pythonot.github.io/master/_modules/ot/solvers.html at master · PythonOT/pythonot.github.io

Repository navigation

History

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

564

565

566

567

568

569

570

571

572

573

574

575

576

577

578

579

580

581

582

583

584

585

586

587

588

589

590

591

592

593

594

595

596

597

598

599

600

601

602

603

604

605

606

607

608

609

610

611

612

613

614

615

616

617

618

619

620

621

622

623

624

625

626

627

628

629

630

631

632

633

634

635

636

637

638

639

640

641

642

643

644

645

646

647

648

649

650

651

652

653

654

655

656

657

658

659

660

661

662

663

664

665

666

667

668

669

670

671

672

673

674

675

676

677

678

679

680

681

682

683

684

685

686

687

688

689

690

691

692

693

694

695

696

697

698

699

700

701

702

703

704

705

706

707

708

709

710

711

712

713

714

715

716

717

718

719

720

721

722

723

724

725

726

727

728

729

730

731

732

733

734

735

736

737

738

739

740

741

742

743

744

745

746

747

748

749

750

751

752

753

754

755

756

757

758

759

760

761

762

763

764

765

766

767

768

769

770

771

772

773

774

775

776

777

778

779

780

781

782

783

784

785

786

787

788

789

790

791

792

793

794

795

796

797

798

799

800

801

802

803

804

805

806

807

808

809

810

811

812

813

814

815

816

817

818

819

820

821

822

823

824

825

826

827

828

829

830

831

832

833

834

835

836

837

838

839

840

841

842

843

844

845

846

847

848

849

850

851

852

853

854

855

856

857

858

859

860

861

862

863

864

865

866

867

868

869

870

871

872

873

874

875

876

877

878

879

880

881

882

883

884

885

886

887

888

889

890

891

892

893

894

895

896

897

898

899

900

901

902

903

904

905

906

907

908

909

910

911

912

913

914

915

916

917

918

919

920

921

922

923

924

925

926

927

928

929

930

931

932

933

934

935

936

937

938

939

940

941

942

943

944

945

946

947

948

949

950

951

952

953

954

955

956

957

958

959

960

961

962

963

964

965

966

967

968

969

970

971

972

973

974

975

976

977

978

979

980

981

982

983

984

985

986

987

988

989

990

991

992

993

994

995

996

997

998

999

1000

<!DOCTYPE html>

<head>

<title>ot.solvers — POT Python Optimal Transport 0.9.7.dev0 documentation</title>

</head>

POT Python Optimal Transport

</a>

</form>

</div>

</div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu">

<ul>

<li class="toctree-l1"><a class="reference internal" href="../../index.html">POT: Python Optimal Transport</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../auto_examples/plot_quickstart_guide.html">Quickstart Guide</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../auto_examples/index.html">Examples gallery</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../user_guide.html">User guide</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../all.html">API and modules</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../releases.html">Releases</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../contributors.html">Contributors</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../contributing.html">Contributing to POT</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../code_of_conduct.html">Code of conduct</a></li>

</ul>

</div>

</nav>

<a href="../../index.html">POT Python Optimal Transport</a>

</nav>

<li class="breadcrumb-item"><a href="../index.html">Module code</a></li>

<li class="breadcrumb-item active">ot.solvers</li>

</li>

</ul>

<hr/>

</div>

<h1>Source code for ot.solvers</h1><div class="highlight"><pre>

# -*- coding: utf-8 -*-

"""

General OT solvers with unified API

"""

# Author: Remi Flamary <remi.flamary@polytechnique.edu>

#

# License: MIT License

from .utils import OTResult, dist

from .lp import emd2, emd2_lazy, wasserstein_1d

from .backend import get_backend

from .unbalanced import mm_unbalanced, sinkhorn_knopp_unbalanced, lbfgsb_unbalanced

from .bregman import (

sinkhorn_log,

empirical_sinkhorn2,

empirical_sinkhorn2_geomloss,

empirical_sinkhorn_nystroem2,

)

