WaBarycenter: Regularized Wasserstein Barycenters
In Barycenter: Regularized Wasserstein Distances and Barycenters
Description
Usage
Arguments
Value
Author(s)
References
Examples
Description
WaBarycenter takes in a list of matrices representing joint measures on the row and column space and outputs the
corresponding Barycenter.
The list has to consist of matrices having all the same dimensions, for instance, each matrix represents the normalized weights of the corresponding pixels of images.
Usage
1
WaBarycenter(images, maxIter = 10, lambda = FALSE, costm = FALSE)
Arguments
images
A list of matrices satisfying the prerequisites described above.
maxIter
Maximum number of iterations.
lambda
Non-negative regularization parameter (for large lambda the regularized Barycenter is close to its true counterpart). If FALSE the algorithm uses a lambda depending on costm.
costm
A matrix of pairwise distances between the locations. If FALSE the algorithm uses the usual euclidean distance matrix on a [0,1]x[0,1] equidistant pixel grid.
Value
The Barycenter of the matrices, represented by a n x m matrix.
Given the MNIST dataset, a Barycenter of the digit three is shown below. The Barycenter is based on 4351 images each represented by
a 28 x 28 pixel grid, respectively. The values for lambda and maxIter were set by default. The dataset is also available in this package (c.f. three).
Author(s)
Marcel Klatt
References
Cuturi, M.: Fast Computation of Wasserstein Barycenters, Proceedings of the International Conference on Machine Learning, Beijing, China, 2014
Examples
1
2
3
4
#Computation of a Barycenter based on five images representing the digit eight, respectively.
WaBarycenter(eight,lambda=10)
#For a more reasonable but longer computation!
## Not run: WaBarycenter(eight)
Example output
Loading required package: Rcpp
user system elapsed
1.717 0.111 1.856
[,1] [,2] [,3] [,4] [,5]
[1,] 8.855358e-05 1.060048e-04 1.256737e-04 1.460480e-04 0.0001653194
[2,] 1.202892e-04 1.490866e-04 1.809473e-04 2.132977e-04 0.0002432104
[3,] 1.668672e-04 2.117140e-04 2.608292e-04 3.101688e-04 0.0003550326
[4,] 2.306307e-04 2.975316e-04 3.710263e-04 4.447984e-04 0.0005110112
[5,] 3.126949e-04 4.095071e-04 5.175997e-04 6.270200e-04 0.0007240267
[6,] 4.121965e-04 5.484372e-04 7.049394e-04 8.665656e-04 0.0010074614
[7,] 5.248879e-04 7.098135e-04 9.301209e-04 1.165741e-03 0.0013651253
[8,] 6.420595e-04 8.803838e-04 1.173715e-03 1.497736e-03 0.0017539356
[9,] 7.515420e-04 1.039347e-03 1.397237e-03 1.782432e-03 0.0020448488
[10,] 8.410897e-04 1.166871e-03 1.570278e-03 1.983520e-03 0.0022328751
[11,] 9.016635e-04 1.250344e-03 1.679376e-03 2.115455e-03 0.0023746155
[12,] 9.287979e-04 1.284975e-03 1.722344e-03 2.174086e-03 0.0024875248
[13,] 9.223865e-04 1.271992e-03 1.700935e-03 2.161275e-03 0.0025373656
[14,] 8.857166e-04 1.217287e-03 1.625749e-03 2.073894e-03 0.0024959808
[15,] 8.242515e-04 1.128679e-03 1.503913e-03 1.931682e-03 0.0023559298
[16,] 7.447406e-04 1.016236e-03 1.351579e-03 1.740718e-03 0.0021415056
[17,] 6.541300e-04 8.895851e-04 1.180691e-03 1.521008e-03 0.0018729897
[18,] 5.588461e-04 7.572303e-04 1.001780e-03 1.286435e-03 0.0015764680
[19,] 4.645497e-04 6.269074e-04 8.261976e-04 1.056999e-03 0.0012895384
[20,] 3.758862e-04 5.049088e-04 6.627859e-04 8.458027e-04 0.0010300826
