pythonot.github.io/_modules/ot/gpu/bregman.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

<!DOCTYPE html>

<html class="no-js" lang="en" >

<head>

<title>ot.gpu.bregman — POT Python Optimal Transport 0.7.0b documentation</title>

</head>

<a href="../../../index.html" class="icon icon-home"> POT Python Optimal Transport

</a>

0.7.0b

</div>

</form>

</div>

<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="../../../quickstart.html">Quick start guide</a></li>

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

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

</ul>

</div>

</nav>

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

</nav>

<li><a href="../../index.html">Module code</a> »</li>

<li>ot.gpu.bregman</li>

</li>

</ul>

<hr/>

</div>

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

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

"""

Bregman projections for regularized OT with GPU

"""

# Author: Remi Flamary <remi.flamary@unice.fr>

# Leo Gautheron <https://github.com/aje>

#

# License: MIT License

import cupy as np # np used for matrix computation

import cupy as cp # cp used for cupy specific operations

from . import utils

def sinkhorn_knopp(a, b, M, reg, numItermax=1000, stopThr=1e-9,

verbose=False, log=False, to_numpy=True, **kwargs):

"""

Solve the entropic regularization optimal transport on GPU

If the input matrix are in numpy format, they will be uploaded to the

GPU first which can incur significant time overhead.

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)

The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in [2]_

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)

verbose : bool, optional

Print information along iterations

log : bool, optional

record log if True

to_numpy : boolean, optional (default True)

If true convert back the GPU array result to numpy format.

Returns

-------

gamma : (ns x nt) ndarray

Optimal transportation matrix for the given parameters

log : dict

log dictionary return only if log==True in parameters

References

----------

.. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport, Advances in Neural Information Processing Systems (NIPS) 26, 2013

See Also

--------

ot.lp.emd : Unregularized OT

ot.optim.cg : General regularized OT

"""

a = cp.asarray(a)

b = cp.asarray(b)

M = cp.asarray(M)

if len(a) == 0:

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

if len(b) == 0:

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

# init data

Nini = len(a)

Nfin = len(b)

if len(b.shape) > 1:

nbb = b.shape[1]

else:

nbb = 0

if log:

log = {'err': []}

# we assume that no distances are null except those of the diagonal of

# distances

if nbb:

u = np.ones((Nini, nbb)) / Nini

v = np.ones((Nfin, nbb)) / Nfin

else:

u = np.ones(Nini) / Nini

v = np.ones(Nfin) / Nfin

# print(reg)

# Next 3 lines equivalent to K= np.exp(-M/reg), but faster to compute

K = np.empty(M.shape, dtype=M.dtype)

np.divide(M, -reg, out=K)

np.exp(K, out=K)

# print(np.min(K))

tmp2 = np.empty(b.shape, dtype=M.dtype)

Kp = (1 / a).reshape(-1, 1) * K

cpt = 0

err = 1

while (err > stopThr and cpt < numItermax):

uprev = u

vprev = v

KtransposeU = np.dot(K.T, u)

v = np.divide(b, KtransposeU)

u = 1. / np.dot(Kp, v)

if (np.any(KtransposeU == 0) or

np.any(np.isnan(u)) or np.any(np.isnan(v)) or

np.any(np.isinf(u)) or np.any(np.isinf(v))):

# we have reached the machine precision

# come back to previous solution and quit loop

print('Warning: numerical errors at iteration', cpt)

u = uprev

v = vprev

break

if cpt % 10 == 0:

# we can speed up the process by checking for the error only all

# the 10th iterations

if nbb:

err = np.sum((u - uprev)**2) / np.sum((u)**2) + \

np.sum((v - vprev)**2) / np.sum((v)**2)

else:

# compute right marginal tmp2= (diag(u)Kdiag(v))^T1

tmp2 = np.sum(u[:, None] * K * v[None, :], 0)

#tmp2=np.einsum('i,ij,j->j', u, K, v)

err = np.linalg.norm(tmp2 - b)**2 # violation of marginal

if log:

log['err'].append(err)

if verbose:

if cpt % 200 == 0:

print(

'{:5s}|{:12s}'.format('It.', 'Err') + '\n' + '-' * 19)

print('{:5d}|{:8e}|'.format(cpt, err))

cpt = cpt + 1

if log:

log['u'] = u

log['v'] = v

if nbb: # return only loss

#res = np.einsum('ik,ij,jk,ij->k', u, K, v, M) (explodes cupy memory)

res = np.empty(nbb)

for i in range(nbb):

res[i] = np.sum(u[:, None, i] * (K * M) * v[None, :, i])

if to_numpy:

res = utils.to_np(res)

if log:

return res, log

else:

return res

else: # return OT matrix

res = u.reshape((-1, 1)) * K * v.reshape((1, -1))

if to_numpy:

if log:

return res, log

else:

return res

# define sinkhorn as sinkhorn_knopp

sinkhorn = sinkhorn_knopp

</pre></div>

</div>

<hr/>

</div>

Built with <a href="http://sphinx-doc.org/">Sphinx</a> using a <a href="https://github.com/rtfd/sphinx_rtd_theme">theme</a> provided by <a href="https://readthedocs.org">Read the Docs</a>.

</footer>

</div>

</section>

</div>

jQuery(function () {

SphinxRtdTheme.Navigation.enable(true);

});

</script>

</body>

</html>

FilesExpand file tree

bregman.html

Latest commit

History

bregman.html

File metadata and controls