Shape losses

Shape losses compare predicted positions with target point clouds.

When cell identity is known

squared_loss applies when each predicted cell has a known target identity and row order is meaningful — the loss compares cell \(i\) against target \(i\) directly. Input shapes must match exactly. Use it for experiments that track cell identities.

When the target is an unordered cloud

For biological shapes sampled from meshes, the rows of the predicted and target clouds are unordered, so the loss must be a distributional distance between point sets. chamfer_distance and make_samples_loss apply here.

chamfer_distance sums the mean nearest-neighbor distance in each direction, giving the same value when clouds with different point counts are exchanged:

\[d_{\mathrm{Ch}}(X,Y) = \frac{1}{|X|}\sum_{x\in X}\min_{y\in Y}\lVert x-y\rVert_2 + \frac{1}{|Y|}\sum_{y\in Y}\min_{x\in X}\lVert x-y\rVert_2.\]

Both clouds must be nonempty rank-2 arrays with the same feature width.

Training sums the selected distance over supervised frames,

\[L_{\text{shape}} = \sum_{k=1}^{K} d\big(X_{\tau_k}, \tilde{X}_{\tau_k}\big),\]

reducing to \(d(X_T, \tilde{X}_T)\) for a single terminal target. PyTorch’s GeomLoss-backed make_samples_loss exposes maximum mean discrepancy, Hausdorff divergence, and debiased Sinkhorn divergence. JAX’s make_sinkhorn_loss exposes debiased OTT Sinkhorn divergence. Numerical values follow each backend’s cost, solver, and option contract. In the PyTorch factory, omitted truncate uses 5 while explicit None is preserved.

Regularizing the trajectory

The training regularizer penalizes the learned position head before prescribed mechanics,

\[L_{\text{reg}} = \sum_t \lVert \Delta t_{\mathrm{GNS}}\,\mathrm{GNS}_{X,t} \rVert_F^2.\]

lambda_reg supplies its weight, so the optimized loss is \(L=L_{\mathrm{shape}}+\lambda_{\mathrm{reg}}L_{\mathrm{reg}}\). This regularizer measures learned displacement; the physics update governs prescribed mechanical displacement.