code: def monte_carlo_kernel_fn(
      init_fn: InitFn,
      apply_fn: ApplyFn,
      key: jax.Array,
      n_samples: Union[int, Iterable[int]],
      batch_size: int = 0,
      device_count: int = -1,
      store_on_device: bool = True,
      trace_axes: Axes = (-1,),
      diagonal_axes: Axes = (),
      vmap_axes: Optional[VMapAxes] = None,
      implementation: Union[
          int, NtkImplementation
      ] = DEFAULT_NTK_IMPLEMENTATION,
      _j_rules: bool = _DEFAULT_NTK_J_RULES,
      _s_rules: bool = _DEFAULT_NTK_S_RULES,
      _fwd: Optional[bool] = _DEFAULT_NTK_FWD,
  ) -> MonteCarloKernelFn:
      r"""Return a Monte Carlo sampler of NTK and NNGP kernels of a given function.
  
      Note that the returned function is appropriately batched / parallelized. You
      don't need to apply the `nt.batch` or `jax.jit` decorators  to it. Further,
      you do not need to apply `jax.jit` to the input `apply_fn` function, as the
      resulting empirical kernel function is JITted internally.
  
      Args:
        init_fn:
          a function initializing parameters of the neural network. From
          :obj:`jax.example_libraries.stax`: "takes an rng key and an input shape
          and returns an `(output_shape, params)` pair".
  
        apply_fn:
          a function computing the output of the neural network.
          From :obj:`jax.example_libraries.stax`: "takes params, inputs, and an
          rng key and applies the layer".
  
        key:
          RNG (`jax.random.PRNGKey`) for sampling random networks. Must have
          shape `(2,)`.
  
        n_samples:
          number of Monte Carlo samples. Can be either an integer or an
          iterable of integers at which the resulting generator will yield
          estimates. Example: use `n_samples=[2**k for k in range(10)]` for the
          generator to yield estimates using 1, 2, 4, ..., 512 Monte Carlo samples.
        batch_size: an integer making the kernel computed in batches of `x1` and
          `x2` of this size. `0` means computing the whole kernel. Must divide
          `x1.shape[0]` and `x2.shape[0]`.
  
        device_count:
          an integer making the kernel be computed in parallel across
          this number of devices (e.g. GPUs or TPU cores). `-1` means use all
          available devices. `0` means compute on a single device sequentially. If
          not `0`, must divide `x1.shape[0]`.
  
        store_on_device:
          a boolean, indicating whether to store the resulting
          kernel on the device (e.g. GPU or TPU), or in the CPU RAM, where larger
          kernels may fit.
  
        trace_axes:
          output axes to trace the output kernel over, i.e. compute only the trace
          of the covariance along the respective pair of axes (one pair for each
          axis in `trace_axes`). This allows to save space and compute if you are
          only interested in the respective trace, but also improve approximation
          accuracy if you know that covariance along these pairs of axes converges
          to a `constant * identity matrix` in the limit of interest (e.g.
          infinite width or infinite `n_samples`). A common use case is the channel
          / feature / logit axis, since activation slices along such axis are i.i.d.
          and the respective covariance along the respective pair of axes indeed
          converges to a constant-diagonal matrix in the infinite width or infinite
          `n_samples` limit.
          Also related to "contracting dimensions" in XLA terms.
          (https://www.tensorflow.org/xla/operation_semantics#dotgeneral)
  
        diagonal_axes:
          output axes to diagonalize the output kernel over, i.e. compute only the
          diagonal of the covariance along the respective pair of axes (one pair for
          each axis in `diagonal_axes`). This allows to save space and compute, if
          off-diagonal values along these axes are not needed, but also improve
          approximation accuracy if their limiting value is known theoretically,
          e.g. if they vanish in the limit of interest (e.g. infinite
          width or infinite `n_samples`). If you further know that on-diagonal
          values converge to the same constant in your limit of interest, you should
          specify these axes in `trace_axes` instead, to save even more compute and
          gain even more accuracy. A common use case is computing the variance
          (instead of covariance) along certain axes.
          Also related to "batch dimensions" in XLA terms.
          (https://www.tensorflow.org/xla/operation_semantics#dotgeneral)
  
        vmap_axes:
          applicable only to NTK. A triple of `(in_axes, out_axes, kwargs_axes)`
          passed to `vmap` to evaluate the empirical NTK in parallel ove these axes.
          Precisely, providing this argument implies that `f(params, x, **kwargs)`
          equals to a concatenation along `out_axes` of `f` applied to slices of
          `x` and `**kwargs` along `in_axes` and `kwargs_axes`, i.e. `f` can be
          evaluated as a `vmap`. This allows to evaluate Jacobians much more
          efficiently. If `vmap_axes` is not a triple, it is interpreted as
          `in_axes = out_axes = vmap_axes, kwargs_axes = {}`. For example a very
          common usecase is `vmap_axes=0` for a neural network with leading (`0`)
          batch dimension, both for inputs and outputs, and no interactions between
          different elements of the batch (e.g. no BatchNorm, and, in the case of
          `nt.stax`, also no Dropout). However, if there is interaction between
          batch elements or no concept of a batch axis at all, `vmap_axes` must be
          set to `None`, to avoid wrong (and potentially silent) results.
  
        implementation:
          Applicable only to NTK, an :class:`NtkImplementation` value (or an
          :class:`int`  `0`, `1`, `2`, or `3`). See the :class:`NtkImplementation`
          docstring for details.
  
