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Gabriel Peyré (CNRS & ENS, Paris, France) Sparsistency for Inverse Optimal Transport

Jia-Jie Zhu
March 27, 2024
59

Gabriel Peyré (CNRS & ENS, Paris, France) Sparsistency for Inverse Optimal Transport

WORKSHOP ON OPTIMAL TRANSPORT
FROM THEORY TO APPLICATIONS
INTERFACING DYNAMICAL SYSTEMS, OPTIMIZATION, AND MACHINE LEARNING
Venue: Humboldt University of Berlin, Dorotheenstraße 24

Berlin, Germany. March 11th - 15th, 2024

Jia-Jie Zhu

March 27, 2024
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  1. OT for Single cell genomics Gromov Wasserstein as metric learning

    Inverse OT for metric learning Wasserstein Singular Vectors for metric learning
  2. Single Cell Multi-omics Understanding cell diversity: many types, many states

    for each type. Applications: cancer mutations, dynamic of adaptation, development, … <latexit 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cells <latexit 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2 Rd Tissue Dissociation isolation RNA amplification sequencing … <latexit 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d ⇠ 102
  3. Single Cell Multi-omics Understanding cell diversity: many types, many states

    for each type. Applications: cancer mutations, dynamic of adaptation, development, … <latexit 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cells <latexit 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2 Rd Tissue Dissociation isolation RNA amplification sequencing … <latexit 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d ⇠ 102 Multi-omics integration: next frontier … Accessability Gene Expression A B C A B C A (n) A (n) A (n) A (n) A (n) Protein A (n) RNA Chromatin Protein Abundance A D E Gene Expression A B C D E F G RNA DNA Protein Transcription Translation A (n) A (n) A (n) A (n) A (n) Protein A (n) RNA Chromatin Protein Abundance A D E Gene Expression A B C D E F G ATAC-seq Different omics spaces: RNA-seq CITE-seq <latexit 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d ⇠ 104 <latexit 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d ⇠ 105
  4. Comparing Distributions for Single Cells • Until recently, samples contained

    many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 carelli et al. 2018 2Lähnemann et al. 2020 Single-cell profiling uncovers cellular heterogeneity • Until recently, samples contained many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 1Mincarelli et al. 2018 2Lähnemann et al. 2020 <latexit sha1_base64="8h9qfn6MXuBvKOcg4p0NkYUZouc=">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</latexit> Bulk <latexit sha1_base64="xfWO8cpwoNz5R1B8gV9utdLBqVI=">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</latexit> Single cell
  5. Comparing Distributions for Single Cells • Until recently, samples contained

    many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 carelli et al. 2018 2Lähnemann et al. 2020 Single-cell profiling uncovers cellular heterogeneity • Until recently, samples contained many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 1Mincarelli et al. 2018 2Lähnemann et al. 2020 <latexit sha1_base64="8h9qfn6MXuBvKOcg4p0NkYUZouc=">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</latexit> Bulk <latexit sha1_base64="xfWO8cpwoNz5R1B8gV9utdLBqVI=">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</latexit> Single cell <latexit sha1_base64="pydPpuNuJAJzk3J9ohxA5exGGlo=">AAACyHicjVHLSsNAFD2Nr1pfVZdugkVwVZIi6s6iG3FVwbSFtkgyndaheZFMlFLc+ANu9cvEP9C/8M6YglpEJyQ5c+49Z+be68W+SKVlvRaMufmFxaXicmlldW19o7y51UyjLGHcYZEfJW3PTbkvQu5IIX3ejhPuBp7PW97oTMVbtzxJRRReyXHMe4E7DMVAMFcS5XS9gXlyXa5YVUsvcxbYOaggX42o/IIu+ojAkCEARwhJ2IeLlJ4ObFiIiethQlxCSOg4xz1KpM0oi1OGS+yIvkPadXI2pL3yTLWa0Sk+vQkpTeyRJqK8hLA6zdTxTDsr9jfvifZUdxvT38u9AmIlboj9SzfN/K9O1SIxwLGuQVBNsWZUdSx3yXRX1M3NL1VJcoiJU7hP8YQw08ppn02tSXXtqreujr/pTMWqPctzM7yrW9KA7Z/jnAXNWtU+rB5c1ir103zURexgF/s0zyPUcY4GHPIWeMQTno0LIzbujPFnqlHINdv4toyHD4JFkLc=</latexit> ? <latexit sha1_base64="QNHLtJL77Xel7kqDwcr8uqkXoFI=">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</latexit> Cluster cells from a single biopsy. <latexit sha1_base64="NAYrO+hblycchNy+rUEIB63nwtY=">AAAC6XicjVHLLgRBFD3a+z1Y2nQMYTXpEcGOYGFJYphkRma6y0Vnqh/prvbIxA/Y2YmtH7DlR8Qf8BdulZ7EI0J1uvvUufecqnuvF0s/VY7z0mV19/T29Q8MDg2PjI6NFyYm99MoSwRVRCSjpOq5KUk/pIrylaRqnJAbeJIOvNamjh+cUZL6UbinLmM6DNyT0D/2hauYahRm64ouWNduNrf4NDcUND9ve6TOiUJbkJTp2lWjUHRKjln2T1DOQRH52okKz6jjCBEEMgQghFCMJVyk/NRQhoOYuUO0mUsY+SZOuMIQazPOIs5wmW3x94R3tZwNea89U6MWfIrkN2GljTnWRJyXMNan2SaeGWfN/ubdNp76bpf893KvgFmFU2b/0nUy/6vTtSgcY9XU4HNNsWF0dSJ3yUxX9M3tT1UpdoiZ0/iI4wljYZSdPttGk5radW9dE381mZrVe5HnZnjTt+QBl7+P8yfYXyyVl0tLu4vF9Y181AOYxgwWeJ4rWMc2dlBh72s84BFPVsu6sW6tu49UqyvXTOHLsu7fAYBOnkY=</latexit> “Distance” between cells? <latexit sha1_base64="yFN6xhMLWSFTo6y4Ic3WSee5g64=">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</latexit> OT on genes’ space. <latexit sha1_base64="pydPpuNuJAJzk3J9ohxA5exGGlo=">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</latexit> ?
  6. Comparing Distributions for Single Cells • Until recently, samples contained

    many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 carelli et al. 2018 2Lähnemann et al. 2020 Single-cell profiling uncovers cellular heterogeneity • Until recently, samples contained many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 1Mincarelli et al. 2018 2Lähnemann et al. 2020 <latexit sha1_base64="8h9qfn6MXuBvKOcg4p0NkYUZouc=">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</latexit> Bulk <latexit sha1_base64="xfWO8cpwoNz5R1B8gV9utdLBqVI=">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</latexit> Single cell <latexit sha1_base64="pydPpuNuJAJzk3J9ohxA5exGGlo=">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</latexit> ? <latexit sha1_base64="QNHLtJL77Xel7kqDwcr8uqkXoFI=">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</latexit> Cluster cells from a single biopsy. <latexit sha1_base64="NAYrO+hblycchNy+rUEIB63nwtY=">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</latexit> “Distance” between cells? <latexit sha1_base64="yFN6xhMLWSFTo6y4Ic3WSee5g64=">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</latexit> OT on genes’ space. <latexit sha1_base64="c8Nzu7i/2HdZbRFNoq24LWiMXCg=">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</latexit> Match cells between two biopsies. <latexit sha1_base64="smAhPpTgHUrzBKRMXffs73pgv0w=">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</latexit> OT on cells’ space. <latexit sha1_base64="pydPpuNuJAJzk3J9ohxA5exGGlo=">AAACyHicjVHLSsNAFD2Nr1pfVZdugkVwVZIi6s6iG3FVwbSFtkgyndaheZFMlFLc+ANu9cvEP9C/8M6YglpEJyQ5c+49Z+be68W+SKVlvRaMufmFxaXicmlldW19o7y51UyjLGHcYZEfJW3PTbkvQu5IIX3ejhPuBp7PW97oTMVbtzxJRRReyXHMe4E7DMVAMFcS5XS9gXlyXa5YVUsvcxbYOaggX42o/IIu+ojAkCEARwhJ2IeLlJ4ObFiIiethQlxCSOg4xz1KpM0oi1OGS+yIvkPadXI2pL3yTLWa0Sk+vQkpTeyRJqK8hLA6zdTxTDsr9jfvifZUdxvT38u9AmIlboj9SzfN/K9O1SIxwLGuQVBNsWZUdSx3yXRX1M3NL1VJcoiJU7hP8YQw08ppn02tSXXtqreujr/pTMWqPctzM7yrW9KA7Z/jnAXNWtU+rB5c1ir103zURexgF/s0zyPUcY4GHPIWeMQTno0LIzbujPFnqlHINdv4toyHD4JFkLc=</latexit> ?
  7. Robust High Dimensional OT Wε c (α, β) = inf

    π {⟨c, π⟩ + εKL(π|α ⊗ β) : π1 = α, π2 = β} Entropic regularization: Erwin Schrödinger ε β α Parallelism Smoothness Marco Cuturi Alfred Galichon +
  8. Robust High Dimensional OT Wε c (α, β) = inf