[21,] 2.963336e-04 3.956879e-04 5.168144e-04 6.574472e-04 0.0008026105
[22,] 2.282297e-04 3.021971e-04 3.919366e-04 4.953502e-04 0.0006073001
[23,] 1.727905e-04 2.259382e-04 2.904782e-04 3.650209e-04 0.0004467845
[24,] 1.300924e-04 1.667879e-04 2.118020e-04 2.642053e-04 0.0003221751
[25,] 9.911791e-05 1.233247e-04 1.535504e-04 1.893349e-04 0.0002296426
[26,] 7.789160e-05 9.318981e-05 1.126106e-04 1.361409e-04 0.0001633573
[27,] 6.386466e-05 7.333887e-05 8.536897e-05 1.001976e-04 0.0001178264
[28,] 5.456979e-05 6.055542e-05 6.796570e-05 7.706301e-05 0.0000880300
[,6] [,7] [,8] [,9] [,10]
[1,] 0.0001817647 0.0001940847 0.0002016128 0.0002043525 0.0002028512
[2,] 0.0002679947 0.0002857530 0.0002957123 0.0002982321 0.0002945203
[3,] 0.0003911284 0.0004156258 0.0004277284 0.0004285887 0.0004206733
[4,] 0.0005625381 0.0005950728 0.0006080712 0.0006043759 0.0005888127
[5,] 0.0007964579 0.0008378366 0.0008486135 0.0008350743 0.0008063357
[6,] 0.0011071756 0.0011567617 0.0011593251 0.0011276202 0.0010781912
[7,] 0.0014958560 0.0015505960 0.0015369907 0.0014784473 0.0014027450
[8,] 0.0018935411 0.0019389994 0.0019167546 0.0018557547 0.0017661508
[9,] 0.0021682790 0.0022260862 0.0022354865 0.0022055998 0.0021341047
[10,] 0.0023270835 0.0023912721 0.0024477964 0.0024795783 0.0024656594
[11,] 0.0024585647 0.0025081318 0.0025703776 0.0026582050 0.0027277377
[12,] 0.0025941774 0.0026329785 0.0026880365 0.0027829631 0.0029133808
[13,] 0.0027169760 0.0027690572 0.0028200179 0.0029089972 0.0030504479
[14,] 0.0027781003 0.0028921638 0.0029442548 0.0030215854 0.0031573957
[15,] 0.0027127469 0.0029219645 0.0030233869 0.0030983380 0.0032204641
[16,] 0.0024973000 0.0027715785 0.0029464799 0.0030607422 0.0031777145
[17,] 0.0021921543 0.0024613308 0.0026700629 0.0028301551 0.0029724212
[18,] 0.0018458447 0.0020778520 0.0022704771 0.0024444230 0.0026006123
[19,] 0.0015083181 0.0017008667 0.0018678514 0.0020226832 0.0021705421
[20,] 0.0012057961 0.0013652866 0.0015068964 0.0016422323 0.0017718037
[21,] 0.0009428970 0.0010736325 0.0011942154 0.0013079649 0.0014177840
[22,] 0.0007186391 0.0008235343 0.0009225898 0.0010174768 0.0011099824
[23,] 0.0005308604 0.0006125197 0.0006917942 0.0007699879 0.0008463969
[24,] 0.0003831323 0.0004448280 0.0005066297 0.0005689259 0.0006324685
[25,] 0.0002731228 0.0003186197 0.0003656385 0.0004143373 0.0004653867
[26,] 0.0001936099 0.0002263016 0.0002611246 0.0002981342 0.0003377100
[27,] 0.0001380712 0.0001606962 0.0001855512 0.0002126423 0.0002421234
[28,] 0.0001009521 0.0001158461 0.0001327129 0.0001515766 0.0001724843
[,11] [,12] [,13] [,14] [,15]
[1,] 0.0001979741 0.0001906571 0.0001817085 0.0001717019 0.0001609690
[2,] 0.0002861898 0.0002748194 0.0002616338 0.0002473530 0.0002322222
[3,] 0.0004068967 0.0003898628 0.0003713777 0.0003522506 0.0003323828
[4,] 0.0005664950 0.0005415924 0.0005166974 0.0004925971 0.0004683653
[5,] 0.0007709763 0.0007353371 0.0007029859 0.0006745655 0.0006476961
[6,] 0.0010241942 0.0009745610 0.0009338997 0.0009023799 0.0008761396
[7,] 0.0013256035 0.0012595414 0.0012104036 0.0011773490 0.0011514258
[8,] 0.0016682027 0.0015866839 0.0015309081 0.0014987318 0.0014745265
[9,] 0.0020372219 0.0019484790 0.0018919127 0.0018654677 0.0018465414
[10,] 0.0024056104 0.0023347602 0.0022880186 0.0022680598 0.0022555739