from .smooth import smooth_ot_dual

from .gromov import (

gromov_wasserstein2,

fused_gromov_wasserstein2,

entropic_gromov_wasserstein2,

entropic_fused_gromov_wasserstein2,

semirelaxed_gromov_wasserstein2,

semirelaxed_fused_gromov_wasserstein2,

entropic_semirelaxed_fused_gromov_wasserstein2,

entropic_semirelaxed_gromov_wasserstein2,

partial_gromov_wasserstein2,

partial_fused_gromov_wasserstein2,

entropic_partial_gromov_wasserstein2,

entropic_partial_fused_gromov_wasserstein2,

fused_unbalanced_gromov_wasserstein,

)

from .gaussian import empirical_bures_wasserstein_distance

from .factored import factored_optimal_transport

from .lowrank import lowrank_sinkhorn

from .optim import cg

import warnings

lst_method_lazy = [

"1d",

"gaussian",

"lowrank",

"nystroem",

"factored",

"geomloss",

"geomloss_auto",

"geomloss_tensorized",

"geomloss_online",

"geomloss_multiscale",

]

def solve(

M,

a=None,

b=None,

reg=None,

c=None,

reg_type="KL",

unbalanced=None,

unbalanced_type="KL",

method=None,

n_threads=1,

max_iter=None,

plan_init=None,

potentials_init=None,

tol=None,

verbose=False,

grad="autodiff",

):

r"""Solve the discrete optimal transport problem and return :any:`OTResult` object

The function solves the following general optimal transport problem

.. math::

\min_{\mathbf{T}\geq 0} \quad \sum_{i,j} T_{i,j}M_{i,j} + \lambda_r R(\mathbf{T}) +

\lambda_1 U(\mathbf{T}\mathbf{1},\mathbf{a}) +

\lambda_2 U(\mathbf{T}^T\mathbf{1},\mathbf{b})

The regularization is selected with `reg` (:math:`\lambda_r`) and `reg_type`. By

default ``reg=None`` and there is no regularization. The unbalanced marginal

penalization can be selected with `unbalanced` (:math:`(\lambda_1, \lambda_2)`) and

`unbalanced_type`. By default ``unbalanced=None`` and the function

solves the exact optimal transport problem (respecting the marginals).

Parameters

----------

M : array-like, shape (dim_a, dim_b)

Loss matrix

a : array-like, shape (dim_a,), optional

Samples weights in the source domain (default is uniform)

b : array-like, shape (dim_b,), optional

Samples weights in the source domain (default is uniform)

reg : float, optional

Regularization weight :math:`\lambda_r`, by default None (no reg., exact

OT)

c : array-like, shape (dim_a, dim_b), optional (default=None)

Reference measure for the regularization.

If None, then use :math:`\mathbf{c} = \mathbf{a} \mathbf{b}^T`.

If :math:`\texttt{reg_type}=`'entropy', then :math:`\mathbf{c} = 1_{dim_a} 1_{dim_b}^T`.

reg_type : str, optional

Type of regularization :math:`R` either "KL", "L2", "entropy",

by default "KL". a tuple of functions can be provided for general

solver (see :any:`cg`). This is only used when ``reg!=None``.

unbalanced : float or indexable object of length 1 or 2

Marginal relaxation term.

If it is a scalar or an indexable object of length 1,

then the same relaxation is applied to both marginal relaxations.

The balanced OT can be recovered using :math:`unbalanced=float("inf")`.

For semi-relaxed case, use either

:math:`unbalanced=(float("inf"), scalar)` or

:math:`unbalanced=(scalar, float("inf"))`.

If unbalanced is an array,

it must have the same backend as input arrays `(a, b, M)`.

unbalanced_type : str, optional

Type of unbalanced penalization function :math:`U` either "KL", "L2",

"TV", by default "KL".

method : str, optional

Method for solving the problem when multiple algorithms are available,

default None for automatic selection.

n_threads : int, optional

Number of OMP threads for exact OT solver, by default 1

max_iter : int, optional

Maximum number of iterations, by default None (default values in each solvers)

plan_init : array-like, shape (dim_a, dim_b), optional

Initialization of the OT plan for iterative methods, by default None

potentials_init : (array-like(dim_a,),array-like(dim_b,)), optional

Initialization of the OT dual potentials for iterative methods (Sinkhorn,

unbalanced) or warmstart for exact EMD solver. For EMD, these potentials

are used to guide initial pivots in the network simplex. By default None

tol : _type_, optional

Tolerance for solution precision, by default None (default values in each solvers)

verbose : bool, optional

Print information in the solver, by default False

grad : str, optional

Type of gradient computation, either or 'autodiff', 'envelope' or 'last_step' used only for

Sinkhorn solver. By default 'autodiff' provides gradients wrt all

outputs (`plan, value, value_linear`) but with important memory cost.

