r""" Computes the partial gradient of the dual optimal transport problem. For each :math:`(i,j)` in a batch of coordinates, the partial gradients are : .. math:: \partial_{\mathbf{u}_i} F = \frac{b_s}{l_v} \mathbf{u}_i - \sum_{j \in B_v} \mathbf{a}_i \mathbf{b}_j
(a, b, M, reg, alpha, beta, batch_size, batch_alpha, batch_beta)
| 376 | |
| 377 | |
| 378 | def batch_grad_dual(a, b, M, reg, alpha, beta, batch_size, batch_alpha, batch_beta): |
| 379 | r""" |
| 380 | Computes the partial gradient of the dual optimal transport problem. |
| 381 | |
| 382 | For each :math:`(i,j)` in a batch of coordinates, the partial gradients are : |
| 383 | |
| 384 | .. math:: |
| 385 | \partial_{\mathbf{u}_i} F = \frac{b_s}{l_v} \mathbf{u}_i - |
| 386 | \sum_{j \in B_v} \mathbf{a}_i \mathbf{b}_j |
| 387 | \exp\left( \frac{\mathbf{u}_i + \mathbf{v}_j - \mathbf{M}_{i,j}}{\mathrm{reg}} \right) |
| 388 | |
| 389 | \partial_{\mathbf{v}_j} F = \frac{b_s}{l_u} \mathbf{v}_j - |
| 390 | \sum_{i \in B_u} \mathbf{a}_i \mathbf{b}_j |
| 391 | \exp\left( \frac{\mathbf{u}_i + \mathbf{v}_j - \mathbf{M}_{i,j}}{\mathrm{reg}} \right) |
| 392 | |
| 393 | Where : |
| 394 | |
| 395 | - :math:`\mathbf{M}` is the (`ns`, `nt`) metric cost matrix |
| 396 | - :math:`\mathbf{u}`, :math:`\mathbf{v}` are dual variables in :math:`\mathbb{R}^{ns} \times \mathbb{R}^{nt}` |
| 397 | - reg is the regularization term |
| 398 | - :math:`B_u` and :math:`B_v` are lists of index |
| 399 | - :math:`b_s` is the size of the batches :math:`B_u` and :math:`B_v` |
| 400 | - :math:`l_u` and :math:`l_v` are the lengths of :math:`B_u` and :math:`B_v` |
| 401 | - :math:`\mathbf{a}` and :math:`\mathbf{b}` are source and target weights (sum to 1) |
| 402 | |
| 403 | |
| 404 | The algorithm used for solving the dual problem is the SGD algorithm |
| 405 | as proposed in :ref:`[19] <references-batch-grad-dual>` [alg.1] |
| 406 | |
| 407 | |
| 408 | Parameters |
| 409 | ---------- |
| 410 | a : ndarray, shape (ns,) |
| 411 | source measure |
| 412 | b : ndarray, shape (nt,) |
| 413 | target measure |
| 414 | M : ndarray, shape (ns, nt) |
| 415 | cost matrix |
| 416 | reg : float |
| 417 | Regularization term > 0 |
| 418 | alpha : ndarray, shape (ns,) |
| 419 | dual variable |
| 420 | beta : ndarray, shape (nt,) |
| 421 | dual variable |
| 422 | batch_size : int |
| 423 | size of the batch |
| 424 | batch_alpha : ndarray, shape (bs,) |
| 425 | batch of index of alpha |
| 426 | batch_beta : ndarray, shape (bs,) |
| 427 | batch of index of beta |
| 428 | |
| 429 | Returns |
| 430 | ------- |
| 431 | grad : ndarray, shape (`ns`,) |
| 432 | partial grad F |
| 433 | |
| 434 | |
| 435 | .. _references-batch-grad-dual: |
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