[MRG] Implement Sinkhorn in log-domain for WDA (#336)

* [MRG] Implement Sinkhorn in log-domain for WDA

* for small values of the regularization parameter (reg) the current implementation runs into numerical issues (nans and infs)

* this can be resolved by using log-domain implementation of the sinkhorn algorithm

* Add feature to RELEASES and contributor name

* Add 'sinkhorn_method' parameter to WDA

* use the standard Sinkhorn solver by default (faster)

* use log-domain Sinkhorn if asked by the user

Co-authored-by: Jakub Zadrożny <jz@qed.ai>
Co-authored-by: Rémi Flamary <remi.flamary@gmail.com>

Author: 3 people

RELEASES.md

@@ -5,6 +5,8 @@
 #### New features
 
 - Better list of related examples in quick start guide with `minigallery` (PR #334)
+- Add optional log-domain Sinkhorn implementation in WDA to support smaller values
+  of the regularization parameter (PR #336)
 
 #### Closed issues
 

ot/dr.py

@@ -11,6 +11,7 @@
 
 # Author: Remi Flamary <remi.flamary@unice.fr>
 #         Minhui Huang <mhhuang@ucdavis.edu>
+#         Jakub Zadrozny <jakub.r.zadrozny@gmail.com>
 #
 # License: MIT License
 
@@ -43,6 +44,28 @@ def sinkhorn(w1, w2, M, reg, k):
     return G
 
 
+def logsumexp(M, axis):
+    r"""Log-sum-exp reduction compatible with autograd (no numpy implementation)
+    """
+    amax = np.amax(M, axis=axis, keepdims=True)
+    return np.log(np.sum(np.exp(M - amax), axis=axis)) + np.squeeze(amax, axis=axis)
+
+
+def sinkhorn_log(w1, w2, M, reg, k):
+    r"""Sinkhorn algorithm in log-domain with fixed number of iteration (autograd)
+    """
+    Mr = -M / reg
+    ui = np.zeros((M.shape[0],))
+    vi = np.zeros((M.shape[1],))
+    log_w1 = np.log(w1)
+    log_w2 = np.log(w2)
+    for i in range(k):
+        vi = log_w2 - logsumexp(Mr + ui[:, None], 0)
+        ui = log_w1 - logsumexp(Mr + vi[None, :], 1)
+    G = np.exp(ui[:, None] + Mr + vi[None, :])
+    return G
+
+
 def split_classes(X, y):
     r"""split samples in :math:`\mathbf{X}` by classes in :math:`\mathbf{y}`
     """
@@ -110,7 +133,7 @@ def proj(X):
     return Popt, proj
 
 
-def wda(X, y, p=2, reg=1, k=10, solver=None, maxiter=100, verbose=0, P0=None, normalize=False):
+def wda(X, y, p=2, reg=1, k=10, solver=None, sinkhorn_method='sinkhorn', maxiter=100, verbose=0, P0=None, normalize=False):
     r"""
     Wasserstein Discriminant Analysis :ref:`[11] <references-wda>`
 
@@ -126,6 +149,14 @@ def wda(X, y, p=2, reg=1, k=10, solver=None, maxiter=100, verbose=0, P0=None, no
     - :math:`W` is entropic regularized Wasserstein distances
     - :math:`\mathbf{X}^i` are samples in the dataset corresponding to class i
 
+    **Choosing a Sinkhorn solver**
+
+    By default and when using a regularization parameter that is not too small
+    the default sinkhorn solver should be enough. If you need to use a small
+    regularization to get sparse cost matrices, you should use the
+    :py:func:`ot.dr.sinkhorn_log` solver that will avoid numerical
+    errors, but can be slow in practice.
+
     Parameters
     ----------
     X : ndarray, shape (n, d)
@@ -139,6 +170,8 @@ def wda(X, y, p=2, reg=1, k=10, solver=None, maxiter=100, verbose=0, P0=None, no
     solver : None | str, optional
         None for steepest descent or 'TrustRegions' for trust regions algorithm
         else should be a pymanopt.solvers
+    sinkhorn_method : str
+        method used for the Sinkhorn solver, either 'sinkhorn' or 'sinkhorn_log'
     P0 : ndarray, shape (d, p)
         Initial starting point for projection.
     normalize : bool, optional
@@ -161,6 +194,13 @@ def wda(X, y, p=2, reg=1, k=10, solver=None, maxiter=100, verbose=0, P0=None, no
             Wasserstein Discriminant Analysis. arXiv preprint arXiv:1608.08063.
     """  # noqa
 
+    if sinkhorn_method.lower() == 'sinkhorn':
+        sinkhorn_solver = sinkhorn
+    elif sinkhorn_method.lower() == 'sinkhorn_log':
+        sinkhorn_solver = sinkhorn_log
+    else:
+        raise ValueError("Unknown Sinkhorn method '%s'." % sinkhorn_method)
+
     mx = np.mean(X)
     X -= mx.reshape((1, -1))
 
@@ -193,7 +233,7 @@ def cost(P):
             for j, xj in enumerate(xc[i:]):
                 xj = np.dot(xj, P)
                 M = dist(xi, xj)
-                G = sinkhorn(wc[i], wc[j + i], M, reg * regmean[i, j], k)
+                G = sinkhorn_solver(wc[i], wc[j + i], M, reg * regmean[i, j], k)
                 if j == 0:
                     loss_w += np.sum(G * M)
                 else:

test/test_dr.py

@@ -60,6 +60,28 @@ def test_wda():
     np.testing.assert_allclose(np.sum(Pwda**2, 0), np.ones(p))
 
 
+@pytest.mark.skipif(nogo, reason="Missing modules (autograd or pymanopt)")
+def test_wda_low_reg():
+
+    n_samples = 100  # nb samples in source and target datasets
+    np.random.seed(0)
+
+    # generate gaussian dataset
+    xs, ys = ot.datasets.make_data_classif('gaussrot', n_samples)
+
+    n_features_noise = 8
+
+    xs = np.hstack((xs, np.random.randn(n_samples, n_features_noise)))
+
+    p = 2
+
+    Pwda, projwda = ot.dr.wda(xs, ys, p, reg=0.01, maxiter=10, sinkhorn_method='sinkhorn_log')
+
+    projwda(xs)
+
+    np.testing.assert_allclose(np.sum(Pwda**2, 0), np.ones(p))
+
+
 @pytest.mark.skipif(nogo, reason="Missing modules (autograd or pymanopt)")
 def test_wda_normalized():