'envelope' provides gradients only for `value` and and other outputs are

detached. This is useful for memory saving when only the value is needed. 'last_step' provides

gradients only for the last iteration of the Sinkhorn solver, but provides gradient for both the OT plan and the objective values.

'detach' does not compute the gradients for the Sinkhorn solver.

Returns

-------

res : OTResult()

Result of the optimization problem. The information can be obtained as follows:

- res.plan : OT plan :math:`\mathbf{T}`

- res.potentials : OT dual potentials

- res.value : Optimal value of the optimization problem

- res.value_linear : Linear OT loss with the optimal OT plan

See :any:`OTResult` for more information.

Notes

-----

The following methods are available for solving the OT problems:

- **Classical exact OT problem [1]** (default parameters) :

.. math::

\min_\mathbf{T} \quad \langle \mathbf{T}, \mathbf{M} \rangle_F

s.t. \ \mathbf{T} \mathbf{1} = \mathbf{a}

\mathbf{T}^T \mathbf{1} = \mathbf{b}

\mathbf{T} \geq 0

can be solved with the following code:

.. code-block:: python

res = ot.solve(M, a, b)

- **Entropic regularized OT [2]** (when ``reg!=None``):

.. math::

\min_\mathbf{T} \quad \langle \mathbf{T}, \mathbf{M} \rangle_F + \lambda R(\mathbf{T})

s.t. \ \mathbf{T} \mathbf{1} = \mathbf{a}

\mathbf{T}^T \mathbf{1} = \mathbf{b}

\mathbf{T} \geq 0

can be solved with the following code:

.. code-block:: python

# default is ``"KL"`` regularization (``reg_type="KL"``)

res = ot.solve(M, a, b, reg=1.0)

# or for original Sinkhorn paper formulation [2]

res = ot.solve(M, a, b, reg=1.0, reg_type='entropy')

# Use envelope theorem differentiation for memory saving

res = ot.solve(M, a, b, reg=1.0, grad='envelope') # M, a, b are torch tensors

res.value.backward() # only the value is differentiable

Note that by default the Sinkhorn solver uses automatic differentiation to

compute the gradients of the values and plan. This can be changed with the

`grad` parameter. The `envelope` mode computes the gradients only

for the value and the other outputs are detached. This is useful for

memory saving when only the gradient of value is needed.

- **Quadratic regularized OT [17]** (when ``reg!=None`` and ``reg_type="L2"``):

.. math::

\min_\mathbf{T} \quad \langle \mathbf{T}, \mathbf{M} \rangle_F + \lambda R(\mathbf{T})

s.t. \ \mathbf{T} \mathbf{1} = \mathbf{a}

\mathbf{T}^T \mathbf{1} = \mathbf{b}

\mathbf{T} \geq 0

can be solved with the following code:

.. code-block:: python

res = ot.solve(M,a,b,reg=1.0,reg_type='L2')

- **Unbalanced OT [41]** (when ``unbalanced!=None``):

.. math::

\min_{\mathbf{T}\geq 0} \quad \sum_{i,j} T_{i,j}M_{i,j} +

\lambda_1 U(\mathbf{T}\mathbf{1},\mathbf{a}) +

\lambda_2 U(\mathbf{T}^T\mathbf{1},\mathbf{b})

can be solved with the following code:

.. code-block:: python

# default is ``"KL"``

res = ot.solve(M,a,b,unbalanced=1.0)

# quadratic unbalanced OT

res = ot.solve(M,a,b,unbalanced=1.0,unbalanced_type='L2')

