An activation function is defined by the formula: Output = σ(Input ⋅ W₁ + b₁) ⊙ (Input ⋅ W₂ + b₂) where Input is a vector, W₁, W₂, b₁, b₂ are learnable parameters, σ is a non-linear function (such as the sigmoid function), and ⊙ denotes the element-wise product. What is the primary functional role of the σ(Input ⋅ W₁ + b₁) component in this architecture?

Calculating the Output of a Gated Activation

In the formula for a gated activation, $\sigma(\mathbf{hW}_1 + \mathbf{b}_1) \odot (\mathbf{hW}_2 + \mathbf{b}_2)$, what is the primary reason that the two resulting vectors, one from each parallel path, must have the same dimensions?