[MRG] Fix the total cost reported by the entropic unbalanced OT solvers

CONTRIBUTORS.md

@@ -63,6 +63,7 @@ The contributors to this library are:
 * Sienna O'Shea (SGOT)
 * [Huy Tran](https://github.com/huytransformer) (Stereographic Spherical Sliced-Wasserstein)
 * [Samuel Vangu](https://github.com/Samuel-Vangu) (Quasi-Monte Carlo Sliced-Wasserstein)
+* [xihaian251](https://github.com/xihaian251) (Unbalanced OT objective correctness and regression tests)
 
 
 ## Acknowledgments

RELEASES.md

@@ -17,6 +17,7 @@
 
 - Allow `NumpyBackend.seed` to adopt an existing `np.random.RandomState` instance and remove NumPy-specific random sampling paths in sliced utilities (PR #849, Issue #848)
 - Remove a leftover debug `print` from `ot.utils.projection_sparse_simplex` with `axis=1`, and make the `ot.datasets.make_gauss_hd` docstring a raw string so importing `ot` no longer emits a `SyntaxWarning` (PR #860)
+- Fix the value reported by the entropic unbalanced OT solvers (`ot.unbalanced.sinkhorn_unbalanced`, `sinkhorn_unbalanced2`, `sinkhorn_knopp_unbalanced`, `sinkhorn_stabilized_unbalanced` and `sinkhorn_unbalanced_translation_invariant`, also exposed as `ot.solve(..., reg=..., unbalanced=...).value`): every penalization is now evaluated with the generalized KL divergence, and with `reg_type="entropy"` the constant `dim_a * dim_b` is removed so that the reported value matches the documented negative entropy regularizer. The transport plans are unchanged. `reg_type` is now handled case insensitively by all the methods and an unknown value raises a `ValueError` instead of silently falling back to `"kl"`. The docstrings now state the generalized KL convention, the regularizer `Omega_reg_type(gamma, c)` with its two cases, the keys of the `log` dictionary and the multi-histogram behaviour (PR #874)
 - Fix `ot.dist` ignoring the weights `w` for `metric="cityblock"`, which returned the unweighted distance although the weights are documented for this metric (PR #859)
 - Fix swapped arguments to `div_to_product` in `ot.gromov.fused_unbalanced_across_spaces_cost`: with `reg_type="independent"` (UCOOT) the entropic terms used the plan marginals as the reference measures and vice versa (PR #855, Issue #854)
 - Fix device placement in `ot.batch.bregman_projection_batch` so `ot.solve_batch(..., method="sinkhorn")` no longer crashes on GPU when the torch default device is CPU (PR #851)