Fixed DEMove Implementation (#480)

* Implemented improved DEMove version

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* fixed bug

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>

src/emcee/moves/de.py

@@ -1,4 +1,5 @@
 # -*- coding: utf-8 -*-
+from functools import lru_cache
 
 import numpy as np
 
@@ -13,7 +14,7 @@ class DEMove(RedBlueMove):
     This `Differential evolution proposal
     <http://www.stat.columbia.edu/~gelman/stuff_for_blog/cajo.pdf>`_ is
     implemented following `Nelson et al. (2013)
-    <https://arxiv.org/abs/1311.5229>`_.
+    <https://doi.org/10.1088/0067-0049/210/1/11>`_.
 
     Args:
         sigma (float): The standard deviation of the Gaussian used to stretch
@@ -27,8 +28,7 @@ class DEMove(RedBlueMove):
     def __init__(self, sigma=1.0e-5, gamma0=None, **kwargs):
         self.sigma = sigma
         self.gamma0 = gamma0
-        kwargs["nsplits"] = 3
-        super(DEMove, self).__init__(**kwargs)
+        super().__init__(**kwargs)
 
     def setup(self, coords):
         self.g0 = self.gamma0
@@ -38,14 +38,40 @@ def setup(self, coords):
             self.g0 = 2.38 / np.sqrt(2 * ndim)
 
     def get_proposal(self, s, c, random):
-        Ns = len(s)
-        Nc = list(map(len, c))
-        ndim = s.shape[1]
-        q = np.empty((Ns, ndim), dtype=np.float64)
-        f = self.sigma * random.randn(Ns)
-        for i in range(Ns):
-            w = np.array([c[j][random.randint(Nc[j])] for j in range(2)])
-            random.shuffle(w)
-            g = np.diff(w, axis=0) * self.g0 + f[i]
-            q[i] = s[i] + g
-        return q, np.zeros(Ns, dtype=np.float64)
+        c = np.concatenate(c, axis=0)
+        ns, ndim = s.shape
+        nc = c.shape[0]
+
+        # Get the pair indices
+        pairs = _get_nondiagonal_pairs(nc)
+
+        # Sample from the pairs
+        indices = random.choice(pairs.shape[0], size=ns, replace=True)
+        pairs = pairs[indices]
+
+        # Compute diff vectors
+        diffs = np.diff(c[pairs], axis=1).squeeze(axis=1)  # (ns, ndim)
+
+        # Sample a gamma value for each walker following Nelson et al. (2013)
+        gamma = self.g0 * (1 + self.sigma * random.randn(ns, 1))  # (ns, 1)
+
+        # In this way, sigma is the standard deviation of the distribution of gamma,
+        # instead of the standard deviation of the distribution of the proposal as proposed by Ter Braak (2006).
+        # Otherwise, sigma should be tuned for each dimension, which confronts the idea of affine-invariance.
+
+        q = s + gamma * diffs
+
+        return q, np.zeros(ns, dtype=np.float64)
+
+
+@lru_cache(maxsize=1)
+def _get_nondiagonal_pairs(n: int) -> np.ndarray:
+    """Get the indices of a square matrix with size n, excluding the diagonal."""
+    rows, cols = np.tril_indices(n, -1)  # -1 to exclude diagonal
+
+    # Combine rows-cols and cols-rows pairs
+    pairs = np.column_stack(
+        [np.concatenate([rows, cols]), np.concatenate([cols, rows])]
+    )
+
+    return pairs