code: def factorized_dot_product_attention(
      query: jnp.ndarray,
      key: jnp.ndarray,
      value: jnp.ndarray,
      bias: Optional[jnp.ndarray] = None,
      broadcast_dropout: bool = True,
      dropout_rng: Optional[Any] = None,
      dropout_rate: float = 0.1,
      deterministic: bool = False,
      dtype: jnp.dtype = jnp.float32,
      precision: Optional[jax.lax.Precision] = None,
  ) -> jnp.ndarray:
      """Applies head-factorized qkv dot-product attention.
  
      This factorizes the dot-product attention by assigning different
      heads to run attention on different axes.
  
  
      Args:
        query: Queries for calculating attention with shape of `[batch...,
          num_heads, qk_depth_per_head]`.
        key: Keys for calculating attention with shape of `[batch..., num_heads,
          qk_depth_per_head]`.
        value: Values to be used in attention with shape of `[batch..., num_heads,
          v_depth_per_head]`.
        bias: Bias for the attention weights. This should be
          broadcastable to the shape: `[batch...]`. This can be used for
            incorporating causal masks, padding masks, proximity bias, etc. Default
            is None, which means no bias is applied on attention matrix.
        broadcast_dropout: Use a broadcasted dropout along batch dims.
        dropout_rng: JAX PRNGKey to be used for dropout.
        dropout_rate: Dropout rate.
        deterministic: Deterministic or not (to apply dropout).
        dtype: The dtype of the computation (default: float32).
        precision: Numerical precision of the computation see `jax.lax.Precision`
          for details.
  
      Returns:
        Output of shape `[bs, ..., num_heads, features]`.
      """
      if query.shape != key.shape:
          raise ValueError(
              "Axial dot product attention only supports "
              "query and key with the same shape."
          )
  
      if bias is not None:
          raise ValueError(
              "Bias is not supported in "
              "factorized_dot_product_attention."
          )
  
      # Normalize the query with the square of its depth.
      query = query / jnp.sqrt(query.shape[-1]).astype(dtype)
      # Shape of query, key, and value: [bs, t, hw, h, c].
  
      prefix_str = "abcdefghijk"
      # Split heads for each axial attention dimension.
      num_attn_dimensions = (
          query.ndim - 3
      )  # all dims but bs, heads, and channel.
      if query.shape[-2] % num_attn_dimensions != 0:
          raise ValueError(
              f"In head-axial dot-product attention, number of "
              f"heads ({query.shape[-2]}) should be divisible by number "
              f"of attention dimensions ({num_attn_dimensions})!"
          )
  
      queries = jnp.split(query, num_attn_dimensions, axis=-2)
      keys = jnp.split(key, num_attn_dimensions, axis=-2)
      values = jnp.split(value, num_attn_dimensions, axis=-2)
      # queries, keys, and values are each a list with two arrays (sinec
      # we have two dims, t and hw) that are made by spliting heads:
      # [(bs, t, hw, h//2, c), (bs, t, hw, h//2, c)].
  
      outputs = []
      for i, (query, key, value) in enumerate(
          zip(queries, keys, values)
      ):
          # Shape of query, key, and value: [bs, t, hw, h//2, c].
          axis = i + 1  # to account for the batch dim
          batch_dims = prefix_str[:axis]
          einsum_str = (
              f"{batch_dims}x...z,{batch_dims}y...z->{batch_dims}x...y"
          )
          # For axis=1 einsum_str (q,k->a): ax...z,ay...z->ax...y
          # For axis=2 einsum_str (q,k->a): abx...z,aby...z->abx...y
          attn_logits = jnp.einsum(
              einsum_str, query, key, precision=precision
          )
          # For axis=1 (attention over t): attn_logits.shape: [bs, t, hw, h//2, t]
          # For axis=2 (attention over hw): attn_logits.shape: [bs, t, hw, h//2, hw]
          attn_weights = jax.nn.softmax(attn_logits, axis=-1)
  
          # Apply dropout.
          if not deterministic and dropout_rate > 0.0:
              if dropout_rng is None:
                  raise ValueError(
                      "Did not provide `rng` to dot_product_attention()."
                  )
              keep_prob = jax.lax.tie_in(
                  attn_weights, 1.0 - dropout_rate
              )
              if broadcast_dropout:
                  # Dropout is broadcast across the batch+head+non-attention dimension.
                  dropout_shape = list(attn_weights.shape)
                  dropout_shape[0] = 1  # Broadcast batch.
                  dropout_shape[-2] = 1  # Broadcast heads.
                  keep = jax.random.bernoulli(
                      dropout_rng, keep_prob, dropout_shape
                  )
              else:
                  keep = jax.random.bernoulli(
                      dropout_rng, keep_prob, attn_weights.shape
                  )
              multiplier = keep.astype(
                  attn_weights.dtype
              ) / jnp.asarray(keep_prob, dtype=attn_weights.dtype)
              attn_weights *= multiplier
  
          einsum_str = (
              f"{batch_dims}x...y,{batch_dims}y...z->{batch_dims}x...z"
          )
          # For axis=1 einsum_str (a,v->o): ax...y,ay...z->ax...z
          # For axis=2 einsum_str (a,v->o): abx...y,aby...z->abx...z
          outputs.append(
              jnp.einsum(
                  einsum_str, attn_weights, value, precision=precision
              )
          )
  
      # Output is list with two arrays [(bs, t, hw, h//2, c), (bs, t, hw, h//2, c)]
      # concatinate the heads.
      return jnp.concatenate(outputs, axis=-2)
  
imports: from typing import Any, Optional, Tuple
  from absl import logging
  import flax
  from flax.linen import linear
  from flax.training import common_utils
  import jax
  import jax.numpy
  import ml_collections
  import numpy
  from scenic.common_lib import debug_utils
  from scenic.model_lib.base_models import model_utils
  import scipy
label_desc: Use image node_id 3 for calling a global function with example usage: factorized_dot_product_attention(query, key, value, bias, broadcast_dropout, dropout_rng, dropout_rate, deterministic, dtype, precision) and returns: jnp