A custom initializer can be defined to draw weight parameters from a complex, non-standard probability distribution. For instance, a neural network's weight parameter $$w$$ could be initialized using the following piecewise distribution:
$$w \sim \begin{cases} U(5, 10) & \textrm{ with probability } \frac{1}{4} \\ 0 & \textrm{ with probability } \frac{1}{2} \\ U(-10, -5) & \textrm{ with probability } \frac{1}{4} \end{cases}$$
This demonstrates the flexibility of defining custom initialization logic beyond standard Gaussian or uniform distributions.

Claude

When standard initialization methods are insufficient, deep learning frameworks allow practitioners to define custom parameter initialization routines. This is achieved by creating a custom function or class that applies a desired mathematical distribution or logic to a given parameter tensor. Once defined, this custom initializer can be applied to the neural network to populate the weights according to the specified custom logic.

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