# TV = partial OT

res = ot.solve(M,a,b,unbalanced=1.0,unbalanced_type='TV')

- **Regularized unbalanced regularized OT [34]** (when ``unbalanced!=None`` and ``reg!=None``):

.. math::

\min_{\mathbf{T}\geq 0} \quad \sum_{i,j} T_{i,j}M_{i,j} + \lambda_r R(\mathbf{T}) +

\lambda_1 U(\mathbf{T}\mathbf{1},\mathbf{a}) +

\lambda_2 U(\mathbf{T}^T\mathbf{1},\mathbf{b})

can be solved with the following code:

.. code-block:: python

# default is ``"KL"`` for both

res = ot.solve(M,a,b,reg=1.0,unbalanced=1.0)

# quadratic unbalanced OT with KL regularization

res = ot.solve(M,a,b,reg=1.0,unbalanced=1.0,unbalanced_type='L2')

# both quadratic

res = ot.solve(M,a,b,reg=1.0, reg_type='L2',unbalanced=1.0,unbalanced_type='L2')

.. _references-solve:

References

----------

.. [1] Bonneel, N., Van De Panne, M., Paris, S., & Heidrich, W.

(2011, December). Displacement interpolation using Lagrangian mass

transport. In ACM Transactions on Graphics (TOG) (Vol. 30, No. 6, p.

158). ACM.

.. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation

of Optimal Transport, Advances in Neural Information Processing

Systems (NIPS) 26, 2013

.. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016).

Scaling algorithms for unbalanced transport problems.

arXiv preprint arXiv:1607.05816.

.. [17] Blondel, M., Seguy, V., & Rolet, A. (2018). Smooth and Sparse

Optimal Transport. Proceedings of the Twenty-First International

Conference on Artificial Intelligence and Statistics (AISTATS).

.. [34] Feydy, J., Séjourné, T., Vialard, F. X., Amari, S. I., Trouvé,

A., & Peyré, G. (2019, April). Interpolating between optimal transport

and MMD using Sinkhorn divergences. In The 22nd International Conference

on Artificial Intelligence and Statistics (pp. 2681-2690). PMLR.

.. [41] Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

"""

# detect backend

nx = get_backend(M, a, b, c)

# create uniform weights if not given

if a is None:

a = nx.ones(M.shape[0], type_as=M) / M.shape[0]

if b is None:

b = nx.ones(M.shape[1], type_as=M) / M.shape[1]

if c is None:

c = a[:, None] * b[None, :]

if reg is None:

reg = 0

# default values for solutions

potentials = None

value = None

value_linear = None

plan = None

status = None

if reg == 0: # exact OT

if unbalanced is None: # Exact balanced OT

# default values for EMD solver

if max_iter is None:

max_iter = 1000000

value_linear, log = emd2(

a,

b,

M,

numItermax=max_iter,

log=True,

return_matrix=True,

numThreads=n_threads,

potentials_init=potentials_init,

)

value = value_linear

potentials = (log["u"], log["v"])

plan = log["G"]

status = log["warning"] if log["warning"] is not None else "Converged"

elif unbalanced_type.lower() in ["kl", "l2"]: # unbalanced exact OT

# default values for exact unbalanced OT

if max_iter is None:

max_iter = 1000

if tol is None:

tol = 1e-12

plan, log = mm_unbalanced(

a,

b,

M,

reg_m=unbalanced,

c=c,

reg=reg,

div=unbalanced_type,

numItermax=max_iter,

stopThr=tol,

log=True,

verbose=verbose,

G0=plan_init,

)

value_linear = log["cost"]

value = log["total_cost"]

if max_iter is None:

max_iter = 1000

if tol is None:

tol = 1e-12

if isinstance(reg_type, str):

reg_type = reg_type.lower()

plan, log = lbfgsb_unbalanced(

a,

b,

M,

reg=reg,

reg_m=unbalanced,

c=c,

reg_div=reg_type,

regm_div=unbalanced_type,

numItermax=max_iter,

stopThr=tol,

verbose=verbose,

log=True,

G0=plan_init,

)

value_linear = log["cost"]

value = log["total_cost"]

else:

raise (

NotImplementedError(

'Unknown unbalanced_type="{}"'.format(unbalanced_type)

