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Vectorization is a technique in programming and linear algebra that allows operations to be applied to entire arrays or matrices of data at once, rather than looping through each element individually. This can lead to significant computational efficiency, especially when using hardware like GPUs that are optimized for parallel processing.

When training a neural network, processing inputs in batches allows us to take advantage of vectorized operations, which can speed up the forward propagation process. Instead of computing the forward pass for each input individually, the computations are performed simultaneously for the entire batch of inputs.

The key benefits of using batch processing and vectorization include:

1. Parallelization: Modern hardware, such as GPUs, can perform parallel computations efficiently. Batch processing enables parallelization, as the forward pass for each input in the batch can be processed simultaneously.

2. Computational Efficiency: Vectorized operations are often more efficient than their non-vectorized counterparts. This is because they leverage optimized linear algebra libraries that are highly efficient.

3. Memory Efficiency: Batch processing can also lead to better utilization of the computer's memory, as it allows for more efficient data storage and retrieval.

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