    π {⟨c, π⟩ + εKL(π|α ⊗ β) : π1 = α, π2 = β} Entropic regularization: Erwin Schrödinger ε β α Parallelism Smoothness Marco Cuturi Alfred Galichon + Sample complexity Richard 
 Dudley Aude Genevay 𝔼 (|W0 c − ̂ W0 c |) = O(n−1/d) 𝔼 (|Wε c − ̂ Wε c |) = O(ε−dn−1/2) ε > 0 +
  9. Robust High Dimensional OT Wε c (α, β) = inf

    π {⟨c, π⟩ + εKL(π|α ⊗ β) : π1 = α, π2 = β} Entropic regularization: Erwin Schrödinger ε β α Parallelism Smoothness Marco Cuturi Alfred Galichon + Sample complexity Richard 
 Dudley Aude Genevay 𝔼 (|W0 c − ̂ W0 c |) = O(n−1/d) 𝔼 (|Wε c − ̂ Wε c |) = O(ε−dn−1/2) ε > 0 + Unbalanced OT: Wε,τ c (α, β) := inf π ⟨c, π⟩ + εKL(π|α ⊗ β) + τKL(π1 |α) + τKL(π2 |β) [Liero et al] [Chizat et al] [Monsaingeon et al] Matthias Liero Lenaic Chizat Bernhard Schmitzer
  10. Robust High Dimensional OT Wε c (α, β) = inf

    π {⟨c, π⟩ + εKL(π|α ⊗ β) : π1 = α, π2 = β} Entropic regularization: Erwin Schrödinger ε β α Parallelism Smoothness Marco Cuturi Alfred Galichon + Sample complexity Richard 
 Dudley Aude Genevay 𝔼 (|W0 c − ̂ W0 c |) = O(n−1/d) 𝔼 (|Wε c − ̂ Wε c |) = O(ε−dn−1/2) ε > 0 + Debiasing: ¯ Wε c (α, β) := Wε c (α, β) − Wε c (α, α)/2 − Wε c (β, β) ε→+∞ ⟶ MMD(α − β)2 [Genevay et al] [Séjourné, Feydy et al] Thibault Séjourné Jean Feydy Unbalanced OT: Wε,τ c (α, β) := inf π ⟨c, π⟩ + εKL(π|α ⊗ β) + τKL(π1 |α) + τKL(π2 |β) [Liero et al] [Chizat et al] [Monsaingeon et al] Matthias Liero Lenaic Chizat Bernhard Schmitzer
  11. Single-cell profiling uncovers cellular heterogeneity • Until recently, samples contained

    many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 1Mincarelli et al. 2018 2Lähnemann et al. 2020 Comparing Two Cells with OT • Until recently, samples contained many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 1Mincarelli et al. 2018 2Lähnemann et al. 2020 <latexit 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Bio-inspired <latexit 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Data-driven <latexit 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(e.g. via regulation networks) <latexit 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Choice of <latexit 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gene’s distance <latexit 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sparse <latexit sha1_base64="hTb29Gui6+TqErQwUSkV1IIo0O4=">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</latexit> noisy Wc (α, β) ∥α − β∥1 α β β α α β π c(xi , yj ) c(xi , yj ) Metric learning: key question.
  12. Paired Multi-omics Integration > pip install mowgli <latexit sha1_base64="D+BTpQDTLsuQxDGkm5bQcuzxV/s=">AABE53ictVzbchTJES3WtzW+sfajX3qtxcE6WCwwvkRsOGJBI4QWLQhmJNhlgOieaQ0Drelhbghm9Q0Ovzgc9pO/xN/hD3CE/eRfcF6quqpnqjurZUyHpOrqOplZ2VVZmVnVJONsOJ1tbv7j3Hvf+Oa3vv2d9797/nvf/8EPf3Thgx8fTvP5pJce9PIsnzxK4mmaDUfpwWw4y9JH40kaHydZ+jB5uYXPHy7SyXSYjzqzN+P0yXE8GA2Phr14BlUPu/F4PMlPnl3Y2LyySf+i9cJVXdhQ+t9+/sGH/1Rd1Ve56qm5OlapGqkZlDMVqylcj9VVtanGUPdELaFuAqUhPU/VqToP2Dm0SqFFDLUv4fcA7h7r2hHcI80poXvAJYOfCSAjdREwObSbQBm5RfR8TpSxtor2kmiibG/gb6JpHUPtTD2HWglnWobisC8zdaR+R30YQp/GVIO962kqc9IKSh45vZoBhTHUYbkPzydQ7hHS6DkizJT6jrqN6fm/qCXW4n1Pt52rf5OUF+GKVFv3Pi8oxGpB9CN6m3N4xvJkwHkAFFLdRyy9Jl0fU+9H0H4J9XfhOqWS0UkC15JqT2uRW3D5kFsicgcuH3JHRO7B5UPuich9uHzIfY1E7IR07se34fLh2yLn+3D5kPdF5AO4fMgHIvIQLh/yUER+BZcP+ZWIvAWXD3lLRN6By4e8IyI7cPmQHRF5AJcPeSAit+HyIbc1snqmTuDKic5QmJU3oFzmgZYig5obonw3yTr6sDcD5nSvAivP6hb89WNbATpNK7DbAePuqAIrj7wdsJF+rGyLbtNq4sPeFrG7MAL82F0R+7l6UYH9PGCmvazAynNtD9r5sbL1/QLu/NgvROxdKPmx8hp1D2r82HsBK8a4ArsvYu+rVxXYEKs/qcDKdr8NdsWPldepDrT3Y0Os6bwCK9vTQ/Bg/Fh5tXoItX7sQxH7SJ1UYB+J2C/BuvuxXwassG8rsGaNPU8ryID8kRRmbB21uJiVWBoDtVjgnxVrS0a+cQL1EmZQYAaEORYROwViJxCxVyD2guWaFnZ0Sv6uzKVdINqBiKRYm7A0E9v3i/ZYygIQrQLRWkHUeaT4rk1fFuRdmBoJOStWLiyF9Ckv7DeWUj0e6i2vQdwrIXhsP6eRf5miJYygUFN11J4XazwjI7qvQ7ym6M300vCQcbPCKrioExGVeFCJiHrjQb0RUXMPai6iFh7UQkTZme/iugEjwOof38WS7ngEsI9cfUXgFdyAVec2zNEIxs8+eIEPqOYe/G1T7C1ddZJhNI/rJGY5npQs8QRKS7UB9TYqbFF8ndEMS0EybnlPx/h4h7mNpZ5zbIVPi5U8KjIm4XSGJM+goIPeYkTzqRmdO1RzSt4dl5rhbxfz3pSa4bdJ46fkxXOpGX6mpZ+dQfaOxnbOgG3DbBpr7dtyUxqcf2EapnyeVl20uPhWj/WYQXonDenv6jeze4b3skUl1o8tN6Mxdfo3LfWvCQ2r56mj52ZU0Htir9eUosY9Gem415abypDTKjrScti7pm8G2/T1mzHlZjT2wePaoph76ZSbjt5x0RtbbkbjUHHe85Q8eVNuRmNA96wPW25GA7MtsY7zbbmpZUcNcOxsy02t+oiywJgD4jHPNdYrmpCfNNfUhuQf1GdrXJ9/fR3DnM3TIkaop2R922o6SbGW1Utk/IUUrNqsoRzoX8wdH6xMY6muifEVyzArre/rdOwaj5rfAy1GMPt5D0DKmWcgoclJoPXOgOJVMeoq98zgrok4HCVHK6iurp2J3qLly1mjct0zqpXiMttbq8cu2espjb0x+YR7pFlJD3uVb7iKoqShvZKGZHpNdPdWz9ey9jdF3HgFMS5GWo92hHgnrT5O9Wm97ej4ot7lmcHFez52/GK2+UhbG4x5crJFKEsdT7edySO5dbiuXlY2x83PInqjaK8WZDWGtCM1FaNQky1mb3xJ95b2Ae3JIQ+m0YP3GGkqY8W7ZphFx3x6RBbVtbcSb9SXydBxeUpW19jjevTAQQ886OYxzhasGHeh1IGY4QDuOgFRzvlCVzlpfKI+KXZHc3qD9RF9VrKQhgbbm7RkIeui7OclKq8BjaOBo/RwGqt0DL67RkmO+n3y2Ni1bPkv0s6t2d+OaYxXj+bqTEyfuF4jrhHNGt7V5btVDizB0vvkGvmv9b1Efk04og2VuD51OLNeRrTjn1IEOybPOKPZJs2Ocms3P7X6xHDaV2bvHHezc7KQEdm/CNannMZkRD/u2QGzg84WISMbGWJ3hoV34/N1huIYs37cUPGpBjveUrJlc+Jv6Lqza0pjkSMGXgdOV8a20cke+YIpcZ1o627ndv3qg0h7TsIdJUzRjpVLxP9j+m1+zDjZWBsRqGF8A1Nt63zvI6eYBXUU0ypfb4NMW1fKjwoZnmqp7fpnZfqoJFmLIi6UB1frPnDu0T3zwlEyIbmna214Ha3L5iLl8YoesbdHFMWz3R/oFRjlvkyr5AbNuS6NkgGMglkRRZi2UhZ5lW89rzL1MNrT/wt1q+uy1pBipGwGlzUk5fdTitZcKTMY1Tx+X9Js8mt9stKqns+IxuKxM5e/htoP4beR29yH0UlKVuEmjQGmYO+sRrgmWmsRxutmiZcZmYaWvbf87Jg0rdyas8TXbN1sjL1oTGWfRs2JzlqY8llovHBovAjUYYf2Gq0WTb2xRM/E2KKjdytD+TXh1mlAeS5Slj0ygxoGSOnGUmFU+yJVOcY3qLcirU2RVgyz1d0NcOd8CNI/11dn99fF6h6pW+Tb9MgD4/ilT7N0SD6Xqa2P1JgCcr6u7as7+7tUg9wTsqBImc9x4ozhXaceXaeFpD/XK1tOdt5aBHNu6bVuY2xsl8q/WkMe05yY0rw0iOvUItXyu3JEKxbpiuNzRJT5j8mnYr+jPmZ2W9t3EpX8CRtv8qyyvDhSGJH+pczb7lr0uuvErxHFhHPtXSdAq/kbRgqMMZkEv2c5pTeEqxzvJLBHm5D9XLdTvIs3ciS6QlIv1e8DbAxHvXasu2PL9Nj07RfQErVu37qvhcwvC+Yo8TvLjl5Mq9qx9lGXK/dnoxXrVa58X6eH+Qpfq485tXEjCxvllTFd9WkwF5aoGRfGhHBp1osm8jeTvInMvDsVStm0NpTLmQa2Mc8pXpLOgSLC591d8npzHwv9SNboJYR1qXGNRAmzcbnOD7iWFrNS0UqE5NZLa1LmrEdV64Xl4a4a1o6zpUzJCmZKyt1wa7cP3VK0ImdjmEJP8cneqjjRpfkpXPg7Ur4o0XAMySG2wc+9obbU9js4FfFKlzmzGVEN2oT+Sgwe636WW9Tr6JVD3aUfwiGcxxB0LUk/pBW1qexMWZbcpR5O/zVZg4lKRelty+Z9cLnIPVnn1KQ/Q7Jwcm+GynyT07QvhkNIT8pcwvnw/obUiyNlvm1q1gdDXe5BmUMTHuY8Q9g7t62b83I51etrnUsoD14HzM6LweEOYHXMYtuFWKiJ80bePQe0Dkc11M1q8b/2w/CxnJrzCuU2pW/OXgS8dW6X6sws+sXN54zlFjKaqzmG88yL3lmvyc+P/b+o0ZvKnd68e/rol9oxYHgtFedDZekY744iK28oFdwf8MmQq/+ov5+Tv0p4VdCokqMJJbNfUU3NtJCpmS8vfb0zz0JksnSqZCpTs/FEm07GbqlddQt+tgoPsOkpUf6mkv8i1v8dbR9qj8h6mGw6ZxC6VJdSFsTupvXp3p6jrZIYz/TyGd8O1OCe+B7V4nnfu9Qez/x2Sn2r/pKE5/oXKlf9UmSyustn51UCPSjvwHEuyHzvG9GZes5m8Qm044A9Rj5HxZGS+fp5SYg+xYWrki4JYUZLHeXESzmhM0lpBe2k1LcejfCx3unHfQc8nx8X2aVI/ZLqYr064EotSbXvkeoxZQYS0v8mRGi/Vpfh72Vd9ku6vybplN5BWaIT51n9SbBT77iwXzNepDyYydQtdLuconq7e1ifiW1VcuET7/X4QQ1+4EjZprf1kuLuiarPHc5raM61TO5+7kiZvCfrAaPZuBgf9fHzoobXIqD/dyrRdxxJd0CWhLLtEe3nTYhepnWzTdLzucr6vO3tGmnNV5tM056stOPAnJGs3xPI9Lirnv18DlLK1aQVdNy5zicypdMiQy8leX6OA05DxAG9lfsa0lOJylyUZB7wJfIiQJZFAJ0jQZojkcJAlETbh2cXNq6u/l8f64XDa1eu/ubK9fvXNz67qf8fkPfVT9XP1CVY+36rPoPxv68OaMX/o/qL+mtr2PpD60+tP3PT985pzE9U6V/rb/8FqTdClw==</latexit> ⇡

