pythonot.github.io/_modules/ot/unbalanced/_sliced.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

<!DOCTYPE html>

<head>

<title>ot.unbalanced._sliced — POT Python Optimal Transport 0.9.7 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.unbalanced._sliced</li>

</li>

</ul>

<hr/>

</div>

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

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

"""

Sliced Unbalanced OT solvers

"""

# Author: Clément Bonet <clement.bonet.mapp@polytechnique.edu>

#

# License: MIT License

from ..backend import get_backend

from ..utils import get_parameter_pair

from ..sliced import get_random_projections

from ._solver_1d import rescale_potentials, uot_1d

from ..lp.solver_1d import emd_1d_dual_backprop, wasserstein_1d

def sliced_unbalanced_ot(

X_s,

X_t,

reg_m,

a=None,

b=None,

n_projections=50,

p=2,

projections=None,

seed=None,

numItermax=10,

log=False,

):

r"""

Compute the Sliced Unbalanced Optimal Transport (SUOT) between two empirical distributions.

The 1D UOT problem is computed with KL regularization and solved with a Frank-Wolfe algorithm, see :ref:`[82] <references-suot>`.

The Sliced Unbalanced Optimal Transport (SUOT) is defined as

.. math::

\mathrm{SUOT}_p^p(\mu, \nu) = \int_{S^{d-1}} \mathrm{UOT}_p^p(P^\theta_\#\mu, P^\theta_\#\nu)\ \mathrm{d}\lambda(\theta)

with :math:`P^\theta(x)=\langle x,\theta\rangle` and :math:`\lambda` the uniform distribution on the unit sphere.

.. warning:: This function only works in pytorch or jax as it uses autodifferentiation to compute the 1D UOT problems. It is not maintained in jax.

Parameters

----------

X_s : ndarray, shape (n_samples_a, dim)

samples in the source domain

X_t : ndarray, shape (n_samples_b, dim)

samples in the target domain

reg_m: float or indexable object of length 1 or 2

Marginal relaxation term.

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

then the same `reg_m` is applied to both marginal relaxations.

The balanced OT can be recovered using `reg_m=float("inf")`.

For semi-relaxed case, use either `reg_m=(float("inf"), scalar)` or `reg_m=(scalar, float("inf"))`.

If `reg_m` is an array, it must have the same backend as input arrays `(X_s, X_t)`.

a : ndarray, shape (n_samples_a,), optional

samples weights in the source domain

b : ndarray, shape (n_samples_b,), optional

samples weights in the target domain

n_projections : int, optional

Number of projections used for the Monte-Carlo approximation

p: float, optional, by default =2

Power p used for computing the sliced Wasserstein

projections: shape (dim, n_projections), optional

Projection matrix (n_projections and seed are not used in this case)

seed: int or RandomState or None, optional

Seed used for random number generator

numItermax: int, optional

log: bool, optional

if True, returns the projections used and their associated UOTs and reweighted marginals.

Returns

-------

loss: float/array-like, shape (...)

SUOT

log: dict, optional

If `log` is True, then returns a dictionary containing the projection directions used, the projected UOTs, and reweighted marginals on each slices.

.. _references-suot:

References

----------

.. [82] Bonet, C., Nadjahi, K., Séjourné, T., Fatras, K., & Courty, N. (2025).

Slicing Unbalanced Optimal Transport. Transactions on Machine Learning Research.

See Also

--------

ot.unbalanced.uot_1d: 1D OT problem

ot.unbalanced.unbalanced_sliced_ot: Unbalanced SOT problem

"""

nx = get_backend(X_s, X_t, a, b, projections)

assert nx.__name__ in ["torch", "jax"], "Function only valid in torch and jax"

n = X_s.shape[0]

m = X_t.shape[0]

if X_s.shape[1] != X_t.shape[1]:

raise ValueError(

"X_s and X_t must have the same number of dimensions {} and {} respectively given".format(

X_s.shape[1], X_t.shape[1]

)

if a is None:

a = nx.full(n, 1 / n, type_as=X_s)

if b is None:

b = nx.full(m, 1 / m, type_as=X_s)

d = X_s.shape[1]

if projections is None:

projections = get_random_projections(

d, n_projections, seed, backend=nx, type_as=X_s

)

else:

n_projections = projections.shape[1]

X_s_projections = nx.dot(X_s, projections) # shape (n, n_projs)

X_t_projections = nx.dot(X_t, projections)

# Compute UOT on each slice

a_reweighted, b_reweighted, projected_uot = uot_1d(

X_s_projections,

X_t_projections,

reg_m,

a,

b,

p,

require_sort=True,

numItermax=numItermax,

returnCost="total",

)

res = nx.mean(projected_uot)

if log:

dico = {

"projections": projections,

"projected_uots": projected_uot,

"a_reweighted": a_reweighted,

"b_reweighted": b_reweighted,

}

return res, dico

return res</div>

def get_reweighted_marginals_usot(

f, g, a, b, reg_m1, reg_m2, X_s_sorter, X_t_sorter, nx

):

r"""

One step of the FW algorithm for the Unbalanced Sliced OT problem, see Algorithm 1 and 3 in :ref:`[82] <references-uot>`.

