code: def nplr(
      measure, N, rank=1, dtype=torch.float, diagonalize_precision=True
  ):
      """Decompose as Normal Plus Low-Rank (NPLR).
  
      Return w, p, q, V, B such that
      (w - p q^*, B) is unitarily equivalent to the original HiPPO A, B by the matrix V
      i.e. A = V[w - p q^*]V^*, B = V B
      """
      assert dtype == torch.float or torch.double
      cdtype = torch.cfloat if dtype == torch.float else torch.cdouble
  
      A, B = transition(measure, N)
      A = torch.as_tensor(A, dtype=dtype)  # (N, N)
      B = torch.as_tensor(B, dtype=dtype)[:, 0]  # (N,)
  
      P = rank_correction(measure, N, rank=rank, dtype=dtype)  # (r N)
      AP = A + torch.sum(P.unsqueeze(-2) * P.unsqueeze(-1), dim=-3)
  
      # We require AP to be nearly skew-symmetric
      _A = AP + AP.transpose(-1, -2)
      # if not torch.allclose(_A - _A[0,0]*torch.eye(N), torch.zeros(N, N), atol=1e-5):
      err = torch.sum((_A - _A[0, 0] * torch.eye(N)) ** 2) / N
      if err > 1e-5:
          print("WARNING: HiPPO matrix not skew symmetric", err)
  
      # Take advantage of identity + skew-symmetric form
      # to calculate real and imaginary parts separately
      # Imaginary part can use eigh instead of eig
      w_re = torch.mean(torch.diagonal(AP), -1, keepdim=True)
  
      # Diagonalize in double precision
      if diagonalize_precision:
          AP = AP.to(torch.double)
      w_im, V = torch.linalg.eigh(AP * -1j)  # (..., N) (..., N, N)
      if diagonalize_precision:
          w_im, V = w_im.to(cdtype), V.to(cdtype)
      w = w_re + 1j * w_im
      # Check: V w V^{-1} = A
      # print("check", V @ torch.diag_embed(w) @ V.conj().transpose(-1, -2))
  
      # Only keep half of each conjugate pair
      _, idx = torch.sort(w.imag)
      w_sorted = w[idx]
      V_sorted = V[:, idx]
  
      # There is an edge case when eigenvalues can be 0,
      # which requires some machinery to handle
      # We use a huge hack here: Assume only one pair is 0,
      # and that it is the first row/column of A (only happens in Fourier case)
      V = V_sorted[:, : N // 2]
      w = w_sorted[: N // 2]
      assert (
          w[-2].abs() > 1e-4
      ), "Only 1 zero eigenvalue allowed in diagonal part of A"
      if w[-1].abs() < 1e-4:
          V[:, -1] = 0.0
          V[0, -1] = 2**-0.5
          V[1, -1] = 2**-0.5 * 1j
  
      _AP = V @ torch.diag_embed(w) @ V.conj().transpose(-1, -2)
      err = torch.sum((2 * _AP.real - AP) ** 2) / N
      if err > 1e-5:
          print(
              "Warning: Diagonalization of A matrix not numerically precise - error",
              err,
          )
      # print("check", V @ torch.diag_embed(w) @ V.conj().transpose(-1, -2))
  
      V_inv = V.conj().transpose(-1, -2)
  
      B = contract("ij, j -> i", V_inv, B.to(V))  # V^* B
      P = contract("ij, ...j -> ...i", V_inv, P.to(V))  # V^* P
  
      return w, P, B, V
  
imports: import logging
  import math
  import os
  from functools import wraps
  from typing import Any, Callable, Optional
  import numpy
  import opt_einsum
  import torch
  import torch.nn
  import torch.nn.functional
  from einops import rearrange, repeat
  from espnet2.asr.state_spaces.components import Activation, DropoutNd, LinearActivation
label_desc: Use image node_id 8 for calling a global function with example usage: nplr(measure, N, rank, dtype, diagonalize_precision) and returns: w,  P,  B,  V