)

else: # regularized OT

if unbalanced is None: # Balanced regularized OT

f, df = reg_type

if max_iter is None:

max_iter = 1000

if tol is None:

tol = 1e-9

plan, log = cg(

a,

b,

M,

reg=reg,

f=f,

df=df,

numItermax=max_iter,

stopThr=tol,

log=True,

verbose=verbose,

G0=plan_init,

)

value_linear = nx.sum(M * plan)

value = log["loss"][-1]

elif reg_type.lower() in ["entropy", "kl"]:

if grad in [

"envelope",

"last_step",

"detach",

]: # if envelope, last_step or detach then detach the input

M0, a0, b0 = M, a, b

M, a, b = nx.detach(M, a, b)

# default values for sinkhorn

if max_iter is None:

max_iter = 1000

if tol is None:

tol = 1e-9

if grad == "last_step":

if max_iter == 0:

raise ValueError(

"The maximum number of iterations must be greater than 0 when using grad=last_step."

)

max_iter = max_iter - 1

plan, log = sinkhorn_log(

a,

b,

M,

reg=reg,

numItermax=max_iter,

stopThr=tol,

log=True,

verbose=verbose,

)

potentials = (log["log_u"], log["log_v"])

# if last_step, compute the last step of the Sinkhorn algorithm with the non-detached inputs

if grad == "last_step":

loga = nx.log(a0)

logb = nx.log(b0)

v = logb - nx.logsumexp(-M0 / reg + potentials[0][:, None], 0)

u = loga - nx.logsumexp(-M0 / reg + potentials[1][None, :], 1)

plan = nx.exp(-M0 / reg + u[:, None] + v[None, :])

potentials = (u, v)

log["niter"] = max_iter + 1

log["log_u"] = u

log["log_v"] = v

log["u"] = nx.exp(u)

log["v"] = nx.exp(v)

if reg_type.lower() == "entropy":

value = value_linear + reg * nx.sum(plan * nx.log(plan + 1e-16))

else:

plan, a[:, None] * b[None, :]

)

if grad == "envelope": # set the gradient at convergence

value = nx.set_gradients(

value,

(M0, a0, b0),

(

plan,

reg * (potentials[0] - potentials[0].mean()),

reg * (potentials[1] - potentials[1].mean()),

),

)

elif reg_type.lower() == "l2":

if max_iter is None:

max_iter = 1000

if tol is None:

tol = 1e-9

plan, log = smooth_ot_dual(

a,

b,

M,

reg=reg,

numItermax=max_iter,

stopThr=tol,

log=True,

verbose=verbose,

)

potentials = (log["alpha"], log["beta"])

else:

raise (

NotImplementedError(

'Not implemented reg_type="{}"'.format(reg_type)

)

else: # unbalanced AND regularized OT

if (

not isinstance(reg_type, tuple)

and reg_type.lower() in ["kl"]

and unbalanced_type.lower() == "kl"

):

if max_iter is None:

max_iter = 1000

if tol is None:

tol = 1e-9

plan, log = sinkhorn_knopp_unbalanced(

a,

b,

M,

reg=reg,

reg_m=unbalanced,

method=method,

reg_type=reg_type,

c=c,

warmstart=potentials_init,

numItermax=max_iter,

stopThr=tol,

verbose=verbose,

log=True,

)

value_linear = log["cost"]

value = log["total_cost"]

potentials = (log["logu"], log["logv"])

elif (

isinstance(reg_type, tuple)

or reg_type.lower() in ["kl", "l2", "entropy"]

) and unbalanced_type.lower() in ["kl", "l2", "tv"]:

if isinstance(reg_type, tuple) and nx.__name__ == "tf":

raise NotImplementedError(

"Custom regularization is not implemented for TensorFlow backend."