[11,] 0.0027510413 0.0027297633 0.0027045524 0.0026944675 0.0026831773
[12,] 0.0030329483 0.0031028007 0.0031196410 0.0031128465 0.0031086475
[13,] 0.0032344734 0.0033897836 0.0034738000 0.0034873319 0.0034942823
[14,] 0.0033449518 0.0035514820 0.0037046374 0.0037623227 0.0037545143
[15,] 0.0033936198 0.0035947789 0.0037769831 0.0038606147 0.0038566161
[16,] 0.0033364227 0.0035229643 0.0036890364 0.0037885875 0.0038095676
[17,] 0.0031256509 0.0032819632 0.0034282511 0.0035482119 0.0036271160
[18,] 0.0027493772 0.0028906598 0.0030300040 0.0031719477 0.0033113126
[19,] 0.0023104533 0.0024459678 0.0025902696 0.0027506564 0.0029286584
[20,] 0.0018965041 0.0020238871 0.0021638703 0.0023284663 0.0025235104
[21,] 0.0015268046 0.0016402968 0.0017684999 0.0019258605 0.0021174681
[22,] 0.0012026704 0.0013011269 0.0014174644 0.0015585172 0.0017276267
[23,] 0.0009253554 0.0010126060 0.0011136556 0.0012330877 0.0013725690
[24,] 0.0006995257 0.0007734355 0.0008574050 0.0009536456 0.0010622777
[25,] 0.0005200616 0.0005800442 0.0006468421 0.0007211205 0.0008019308
[26,] 0.0003804867 0.0004271795 0.0004782529 0.0005334855 0.0005914676
[27,] 0.0002742280 0.0003091334 0.0003467520 0.0003864648 0.0004268284
[28,] 0.0001954627 0.0002204284 0.0002470598 0.0002746443 0.0003019289
[,16] [,17] [,18] [,19] [,20]
[1,] 0.0001496726 0.0001379140 0.0001258312 0.0001136567 0.0001017237
[2,] 0.0002161896 0.0001991445 0.0001811108 0.0001623556 0.0001434061
[3,] 0.0003111538 0.0002879491 0.0002625217 0.0002351397 0.0002066269
[4,] 0.0004420738 0.0004120656 0.0003776465 0.0003390262 0.0002975970
[5,] 0.0006175841 0.0005804995 0.0005355352 0.0004823550 0.0004233817
[6,] 0.0008434459 0.0007993172 0.0007428832 0.0006729982 0.0005912917
[7,] 0.0011156606 0.0010648737 0.0009965193 0.0009101180 0.0008070301
[8,] 0.0014395825 0.0013829372 0.0013014512 0.0011967275 0.0010711471
[9,] 0.0018158571 0.0017554356 0.0016581614 0.0015338956 0.0013836376
[10,] 0.0022257461 0.0021631274 0.0020619089 0.0019231546 0.0017458783
[11,] 0.0026590436 0.0026007269 0.0024999481 0.0023541073 0.0021524283
[12,] 0.0030940734 0.0030370685 0.0029423079 0.0027960721 0.0025825465
[13,] 0.0034712581 0.0034006097 0.0033035732 0.0031775877 0.0029789791
[14,] 0.0037099481 0.0036124771 0.0035054340 0.0033974616 0.0032365410
[15,] 0.0037805393 0.0036778571 0.0035677101 0.0034628875 0.0033416823
[16,] 0.0037519454 0.0036493218 0.0035402139 0.0034442015 0.0033506169
[17,] 0.0036362187 0.0035705406 0.0034739534 0.0033731947 0.0032770288
[18,] 0.0034177180 0.0034400258 0.0033756747 0.0032707313 0.0031600358
[19,] 0.0031080659 0.0032233604 0.0032268886 0.0031412902 0.0030289598
[20,] 0.0027310418 0.0029143990 0.0029985867 0.0029655967 0.0028716319
[21,] 0.0023271281 0.0025335019 0.0026794713 0.0027262271 0.0026631552
[22,] 0.0019196875 0.0021092808 0.0022745003 0.0023706060 0.0023557386
[23,] 0.0015288411 0.0016886891 0.0018277096 0.0019177281 0.0019266365
[24,] 0.0011797556 0.0012972112 0.0013991233 0.0014580071 0.0014560501
[25,] 0.0008857680 0.0009655520 0.0010296524 0.0010630400 0.0010528154
[26,] 0.0006490590 0.0007009228 0.0007394817 0.0007561214 0.0007436770
[27,] 0.0004653117 0.0004981714 0.0005206835 0.0005279685 0.0005163392
[28,] 0.0003270198 0.0003474080 0.0003602151 0.0003627209 0.0003530977
[,21] [,22] [,23] [,24] [,25]
