8 Chain Rule Optimal Transport
215
References
1. Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving,
G., Isard, M., et al.: Tensorflow: a system for large-scale machine learning. In: 12th USENIX
Symposium on Operating Systems Design and Implementation (OSDI 16), pp. 265–283 (2016)
2. Amari, S.-I.: Information Geometry and Its Applications. Applied Mathematical Sciences.
Springer, Japan (2016)
3. Bauschke, H.H., Borwein, J.M.: Joint and separate convexity of the Bregman distance. In:
Studies in Computational Mathematics, vol. 8, pp. 23–36. Elsevier (2001)
4. Bonneel, N., Rabin, J., Peyré, G., Pfister, H.: Sliced and radon Wasserstein barycenters of
measures. J. Math. Imaging Vis. 51(1), 22–45 (2015)
5. Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations and
Counterexamples, vol. 109. Cambridge University Press, Cambridge (2010)
6. Chang, K.-C., Sun, W.: Scalable fusion with mixture distributions in sensor networks. In: 11th
International Conference on Control Automation Robotics & Vision (ICARCV), pp. 1251–
1256 (2010)
7. Chen, Y., Georgiou, T.T., Tannenbaum, A.: Optimal transport for Gaussian mixture models.
IEEE Access 7, 6269–6278 (2019)
8. Cuturi, M.: Sinkhorn distances: lightspeed computation of optimal transport. In: NIPS, pp.
2292–2300 (2013)
9. Cuturi, M., Teboul, O., Vert, J.: Differentiable sorting using optimal transport: the Sinkhorn
CDF and quantile operator (2019). CoRR arXiv:abs/1905.11885
10. Dacorogna, B., Maréchal, P.: The role of perspective functions in convexity, polyconvexity,
rank-one convexity and separate convexity. J. Convex Anal. 15(2), 271 (2008)
11. Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the
EM algorithm. J. R. Stat. Soc. Ser. B (Methodol.), pp. 1–38 (1977)
12. Do, M.N.: Fast approximation of Kullback–Leibler distance for dependence trees and hidden
Markov models. IEEE Signal Process. Lett. 10(4), 115–118 (2003)
13. Dowson, D.C., Landau, B.: The Fréchet distance between multivariate normal distributions. J.
Multivar. Anal. 12(3), 450–455 (1982)
14. Dragomir, S.S.: Inequalities for Csiszár f-divergence in information theory. Victoria University,
Melbourne, Australia (2000)
15. Durrieu, J.-L., Thiran, J.-P., Kelly, F.: Lower and upper bounds for approximation of the
Kullback–Leibler divergence between Gaussian mixture models. In: 2012 IEEE International
Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4833–4836. IEEE
(2012)
16. Everett, B.: An Introduction to Latent Variable Models. Springer Science & Business Media
(2013)
17. Feydy, J., Séjourné, T., Vialard, F.-X., Amari, S.-I., Trouvé, A., Peyré, G.: Interpolating between
optimal transport and MMD using Sinkhorn divergences (2018). arXiv:1810.08278
18. Flamary, R., Courty, N.: POT python optimal transport library (2017)
19. Fuglede, B., Topsoe, F.: Jensen-Shannon divergence and Hilbert space embedding. In: International Symposium on Information Theory (ISIT 2004), p. 31. IEEE (2004)
20. Gangbo, W., McCann, R.J.: The geometry of optimal transportation. Acta Math. 177(2), 113–
161 (1996)
21. Gelbrich, M.: On a formula for the L2 Wasserstein metric between measures on Euclidean and
Hilbert spaces. Mathematische Nachrichten 147(1), 185–203 (1990)
22. Ghaffari, N., Walker, S.: On multivariate optimal transportation (2018)
23. Goldberger, J., Aronowitz, H.: A distance measure between GMMs based on the unscented
transform and its application to speaker recognition. In: INTERSPEECH European Conference
on Speech Communication and Technology, pp. 1985–1988 (2005)
24. Goldberger, J., Gordon, S., Greenspan, H.: An efficient image similarity measure based on
approximations of KL-divergence between two Gaussian mixtures. In: IEEE International
Conference on Computer Vision (ICCV), p. 487. IEEE (2003)
215
References
1. Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving,
G., Isard, M., et al.: Tensorflow: a system for large-scale machine learning. In: 12th USENIX
Symposium on Operating Systems Design and Implementation (OSDI 16), pp. 265–283 (2016)
2. Amari, S.-I.: Information Geometry and Its Applications. Applied Mathematical Sciences.
Springer, Japan (2016)
3. Bauschke, H.H., Borwein, J.M.: Joint and separate convexity of the Bregman distance. In:
Studies in Computational Mathematics, vol. 8, pp. 23–36. Elsevier (2001)
4. Bonneel, N., Rabin, J., Peyré, G., Pfister, H.: Sliced and radon Wasserstein barycenters of
measures. J. Math. Imaging Vis. 51(1), 22–45 (2015)
5. Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations and
Counterexamples, vol. 109. Cambridge University Press, Cambridge (2010)
6. Chang, K.-C., Sun, W.: Scalable fusion with mixture distributions in sensor networks. In: 11th
International Conference on Control Automation Robotics & Vision (ICARCV), pp. 1251–
1256 (2010)
7. Chen, Y., Georgiou, T.T., Tannenbaum, A.: Optimal transport for Gaussian mixture models.
IEEE Access 7, 6269–6278 (2019)
8. Cuturi, M.: Sinkhorn distances: lightspeed computation of optimal transport. In: NIPS, pp.
2292–2300 (2013)
9. Cuturi, M., Teboul, O., Vert, J.: Differentiable sorting using optimal transport: the Sinkhorn
CDF and quantile operator (2019). CoRR arXiv:abs/1905.11885
10. Dacorogna, B., Maréchal, P.: The role of perspective functions in convexity, polyconvexity,
rank-one convexity and separate convexity. J. Convex Anal. 15(2), 271 (2008)
11. Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the
EM algorithm. J. R. Stat. Soc. Ser. B (Methodol.), pp. 1–38 (1977)
12. Do, M.N.: Fast approximation of Kullback–Leibler distance for dependence trees and hidden
Markov models. IEEE Signal Process. Lett. 10(4), 115–118 (2003)
13. Dowson, D.C., Landau, B.: The Fréchet distance between multivariate normal distributions. J.
Multivar. Anal. 12(3), 450–455 (1982)
14. Dragomir, S.S.: Inequalities for Csiszár f-divergence in information theory. Victoria University,
Melbourne, Australia (2000)
15. Durrieu, J.-L., Thiran, J.-P., Kelly, F.: Lower and upper bounds for approximation of the
Kullback–Leibler divergence between Gaussian mixture models. In: 2012 IEEE International
Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4833–4836. IEEE
(2012)
16. Everett, B.: An Introduction to Latent Variable Models. Springer Science & Business Media
(2013)
17. Feydy, J., Séjourné, T., Vialard, F.-X., Amari, S.-I., Trouvé, A., Peyré, G.: Interpolating between
optimal transport and MMD using Sinkhorn divergences (2018). arXiv:1810.08278
18. Flamary, R., Courty, N.: POT python optimal transport library (2017)
19. Fuglede, B., Topsoe, F.: Jensen-Shannon divergence and Hilbert space embedding. In: International Symposium on Information Theory (ISIT 2004), p. 31. IEEE (2004)
20. Gangbo, W., McCann, R.J.: The geometry of optimal transportation. Acta Math. 177(2), 113–
161 (1996)
21. Gelbrich, M.: On a formula for the L2 Wasserstein metric between measures on Euclidean and
Hilbert spaces. Mathematische Nachrichten 147(1), 185–203 (1990)
22. Ghaffari, N., Walker, S.: On multivariate optimal transportation (2018)
23. Goldberger, J., Aronowitz, H.: A distance measure between GMMs based on the unscented
transform and its application to speaker recognition. In: INTERSPEECH European Conference
on Speech Communication and Technology, pp. 1985–1988 (2005)
24. Goldberger, J., Gordon, S., Greenspan, H.: An efficient image similarity measure based on
approximations of KL-divergence between two Gaussian mixtures. In: IEEE International
Conference on Computer Vision (ICCV), p. 487. IEEE (2003)