[WIP] Add backend dual loss and plan computation for stochastic optimization or regularized OT (#360)

* add losses and plan computations and exmaple for dual oiptimization

* pep8

* add nice exmaple

* update awesome example stochasti dual

* add all tests

* pep8 + speedup exmaple

* add release info

1 files changed, 114 insertions, 1 deletions

diff --git a/test/test_stochastic.py b/test/test_stochastic.py
index 736df32..2b5c0fb 100644
--- a/test/test_stochastic.py
+++ b/test/test_stochastic.py
@@ -8,7 +8,8 @@ for descrete and semicontinous measures from the POT library.
 
 """
 
-# Author: Kilian Fatras <kilian.fatras@gmail.com>
+# Authors: Kilian Fatras <kilian.fatras@gmail.com>
+#          Rémi Flamary <remi.flamary@polytechnique.edu>
 #
 # License: MIT License
 
@@ -213,3 +214,115 @@ def test_dual_sgd_sinkhorn():
         G_sgd.sum(0), G_sinkhorn.sum(0), atol=1e-03)
     np.testing.assert_allclose(
         G_sgd, G_sinkhorn, atol=1e-03)  # cf convergence sgd
+
+
+def test_loss_dual_entropic(nx):
+
+    nx.seed(0)
+
+    xs = nx.randn(50, 2)
+    xt = nx.randn(40, 2) + 2
+
+    ws = nx.rand(50)
+    ws = ws / nx.sum(ws)
+
+    wt = nx.rand(40)
+    wt = wt / nx.sum(wt)
+
+    u = nx.randn(50)
+    v = nx.randn(40)
+
+    def metric(x, y):
+        return -nx.dot(x, y.T)
+
+    ot.stochastic.loss_dual_entropic(u, v, xs, xt)
+
+    ot.stochastic.loss_dual_entropic(u, v, xs, xt, ws=ws, wt=wt, metric=metric)
+
+
+def test_plan_dual_entropic(nx):
+
+    nx.seed(0)
+
+    xs = nx.randn(50, 2)
+    xt = nx.randn(40, 2) + 2
+
+    ws = nx.rand(50)
+    ws = ws / nx.sum(ws)
+
+    wt = nx.rand(40)
+    wt = wt / nx.sum(wt)
+
+    u = nx.randn(50)
+    v = nx.randn(40)
+
+    def metric(x, y):
+        return -nx.dot(x, y.T)
+
+    G1 = ot.stochastic.plan_dual_entropic(u, v, xs, xt)
+
+    assert np.all(nx.to_numpy(G1) >= 0)
+    assert G1.shape[0] == 50
+    assert G1.shape[1] == 40
+
+    G2 = ot.stochastic.plan_dual_entropic(u, v, xs, xt, ws=ws, wt=wt, metric=metric)
+
+    assert np.all(nx.to_numpy(G2) >= 0)
+    assert G2.shape[0] == 50
+    assert G2.shape[1] == 40
+
+
+def test_loss_dual_quadratic(nx):
+
+    nx.seed(0)
+
+    xs = nx.randn(50, 2)
+    xt = nx.randn(40, 2) + 2
+
+    ws = nx.rand(50)
+    ws = ws / nx.sum(ws)
+
+    wt = nx.rand(40)
+    wt = wt / nx.sum(wt)
+
+    u = nx.randn(50)
+    v = nx.randn(40)
+
+    def metric(x, y):
+        return -nx.dot(x, y.T)
+
+    ot.stochastic.loss_dual_quadratic(u, v, xs, xt)
+
+    ot.stochastic.loss_dual_quadratic(u, v, xs, xt, ws=ws, wt=wt, metric=metric)
+
+
+def test_plan_dual_quadratic(nx):
+
+    nx.seed(0)
+
+    xs = nx.randn(50, 2)
+    xt = nx.randn(40, 2) + 2
+
+    ws = nx.rand(50)
+    ws = ws / nx.sum(ws)
+
+    wt = nx.rand(40)
+    wt = wt / nx.sum(wt)
+
+    u = nx.randn(50)
+    v = nx.randn(40)
+
+    def metric(x, y):
+        return -nx.dot(x, y.T)
+
+    G1 = ot.stochastic.plan_dual_quadratic(u, v, xs, xt)
+
+    assert np.all(nx.to_numpy(G1) >= 0)
+    assert G1.shape[0] == 50
+    assert G1.shape[1] == 40
+
+    G2 = ot.stochastic.plan_dual_quadratic(u, v, xs, xt, ws=ws, wt=wt, metric=metric)
+
+    assert np.all(nx.to_numpy(G2) >= 0)
+    assert G2.shape[0] == 50
+    assert G2.shape[1] == 40