[1,] 9.041997e-05 8.010965e-05 7.105494e-05 6.336546e-05 5.699294e-05
[2,] 1.249660e-04 1.077705e-04 9.244652e-05 7.940184e-05 6.875500e-05
[3,] 1.781748e-04 1.510322e-04 1.263096e-04 1.048654e-04 8.718840e-05
[4,] 2.554868e-04 2.146977e-04 1.769144e-04 1.435080e-04 1.154281e-04
[5,] 3.628129e-04 3.036253e-04 2.482511e-04 1.987061e-04 1.564138e-04
[6,] 5.067182e-04 4.230792e-04 3.442424e-04 2.733680e-04 2.124461e-04
[7,] 6.941723e-04 5.784463e-04 4.687467e-04 3.700225e-04 2.851586e-04
[8,] 9.283483e-04 7.741934e-04 6.249165e-04 4.905711e-04 3.754843e-04
[9,] 1.206557e-03 1.011560e-03 8.140884e-04 6.357185e-04 4.833449e-04
[10,] 1.529861e-03 1.286401e-03 1.035878e-03 8.042398e-04 6.070331e-04
[11,] 1.895565e-03 1.598218e-03 1.286586e-03 9.911211e-04 7.423402e-04
[12,] 2.289985e-03 1.932382e-03 1.549143e-03 1.185788e-03 8.821593e-04
[13,] 2.665390e-03 2.253707e-03 1.803000e-03 1.374665e-03 1.017196e-03
[14,] 2.950753e-03 2.519832e-03 2.030868e-03 1.542099e-03 1.136781e-03
[15,] 3.105900e-03 2.703921e-03 2.194404e-03 1.670144e-03 1.230349e-03
[16,] 3.163336e-03 2.802741e-03 2.288830e-03 1.749222e-03 1.290464e-03
[17,] 3.124980e-03 2.813609e-03 2.315965e-03 1.777737e-03 1.313957e-03
[18,] 3.019262e-03 2.750994e-03 2.285289e-03 1.759061e-03 1.300721e-03
[19,] 2.885970e-03 2.641359e-03 2.209601e-03 1.695217e-03 1.251150e-03
[20,] 2.724351e-03 2.476924e-03 2.068070e-03 1.581050e-03 1.165352e-03
[21,] 2.502784e-03 2.236521e-03 1.850619e-03 1.417582e-03 1.046881e-03
[22,] 2.214032e-03 1.949175e-03 1.590359e-03 1.219890e-03 9.048763e-04
[23,] 1.817638e-03 1.588787e-03 1.302246e-03 1.004786e-03 7.512971e-04
[24,] 1.372482e-03 1.213501e-03 1.007491e-03 7.900757e-04 5.988077e-04
[25,] 9.911363e-04 8.824361e-04 7.432256e-04 5.955776e-04 4.592523e-04
[26,] 6.990873e-04 6.257772e-04 5.333898e-04 4.345780e-04 3.407537e-04
[27,] 4.847320e-04 4.356236e-04 3.747109e-04 3.093029e-04 2.462503e-04
[28,] 3.311175e-04 2.985130e-04 2.587349e-04 2.161351e-04 1.749345e-04
[,26] [,27] [,28]
[1,] 5.177216e-05 4.748707e-05 4.392990e-05
[2,] 6.033711e-05 5.377985e-05 4.865018e-05
[3,] 7.330889e-05 6.281631e-05 5.501978e-05
[4,] 9.307093e-05 7.619342e-05 6.397340e-05
[5,] 1.221209e-04 9.583541e-05 7.678648e-05
[6,] 1.624527e-04 1.234896e-04 9.482897e-05
[7,] 2.152656e-04 1.602493e-04 1.191455e-04
[8,] 2.809406e-04 2.063628e-04 1.501134e-04
[9,] 3.589519e-04 2.612196e-04 1.873258e-04
[10,] 4.475804e-04 3.233105e-04 2.296218e-04
[11,] 5.435568e-04 3.901462e-04 2.751511e-04
[12,] 6.419515e-04 4.582645e-04 3.214630e-04
[13,] 7.364871e-04 5.234262e-04 3.656648e-04
[14,] 8.201559e-04 5.810113e-04 4.046736e-04
[15,] 8.862066e-04 6.265927e-04 4.355505e-04
[16,] 9.293955e-04 6.565665e-04 4.558524e-04
[17,] 9.466948e-04 6.685791e-04 4.639017e-04
[18,] 9.370513e-04 6.615969e-04 4.589150e-04
[19,] 9.006503e-04 6.357738e-04 4.410296e-04
[20,] 8.388413e-04 5.924690e-04 4.113265e-04
[21,] 7.551384e-04 5.344810e-04 3.718545e-04
[22,] 6.557351e-04 4.660759e-04 3.255392e-04
[23,] 5.485980e-04 3.925216e-04 2.758899e-04
[24,] 4.421221e-04 3.193195e-04 2.265561e-04
[25,] 3.438202e-04 2.514005e-04 1.808533e-04
[26,] 2.590554e-04 1.924661e-04 1.413522e-04
[27,] 1.904760e-04 1.446551e-04 1.095777e-04
[28,] 1.384131e-04 1.085224e-04 8.582502e-05
Barycenter documentation built on May 2, 2019, 6:47 a.m.
