chainer.functions.bilinear

chainer.functions.bilinear(e1, e2, W, V1=None, V2=None, b=None)

Applies a bilinear function based on given parameters.

This is a building block of Neural Tensor Network (see the reference paper below). It takes two input variables and one or four parameters, and outputs one variable.

To be precise, denote six input arrays mathematically by $e^1\in \mathbb{R}^{I\cdot J}$, $e^2\in \mathbb{R}^{I\cdot K}$, $W\in \mathbb{R}^{J \cdot K \cdot L}$, $V^1\in \mathbb{R}^{J \cdot L}$, $V^2\in \mathbb{R}^{K \cdot L}$, and $b\in \mathbb{R}^{L}$, where $I$ is mini-batch size. In this document, we call $V^1$, $V^2$, and $b$ linear parameters.

The output of forward propagation is calculated as

$$y_{il} = \sum_{jk} e^1_{ij} e^2_{ik} W_{jkl} + \ \sum_{j} e^1_{ij} V^1_{jl} + \sum_{k} e^2_{ik} V^2_{kl} + b_{l}.$$

Note that V1, V2, b are optional. If these are not given, then this function omits the last three terms in the above equation.

Note

This function accepts an input variable e1 or e2 of a non-matrix array. In this case, the leading dimension is treated as the batch dimension, and the other dimensions are reduced to one dimension.

Note

In the original paper, $J$ and $K$ must be equal and the author denotes $[V^1 V^2]$ (concatenation of matrices) by $V$.

Parameters

• e1 (Variable or N-dimensional array) – Left input variable.

• e2 (Variable or N-dimensional array) – Right input variable.

• W (Variable or N-dimensional array) – Quadratic weight variable.

• V1 (Variable or N-dimensional array) – Left coefficient variable.

• V2 (Variable or N-dimensional array) – Right coefficient variable.

• b (Variable or N-dimensional array) – Bias variable.

Returns

Output variable.

Return type

Variable

See:

Reasoning With Neural Tensor Networks for Knowledge Base Completion [Socher+, NIPS2013].