This function computes the reweighted marginals given the current potentials and the translation term.

It returns the current potentials, and the reweighted marginals (normalized by the mass, so that they sum to 1).

Parameters

----------

f: array-like shape (n, ...)

Current potential on the source samples

g: array-like shape (m, ...)

Current potential on the target samples

a: array-like shape (n, ...)

Current weights on the source samples

b: array-like shape (m, ...)

Current weights on the target samples

reg_m1: float

Marginal relaxation term for the source distribution

reg_m2: float

Marginal relaxation term for the target distribution

X_s_sorter: array-like shape (n_projs, n)

Sorter for the projected source samples

X_t_sorter: array-like shape (n_projs, m)

Sorter for the projected target samples

nx: module

backend module

Returns

-------

f: array-like shape (n, ...)

Current potential on the source samples

g: array-like shape (m, ...)

Current potential on the target samples

a_reweighted: array-like shape (n, ...)

Reweighted weights on the source samples (normalized by the mass)

b_reweighted: array-like shape (m, ...)

Reweighted weights on the target samples (normalized by the mass)

full_mass: array-like shape (...)

Mass of the reweighted measures

.. _references-uot:

References

----------

[82] Bonet, C., Nadjahi, K., Séjourné, T., Fatras, K., & Courty, N. (2025).

Slicing Unbalanced Optimal Transport. Transactions on Machine Learning Research.

"""

# translate potentials

transl = rescale_potentials(f, g, a, b, reg_m1, reg_m2, nx)

f = f + transl

g = g - transl

# update measures

if reg_m1 != float("inf"):

a_reweighted = (a * nx.exp(-f / reg_m1))[..., X_s_sorter]

else:

a_reweighted = a[..., X_s_sorter]

if reg_m2 != float("inf"):

b_reweighted = (b * nx.exp(-g / reg_m2))[..., X_t_sorter]

else:

b_reweighted = b[..., X_t_sorter]

full_mass = nx.sum(a_reweighted, axis=1)

# normalize the weights for compatibility with wasserstein_1d

a_reweighted = a_reweighted / nx.sum(a_reweighted, axis=1, keepdims=True)

b_reweighted = b_reweighted / nx.sum(b_reweighted, axis=1, keepdims=True)

return f, g, a_reweighted, b_reweighted, full_mass

def unbalanced_sliced_ot(

X_s,

X_t,

reg_m,

a=None,

b=None,

n_projections=50,

p=2,

projections=None,

seed=None,

numItermax=10,

log=False,

):

r"""

Compute the Unbalanced Sliced Optimal Transpot (USOT) with KL regularization between two empirical distributions.

The Unbalanced SOT problem reads as

.. math::

\mathrm{USOT}_p^p(\mu, \nu) = \inf_{\pi_1,\pi_2} \mathrm{SW}_p^p(\pi_1, \pi_2) + \mathrm{reg_{m}}_1 \mathrm{KL}(\pi_1||\mu) + \mathrm{reg_{m}}_2 \mathrm{KL}(\pi_2||\nu).

The USOT problem is solved with a Frank-Wolfe algorithm as proposed in :ref:`[82] <references-usot>`.

.. warning:: This function only works in pytorch or jax as it uses autodifferentiation to compute the 1D potentials. It is not maintained in jax.

Parameters

----------

X_s : ndarray, shape (n_samples_a, dim)

samples in the source domain

X_t : ndarray, shape (n_samples_b, dim)

samples in the target domain

reg_m: float or indexable object of length 1 or 2

Marginal relaxation term.

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

then the same `reg_m` is applied to both marginal relaxations.

The balanced OT can be recovered using `reg_m=float("inf")`.

For semi-relaxed case, use either `reg_m=(float("inf"), scalar)` or `reg_m=(scalar, float("inf"))`.

If `reg_m` is an array, it must have the same backend as input arrays `(X_s, X_t)`.

a : ndarray, shape (n_samples_a,), optional

samples weights in the source domain

b : ndarray, shape (n_samples_b,), optional

samples weights in the target domain

n_projections : int, optional

Number of projections used for the Monte-Carlo approximation

p: float, optional, by default =2

Power p used for computing the sliced Wasserstein

projections: shape (dim, n_projections), optional

Projection matrix (n_projections and seed are not used in this case)

seed: int or RandomState or None, optional

Seed used for random number generator

numItermax: int, optional

log: bool, optional

if True, returns the sot loss, the projections used, their associated EMD and the full mass of the reweighted marginals.

Returns

-------

a_reweighted: array-like shape (n, ...)

First marginal reweighted

b_reweighted: array-like shape (m, ...)

Second marginal reweighted

loss: float/array-like, shape (...)

USOT

log: dict, optional

If `log` is True, then returns a dictionary containing the projection directions used, the 1D OT losses, the SOT loss and the full mass of reweighted marginals.