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Description Usage Arguments Value Author(s) References Examples
Description
WaBarycenter takes in a list of matrices representing joint measures on the row and column space and outputs the
corresponding Barycenter.
The list has to consist of matrices having all the same dimensions, for instance, each matrix represents the normalized weights of the corresponding pixels of images.
Usage
1 | WaBarycenter(images, maxIter = 10, lambda = FALSE, costm = FALSE)
|
Arguments
images |
A list of matrices satisfying the prerequisites described above. |
maxIter |
Maximum number of iterations. |
lambda |
Non-negative regularization parameter (for large lambda the regularized Barycenter is close to its true counterpart). If FALSE the algorithm uses a lambda depending on |
costm |
A matrix of pairwise distances between the locations. If FALSE the algorithm uses the usual euclidean distance matrix on a [0,1]x[0,1] equidistant pixel grid. |
Value
The Barycenter of the matrices, represented by a n x m matrix.
Given the MNIST dataset, a Barycenter of the digit three is shown below. The Barycenter is based on 4351 images each represented by
a 28 x 28 pixel grid, respectively. The values for lambda and maxIter were set by default. The dataset is also available in this package (c.f. three).
Author(s)
Marcel Klatt
References
Cuturi, M.: Fast Computation of Wasserstein Barycenters, Proceedings of the International Conference on Machine Learning, Beijing, China, 2014
Examples
1 2 3 4 | #Computation of a Barycenter based on five images representing the digit eight, respectively.
WaBarycenter(eight,lambda=10)
#For a more reasonable but longer computation!
## Not run: WaBarycenter(eight)
|
Example output
Loading required package: Rcpp
user system elapsed
1.717 0.111 1.856
[,1] [,2] [,3] [,4] [,5]
[1,] 8.855358e-05 1.060048e-04 1.256737e-04 1.460480e-04 0.0001653194
[2,] 1.202892e-04 1.490866e-04 1.809473e-04 2.132977e-04 0.0002432104
[3,] 1.668672e-04 2.117140e-04 2.608292e-04 3.101688e-04 0.0003550326
[4,] 2.306307e-04 2.975316e-04 3.710263e-04 4.447984e-04 0.0005110112
[5,] 3.126949e-04 4.095071e-04 5.175997e-04 6.270200e-04 0.0007240267
[6,] 4.121965e-04 5.484372e-04 7.049394e-04 8.665656e-04 0.0010074614
[7,] 5.248879e-04 7.098135e-04 9.301209e-04 1.165741e-03 0.0013651253
[8,] 6.420595e-04 8.803838e-04 1.173715e-03 1.497736e-03 0.0017539356
[9,] 7.515420e-04 1.039347e-03 1.397237e-03 1.782432e-03 0.0020448488
[10,] 8.410897e-04 1.166871e-03 1.570278e-03 1.983520e-03 0.0022328751
[11,] 9.016635e-04 1.250344e-03 1.679376e-03 2.115455e-03 0.0023746155
[12,] 9.287979e-04 1.284975e-03 1.722344e-03 2.174086e-03 0.0024875248
[13,] 9.223865e-04 1.271992e-03 1.700935e-03 2.161275e-03 0.0025373656
[14,] 8.857166e-04 1.217287e-03 1.625749e-03 2.073894e-03 0.0024959808
[15,] 8.242515e-04 1.128679e-03 1.503913e-03 1.931682e-03 0.0023559298
[16,] 7.447406e-04 1.016236e-03 1.351579e-03 1.740718e-03 0.0021415056
[17,] 6.541300e-04 8.895851e-04 1.180691e-03 1.521008e-03 0.0018729897
[18,] 5.588461e-04 7.572303e-04 1.001780e-03 1.286435e-03 0.0015764680
[19,] 4.645497e-04 6.269074e-04 8.261976e-04 1.056999e-03 0.0012895384
[20,] 3.758862e-04 5.049088e-04 6.627859e-04 8.458027e-04 0.0010300826
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