diff options
| -rw-r--r-- | README.md | 5 | ||||
| -rw-r--r-- | ot/bregman.py | 153 | ||||
| -rw-r--r-- | test/test_bregman.py | 3 |
3 files changed, 159 insertions, 2 deletions
@@ -14,7 +14,7 @@ This open source Python library provide several solvers for optimization problem It provides the following solvers: * OT Network Flow solver for the linear program/ Earth Movers Distance [1]. -* Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2] and stabilized version [9][10] with optional GPU implementation (requires cudamat). +* Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2] and stabilized version [9][10] and greedy SInkhorn [21] with optional GPU implementation (requires cudamat). * Smooth optimal transport solvers (dual and semi-dual) for KL and squared L2 regularizations [17]. * Non regularized Wasserstein barycenters [16] with LP solver (only small scale). * Bregman projections for Wasserstein barycenter [3] and unmixing [4]. @@ -165,6 +165,7 @@ The contributors to this library are: * [Antoine Rolet](https://arolet.github.io/) * Erwan Vautier (Gromov-Wasserstein) * [Kilian Fatras](https://kilianfatras.github.io/) +* [Alain Rakotomamonjy](https://sites.google.com/site/alainrakotomamonjy/home) This toolbox benefit a lot from open source research and we would like to thank the following persons for providing some code (in various languages): @@ -230,3 +231,5 @@ You can also post bug reports and feature requests in Github issues. Make sure t [20] Cuturi, M. and Doucet, A. (2014) [Fast Computation of Wasserstein Barycenters](http://proceedings.mlr.press/v32/cuturi14.html). International Conference in Machine Learning [21] Solomon, J., De Goes, F., Peyré, G., Cuturi, M., Butscher, A., Nguyen, A. & Guibas, L. (2015). [Convolutional wasserstein distances: Efficient optimal transportation on geometric domains](https://dl.acm.org/citation.cfm?id=2766963). ACM Transactions on Graphics (TOG), 34(4), 66. + +[21] J. Altschuler, J.Weed, P. Rigollet, (2017) [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://papers.nips.cc/paper/6792-near-linear-time-approximation-algorithms-for-optimal-transport-via-sinkhorn-iteration.pdf), Advances in Neural Information Processing Systems (NIPS) 31 diff --git a/ot/bregman.py b/ot/bregman.py index 35e51f8..418de57 100644 --- a/ot/bregman.py +++ b/ot/bregman.py @@ -47,7 +47,7 @@ def sinkhorn(a, b, M, reg, method='sinkhorn', numItermax=1000, reg : float Regularization term >0 method : str - method used for the solver either 'sinkhorn', 'sinkhorn_stabilized' or + method used for the solver either 'sinkhorn', 'greenkhorn', 'sinkhorn_stabilized' or 'sinkhorn_epsilon_scaling', see those function for specific parameters numItermax : int, optional Max number of iterations @@ -103,6 +103,10 @@ def sinkhorn(a, b, M, reg, method='sinkhorn', numItermax=1000, def sink(): return sinkhorn_knopp(a, b, M, reg, numItermax=numItermax, stopThr=stopThr, verbose=verbose, log=log, **kwargs) + elif method.lower() == 'greenkhorn': + def sink(): + return greenkhorn(a, b, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, log=log) elif method.lower() == 'sinkhorn_stabilized': def sink(): return sinkhorn_stabilized(a, b, M, reg, numItermax=numItermax, @@ -197,6 +201,8 @@ def sinkhorn2(a, b, M, reg, method='sinkhorn', numItermax=1000, .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816. + [21] Altschuler J., Weed J., Rigollet P. : Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration, Advances in Neural Information Processing Systems (NIPS) 31, 2017 + See Also @@ -204,6 +210,7 @@ def sinkhorn2(a, b, M, reg, method='sinkhorn', numItermax=1000, ot.lp.emd : Unregularized OT ot.optim.cg : General regularized OT ot.bregman.sinkhorn_knopp : Classic Sinkhorn [2] + ot.bregman.greenkhorn : Greenkhorn [21] ot.bregman.sinkhorn_stabilized: Stabilized sinkhorn [9][10] ot.bregman.sinkhorn_epsilon_scaling: Sinkhorn with epslilon scaling [9][10] @@ -410,6 +417,150 @@ def sinkhorn_knopp(a, b, M, reg, numItermax=1000, return u.reshape((-1, 1)) * K * v.reshape((1, -1)) +def greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log=False): + """ + Solve the entropic