Adapt centroids' initialisation in BallTree

[jjerphan]

As reported by @Witiko in #20097 (reply in thread), the centroids' initialization in BallTree is tight to the case of the euclidean distance regardless of the DistanceMetric set indirectly by the user provided metric argument.

scikit-learn/sklearn/neighbors/_ball_tree.pyx

Lines 71 to 104 in 7b71511

	# determine Node centroid
	for j in range(n_features):
	centroid[j] = 0
	if with_sample_weight:
	sum_weight_node = 0
	for i in range(idx_start, idx_end):
	sum_weight_node += sample_weight[idx_array[i]]
	this_pt = data + n_features * idx_array[i]
	for j from 0 <= j < n_features:
	centroid[j] += this_pt[j] * sample_weight[idx_array[i]]
	for j in range(n_features):
	centroid[j] /= sum_weight_node
	else:
	for i in range(idx_start, idx_end):
	this_pt = data + n_features * idx_array[i]
	for j from 0 <= j < n_features:
	centroid[j] += this_pt[j]
	for j in range(n_features):
	centroid[j] /= n_points
	# determine Node radius
	radius = 0
	for i in range(idx_start, idx_end):
	radius = fmax(radius,
	tree.rdist(centroid,
	data + n_features * idx_array[i],
	n_features))
	tree.node_data[i_node].radius = tree.dist_metric._rdist_to_dist(radius)
	tree.node_data[i_node].idx_start = idx_start
	tree.node_data[i_node].idx_end = idx_end

The initialisation of those centroids should depends on the effective DistanceMetric. Apart from the euclidean case, are there closed form solutions for computing them?