        _j_rules:
          Internal debugging parameter, applicable only to NTK when
          `implementation` is :attr:`~NtkImplementation.STRUCTURED_DERIVATIVES`
          (`3`) or :attr:`~NtkImplementation.AUTO` (`0`). Set to `True` to allow
          custom Jacobian rules for intermediary primitive `dy/dw` computations for
          MJJMPs (matrix-Jacobian-Jacobian-matrix products). Set to `False` to use
          JVPs or VJPs, via JAX's :obj:`jax.jacfwd` or :obj:`jax.jacrev`. Custom
          Jacobian rules (`True`) are expected to be not worse, and sometimes better
          than automated alternatives, but in case of a suboptimal implementation
          setting it to `False` could improve performance.
  
        _s_rules:
          Internal debugging parameter, applicable only to NTK when
          `implementation` is :attr:`~NtkImplementation.STRUCTURED_DERIVATIVES`
          (`3`) or :attr:`~NtkImplementation.AUTO` (`0`). Set to `True` to allow
          efficient MJJMp rules for structured `dy/dw` primitive Jacobians. In
          practice should be set to `True`, and setting it to `False` can lead to
          dramatic deterioration of performance.
  
        _fwd:
          Internal debugging parameter, applicable only to NTK when
          `implementation` is :attr:`~NtkImplementation.STRUCTURED_DERIVATIVES`
          (`3`) or :attr:`~NtkImplementation.AUTO` (`0`). Set to `True` to allow
          :obj:`jax.jvp` in intermediary primitive Jacobian `dy/dw` computations,
          `False` to always use :obj:`jax.vjp`. `None` to decide automatically
          based on input/output sizes. Applicable when `_j_rules=False`, or when a
          primitive does not have a Jacobian rule. Should be set to `None` for best
          performance.
  
      Returns:
        If `n_samples` is an integer, returns a function of signature
        `kernel_fn(x1, x2, get)` that returns an MC estimation of the kernel using
        `n_samples`. If `n_samples` is a collection of integers,
        `kernel_fn(x1, x2, get)` returns a generator that yields estimates using
        `n` samples for `n in n_samples`.
  
      Example:
        >>> from jax import random
        >>> import neural_tangents as nt
        >>> from neural_tangents import stax
        >>> #
        >>> key1, key2 = random.split(random.PRNGKey(1), 2)
        >>> x_train = random.normal(key1, (20, 32, 32, 3))
        >>> y_train = random.uniform(key1, (20, 10))
        >>> x_test = random.normal(key2, (5, 32, 32, 3))
        >>> #
        >>> init_fn, apply_fn, _ = stax.serial(
        >>>     stax.Conv(128, (3, 3)),
        >>>     stax.Relu(),
        >>>     stax.Conv(256, (3, 3)),
        >>>     stax.Relu(),
        >>>     stax.Conv(512, (3, 3)),
        >>>     stax.Flatten(),
        >>>     stax.Dense(10)
        >>> )
        >>> #
        >>> n_samples = 200
        >>> kernel_fn = nt.monte_carlo_kernel_fn(init_fn, apply_fn, key1, n_samples)
        >>> kernel = kernel_fn(x_train, x_test, get=('nngp', 'ntk'))
        >>> # `kernel` is a tuple of NNGP and NTK MC estimate using `n_samples`.
        >>> #
        >>> n_samples = [1, 10, 100, 1000]
        >>> kernel_fn_generator = nt.monte_carlo_kernel_fn(init_fn, apply_fn, key1,
        >>>                                                n_samples)
        >>> kernel_samples = kernel_fn_generator(x_train, x_test,
        >>>                                      get=('nngp', 'ntk'))
        >>> for n, kernel in zip(n_samples, kernel_samples):
        >>>   print(n, kernel)
        >>>   # `kernel` is a tuple of NNGP and NTK MC estimate using `n` samples.
      """
      kwargs = dict(
          f=apply_fn,
          trace_axes=trace_axes,
          diagonal_axes=diagonal_axes,
          vmap_axes=vmap_axes,
          implementation=implementation,
          _s_rules=_s_rules,
          _j_rules=_j_rules,
          _fwd=_fwd,
      )
  
      kernel_fn = empirical_kernel_fn(**kwargs)
  
      kernel_fn_sample_once = _sample_once_kernel_fn(
          kernel_fn=kernel_fn,
          init_fn=init_fn,
          batch_size=batch_size,
          device_count=device_count,
          store_on_device=store_on_device,
      )
  
      n_samples, get_generator = _canonicalize_n_samples(n_samples)
      kernel_fn = _sample_many_kernel_fn(
          kernel_fn_sample_once=kernel_fn_sample_once,
          key=key,
          n_samples=n_samples,
          get_generator=get_generator,
      )
      return kernel_fn
  
imports: from functools import partial
  import operator
  from typing import Generator, Iterable, Optional, Union
  import jax
  from jax import random
  import jax.numpy
  from jax.tree_util import tree_map
  from .batching import batch
  from .empirical import _DEFAULT_NTK_FWD
  from .empirical import _DEFAULT_NTK_J_RULES
  from .empirical import _DEFAULT_NTK_S_RULES
  from .empirical import DEFAULT_NTK_IMPLEMENTATION
  from .empirical import empirical_kernel_fn
  from .empirical import NtkImplementation
  from .utils import utils
  from .utils.typing import ApplyFn
  from .utils.typing import Axes
  from .utils.typing import EmpiricalGetKernelFn
  from .utils.typing import Get
  from .utils.typing import InitFn
  from .utils.typing import MonteCarloKernelFn
  from .utils.typing import NTTree
  from .utils.typing import PyTree
  from .utils.typing import VMapAxes
label_desc: Use image node_id 3 for calling a global function with example usage: monte_carlo_kernel_fn(init_fn, apply_fn, key, n_samples, batch_size, device_count, store_on_device, trace_axes, diagonal_axes, vmap_axes, implementation, _j_rules, _s_rules, _fwd) and returns: kernel_fn