code: def estimate_largest_eigenvalue_of_covariance(x, remove_mean=True):
      """Estimate the largest eigenvalue of a covariance matrix given a sample.
  
      Implements Algorithm 2 from [1].
  
      We assume the rows of the input matrix `x` are i.i.d. draws from some
      distribution with (unknown) covariance matrix `sigma`. We want an estimate of
      the largest eigenvalue of `sigma`, but we cannot directly observe `sigma`.
      The naive estimator
      ```
      sigma_hat = x.T.dot(x) / (x.shape[0] - 1)
      max_eig = np.linalg.eigvalsh(sigma_hat).max()
      ```
      can be quite biased if the number of rows in x is not quite large, even
      though `sigma_hat` is unbiased. Instead, we compute the ratio of the sum
      of `sigma_hat`'s squared eigenvalues of to the sum of its eigenvalues.
      Although this estimator is asymptotically biased, it tends to be much more
      accurate than the naive estimator above unless `x` has a very large number
      of rows.
  
      [1] M.D. Hoffman and P. Sountsov, "Tuning-Free Generalized Hamiltonian Monte
        Carlo," AISTATS 2022.
  
      Args:
        x: A matrix whose rows are independent draws from the distribution whose
          covariance we are interested in.
        remove_mean: A boolean flag indicating whether or not to replace `x` with `x
          - x.mean(0)`.
  
      Returns:
        max_eig: An estimate of the largest eigenvalue of the covariance of the
          distribution that generated `x`.
      """
      x = x - remove_mean * x.mean(0)
      trace_est = (x**2).sum() / x.shape[0]
      # Note that this has a cost that's quadratic in num_chains. This should only
      # be an issue if cost_per_gradient < num_chains*num_dimensions. In this
      # presumably uncommon event, we could potentially use the Hutchinson
      # trace estimator.
      trace_sq_est = (x.dot(x.T) ** 2).sum() / x.shape[0] ** 2
      return trace_sq_est / trace_est
  
imports: import functools
  import os
  from typing import Any, Callable, Dict, Optional, Mapping, NamedTuple, Tuple, Union
  from etils import epath
  import gin
  import immutabledict
  import jax
  import jax.numpy
  import numpy
  from discussion.adaptive_malt import utils
  from fun_mc import using_jax
  from inference_gym import using_jax
  import tensorflow_probability.substrates.jax
label_desc: Use image node_id 13 for calling a global function with example usage: estimate_largest_eigenvalue_of_covariance(x, remove_mean) and returns: unknown