    <latexit 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⇥ <latexit 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⇡ <latexit 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clustering <latexit sha1_base64="lIRSBtvJcg/dPYE+yx9m6aNrdEs=">AABE6XictVzbchTJES3WtzW+sfajX3qtxcE6WHnQwhpiwxELGiG0CBDMSMAiIObSGgb1XJieGS6z+giHXxwO+8kf4u/wBzjCfvIvOC9VXdUz1Z3VMqZDUnV1ncys7KqszKxq2uOkn05rtX+c+eA73/3e93/w4Q/P/ujHP/npz8599PODdDSbdOL9zigZTR61W2mc9Ifx/rQ/TeJH40ncGrST+GH7eBOfP5zHk7Q/Gjanb8fx00GrN+wf9TutKVQ93o6HcZTG0+j5ubXa+rVrtcuXrkW19Su12sbVL6BQ+3zj6pUr0aX1Gv1bU/rf3uijj/+pDlVXjVRHzdRAxWqoplBOVEulcD1Rl1RNjaHuqVpA3QRKfXoeqxN1FrAzaBVDixbUHsPvHtw90bVDuEeaKaE7wCWBnwkgI3UeMCNoN4Eycovo+YwoY20R7QXRRNnewt+2pjWA2ql6AbUSzrQMxWFfpupIXaU+9KFPY6rB3nU0lRlpBSWPnF5NgcIY6rDchecTKHcIafQcESalvqNuW/T8X9QSa/G+o9vO1L9JyvNwRaqhez/KKLTUnOhH9DZn8IzlSYBzDyjEuo9Yek26HlDvh9B+AfV34TqhktFJG64F1Z6UIjfh8iE3ReQ2XD7ktojchcuH3BWRe3D5kHsaidgJ6dyPb8DlwzdEzvfh8iHvi8gHcPmQD0TkAVw+5IGI/AYuH/IbEXkTLh/ypoi8DZcPeVtENuHyIZsich8uH3JfRG7B5UNuaWTxTJ3ANSI6fWFWXodyngdaigRqrovy3SDr6MPeCJjTnQKsPKvr8NePrQfoNC7AbgWMu6MCrDzytsFG+rGyLbpFq4kPe0vE7sAI8GN3ROzX6mUB9uuAmXZcgJXn2i6082Nl63sH7vzYOyL2LpT8WHmNugc1fuy9gBVjXIDdE7H31asCbIjVnxRgZbvfALvix8rrVBPa+7Eh1nRWgJXt6QF4MH6svFo9hFo/9qGIfaTeFGAfidjHYN392McBK+y7AqxZY8/SCtIjfySGGVtGrZXNSiyNgVpL4J9ka0tCvnEb6iVML8P0CDMQEdsZYjsQsZshdoPlSjM7mpK/K3NpZIhGIKKdrU1Ymortu1l7LCUBiHqGqC8hyjxSfNemL3PyLkyNhJxmKxeWQvo0yuw3lmI9Hsotr0HcyyF4bL+gkX+RoiWMoFBTZdReZGs8IyO6L0O8pujN9NLwkHHTzCq4qDciqu1BtUXUWw/qrYiaeVAzETX3oOYiys58F3cYMAKs/vFdLOiORwD7yMVXBF7BdVh1bsEcjWD87IEX+IBq7sHfBsXe0lUmGUbzuE5iluNpzhJPoLRQa1Bvo8I6xdcJzbAYJOOW93SMj3eY21joOcdW+CRbyaMsYxJOp0/y9DI66C1GNJ+q0blNNSfk3XGpGv5WNu9NqRp+izR+Ql48l6rhp1r66Slkb2ps8xTYBsymsda+LVelwfkXpmHKZ2nVRYuLb3WgxwzSe1OR/o5+MzuneC+bVGL92HI1GqnTvzTXvyo0rJ5TR8/VqKD3xF6vKUWVezLUca8tV5VhRKvoUMth76q+GWzT1W/GlKvR2AOPa5Ni7oVTrjp6x1lvbLkajQPFec8T8uRNuRqNHt2zPmy5Gg3MtrR0nG/LVS07aoBjZ1uuatWHlAXGHBCPea6xXtGE/KSZptYn/6A8W+P6/KvrGOZsnmUxQjkl69sW02lna1m5RMZfiMGqTSvKgf7FzPHB8jQWakOMr1iGaW59X6Vj13jU/C5oMYLZz3sAUs48AQlNTgKtdwIUL4lRV75nBrch4nCUHC2hDnXtVPQWLV/OGuXrnlOtFJfZ3lo9HpK9Tmnsjckn3CXNSnrYLXzDRRQlDe3mNCTTq6K7d3q+5rVfE3HjJcQ4G2kd2hHinbTyONWn9Yaj4/N6l2cKF+/52PGL2eYjbW0w5hmRLUJZyni67Uweya3DdfWisjlufhbRG0V7NSer0acdqVSMQk22mL3xBd1b2vu0J4c8mEYH3mOkqYwV75phFh3z6RFZVNfeSrxRXyZDx+WUrK6xx+XonoPuedDVY5xNWDHuQqkJMcM+3DUDopyzma5GpPGJ+izbHR3RGyyP6JOchTQ02N7EOQtZFmW/yFF5DWgcDRylh9NYpmPwhyuU5KjfJ4+NXfOW/zzt3Jr97RaN8eLRXJyJ6RLXDeIa0azhXV2+W+bAEiy8TzbIfy3vJfKrwhFtqMT1mcOZ9TKkHf+YItgxecYJzTZpduRbu/mp5SeG054ye+e4mz0iCxmR/YtgfRrRmIzoxz07YHbQ2SIkZCND7E4/8258vk5fHGPWj+srPtVgx1tMtmxG/A1dd3alNBY5YuB14GRpbBud7JIvGBPXibbudm6Xrz6ItOck3FHCFO1YuUD8P6Xf5seMk7WVEYEaxjeQalvnex8jillQRy1a5cttkGnrSvlJJsMzLbVd/6xMn+Qkq1PEhfLgat0Fzh26Z144SiYkd7rShtfRsmwuUh4v6RF7e0RRPNv9nl6BUe6LtEqu0Zw7pFHSg1EwzaII01bKIi/zLeeVpx5GO/2/ULe6zmsNKUbKZnBZQ1J+P6ZozZUygVHN4/eYZpNf65OlVuV8hjQWB85c/hZqP4bfRm5zH0annbMKN2gMMAV7ZzXCNdFKizBeN3K8zMg0tOy95WfHpGnl1pwmvmbrZmPseWUqezRq3uishSmfhsZLh8bLQB02aa/RatHUG0v0XIwtmnq3MpRfFW7NCpRnImXZIzOofoCUbiwVRrUrUpVjfIN6J9KqibRaMFvd3QB3zocg/XN9eXZ/m63ukbpJvk2HPDCOX7o0S/vkc5na8kiNKSDny9q+urP/kGqQe5ssKFLmc5w4Y3jXqUPXSSbpr/XKNiI7by2CObf0WrcxNvaQyp+vIAc0J1KalwZxmVrEWn5XjmjJIq07PkdEmf8W+VTsd5THzG5r+06inD9h402eVZYXRwpD0r+UedtZiV53nPg1ophwpr3rNtCq/oaRAmNMJsHvWab0hnCV450E9mjbZD9X7RTv4g0didZJ6oX6fYCN4ajXjnV3bJkem779Blqi1u1b97WQ+SXBHCV+p9nRa9GqNtA+6mLp/nS0WnqVy9+X6WG2xNfqY0Zt3MjCRnl5zKH6MpgLS1SNC2NCuFTrRRX5q0leRWbenQqlbFobyvlMA9uYFxQvSedAEeHz7i54vblPhX60V+i1CetS4xqJEmbjRjo/4FpazEpFSxGSWy+tSYmzHhWtF5aHu2pYO86WMiYrmCgpd8Ot3T4c5qIVORvDFDqKT/YWxYkuzS/hwt+R8kWJhmNIDrEBfu51tam23sOpiFe6zJnNiGrQJnSXYvCW7me+RbmOXjnUXfohHMJ59EHXkvR9WlGrys6UZcld6uH0X5M1mKhYlN62rN4Hl4vck1VOVfrTJwsn96avzDc5VftiOIT0JM8lnA/vb0i9OFLm26ZqfTDU5R7kOVThYc4zhL1z27o6L5dTub5WuYTy4HXA7LwYHO4AFscstl2IhZo4b+T9c0DrcFRC3awW/2s/DB/LqTqvUG4pfXP2MuCtc7tYZ2bRL64+Zyy3kNFczDGc5yjrnfWa/PzY/4sqvamR05v3Tx/9UjsGDK+F4nyoLB3j3VFk5Q2lgvsDPhlG6j/q72fkrxJeZTSK5KhCyexXFFMzLWRq5stLX+/MsxCZLJ0imfLUbDzRoJOxm2pH3YSfzcwDrHpKlL+p5L+I9X9H24XaI7IeJpvOGYRDqospC2J307p0b8/RFkmMZ3r5jG8TanBPfJdq8bzvXWqPZ36bub4Vf0nCc/2OGqluLjJZ3uWz86oNPcjvwHEuyHzvG9GZes5m8Qm0QcAeI5+j4kjJfP28IESX4sJlSReEMKOljHLbS7lNZ5LiAtrtXN86NMLHeqcf9x3wfH4ryy5F6rdU19KrA67UklR7HqmeUGagTfqvQYR2RV2Evxd12S/p3oqkKb2DvERvnGflJ8FOvOPCfs14nvJgJlM31+1GFNXb3cPyTGy9kAufeC/H90rwPUfKBr2tY4q7J6o8dzgroTnTMrn7uUNl8p6sB4xmW9n4KI+f5yW85gH9v12Ivu1Iug2ytCnbHtF+3oToJVo3WyQ9n6ssz9veKpHWfLXJNO3JSjsOzBnJ8j2BRI+74tnP5yClXE1cQMed63wiUzot0vdSkufnOOA0RCugt3JfQ3oqUZmJkswCvkSeB8gyD6BzJEhzJFLoiZJo+/D83Jr5Lz6i4sLBxvqlL9Yv399Y++qG/n9APlS/VL9SF2Dt+536Csb/ntoHTgP1R/UX9df6cf0P9T/V/8xNPzijMb9QuX/1v/0XbJNDCg==</latexit> Gene set <latexit sha1_base64="p9qBAtT0sQ9OxWrZqTMkFBJLPXA=">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</latexit> enrichment Geert-Jan Huizing Laura Cantini min (wk ),(Hm ) {∑ m ∑ k Wcm (Hm wk , αm k ) : wk ≥ 0,Hm ≥ 0} wk αk ATAC ATAC H αk RNA RNA H
  13. (Unbalanced) OT Across Cells <latexit 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day 1 <latexit 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    day 18 <latexit sha1_base64="vh31vAG5AzHYBeJ+kRgiWs/Cvbc=">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</latexit> [Schiebinger et al 2019] <latexit sha1_base64="E7xwgtYw1POYNmjFBGWIgnslFAY=">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</latexit> [Waddington, 1936] Conrad Waddington Geoffrey Schiebinger
  14. (Unbalanced) OT Across Cells <latexit 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day 1 <latexit 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    day 18 <latexit sha1_base64="vh31vAG5AzHYBeJ+kRgiWs/Cvbc=">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</latexit> [Schiebinger et al 2019] <latexit sha1_base64="E7xwgtYw1POYNmjFBGWIgnslFAY=">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</latexit> [Waddington, 1936] Conrad Waddington Geoffrey Schiebinger
  15. OT for Single cell genomics Gromov Wasserstein as metric learning