)

if max_iter is None:

max_iter = 1000

if tol is None:

tol = 1e-12

if isinstance(reg_type, str):

reg_type = reg_type.lower()

plan, log = lbfgsb_unbalanced(

a,

b,

M,

reg=reg,

reg_m=unbalanced,

c=c,

reg_div=reg_type,

regm_div=unbalanced_type,

numItermax=max_iter,

stopThr=tol,

verbose=verbose,

log=True,

G0=plan_init,

)

value_linear = log["cost"]

value = log["total_cost"]

else:

raise (

NotImplementedError(

'Not implemented reg_type="{}" and unbalanced_type="{}"'.format(

reg_type, unbalanced_type

)

res = OTResult(

potentials=potentials,

value=value,

value_linear=value_linear,

plan=plan,

status=status,

backend=nx,

)

return res</div>

def solve_gromov(

Ca,

Cb,

M=None,

a=None,

b=None,

loss="L2",

symmetric=None,

alpha=0.5,

reg=None,

reg_type="entropy",

unbalanced=None,

unbalanced_type="KL",

n_threads=1,

method=None,

max_iter=None,

plan_init=None,

tol=None,

verbose=False,

):

r"""Solve the discrete (Fused) Gromov-Wasserstein and return :any:`OTResult` object

The function solves the following optimization problem:

.. math::

\min_{\mathbf{T}\geq 0} \quad (1 - \alpha) \langle \mathbf{T}, \mathbf{M} \rangle_F +

\alpha \sum_{i,j,k,l} L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l}) \mathbf{T}_{i,j} \mathbf{T}_{k,l} + \lambda_r R(\mathbf{T}) + \lambda_u U(\mathbf{T}\mathbf{1},\mathbf{a}) + \lambda_u U(\mathbf{T}^T\mathbf{1},\mathbf{b})

The regularization is selected with `reg` (:math:`\lambda_r`) and

`reg_type`. By default ``reg=None`` and there is no regularization. The

unbalanced marginal penalization can be selected with `unbalanced`

(:math:`\lambda_u`) and `unbalanced_type`. By default ``unbalanced=None``

and the function solves the exact optimal transport problem (respecting the

marginals).

Parameters

----------

Ca : array-like, shape (dim_a, dim_a)

Cost matrix in the source domain

Cb : array-like, shape (dim_b, dim_b)

Cost matrix in the target domain

M : array-like, shape (dim_a, dim_b), optional

Linear cost matrix for Fused Gromov-Wasserstein (default is None).

a : array-like, shape (dim_a,), optional

Samples weights in the source domain (default is uniform)

b : array-like, shape (dim_b,), optional

Samples weights in the source domain (default is uniform)

loss : str, optional

Type of loss function, either ``"L2"`` or ``"KL"``, by default ``"L2"``

symmetric : bool, optional

Use symmetric version of the Gromov-Wasserstein problem, by default None

tests whether the matrices are symmetric or True/False to avoid the test.

reg : float, optional

Regularization weight :math:`\lambda_r`, by default None (no reg., exact

OT)

reg_type : str, optional

Type of regularization :math:`R`, by default "entropy" (only used when

``reg!=None``)

alpha : float, optional

Weight the quadratic term (alpha*Gromov) and the linear term

((1-alpha)*Wass) in the Fused Gromov-Wasserstein problem. Not used for

Gromov problem (when M is not provided). By default ``alpha=None``

corresponds to ``alpha=1`` for Gromov problem (``M==None``) and

``alpha=0.5`` for Fused Gromov-Wasserstein problem (``M!=None``)

unbalanced : float, optional

Unbalanced penalization weight :math:`\lambda_u`, by default None

(balanced OT). Not implemented yet for "KL" unbalanced penalization

function :math:`U`. Corresponds to the total transport mass for partial OT.

unbalanced_type : str, optional

Type of unbalanced penalization function :math:`U` either "KL", "semirelaxed",

"partial", by default "KL".

n_threads : int, optional

Number of OMP threads for exact OT solver, by default 1

method : str, optional

Method for solving the problem when multiple algorithms are available,

default None for automatic selection.

max_iter : int, optional

Maximum number of iterations, by default None (default values in each

solvers)

plan_init : array-like, shape (dim_a, dim_b), optional

Initialization of the OT plan for iterative methods, by default None

tol : float, optional

Tolerance for solution precision, by default None (default values in

each solvers)

verbose : bool, optional

Print information in the solver, by default False

Returns

-------

res : OTResult()

Result of the optimization problem. The information can be obtained as follows:

- res.plan : OT plan :math:`\mathbf{T}`

- res.potentials : OT dual potentials

- res.value : Optimal value of the optimization problem

- res.value_linear : Linear OT loss with the optimal OT plan

- res.value_quad : Quadratic (GW) part of the OT loss with the optimal OT plan

See :any:`OTResult` for more information.

