[MRG] Faster gromov-wasserstein linesearch for symmetric matrices (#607)

* improved gw linsearch for symmetric case

* add a demo in examples

* remove example

* releases.md updated

---------

Co-authored-by: Rémi Flamary <remi.flamary@gmail.com>

Author: KrzakalaPaul

RELEASES.md

@@ -2,6 +2,9 @@
 
 ## 0.9.3
 
+#### New features
++ `ot.gromov._gw.solve_gromov_linesearch` now has an argument to specifify if the matrices are symmetric in which case the computation can be done faster.
+
 #### Closed issues
 - Fixed an issue with cost correction for mismatched labels in `ot.da.BaseTransport` fit methods. This fix addresses the original issue introduced PR #587 (PR #593)
 - Fix gpu compatibility of sr(F)GW solvers when `G0 is not None`(PR #596)

ot/gromov/_gw.py

@@ -171,7 +171,7 @@ def line_search(cost, G, deltaG, Mi, cost_G, **kwargs):
             return line_search_armijo(cost, G, deltaG, Mi, cost_G, nx=np_, **kwargs)
     else:
         def line_search(cost, G, deltaG, Mi, cost_G, **kwargs):
-            return solve_gromov_linesearch(G, deltaG, cost_G, hC1, hC2, M=0., reg=1., nx=np_, **kwargs)
+            return solve_gromov_linesearch(G, deltaG, cost_G, hC1, hC2, M=0., reg=1., nx=np_, symmetric=symmetric, **kwargs)
 
     if not nx.is_floating_point(C10):
         warnings.warn(
@@ -479,7 +479,7 @@ def line_search(cost, G, deltaG, Mi, cost_G, **kwargs):
             return line_search_armijo(cost, G, deltaG, Mi, cost_G, nx=np_, **kwargs)
     else:
         def line_search(cost, G, deltaG, Mi, cost_G, **kwargs):
-            return solve_gromov_linesearch(G, deltaG, cost_G, hC1, hC2, M=(1 - alpha) * M, reg=alpha, nx=np_, **kwargs)
+            return solve_gromov_linesearch(G, deltaG, cost_G, hC1, hC2, M=(1 - alpha) * M, reg=alpha, nx=np_, symmetric=symmetric, **kwargs)
     if not nx.is_floating_point(M0):
         warnings.warn(
             "Input feature matrix consists of integer. The transport plan will be "
@@ -647,7 +647,7 @@ def fused_gromov_wasserstein2(M, C1, C2, p=None, q=None, loss_fun='square_loss',
 
 
 def solve_gromov_linesearch(G, deltaG, cost_G, C1, C2, M, reg,
-                            alpha_min=None, alpha_max=None, nx=None, **kwargs):
+                            alpha_min=None, alpha_max=None, nx=None, symmetric=False, **kwargs):
     """
     Solve the linesearch in the FW iterations for any inner loss that decomposes as in Proposition 1 in :ref:`[12] <references-solve-linesearch>`.
 
@@ -676,6 +676,10 @@ def solve_gromov_linesearch(G, deltaG, cost_G, C1, C2, M, reg,
         Maximum value for alpha
     nx : backend, optional
         If let to its default value None, a backend test will be conducted.
+    symmetric : bool, optional
+        Either structures are to be assumed symmetric or not. Default value is False.
+        Else if set to True (resp. False), C1 and C2 will be assumed symmetric (resp. asymmetric).
+
     Returns
     -------
     alpha : float
@@ -708,7 +712,10 @@ def solve_gromov_linesearch(G, deltaG, cost_G, C1, C2, M, reg,
 
     dot = nx.dot(nx.dot(C1, deltaG), C2.T)
     a = - reg * nx.sum(dot * deltaG)
-    b = nx.sum(M * deltaG) - reg * (nx.sum(dot * G) + nx.sum(nx.dot(nx.dot(C1, G), C2.T) * deltaG))
+    if symmetric:
+        b = nx.sum(M * deltaG) - 2 * reg * nx.sum(dot * G)
+    else:
+        b = nx.sum(M * deltaG) - reg * (nx.sum(dot * G) + nx.sum(nx.dot(nx.dot(C1, G), C2.T) * deltaG))
 
     alpha = solve_1d_linesearch_quad(a, b)
     if alpha_min is not None or alpha_max is not None: