Monotone 1-D transport map fitted to the quantile coupling
Source:R/verification_ports_native.R
morie_neural_kantorovich_map.RdFits a monotone softplus-sigmoid basis to the known 1-D optimal coupling \(T = F_{target}^{-1}\circ F_{source}\) by least squares. Monotone by construction (softplus weights are non-negative). This is a monotone fit to the closed-form coupling, not a solve of the Kantorovich problem.
Usage
morie_neural_kantorovich_map(
source,
target,
n_iter = 400L,
lr = 0.05,
n_basis = 12L,
seed = 0L
)References
Makkuva, A., Taghvaei, A., Oh, S. and Lee, J. (2020). Optimal transport mapping via input convex neural networks. In ICML. Brenier, Y. (1991). Polar factorization and monotone rearrangement of vector-valued functions. Communications on Pure and Applied Mathematics, 44(4), 375-417.
Examples
V <- c(1, 2, 3, 4, 5, 6, 7, 8)
morie_neural_kantorovich_map(V, V)
#> $map_at_source
#> [1] 1.728938 2.377740 3.137907 3.987659 4.892472 5.808326 6.687427 7.486046
#>
#> $exact_map
#> [1] 1.4375 2.3125 3.1875 4.0625 4.9375 5.8125 6.6875 7.5625
#>
#> $monotone
#> [1] TRUE
#>
#> $rmse_vs_exact
#> [1] 0.1146424
#>
#> $w2
#> [1] 0.13792
#>
#> $w2_exact
#> [1] 0.08203125
#>
#> $loss_history
#> [1] 18.46997242 12.53941814 8.81381467 6.38047680 4.74693426 3.62800874
#> [7] 2.84974501 2.30182410 1.91218643 1.63268021 1.43055368 1.28321643
#> [13] 1.17491289 1.09455630 1.03428932 0.98851107 0.95321048 0.92550415
#> [19] 0.90331304 0.88513442 0.86988002 0.85676055 0.84520291 0.83479063
#> [25] 0.82522090 0.81627351 0.80778829 0.79964876 0.79177026 0.78409128
#> [31] 0.77656720 0.76916566 0.76186324 0.75464304 0.74749285 0.74040394
#> [37] 0.73337003 0.72638669 0.71945076 0.71256005 0.70571305 0.69890871
#> [43] 0.69214637 0.68542560 0.67874615 0.67210789 0.66551079 0.65895488
#> [49] 0.65244020 0.64596685 0.63953494 0.63314457 0.62679586 0.62048892
#> [55] 0.61422387 0.60800082 0.60181986 0.59568110 0.58958464 0.58353054
#> [61] 0.57751889 0.57154975 0.56562320 0.55973928 0.55389804 0.54809953
#> [67] 0.54234377 0.53663079 0.53096062 0.52533325 0.51974871 0.51420699
#> [73] 0.50870809 0.50325198 0.49783865 0.49246807 0.48714022 0.48185504
#> [79] 0.47661251 0.47141256 0.46625515 0.46114020 0.45606766 0.45103744
#> [85] 0.44604948 0.44110368 0.43619996 0.43133822 0.42651836 0.42174029
#> [91] 0.41700388 0.41230903 0.40765561 0.40304351 0.39847259 0.39394273
#> [97] 0.38945378 0.38500560 0.38059805 0.37623098 0.37190424 0.36761767
#> [103] 0.36337110 0.35916438 0.35499733 0.35086978 0.34678156 0.34273249
#> [109] 0.33872239 0.33475107 0.33081835 0.32692402 0.32306791 0.31924981
#> [115] 0.31546952 0.31172685 0.30802158 0.30435351 0.30072243 0.29712814
#> [121] 0.29357040 0.29004902 0.28656377 0.28311444 0.27970080 0.27632262
#> [127] 0.27297969 0.26967177 0.26639864 0.26316008 0.25995584 0.25678569
#> [133] 0.25364941 0.25054676 0.24747750 0.24444140 0.24143821 0.23846771
#> [139] 0.23552964 0.23262377 0.22974987 0.22690768 0.22409697 0.22131750
#> [145] 0.21856901 0.21585128 0.21316405 0.21050708 0.20788013 0.20528295
#> [151] 0.20271530 0.20017694 0.19766762 0.19518710 0.19273513 0.19031147
#> [157] 0.18791588 0.18554811 0.18320792 0.18089507 0.17860931 0.17635041
#> [163] 0.17411812 0.17191221 0.16973243 0.16757854 0.16545030 0.16334749
#> [169] 0.16126985 0.15921715 0.15718916 0.15518564 0.15320636 0.15125108
#> [175] 0.14931958 0.14741161 0.14552696 0.14366539 0.14182666 0.14001057
#> [181] 0.13821687 0.13644534 0.13469577 0.13296792 0.13126158 0.12957652
#> [187] 0.12791252 0.12626937 0.12464685 0.12304475 0.12146284 0.11990093
#> [193] 0.11835878 0.11683620 0.11533298 0.11384891 0.11238377 0.11093738
#> [199] 0.10950951 0.10809998 0.10670858 0.10533511 0.10397937 0.10264116
#> [205] 0.10132030 0.10001658 0.09872982 0.09745983 0.09620641 0.09496937
#> [211] 0.09374854 0.09254373 0.09135474 0.09018141 0.08902355 0.08788098
#> [217] 0.08675353 0.08564101 0.08454326 0.08346010 0.08239136 0.08133687
#> [223] 0.08029646 0.07926996 0.07825721 0.07725805 0.07627232 0.07529984
#> [229] 0.07434046 0.07339403 0.07246039 0.07153938 0.07063085 0.06973464
#> [235] 0.06885061 0.06797861 0.06711849 0.06627010 0.06543330 0.06460794
#> [241] 0.06379388 0.06299099 0.06219912 0.06141813 0.06064789 0.05988827
#> [247] 0.05913912 0.05840033 0.05767175 0.05695325 0.05624472 0.05554602
#> [253] 0.05485703 0.05417761 0.05350766 0.05284705 0.05219566 0.05155337
#> [259] 0.05092006 0.05029561 0.04967992 0.04907287 0.04847434 0.04788422
#> [265] 0.04730241 0.04672880 0.04616327 0.04560572 0.04505605 0.04451416
#> [271] 0.04397993 0.04345328 0.04293409 0.04242227 0.04191773 0.04142035
#> [277] 0.04093006 0.04044675 0.03997034 0.03950072 0.03903780 0.03858151
#> [283] 0.03813174 0.03768842 0.03725144 0.03682074 0.03639621 0.03597778
#> [289] 0.03556537 0.03515890 0.03475827 0.03436342 0.03397426 0.03359072
#> [295] 0.03321271 0.03284017 0.03247302 0.03211118 0.03175458 0.03140315
#> [301] 0.03105682 0.03071551 0.03037916 0.03004770 0.02972105 0.02939916
#> [307] 0.02908196 0.02876938 0.02846136 0.02815783 0.02785873 0.02756400
#> [313] 0.02727357 0.02698740 0.02670541 0.02642755 0.02615376 0.02588398
#> [319] 0.02561816 0.02535624 0.02509817 0.02484389 0.02459334 0.02434648
#> [325] 0.02410325 0.02386360 0.02362748 0.02339484 0.02316563 0.02293980
#> [331] 0.02271730 0.02249809 0.02228211 0.02206932 0.02185969 0.02165315
#> [337] 0.02144967 0.02124919 0.02105169 0.02085712 0.02066543 0.02047658
#> [343] 0.02029053 0.02010724 0.01992667 0.01974879 0.01957354 0.01940091
#> [349] 0.01923083 0.01906329 0.01889824 0.01873565 0.01857547 0.01841768
#> [355] 0.01826225 0.01810912 0.01795828 0.01780969 0.01766332 0.01751913
#> [361] 0.01737709 0.01723717 0.01709934 0.01696357 0.01682983 0.01669809
#> [367] 0.01656831 0.01644048 0.01631455 0.01619051 0.01606833 0.01594797
#> [373] 0.01582942 0.01571264 0.01559761 0.01548430 0.01537269 0.01526275
#> [379] 0.01515446 0.01504779 0.01494272 0.01483922 0.01473728 0.01463686
#> [385] 0.01453796 0.01444053 0.01434457 0.01425004 0.01415693 0.01406522
#> [391] 0.01397489 0.01388591 0.01379827 0.01371194 0.01362691 0.01354316
#> [397] 0.01346066 0.01337940 0.01329936 0.01322053
#>
#> $converged
#> [1] FALSE
#>
#> $n_source
#> [1] 8
#>
#> $n_target
#> [1] 8
#>