//  [python]
def rmi_lower_bound(self, labels_4D, probs_4D):
        """
        calculate the lower bound of the region mutual information.
        Args:
            labels_4D 	:	[N, C, H, W], dtype=float32
            probs_4D 	:	[N, C, H, W], dtype=float32
        """
        assert labels_4D.size() == probs_4D.size()

        p, s = self.rmi_pool_size, self.rmi_pool_stride
        if self.rmi_pool_stride > 1:
            if self.rmi_pool_way == 0:
                labels_4D = F.max_pool2d(labels_4D, kernel_size=p, stride=s, padding=self.kernel_padding)
                probs_4D = F.max_pool2d(probs_4D, kernel_size=p, stride=s, padding=self.kernel_padding)
            elif self.rmi_pool_way == 1:
                labels_4D = F.avg_pool2d(labels_4D, kernel_size=p, stride=s, padding=self.kernel_padding)
                probs_4D = F.avg_pool2d(probs_4D, kernel_size=p, stride=s, padding=self.kernel_padding)
            elif self.rmi_pool_way == 2:
                # interpolation
                shape = labels_4D.size()
                new_h, new_w = shape[2] // s, shape[3] // s
                labels_4D = t.nn.functional.interpolate(labels_4D, size=(new_h, new_w), mode='nearest')
                probs_4D = t.nn.functional.interpolate(probs_4D, size=(new_h, new_w), mode='bilinear', align_corners=True)
            else:
                raise NotImplementedError("Pool way of RMI is not defined!")
        # we do not need the gradient of label.
        label_shape = labels_4D.size()
        n, c = label_shape[0], label_shape[1]

        # combine the high dimension points from label and probability map. new shape [N, C, radius * radius, H, W]
        la_vectors, pr_vectors = map_get_pairs(labels_4D, probs_4D, radius=self.rmi_radius, is_combine=0)

        la_vectors = la_vectors.view([n, c, self.half_d, -1]).type(torch.cuda.DoubleTensor).requires_grad_(False)
        pr_vectors = pr_vectors.view([n, c, self.half_d, -1]).type(torch.cuda.DoubleTensor)

        # small diagonal matrix, shape = [1, 1, radius * radius, radius * radius]
        diag_matrix = torch.eye(self.half_d).unsqueeze(dim=0).unsqueeze(dim=0)

        # the mean and covariance of these high dimension points
        # Var(X) = E(X^2) - E(X) E(X), N * Var(X) = X^2 - X E(X)
        la_vectors = la_vectors - la_vectors.mean(dim=3, keepdim=True)
        la_cov = torch.matmul(la_vectors, la_vectors.transpose(2, 3))

        pr_vectors = pr_vectors - pr_vectors.mean(dim=3, keepdim=True)
        pr_cov = torch.matmul(pr_vectors, pr_vectors.transpose(2, 3))
        # https://github.com/pytorch/pytorch/issues/7500
        # waiting for batched torch.cholesky_inverse()
        pr_cov_inv = torch.inverse(pr_cov + diag_matrix.type_as(pr_cov) * _POS_ALPHA)
        # if the dimension of the point is less than 9, you can use the below function
        # to acceleration computational speed.
        # pr_cov_inv = utils.batch_cholesky_inverse(pr_cov + diag_matrix.type_as(pr_cov) * _POS_ALPHA)

        la_pr_cov = torch.matmul(la_vectors, pr_vectors.transpose(2, 3))
        # the approxiamation of the variance, det(c A) = c^n det(A), A is in n x n shape;
        # then log det(c A) = n log(c) + log det(A).
        # appro_var = appro_var / n_points, we do not divide the appro_var by number of points here,
        # and the purpose is to avoid underflow issue.
        # If A = A^T, A^-1 = (A^-1)^T.
        appro_var = la_cov - torch.matmul(la_pr_cov.matmul(pr_cov_inv), la_pr_cov.transpose(-2, -1))
        # appro_var = la_cov - torch.chain_matmul(la_pr_cov, pr_cov_inv, la_pr_cov.transpose(-2, -1))
        # appro_var = torch.div(appro_var, n_points.type_as(appro_var)) + diag_matrix.type_as(appro_var) * 1e-6

        # The lower bound. If A is nonsingular, ln( det(A) ) = Tr( ln(A) ).
        rmi_now = 0.5 * log_det_by_cholesky(appro_var + diag_matrix.type_as(appro_var) * _POS_ALPHA)
        # rmi_now = 0.5 * torch.logdet(appro_var + diag_matrix.type_as(appro_var) * _POS_ALPHA)

        # mean over N samples. sum over classes.
        rmi_per_class = rmi_now.view([-1, self.seg_num_classes]).mean(dim=0).float()
        # is_half = False
        # if is_half:
        #	rmi_per_class = torch.div(rmi_per_class, float(self.half_d / 2.0))
        # else:
        rmi_per_class = torch.div(rmi_per_class, float(self.half_d))

        rmi_loss = torch.sum(rmi_per_class) if _IS_SUM else torch.mean(rmi_per_class)
        return rmi_loss