Notes

-----

The following methods are available for solving the Gromov-Wasserstein

problem:

- **Classical Gromov-Wasserstein (GW) problem [3]** (default parameters):

.. math::

\min_{\mathbf{T}\geq 0} \sum_{i,j,k,l} L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l}) \mathbf{T}_{i,j}\mathbf{T}_{k,l}

s.t. \ \mathbf{T} \mathbf{1} = \mathbf{a}

\mathbf{T}^T \mathbf{1} = \mathbf{b}

\mathbf{T} \geq 0

can be solved with the following code:

.. code-block:: python

res = ot.solve_gromov(Ca, Cb) # uniform weights

res = ot.solve_gromov(Ca, Cb, a=a, b=b) # given weights

res = ot.solve_gromov(Ca, Cb, loss='KL') # KL loss

plan = res.plan # GW plan

value = res.value # GW value

- **Fused Gromov-Wasserstein (FGW) problem [24]** (when ``M!=None``):

.. math::

\min_{\mathbf{T}\geq 0} \quad (1 - \alpha) \langle \mathbf{T}, \mathbf{M} \rangle_F +

\alpha \sum_{i,j,k,l} L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l}) \mathbf{T}_{i,j}\mathbf{T}_{k,l}

s.t. \ \mathbf{T} \mathbf{1} = \mathbf{a}

\mathbf{T}^T \mathbf{1} = \mathbf{b}

\mathbf{T} \geq 0

can be solved with the following code:

.. code-block:: python

res = ot.solve_gromov(Ca, Cb, M) # uniform weights, alpha=0.5 (default)

res = ot.solve_gromov(Ca, Cb, M, a=a, b=b, alpha=0.1) # given weights and alpha

plan = res.plan # FGW plan

loss_linear_term = res.value_linear # Wasserstein part of the loss

loss_quad_term = res.value_quad # Gromov part of the loss

loss = res.value # FGW value

- **Regularized (Fused) Gromov-Wasserstein (GW) problem [12]** (when ``reg!=None``):

.. math::

\min_{\mathbf{T}\geq 0} \quad (1 - \alpha) \langle \mathbf{T}, \mathbf{M} \rangle_F +

\alpha \sum_{i,j,k,l} L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l}) \mathbf{T}_{i,j}\mathbf{T}_{k,l} + \lambda_r R(\mathbf{T})

s.t. \ \mathbf{T} \mathbf{1} = \mathbf{a}

\mathbf{T}^T \mathbf{1} = \mathbf{b}

\mathbf{T} \geq 0

can be solved with the following code:

.. code-block:: python

res = ot.solve_gromov(Ca, Cb, reg=1.0) # GW entropy regularization (default)

res = ot.solve_gromov(Ca, Cb, M, a=a, b=b, reg=10, alpha=0.1) # FGW with entropy

plan = res.plan # FGW plan

loss_linear_term = res.value_linear # Wasserstein part of the loss

loss_quad_term = res.value_quad # Gromov part of the loss

loss = res.value # FGW value (including regularization)

- **Semi-relaxed (Fused) Gromov-Wasserstein (GW) [48]** (when ``unbalanced='semirelaxed'``):

.. math::

\min_{\mathbf{T}\geq 0} \quad (1 - \alpha) \langle \mathbf{T}, \mathbf{M} \rangle_F +

\alpha \sum_{i,j,k,l} L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l}) \mathbf{T}_{i,j}\mathbf{T}_{k,l}

s.t. \ \mathbf{T} \mathbf{1} = \mathbf{a}

\mathbf{T} \geq 0

can be solved with the following code:

.. code-block:: python

res = ot.solve_gromov(Ca, Cb, unbalanced='semirelaxed') # semirelaxed GW

res = ot.solve_gromov(Ca, Cb, unbalanced='semirelaxed', reg=1) # entropic semirelaxed GW

res = ot.solve_gromov(Ca, Cb, M, unbalanced='semirelaxed', alpha=0.1) # semirelaxed FGW

plan = res.plan # FGW plan

right_marginal = res.marginal_b # right marginal of the plan

- **Partial (Fused) Gromov-Wasserstein (GW) problem [29]** (when ``unbalanced='partial'``):

.. math::

\min_{\mathbf{T}\geq 0} \quad (1 - \alpha) \langle \mathbf{T}, \mathbf{M} \rangle_F +

\alpha \sum_{i,j,k,l} L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l}) \mathbf{T}_{i,j}\mathbf{T}_{k,l}

s.t. \ \mathbf{T} \mathbf{1} \leq \mathbf{a}

\mathbf{T}^T \mathbf{1} \leq \mathbf{b}

\mathbf{T} \geq 0

\mathbf{1}^T\mathbf{T}\mathbf{1} = m

can be solved with the following code:

.. code-block:: python

res = ot.solve_gromov(Ca, Cb, unbalanced_type='partial', unbalanced=0.8) # partial GW with m=0.8

res = ot.solve_gromov(Ca, Cb, M, unbalanced_type='partial', unbalanced=0.8, alpha=0.5) # partial FGW with m=0.8

.. _references-solve-gromov:

References

----------

.. [3] Mémoli, F. (2011). Gromov–Wasserstein distances and the metric

approach to object matching. Foundations of computational mathematics,

11(4), 417-487.

.. [12] Gabriel Peyré, Marco Cuturi, and Justin Solomon (2016),

Gromov-Wasserstein averaging of kernel and distance matrices

International Conference on Machine Learning (ICML).

.. [24] Vayer, T., Chapel, L., Flamary, R., Tavenard, R. and Courty, N.

(2019). Optimal Transport for structured data with application on graphs

Proceedings of the 36th International Conference on Machine Learning

(ICML).

.. [48] Cédric Vincent-Cuaz, Rémi Flamary, Marco Corneli, Titouan Vayer,

Nicolas Courty (2022). Semi-relaxed Gromov-Wasserstein divergence and

applications on graphs. International Conference on Learning

Representations (ICLR), 2022.

.. [29] Chapel, L., Alaya, M., Gasso, G. (2020). Partial Optimal Transport

with Applications on Positive-Unlabeled Learning, Advances in Neural

Information Processing Systems (NeurIPS), 2020.

"""

# detect backend

nx = get_backend(Ca, Cb, M, a, b)

# create uniform weights if not given

if a is None:

a = nx.ones(Ca.shape[0], type_as=Ca) / Ca.shape[0]

if b is None:

b = nx.ones(Cb.shape[1], type_as=Cb) / Cb.shape[1]

# default values for solutions

potentials = None

value = None

value_linear = None

value_quad = None

plan = None

status = None

log = None

loss_dict = {"l2": "square_loss", "kl": "kl_loss"}

if loss.lower() not in loss_dict.keys():

raise (NotImplementedError('Not implemented GW loss="{}"'.format(loss)))

loss_fun = loss_dict[loss.lower()]

if reg is None or reg == 0: # exact OT

if unbalanced is None and unbalanced_type.lower() not in [

"semirelaxed",

]: # Exact balanced OT

if unbalanced_type.lower() in ["partial"]:

warnings.warn(

"Exact balanced OT is computed as `unbalanced=None` even though "

f"unbalanced_type = {unbalanced_type}.",

stacklevel=2,

)

if M is None or alpha == 1: # Gromov-Wasserstein problem

# default values for solver

if max_iter is None:

max_iter = 10000

if tol is None:

tol = 1e-9

value, log = gromov_wasserstein2(

Ca,

Cb,

a,

b,

loss_fun=loss_fun,

log=True,

symmetric=symmetric,

max_iter=max_iter,

G0=plan_init,

tol_rel=tol,

View remainder of file in raw view

FilesExpand file tree

solvers.html

Latest commit

History

solvers.html

File metadata and controls