    Inverse OT for metric learning Wasserstein Singular Vectors for metric learning
  16. Supervised Ground Metric Learning c ↦ Wc (α, β) =

    infπ1 =α,π2 =β ⟨c, π⟩ + εKL(π|α ⊗ β) α ↦ Wc (α, β) = supf,g ⟨f, α⟩ + ⟨g, β⟩ − ε⟨ef ⊕ g−c ε , α ⊗ β⟩ convex ⟶ concave ⟶
  17. Supervised Ground Metric Learning c ↦ Wc (α, β) =

    infπ1 =α,π2 =β ⟨c, π⟩ + εKL(π|α ⊗ β) α ↦ Wc (α, β) = supf,g ⟨f, α⟩ + ⟨g, β⟩ − ε⟨ef ⊕ g−c ε , α ⊗ β⟩ convex ⟶ concave ⟶ inf c∈ 𝒞 ∑ i,j λi,j Wc (αi , αj ) Ground metric learning: [Cuturi, Avis 2014] α2 α3 λ1,2 > 0 α1 λ1,3 > 0 λ1,2 > 0
  18. Supervised Ground Metric Learning c ↦ Wc (α, β) =

    infπ1 =α,π2 =β ⟨c, π⟩ + εKL(π|α ⊗ β) α ↦ Wc (α, β) = supf,g ⟨f, α⟩ + ⟨g, β⟩ − ε⟨ef ⊕ g−c ε , α ⊗ β⟩ convex ⟶ concave ⟶ If : convex, congested optimal transport ∀(i, j), λi,j ≤ 0 xt xs 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5: 2D and 3D display of the optimal metric −2 −1.5 −1 −0.5 72 74 76 78 xt xs 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5: 2D and 3D display of the optimal metric ⇠?. 0 200 400 600 800 1000 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 200 400 600 800 1000 64 66 68 70 72 74 76 78 [Benmansour, Carlier, Peyré and Santambrogio, 2010] α2 α1 inf c∈ 𝒞 ∑ i,j λi,j Wc (αi , αj ) Ground metric learning: [Cuturi, Avis 2014] α2 α3 λ1,2 > 0 α1 λ1,3 > 0 λ1,2 > 0
  19. Supervised Ground Metric Learning c ↦ Wc (α, β) =

    infπ1 =α,π2 =β ⟨c, π⟩ + εKL(π|α ⊗ β) α ↦ Wc (α, β) = supf,g ⟨f, α⟩ + ⟨g, β⟩ − ε⟨ef ⊕ g−c ε , α ⊗ β⟩ convex ⟶ concave ⟶ If : convex, congested optimal transport ∀(i, j), λi,j ≤ 0 xt xs 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5: 2D and 3D display of the optimal metric −2 −1.5 −1 −0.5 72 74 76 78 xt xs 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5: 2D and 3D display of the optimal metric ⇠?. 0 200 400 600 800 1000 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 200 400 600 800 1000 64 66 68 70 72 74 76 78 [Benmansour, Carlier, Peyré and Santambrogio, 2010] α2 α1 inf c∈ 𝒞 ∑ i,j λi,j Wc (αi , αj ) Ground metric learning: [Cuturi, Avis 2014] α2 α3 λ1,2 > 0 α1 λ1,3 > 0 λ1,2 > 0 14 Matthieu Heitz et al. infc∈ 𝒞 ∫1 0 Wc (αt , ̂ αt ) Approximating curves by geodesics : ̂ αt c αt [Heitz, Bonneel, Coeurjolly, Cuturi and G. Peyré, 2021]
  20. Gromov-Wasserstein ↵ 2 M1 + (X) and dX distance on

    X. <latexit 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<latexit 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<latexit 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<latexit 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Metric measure space X , (X, ↵, dX ): [Memoli 2011]<latexit 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[Sturm 2011] k(x, y, x0, y0) def. = |dX (x, x0) dY (y, y0)|2 <latexit 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<latexit 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<latexit 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GW2 2 (X, Y) def. = min ⇡1=↵,⇡2= Z (X⇥Y)2 k d⇡ ⌦ ⇡ X <latexit 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<latexit 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x <latexit 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x0 Facundo Memoli Karl-Theodor Sturm
  21. Gromov-Wasserstein ↵ 2 M1 + (X) and dX distance on

    X. <latexit 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<latexit 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<latexit 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<latexit 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Metric measure space X , (X, ↵, dX ): X <latexit 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[Memoli 2011]<latexit 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[Sturm 2011] k(x, y, x0, y0) def. = |dX (x, x0) dY (y, y0)|2 <latexit 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<latexit 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<latexit 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GW2 2 (X, Y) def. = min ⇡1=↵,⇡2= Z (X⇥Y)2 k d⇡ ⌦ ⇡ X <latexit 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<latexit 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x0 Facundo Memoli Karl-Theodor Sturm <latexit 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up to isometries. <latexit 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Theorem: GW is a distance ! non-convex, NP-hard . . . <latexit 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<latexit 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if dX = || · ||, dY = || · ||: concave!
  22. Multi-Omics Integration <latexit 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Gromov-Wasserstein <latexit 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registration <latexit 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    [Othmane Sebbouh, 2021] <latexit 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ATAC-seq <latexit 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RNA-seq <latexit 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↵ <latexit 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<latexit sha1_base64="8H4xN4xGPG5ztU4N+maxzHTbjXo=">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</latexit> dim ⇠ 103(genes) <latexit sha1_base64="NRLid9ocOcEHNQq7hjUXkZi7pNg=">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</latexit> dim ⇠ 102(peaks) <latexit sha1_base64="SMZnFdBIvXrGaOn22tljzC0pXcE=">AABFi3ictVxbdxPJEW42l92QbMImD3nIy2QNOewe4jVecjnZ5JwFWxgvBgySDSwCjkYaywJJI3TjotWvyWvyG/I78g+Sp/yF1KV7ukfqmepxCHNs9/T0V1Vd011dVd1DPOr3JtOtrX+e++B73//BDz/86Efnf/yTj3/6swuf/Px4ks7G7eSonfbT8aO4NUn6vWFyNO1N+8mj0ThpDeJ+8jB+uYPPH86T8aSXDhvTt6Pk6aDVHfZOeu3WFKqeX/jlxSYRWYyTzrI5aE1P43jxaHnx+YWNrc0t+hetF67qwobS/w7TTz7dVE3VUalqq5kaqEQN1RTKfdVSE7ieqKtqS42g7qlaQN0YSj16nqilOg/YGbRKoEULal/C7y7cPdG1Q7hHmhNCt4FLH37GgIzUJcCk0G4MZeQW0fMZUcbaItoLoomyvYW/saY1gNqpOoVaCWdahuKwL1N1ov5IfehBn0ZUg71rayoz0gpKHjm9mgKFEdRhuQPPx1BuE9LoOSLMhPqOum3R839RS6zF+7ZuO1P/JikvwRWpuu59mlFoqTnRj+htzuAZy9MHzl2gkOg+Yuk16XpAvR9C+wXU34VrSSWjkxiuBdUuS5E7cPmQOyJyDy4fck9EHsDlQx6IyEO4fMhDjUTsmHTux9fh8uHrIuf7cPmQ90XkA7h8yAci8hguH/JYRH4Llw/5rYi8CZcPeVNE3obLh7wtIhtw+ZANEXkElw95JCJrcPmQNY0snqljuFKi0xNm5XUo53mgpehDzXVRvhtkHX3YGwFzul2AlWf1Lvz1Y3cDdJoUYGsB4+6kACuPvD2wkX6sbItu0Wriw94SsfswAvzYfRH7jXpRgP0mYKa9LMDKc+0A2vmxsvW9A3d+7B0RexdKfqy8Rt2DGj/2XsCKMSrAHorY++pVATbE6o8LsLLdr4Nd8WPldaoB7f3YEGs6K8DK9vQYPBg/Vl6tHkKtH/tQxD5Sbwqwj0TsY7DufuzjgBX2XQHWrLHnaQXpkj+SwIwto9bKZiWWRkCtJfDvZ2tLn3zjGOolTDfDdAkzEBF7GWIvEHGQIQ6C5ZpkdnRC/q7MpZ4h6oGIOFubsDQV23ey9ljqByB2M8TuCqLMI8V3bfoyJ+/C1EjIabZyYSmkT2lmv7GU6PFQbnkN4l4OwWP7lEb+FYqWMIJCTZVRO83WeEZGdF+GeE3Rm+ml4SHjpplVcFFvRFTsQcUi6q0H9VZEzTyomYiae1BzEWVnvotrBowAq398Fwu64xHAPnLxFYFXcB1WnVswRyMYP4fgBT6gmnvwt06xt3SVSYbRPK6TmOV4mrPEYygt1AbU26hwl+LrPs2wBCTjlvd0jI93mNtY6DnHVniZreRRljEJp9MjeboZHfQWI5pP1ejcppoleXdcqoa/lc17U6qGr5HGl+TFc6kafqqln55B9obGNs6ArcNsGmnt23JVGpx/YRqmfJ5WXbS4+FYHeswgvTcV6e/rN7N/hveyQyXWjy1XozFx+jfJ9a8KDavniaPnalTQe2Kv15Siyj0Z6rjXlqvKkNIqOtRy2LuqbwbbdPSbMeVqNA7B49qhmHvhlKuO3lHWG1uuRuNYcd5zSZ68KVej0aV71octV6OB2ZaWjvNtuaplRw1w7GzLVa36kLLAmAPiMc811isak58009R65B+UZ2tcn399HcOczbMsRiinZH3bYjpxtpaVS2T8hQSs2rSiHOhfzBwfLE9jobbF+IplmObW93U6do1HzR+AFiOY/bwHIOXM+yChyUmg9e4Dxati1JXvmcFtizgcJScrqKaunYreouXLWaN83XOqleIy21urxybZ6wmNvRH5hAekWUkPB4VvuIiipKGDnIZkelV0907P17z2t0TcaAUxykZam3aEeCetPE71ab3u6PiS3uWZwsV7Pnb8Yrb5RFsbjHlSskUoSxlPt53JI7l1uK5eUTbHzc8ieqNor+ZkNXq0IzURo1CTLWZvfEH3lvYR7ckhD6bRhvcYaSojxbtmmEXHfHpEFtW1txJv1JfJ0HF5QlbX2ONydNdBdz3o6jHODqwYd6HUgJjhCO4aAVHO+UxXKWl8rH6b7Y6m9AbLI/p+zkIaGmxvkpyFLIuyT3NUXgMaRwNH6eE0VukYfHONkhz1++SxsWve8l+inVuzv92iMV48moszMR3iuk1cI5o1vKvLd6scWIKF98k2+a/lvUR+VTiiDZW4PnM4s16GtOOfUAQ7Is+4T7NNmh351m5+avWJ4XSozN457manZCEjsn8RrE8pjcmIftyzA2YHnS1Cn2xkiN3pZd6Nz9fpiWPM+nE9xaca7HhLyJbNiL+h686uCY1Fjhh4HViujG2jkwPyBRPiOtbW3c7t8tUHkfachDtKmKIdK5eJ/2f02/yYcbKxNiJQw/gGJtrW+d5HSjEL6qhFq3y5DTJtXSkvZjI801Lb9c/KdDEn2S5FXCgPrtYd4Nyme+aFo2RMck/W2vA6WpbNRcqjFT1ib08oime739UrMMp9hVbJDZpzTRolXRgF0yyKMG2lLPIq33JeeephtCf/F+pW13mtIcVI2Qwua0jK7ycUrblS9mFU8/h9SbPJr/XxSqtyPkMaiwNnLn8Htb+G30Zucx9GJ85ZhRs0BpiCvbMa4ZporUUYrxs5XmZkGlr23vKzY9K0cmvOEl+zdbMx9rwylUMaNW901sKUz0LjhUPjRaAOG7TXaLVo6o0lei7GFg29WxnKrwq3RgXKM5Gy7JEZVC9ASjeWCqPaEanKMb5BvRNpbYm0WjBb3d0Ad86HIP1zfXV2f5et7pG6Sb5Nmzwwjl86NEt75HOZ2vJIjSkg52vavrqzv0k1yD0mC4qU+RwnzhjedWrTtcwk/Y1e2VKy89YimHNLr3UbY2ObVP5yDTmgOTGheWkQ16hFouV35YhWLNKm43NElPlvkU/Ffkd5zOy2tu8kyvkTNt7kWWV5caQwJP1Lmbf9teh134lfI4oJZ9q7joFW9TeMFBhjMgl+z3JCbwhXOd5JYI82Jvu5bqd4F2/oSLRJUi/UXwJsDEe9dqy7Y8v02PTtc2iJWrdv3ddC5tcP5ijxO8uOXotWtYH2URcr92ej1dKrXP6+TA+zFb5WHzNq40YWNsrLY5rqq2AuLFE1LowJ4VKtF1XkryZ5FZl5dyqUsmltKOczDWxjTileks6BIsLn3V32enOfCf2I1+jFhHWpcY1ECbNxqc4PuJYWs1LRSoTk1ktrUt9Zj4rWC8vDXTWsHWdLmZAV7Cspd8Ot3T40c9GKnI1hCm3FJ3uL4kSX5ldw4e9I+aJEwzEkh1gHP/e62lG193Aq4pUuc2Yzohq0CZ2VGLyl+5lvUa6jVw51l34Ih3AePdC1JH2PVtSqsjNlWXKXejj912QNxioRpbctq/fB5SL3ZJ1Tlf70yMLJvekp801O1b4YDiE9yXMJ58P7G1IvTpT5tqlaHwx1uQd5DlV4mPMMYe/ctq7Oy+VUrq91LqE8eB0wOy8GhzuAxTGLbRdiocbOG3n/HNA6nJRQN6vF/9oPw8dyqs4rlNuEvjl7EfDWuV2iM7PoF1efM5ZbyGgu5hjOM816Z70mPz/2/6JKbyp1evP+6aNfaseA4bVQnA+VpWO8O4qsvKFUcH/AJ0Oq/qP+cU7+KuFVRqNIjiqUzH5FMTXTQqZmvrz09c48C5HJ0imSKU/NxhN1Ohm7o/bVTfjZyTzAqqdE+ZtK/otY/3e0Hag9IethsumcQWhSXUJZELub1qF7e462SGI808tnfBtQg3viB1SL533vUns889vI9a34SxKe63dUqjq5yGR1l8/Oqxh6kN+B41yQ+d43ojP1nM3iE2iDgD1GPkfFkZL5+nlBiA7FhauSLghhRksZ5dhLOaYzSUkB7TjXtzaN8JHe6cd9Bzyf38qyS5H6gupaenXAlVqS6tAj1RPKDMSk/y2I0H6nrsDfK7rsl/RwTdIJvYO8RG+cZ+UnwZbecWG/ZrxEeTCTqZvrdilF9Xb3sDwTu1vIhU+8l+O7JfiuI2Wd3tZLirvHqjx3OCuhOdMyufu5Q2XynqwHjGZb2fgoj5/nJbzmAf2/XYi+7Ui6B7LElG2PaD9vTPT6Wjc1kp7PVZbnbW+VSGu+2mSa9mSlHQfmjGQZB/yqbEeY/fzlWfnKgF+Y+em4c/1xwL4L9kSSiE9mStmjJEAiPiMqnV/peSnJFmMUcD6jFdBbua8hPZWozERJZgHfRs8DZJkH0DkRpDkRKXRFSbTFen5h4+rq/z6yXjje3rz6+81r97c3vr6h/2eSj9Sv1KfqMqzGf1Bfw4w8VEfk9/xV/U39vfZx7cvan2p/5qYfnNOYX6jcv1rtv5SpZJo=</latexit> X <latexit 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Y Unbalanced Gromov-Wasserstein « Monge map » → SCOT: single-cell alignment with optimal transport [Demetci et al 2022] Pinar Demetci
  23. Gromov-Wasserstein as Metric Learning inf π1 =α,π2 =β 𝒬 (π)

    := ⟨Qπ, π⟩ local minimizer π⋆ ⟺ Q(π)(x, y) := ∫ |dX (x, x′  )p − dY (y, y′  )p |2 dπ(x′  , y′  ) π⋆ ∈ argmin π1 =α,π2 =β ∫ c⋆dπ c⋆ = Qπ⋆ X <latexit 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x <latexit 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x0 Quadratic minimization: Proposition:
  24. Gromov-Wasserstein as Metric Learning inf π1 =α,π2 =β 𝒬 (π)

    := ⟨Qπ, π⟩ local minimizer π⋆ ⟺ Q(π)(x, y) := ∫ |dX (x, x′  )p − dY (y, y′  )p |2 dπ(x′  , y′  ) π⋆ ∈ argmin π1 =α,π2 =β ∫ c⋆dπ c⋆ = Qπ⋆ X <latexit 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x <latexit 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x0 Quadratic minimization: Proposition: John Toland Guillaume Carlier If dX (x, x′  ) = ∥x − x′  ∥, dY (y, y′  ) = ∥y − y′  ∥, 0 ≤ p ≤ 2, inf π1 =α,π2 =β ⟨Qπ, π⟩ = inf c Wc (α, β) + ∥c∥2 * (Sobolev norm) ∥c∥2 * := (− 𝒬 )*(c) Proposition: then is concave, and let 𝒬 (Toland duality) Then
  25. Gromov-Wasserstein as Metric Learning inf π1 =α,π2 =β 𝒬 (π)

    := ⟨Qπ, π⟩ local minimizer π⋆ ⟺ Q(π)(x, y) := ∫ |dX (x, x′  )p − dY (y, y′  )p |2 dπ(x′  , y′  ) π⋆ ∈ argmin π1 =α,π2 =β ∫ c⋆dπ c⋆ = Qπ⋆ X <latexit 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x0 Quadratic minimization: Proposition: John Toland Guillaume Carlier Example: for p = 2, ∥c∥2 * := { ∥M∥2 F if c(x, y) = − ⟨Mx, y⟩ − ∥x∥2∥y∥2 +∞ otherwise Titouan Vayer Extension: sparsity regularizers R(c) = ∥M∥1 , … Othmane Sebbouh If dX (x, x′  ) = ∥x − x′  ∥, dY (y, y′  ) = ∥y − y′  ∥, 0 ≤ p ≤ 2, inf π1 =α,π2 =β ⟨Qπ, π⟩ = inf c Wc (α, β) + ∥c∥2 * (Sobolev norm) ∥c∥2 * := (− 𝒬 )*(c) Proposition: then is concave, and let 𝒬 (Toland duality) Then
  26. OT for Single cell genomics Gromov Wasserstein as metric learning

    Inverse OT for metric learning Wasserstein Singular Vectors for metric learning
  27. Wasserstein Singular Vectors α1 , α2 , α3 , …

    β1 β2 β3 … Goal: find costs and such that c d c(α, α′  ) ∝ Wd (α, α′  ) d(β, β′  ) ∝ Wc (β, β′  ) M βj = ∑ i Mi,j δαi αi = ∑ j Mi,j δβj
  28. Wasserstein Singular Vectors α1 , α2 , α3 , …

    β1 β2 β3 … Goal: find costs and such that c d c(α, α′  ) ∝ Wd (α, α′  ) d(β, β′  ) ∝ Wc (β, β′  ) M βj = ∑ i Mi,j δαi αi = ∑ j Mi,j δβj Φ : d ↦ c, Ψ : c ↦ d, Wasserstein transfert operators: c(α, α′  ) := Wd (α, α′  ) + τ∥α − α′  ∥ d(β, β′  ) := Wc (β, β′  ) + τ∥β − β′  ∥ and costs (λ, μ) ∈ ℝ+ * (c, d) Φ(d) = λc and Ψ(c) = μd Wasserstein singular values/vectors: Stéphane Gaubert c d Ψ Φ
  29. Wasserstein Singular Vectors α1 , α2 , α3 , …