.. _references-usot:

References

----------

.. [82] Bonet, C., Nadjahi, K., Séjourné, T., Fatras, K., & Courty, N. (2025).

Slicing Unbalanced Optimal Transport. Transactions on Machine Learning Research.

See Also

--------

ot.unbalanced.uot_1d: 1D OT problem

ot.unbalanced.sliced_unbalanced_ot: SUOT problem

"""

reg_m1, reg_m2 = get_parameter_pair(reg_m)

raise ValueError(

)

if a is None:

if b is None:

if projections is None:

projections = get_random_projections(

)

else:

# Compute projections of the samples, and sort them for later use in the FW algorithm

X_s_sorter = nx.argsort(X_s_projections, -1)

X_s_rev_sorter = nx.argsort(X_s_sorter, -1)

X_s_sorted = nx.take_along_axis(X_s_projections, X_s_sorter, -1)

X_t_sorter = nx.argsort(X_t_projections, -1)

X_t_rev_sorter = nx.argsort(X_t_sorter, -1)

X_t_sorted = nx.take_along_axis(X_t_projections, X_t_sorter, -1)

# Initialize potentials - WARNING: They correspond to non-sorted samples

f = nx.zeros(a.shape, type_as=a)

g = nx.zeros(b.shape, type_as=b)

for i in range(numItermax):

f, g, a_reweighted, b_reweighted, _ = get_reweighted_marginals_usot(

)

fd, gd, _ = emd_1d_dual_backprop(

X_s_sorted.T,

X_t_sorted.T,

u_weights=a_reweighted.T,

v_weights=b_reweighted.T,

p=p,

require_sort=False,

)

fd, gd = fd.T, gd.T

# default step for FW

t = 2.0 / (2.0 + i)

f = f + t * (nx.mean(nx.take_along_axis(fd, X_s_rev_sorter, 1), axis=0) - f)

g = g + t * (nx.mean(nx.take_along_axis(gd, X_t_rev_sorter, 1), axis=0) - g)

f, g, a_reweighted, b_reweighted, full_mass = get_reweighted_marginals_usot(

)

ot_loss = wasserstein_1d(

X_s_sorted.T,

X_t_sorted.T,

u_weights=a_reweighted.T,

v_weights=b_reweighted.T,

p=p,

require_sort=False,

)

sot_loss = nx.mean(ot_loss * full_mass)

if reg_m1 != float("inf"):

a_reweighted = a * nx.exp(-f / reg_m1)

else:

a_reweighted = a

if reg_m2 != float("inf"):

b_reweighted = b * nx.exp(-g / reg_m2)

else:

b_reweighted = b

if reg_m1 == float("inf") and reg_m2 == float("inf"):

uot_loss = sot_loss

elif reg_m1 == float("inf"):

uot_loss = sot_loss + reg_m2 * nx.kl_div(b_reweighted, b, mass=True)

elif reg_m2 == float("inf"):

uot_loss = sot_loss + reg_m1 * nx.kl_div(a_reweighted, a, mass=True)

else:

uot_loss = (

sot_loss

+ reg_m1 * nx.kl_div(a_reweighted, a, mass=True)

+ reg_m2 * nx.kl_div(b_reweighted, b, mass=True)

)

if log:

dico = {

"projections": projections,

"sot_loss": sot_loss,

"1d_losses": ot_loss,

"full_mass": full_mass,

}

return a_reweighted, b_reweighted, uot_loss, dico

return a_reweighted, b_reweighted, uot_loss</div>

</pre></div>

</div>

<hr/>

</div>

Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a

<a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a>

provided by <a href="https://readthedocs.org">Read the Docs</a>.

</footer>

</div>

</section>

</div>

<div class="rst-versions" data-toggle="rst-versions" role="note"

aria-label="versions">

<!-- add shift_up to the class for force viewing ,

data-toggle="rst-current-version" -->

Python Optimal Transport 0.9.7

<!-- versions

-->

<span class="rst-current-version" style="display: inline-block;padding:

0px;color:#fcfcfcab;float:left;font-size: 100%;">

Versions:

<a href="https://pythonot.github.io/"

style="padding: 3px;color:#fcfcfc;font-size: 100%;">Release</a>

<a href="https://pythonot.github.io/master"

style="padding: 3px;color:#fcfcfc;font-size: 100%;">Development</a>

<a href="https://github.com/PythonOT/POT"

style="padding: 3px;color:#fcfcfc;font-size: 100%;">Code</a>

<!--

<dl>

<dt>Versions</dt>

<dd><a href="https://pythonot.github.io/">Release</a></dd>

<dd><a href="https://pythonot.github.io/master">Development</a></dd>

<dt><a href="https://github.com/PythonOT/POT">Code on Github</a></dt>

</dl>

<hr>

</div>

</div>-->

</div><script>

jQuery(function () {

SphinxRtdTheme.Navigation.enable(true);

});

</script>

</body>

</html>

FilesExpand file tree

_sliced.html

Latest commit

History

_sliced.html

File metadata and controls