regularization optimal transport problem and return the OT matrix + + The algorithm used is based on the paper + + Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration + by Jason Altschuler, Jonathan Weed, Philippe Rigollet + appeared at NIPS 2017 + + which is a stochastic version of the Sinkhorn-Knopp algorithm [2]. + + The function solves the following optimization problem: + + .. math:: + \gamma = arg\min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma) + + s.t. \gamma 1 = a + + \gamma^T 1= b + + \gamma\geq 0 + where : + + - M is the (ns,nt) metric cost matrix + - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` + - a and b are source and target weights (sum to 1) + + + + Parameters + ---------- + a : np.ndarray (ns,) + samples weights in the source domain + b : np.ndarray (nt,) or np.ndarray (nt,nbb) + samples in the target domain, compute sinkhorn with multiple targets + and fixed M if b is a matrix (return OT loss + dual variables in log) + M : np.ndarray (ns,nt) + loss matrix + reg : float + Regularization term >0 + numItermax : int, optional + Max number of iterations + stopThr : float, optional + Stop threshol on error (>0) + log : bool, optional + record log if True + + + Returns + ------- + gamma : (ns x nt) ndarray + Optimal transportation matrix for the given parameters + log : dict + log dictionary return only if log==True in parameters + + Examples + -------- + + >>> import ot + >>> a=[.5,.5] + >>> b=[.5,.5] + >>> M=[[0.,1.],[1.,0.]] + >>> ot.bregman.greenkhorn(a,b,M,1) + array([[ 0.36552929, 0.13447071], + [ 0.13447071, 0.36552929]]) + + + References + ---------- + + .. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport, Advances in Neural Information Processing Systems (NIPS) 26, 2013 + [21] J. Altschuler, J.Weed, P. Rigollet : Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration, Advances in Neural Information Processing Systems (NIPS) 31, 2017 + + + See Also + -------- + ot.lp.emd : Unregularized OT + ot.optim.cg : General regularized OT + + """ + + i = 0 + + n = a.shape[0] + m = b.shape[0] + + # Next 3 lines equivalent to K= np.exp(-M/reg), but faster to compute + K = np.empty_like(M) + np.divide(M, -reg, out=K) + np.exp(K, out=K) + + u = np.full(n, 1. / n) + v = np.full(m, 1. / m) + G = u[:, np.newaxis] * K * v[np.newaxis, :] + + viol = G.sum(1) - a + viol_2 = G.sum(0) - b + stopThr_val = 1 + if log: + log['u'] = u + log['v'] = v + + for i in range(numItermax): + i_1 = np.argmax(np.abs(viol)) + i_2 = np.argmax(np.abs(viol_2)) + m_viol_1 = np.abs(viol[i_1]) + m_viol_2 = np.abs(viol_2[i_2]) + stopThr_val = np.maximum(m_viol_1, m_viol_2) + + if m_viol_1 > m_viol_2: + old_u = u[i_1] + u[i_1] = a[i_1] / (K[i_1, :].dot(v)) + G[i_1, :] = u[i_1] * K[i_1, :] * v + + viol[i_1] = u[i_1] * K[i_1, :].dot(v) - a[i_1] + viol_2 += (K[i_1, :].T * (u[i_1] - old_u) * v) + + else: + old_v = v[i_2] + v[i_2] = b[i_2] / (K[:, i_2].T.dot(u)) + G[:, i_2] = u * K[:, i_2] * v[i_2] + #aviol = (G@one_m - a) + #aviol_2 = (G.T@one_n - b) + viol += (-old_v + v[i_2]) * K[:, i_2] * u + viol_2[i_2] = v[i_2] * K[:, i_2].dot(u) - b[i_2] + + #print('b',np.max(abs(aviol -viol)),np.max(abs(aviol_2 - viol_2))) + + if stopThr_val <= stopThr: + break + else: + print('Warning: Algorithm did not converge') + + if log: + log['u'] = u + log['v'] = v + + if log: + return G, log + else: + return G + + def sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, warmstart=None, verbose=False, print_period=20, log=False, **kwargs): """ diff --git a/test/test_bregman.py b/test/test_bregman.py index 01ec655..58afd7a 100644 --- a/test/test_bregman.py +++ b/test/test_bregman.py @@ -71,11 +71,14 @@ def test_sinkhorn_variants(): Ges = ot.sinkhorn( u, u, M, 1, method='sinkhorn_epsilon_scaling', stopThr=1e-10) Gerr = ot.sinkhorn(u, u, M, 1, method='do_not_exists', stopThr=1e-10) + G_green = ot.sinkhorn(u, u, M, 1, method='greenkhorn', stopThr=1e-10) # check values np.testing.assert_allclose(G0, Gs, atol=1e-05) np.testing.assert_allclose(G0, Ges, atol=1e-05) np.testing.assert_allclose(G0, Gerr) + np.testing.assert_allclose(G0, G_green, atol=1e-5) + print(G0, G_green) def test_bary(): |