bilinear

→Tensor

bilinear(x1: Tensor, x2: Tensor, weight: Tensor, bias: Tensor | None = None)

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Bilinear transformation applied to two inputs.

Computes a quadratic interaction between every pair of features from x1 and x2. Common in fine-grained recognition (bilinear pooling), tensor-decomposition attention, and gated cross-modal fusion layers.

Parameters

x1Tensor

First input, shape (..., in1_features).

x2Tensor

Second input, shape (..., in2_features).

weightTensor

Weight tensor, shape (out_features, in1_features, in2_features).

biasTensor= None

Bias vector, shape (out_features,). Default None.

Returns

Tensor

Output tensor, shape (..., out_features).

Notes

For each output channel $k$ ,

y_k = x_1^\top W_k x_2 + b_k,

where $W_k$ is the $k$ -th slice of the weight tensor. The parameter count grows multiplicatively in $\text{in1} \times \text{in2}$ , so bilinear layers are usually kept narrow. Reduces to two ordinary linear layers when $W_k = u_k v_k^\top$ is rank-one (the factorised bilinear trick).

Examples

>>> import lucid
>>> from lucid.nn.functional import bilinear
>>> x1 = lucid.randn(2, 3)
>>> x2 = lucid.randn(2, 4)
>>> w = lucid.randn(5, 3, 4)
>>> bilinear(x1, x2, w).shape
(2, 5)

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