled higher-level designs of machine learning in the form of tensor graphs. This leads to new architectures, such as tensor-graph convolutional networks (TGCN), which identify highly non-linear associations in data, combine multiple relations, and scale gracefully, while remaining robust and performant.[31]

These developments are impacting all areas of machine learning, such as text mining and clustering, time varying data, and neural networks wherein the input data is a social graph and the data changes dynamically.[32][33][34][35]