    β1 β2 β3 … Goal: find costs and such that c d c(α, α′  ) ∝ Wd (α, α′  ) d(β, β′  ) ∝ Wc (β, β′  ) M βj = ∑ i Mi,j δαi αi = ∑ j Mi,j δβj Φ : d ↦ c, Ψ : c ↦ d, Wasserstein transfert operators: c(α, α′  ) := Wd (α, α′  ) + τ∥α − α′  ∥ d(β, β′  ) := Wc (β, β′  ) + τ∥β − β′  ∥ and costs (λ, μ) ∈ ℝ+ * (c, d) Φ(d) = λc and Ψ(c) = μd Wasserstein singular values/vectors: Stéphane Gaubert c d Ψ Φ Proposition: for , and : τ ≥ 0 Φ Ψ Theorem: for there is a unique pair of singular vectors τ > 0 • maps distances to distances, • are 1-homogenous monotone. non-linear Perron-Frobenius → Algorithm: power iterations, ct+1 := Φ(dt ), dt+1 := Ψ(ct+1 ) Geert-Jan Huizing
  30. Examples of Wasserstein Singular Vectors 0 20 40 60 80

    Figure 2. Illustration on the 1-D torus. (top, left) histograms whose t associated to the singular vectors B1 , B2 , B3 for varying values of ⌧ right) convergence rate of the power iterations for ⌧ = 0.1, accordi proposition, proved in Appendix C, states that for large enough regularization, uniqueness and linear convergence are maintained. Proposition 2.6. For ⌧ large enough, the singular vectors are unique and the power iterations (4) converge linearly for k·k . When ⌧ ! 1, the singular vectors converge to Translated histograms: Mi,j = h(i − j) Unsupervised Ground Metric Learning Using Wasserstein Singular 0 20 40 60 80 Figure 2. Illustration on the 1-D torus. (top, left) histograms whose translations form B1 , B2 , B3 ; (b associated to the singular vectors B1 , B2 , B3 for varying values of ⌧ ; (top, right) functions h1 , h2 , right) convergence rate of the power iterations for ⌧ = 0.1, according to the d H metric. α0 αi i c(0,⋅) Input histograms Singular vector c M
  31. Examples of Wasserstein Singular Vectors 0 20 40 60 80

    Figure 2. Illustration on the 1-D torus. (top, left) histograms whose t associated to the singular vectors B1 , B2 , B3 for varying values of ⌧ right) convergence rate of the power iterations for ⌧ = 0.1, accordi proposition, proved in Appendix C, states that for large enough regularization, uniqueness and linear convergence are maintained. Proposition 2.6. For ⌧ large enough, the singular vectors are unique and the power iterations (4) converge linearly for k·k . When ⌧ ! 1, the singular vectors converge to Translated histograms: Mi,j = h(i − j) Unsupervised Ground Metric Learning Using Wasserstein Singular 0 20 40 60 80 Figure 2. Illustration on the 1-D torus. (top, left) histograms whose translations form B1 , B2 , B3 ; (b associated to the singular vectors B1 , B2 , B3 for varying values of ⌧ ; (top, right) functions h1 , h2 , right) convergence rate of the power iterations for ⌧ = 0.1, according to the d H metric. α0 αi i c(0,⋅) Input histograms Singular vector c are singular vectors of ( 1 A , 1 B ), with singular value 2 2. This proposition shows that for " = +1 a set of positive singular vectors is obtained as simply squared Euclidean distances over 1-D principal component embeddings of the data. Entropic regularization thus draws a link between our novel set of OT-based metric learning techniques and classical dimensionality reduction methods. This frames Sinkhorn singular vectors as a well-posed problem regard- less of the value of ". 5. Metric Learning for Single-Cell Genomics between precomputed Gene2Vec (Du et al., 2019) embed- dings. (Huizing et al., 2021) use a Sinkhorn divergence with a cosine distance between genes (i.e. vectors of cells) as a ground cost. In the present paper we compute OT distances using the Python package POT (Flamary et al., 2021). Dataset A commonly analyzed scRNA-seq dataset is the “PBMC 3k” dataset produced by 10X Genomics, obtained through the function pbmc3k of Scanpy (Wolf et al., 2018). Details on preprocessing and cell type annotation are given in Appendix H. The processed dataset contains m = 1030 genes and n = 2043 cells, each belonging to one of 6 immune cell types: ‘B cell’, ‘Natural Killer’, ‘CD4+ T cell’, ‘CD8+ T cell’, ‘Dendritic cell’ and ‘Monocyte’. The cell populations are heavily unbalanced. In addition, for each cell type we consider the set of canonical marker genes given by Azimuth (Hao et al., 2021), i.e. genes whose expression is characteristic of a certain cell type. Evaluation We use the annotation on cells (resp. on marker genes) to evaluate the quality of distances between cells (resp. between marker genes). We report in Table 1 and Table 2 the Average Silhouette Width (ASW), computed us- ing the function silhouette score of Scikit-learn (Pe- Unsupervised Ground Metric Learning Using Wasserstein Singular Vectors RNA-seq expression data : W Singular vector on cells c Singular vector on genes d genes genes and to a single-cell RNA sequencing dataset. In all cases, the ground metric learned iteratively is intu- itively interpretable. In particular, the ground metric learned on biological data not only leads to improved clustering, but also encodes biologically relevant infor- mation. Theoretical perspectives include further results on the existence of positive eigenvectors, in particular for ⌧ = 0 and for " > 0. In addition, integrating un- balanced optimal transport [38, 9] into the method could avoid the need for the step of normalization to histograms. Applying our method to large single cell datasets is also a promising avenue to extend the appli- cability of OT to new classes of problems in genomics. d Figure 9: Dataset, with genes arranged according to clustering of singular vector C Aur´ elien Bellet, Amaury Habrard, and Marc ebban. A survey on metric learning for fea- ure vectors and structured data. arXiv preprint rXiv:1306.6709, 2013. ethallah Benmansour, Guillaume Carlier, Gabriel Peyr´ e, and Filippo Santambrogio. Derivatives ith respect to metrics and applications: subgradi- nt marching algorithm. Numerische Mathematik, 16(3):357–381, 2010. Guillaume Carlier, Arnaud Dupuy, Alfred Gali- hon, and Yifei Sun. Sista: learning optimal ransport costs under sparsity constraints. arXiv reprint arXiv:2009.08564, 2020. Guillaume Carlier, Vincent Duval, Gabriel Peyr´ e, nd Bernhard Schmitzer. Convergence of en- ropic schemes for optimal transport and gradient ows. SIAM Journal on Mathematical Analysis, 9(2):1385–1418, 2017. ´ ena¨ ıc Chizat, Gabriel Peyr´ e, Bernhard Schmitzer, nd Fran¸ cois-Xavier Vialard. Unbalanced optimal ransport: Dynamic and kantorovich formulations. ournal of Functional Analysis, 274(11):3090–3123, 018. enaic Chizat, Pierre Roussillon, Flavien L´ eger, ran¸ cois-Xavier Vialard, and Gabriel Peyr´ e. Faster asserstein distance estimation with the sinkhorn ivergence. In Proc. NeurIPS’20, 2020. Marco Cuturi. Sinkhorn distances: Lightspeed omputation of optimal transport. In Adv. in Neu- al Information Processing Systems, pages 2292– 300, 2013. Marco Cuturi and David Avis. Ground metric earning. The Journal of Machine Learning Re- earch, 15(1):533–564, 2014. ason V Davis and Inderjit S Dhillon. Structured metric learning for high dimensional problems. In Proceedings of the 14th ACM SIGKDD tional conference on Knowledge discovery a mining, pages 195–203, 2008. [14] Arnaud Dupuy, Alfred Galichon, and Y Estimating matching a nity matrices un rank constraints. Information and Infer Journal of the IMA, 8(4):677–689, 2019. [15] Jean Feydy, Thibault S´ ejourn´ e, Fran¸ cois Vialard, Shun-ichi Amari, Alain Trou Gabriel Peyr´ e. Interpolating between transport and mmd using sinkhorn diverge The 22nd International Conference on A Intelligence and Statistics, pages 2681–269 [16] R´ emi Flamary and Nicolas Courty. Pot optimal transport library, 2017. [17] R´ emi Flamary, Marco Cuturi, Nicolas Cou Alain Rakotomamonjy. Wasserstein discr analysis. Machine Learning, 107(12):192 2018. [18] Alfred Galichon and Bernard Salani´ e. invisible hand: Social surplus and ident in matching models. Available at SSRN 1 2020. [19] A. Genevay, G. Peyr´ e, and M. Cuturi. L generative models with sinkhorn diverge Proc. AISTATS’18, pages 1608–1617, 201 [20] Aude Genevay, L´ enaic Chizat, Franci Marco Cuturi, and Gabriel Peyr´ e. Samp plexity of sinkhorn divergences. In The 2 ternational Conference on Artificial Inte and Statistics, pages 1574–1583. PMLR, [21] Alison L Gibbs and Francis Edward Su. O ing and bounding probability metrics. tional statistical review, 70(3):419–435, 20 [22] Alexandre Gramfort, Gabriel Peyr´ e, and Cuturi. Fast optimal transport averaging roimaging data. In International Confer 10 M M
  32. OT for Single cell genomics Gromov Wasserstein as metric learning

    Inverse OT for metric learning Wasserstein Singular Vectors for metric learning
  33. Inverse Optimal Transport Schrodinger problem: πϵ (c) := argminπ {⟨c,

