src/geo_metrics/land.py

### Adapted from Metric Flow Matching (https://github.com/kkapusniak/metric-flow-matching)

import torch

def weighting_function(x, samples, gamma):
    pairwise_sq_diff = (x[:, None, :] - samples[None, :, :]) ** 2
    pairwise_sq_dist = pairwise_sq_diff.sum(-1)
    weights = torch.exp(-pairwise_sq_dist / (2 * gamma**2))
    return weights


def land_metric_tensor(x, samples, gamma, rho):
    weights = weighting_function(x, samples, gamma)  # Shape [B, N]
    differences = samples[None, :, :] - x[:, None, :]  # Shape [B, N, D]
    squared_differences = differences**2  # Shape [B, N, D]

    # Compute the sum of weighted squared differences for each dimension
    M_dd_diag = torch.einsum("bn,bnd->bd", weights, squared_differences) + rho

    # Invert the metric tensor diagonal for each x_t
    M_dd_inv_diag = 1.0 / M_dd_diag  # Shape [B, D] since it's diagonal
    return M_dd_inv_diag


def weighting_function_dt(x, dx_dt, samples, gamma, weights):
    pairwise_sq_diff_dt = (x[:, None, :] - samples[None, :, :]) * dx_dt[:, None, :]
    return -pairwise_sq_diff_dt.sum(-1) * weights / (gamma**2)