    π⟩ + εKL(π|α ⊗ β) : π1 = α, π2 = β} c(x, y) πε (c) ̂ π Forward OT Sampling Inverse OT Alfred Galichon
  34. Inverse Optimal Transport Schrodinger problem: πϵ (c) := argminπ {⟨c,

    π⟩ + εKL(π|α ⊗ β) : π1 = α, π2 = β} c(x, y) πε (c) ̂ π Forward OT Sampling Inverse OT Alfred Galichon Application: gene-gene regulation networks estimation Cells , cost on genes space. x, y ∈ ℝgenes c(x, y) [Weinreb et al., 2020] (undifferentiated) t = 0 ̂ π day (differentiated) t = 1 barcoding differentiation RNA-seq
  35. Fenchel-Young Losses Cross loss function: L(c| ̂ π) := ⟨c,

    ̂ π⟩ + Φ( ̂ π) − inf π ⟨c, π⟩ + Φ(π) ≥ 0 L(c| ̂ π) = 0 ⇔ ̂ π = πε (c) Proposition: if , ε > 0 πε (c) := argminπ ⟨c, π⟩ + Φ(π) Schrodinger problem: Φ(π) := { εKL(π|α ⊗ β) if π1 = α, π2 = β +∞ otherwise where is convex! → L( ⋅ | ̂ π)
  36. Fenchel-Young Losses Cross loss function: L(c| ̂ π) := ⟨c,

    ̂ π⟩ + Φ( ̂ π) − inf π ⟨c, π⟩ + Φ(π) ≥ 0 L(c| ̂ π) = 0 ⇔ ̂ π = πε (c) Proposition: if , ε > 0 πε (c) := argminπ ⟨c, π⟩ + Φ(π) Schrodinger problem: Φ(π) := { εKL(π|α ⊗ β) if π1 = α, π2 = β +∞ otherwise where is convex! → L( ⋅ | ̂ π) Proposition: L(c| ̂ π) := Φ( ̂ π) + Φ*(−c) − ⟨−c, ̂ π⟩ Fenchel-Young loss of Mathieu Blondel. → Mathieu Blondel
  37. Fenchel-Young Losses Cross loss function: L(c| ̂ π) := ⟨c,

    ̂ π⟩ + Φ( ̂ π) − inf π ⟨c, π⟩ + Φ(π) ≥ 0 L(c| ̂ π) = 0 ⇔ ̂ π = πε (c) Proposition: if , ε > 0 πε (c) := argminπ ⟨c, π⟩ + Φ(π) Schrodinger problem: Φ(π) := { εKL(π|α ⊗ β) if π1 = α, π2 = β +∞ otherwise where is convex! → L( ⋅ | ̂ π) Alternate interpretation [Galichon]: maximum likelihood estimator. L(c| ̂ π) ∝ − ∑ i log( dπε (c) dαdβ (xi , yi )) for ̂ π = ∑ i δ(xi ,yi ) Proposition: L(c| ̂ π) := Φ( ̂ π) + Φ*(−c) − ⟨−c, ̂ π⟩ Fenchel-Young loss of Mathieu Blondel. → Mathieu Blondel
  38. Regularized Inverse Optimal Transport Regularized inversion: min θ L(cθ |

    ̂ π) + λR(θ) R(θ) = ∥θ∥1 cθ (x, y) = ⟨θx, y⟩ SISTA [Dupuis, Galichon, Carlier]
  39. Regularized Inverse Optimal Transport Regularized inversion: min θ L(cθ |

    ̂ π) + λR(θ) R(θ) = ∥θ∥1 cθ (x, y) = ⟨θx, y⟩ Estimating from samples: θ⋆ ̂ π := ∑n i=1 δ(xi ,yi ) (xi , yi )i ∼ πϵ (cθ⋆ ) SISTA [Dupuis, Galichon, Carlier] πϵ ̂ π
  40. Regularized Inverse Optimal Transport Regularized inversion: min θ L(cθ |

    ̂ π) + λR(θ) R(θ) = ∥θ∥1 cθ (x, y) = ⟨θx, y⟩ Estimating from samples: θ⋆ ̂ π := ∑n i=1 δ(xi ,yi ) (xi , yi )i ∼ πϵ (cθ⋆ ) SISTA [Dupuis, Galichon, Carlier] Intuition: ∥η⋆∥∞ ≤ 1 ⇔ primal-dual solutions as (θ⋆, η⋆) λ → 0 J(θ) := L(cθ |πϵ (cθ⋆ )) Dual certificate: η⋆ := argminη {⟨J′  ′  (θ⋆)η, η⟩ : ηI = sign(θ⋆ I )} θ θ⋆ J(θ) I := supp(θ⋆) +1 −1 I non-degenerated degenerated πϵ ̂ π
  41. Regularized Inverse Optimal Transport Regularized inversion: min θ L(cθ |

    ̂ π) + λR(θ) R(θ) = ∥θ∥1 cθ (x, y) = ⟨θx, y⟩ Estimating from samples: θ⋆ ̂ π := ∑n i=1 δ(xi ,yi ) (xi , yi )i ∼ πϵ (cθ⋆ ) SISTA [Dupuis, Galichon, Carlier] Intuition: ∥η⋆∥∞ ≤ 1 ⇔ primal-dual solutions as (θ⋆, η⋆) λ → 0 J(θ) := L(cθ |πϵ (cθ⋆ )) Dual certificate: η⋆ := argminη {⟨J′  ′  (θ⋆)η, η⟩ : ηI = sign(θ⋆ I )} θ θ⋆ J(θ) I := supp(θ⋆) +1 −1 I non-degenerated degenerated πϵ ̂ π Theorem: if there exists , so that for ∥η⋆ Ic ∥∞ < 1, (A, B) n− 1 2 eA ε log(1/δ) ≤ λ ≤ B supp(θλ ) = supp(θ⋆) ∥θλ − θ⋆∥2 = 𝒪 ( exp(A/ε)log(1/δ) n ) one has with probability 1 − δ and Clarice Poon Francisco Andrade
  42. Gaussian Case α = 𝒩 (mα , Σα ) β

    = 𝒩 (mβ , Σβ ) Σ = Σα θΔ(Δθ⊤Σα θΔ)−1 2 Δ − εθ†,⊤ Δ = (Σβ +ε2θ†Σα −1θ†,⊤)1 2 [Bojilov & Galichon’16] πϵ (cθ ) = 𝒩 ( mα mβ , ( Σα Σ Σ Σβ )) mα mβ ∂2J(θ⋆) = 2ε[4ε2(Σβ −ΣTΣα Σ)−1 ⊗ (Σα − ΣΣβ −1Σ⊤)−1 + (θ⋆⊤ ⊗ θ⋆)]−1 Proposition: J(θ) := L(cθ |πϵ (cθ⋆ )) θ θ⋆ J(θ) Easy to check when is non-degenerated. → η⋆
  43. Gaussian Case α = 𝒩 (mα , Σα ) β

    = 𝒩 (mβ , Σβ ) Σ = Σα θΔ(Δθ⊤Σα θΔ)−1 2 Δ − εθ†,⊤ Δ = (Σβ +ε2θ†Σα −1θ†,⊤)1 2 [Bojilov & Galichon’16] πϵ (cθ ) = 𝒩 ( mα mβ , ( Σα Σ Σ Σβ )) mα mβ ∂2J(θ⋆) = 2ε[4ε2(Σβ −ΣTΣα Σ)−1 ⊗ (Σα − ΣΣβ −1Σ⊤)−1 + (θ⋆⊤ ⊗ θ⋆)]−1 Proposition: J(θ) := L(cθ |πϵ (cθ⋆ )) θ θ⋆ J(θ) Easy to check when is non-degenerated. → η⋆ min θ L(cθ | ̂ π) + λ∥θ∥1 min θ 1 2 ∥(Σ1 2 β ⊗ Σ1 2 α )(θ − ̂ θ)∥2 F + λ0 ∥θ∥1 λ = λ0 /ε ε → + ∞ min θ≻0 1 2 log det(θ)+ 1 2 ⟨θ, ̂ θ−1⟩ + λ0 ∥θ∥1 ε → 0 λ = λ0 ε Lasso Graphical-Lasso
  44. Graph Estimation Experiments θ⋆ = δI + diag(A1) − A

    dgeod (i, j) dgeod (i, j) dgeod (i, j) = 2 Graph Laplacian η⋆ (i,j) η⋆ (i,j) Adjacency A
  45. Better modeling the dynamics: use gradient flows Luigi Ambrosio Nicola

    Gigli Giuseppe Savare <latexit 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↵0 <latexit 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↵1 Conclusion: gradient flows for genomics αt+1 = argmin β W(αt , β) + τf(β)
  46. Better modeling the dynamics: use gradient flows Luigi Ambrosio Nicola

    Gigli Giuseppe Savare <latexit 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Cost c <latexit 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RNA$RNA <latexit 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Space$ATAC <latexit 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Time t <latexit 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Time t + 1 <latexit 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Potential f Key issues: learning the potential f(β) Integrate several omics Conclusion: gradient flows for genomics αt+1 = argmin β W(αt , β) + τf(β)