Sinkhorn Permutation Variational Marginal...
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Gonzalo Mena*, Erdem Varol**, Amin Nejatbakhsh**, Eviatar Yemini**, Liam Paninski** *Harvard University, **Columbia University Sinkhorn Permutation Variational Marginal Inference [email protected] This work was supported by the Harvard Data Science Initiative Summary • Statistical Inference for exponential families defined over permutation matrices quickly becomes intractable, since it involves the computation of the permanent of a matrix, a #P-hard problem (Valiant,1979) • Here: marginal inference. We appeal to Variational Inference (Wainwright and Jordan, 2007) to show that the Sinkhorn algorithm provides an accurate and computationally efficient approximation to the permanent, and hence, to the marginal. • We successfully apply our technique for probabilistic neural inference in C.elegans in the context of the novel NeuroPAL technology. References: 1. Wainwright, Martin and Jordan, Michael (2008) Graphical Models, Exponential Families, and Variational Inference . Foundations and Trends in Machine Learning 2. Valiant, Leslie. (1979) The complexity of computing the permanent. Theoretical computer Science 3. Mena, G., Belanger, D., Linderman, S., & Snoek, J.” (2018) Learning Latent Permutations with Gumbel-Sinkhorn Networks." ICLR 4. Yemini, Eviatar, Albert Lin, Amin Nejatbakhsh, Erdem Varol, Ruoxi Sun, Gonzalo E. Mena, Aravinthan DT Samuel, Liam Paninski, Vivek Venkatachalam, and Oliver Hobert. (2019) “NeuroPAL: A Neuronal Polychromatic Atlas of Landmarks for Whole-Brain Imaging in C. elegans." bioRxiv 5. Vontobel, Pascal O. "The Bethe permanent of a nonnegative matrix.” (2012) IEEE Transactions on Information Theory p(P |L)= 1 Z L exp (hlog L, P i F ) Problem statement For a permutation matrix P consider the exponential family distribution How to compute the matrix of marginals ? Requires access to , the intractable permanent. ⇢ = E (P ) Z L Variational Inference log Z L = sup μ2M hlog L, μi- A ⇤ (μ) The permanent admits a variational representation Where is the set of doubly stochastic matrices and is the intractable dual function, or entropy. Intractable to compute. Key fact: , the matrix achieving the maximum above. A ⇤ (μ) M ⇢ = μ opt Our Approximation: solve instead The solution is realized through Sinkhorn algorithm (Mena et al., 2018) S (L) := μ opt Input Output Iterate until convergence Normalize each row of Normalize each column of L S (L) L L L L S (L) L log S Z = sup μ2M hlog L, μi- X i,j μ i,j log μ i,j Application to neural identification in C. Elegans • The worm C.elegans is a unique species since its nervous system is stereotypical, neurons (~300) and connections remain the same from animal to animal. • Now it is possible to image worm’s whole brains to better understand mind (Fig. 1) • However, before that, a technical problem has to be solved: given colored volumetric images of the worm we need to identify the neurons, that is, assign a canonical label (a name) to each of them. • Further, we prefer to have probabilistic identities to acknowledge the possibility of errors. Fig 1. NeuroPAL technology (Yemini et al., 2019) enables the deterministic coloring of C.elegans neurons to facilitate neural identification w 1 = f 1 (z 1 ) w 2 = f 2 (z 2 ) z 1 z 2 • Given labeled worms we considered a gaussian to represent the variability in color and position of each of neurons. • We also inferred afine mappings to a canonical space to account for rotations, stretching, etc (Fig. 2) • For an unlabeled worm, the the above model induces a posterior distribution over assignments between observed neurons and canonical identities. Is the likelihood of observed neuron under the model of neuron . The resulting is the likelihood that observed neuron has identity p(z )= N Y i=1 N (z i ; μ i , ⌃ i ) M N p(P |L) L i,j i j Fig 2. Observations are mapped to canonical space Results ⇢ i,j j i • We used to efficiently guide annotation. • We iteratively look for neurons where model is most uncertain, and ask the human to annotate manually. After, is updated. • This strategy is efficient, accuracy grows fast (Fig 3) • The so-called Bethe (Vontobel, 2014) approximation does similarly, but it is ~100 times slower ⇢ ⇢ Fig 3. Identification accuracy as a function of human-labeled neurons f
Transcript of Sinkhorn Permutation Variational Marginal...
Gonzalo Mena*, Erdem Varol**, Amin Nejatbakhsh**, Eviatar Yemini**, Liam Paninski***Harvard University, **Columbia University
Sinkhorn Permutation Variational Marginal Inference
[email protected] This work was supported by the Harvard Data Science Initiative
Summary• Statistical Inference for exponential families defined over permutation
matrices quickly becomes intractable, since it involves the computation of the permanent of a matrix, a #P-hard problem (Valiant,1979)
• Here: marginal inference. We appeal to Variational Inference (Wainwright and Jordan, 2007) to show that the Sinkhorn algorithm provides an accurate and computationally efficient approximation to the permanent, and hence, to the marginal.
• We successfully apply our technique for probabilistic neural inference in C.elegans in the context of the novel NeuroPAL technology.
References:1. Wainwright, Martin and Jordan, Michael (2008) Graphical Models, Exponential Families, and Variational Inference . Foundations and Trends in Machine Learning2. Valiant, Leslie. (1979) The complexity of computing the permanent. Theoretical computer Science3. Mena, G., Belanger, D., Linderman, S., & Snoek, J.” (2018) Learning Latent Permutations with Gumbel-Sinkhorn Networks." ICLR 4. Yemini, Eviatar, Albert Lin, Amin Nejatbakhsh, Erdem Varol, Ruoxi Sun, Gonzalo E. Mena, Aravinthan DT Samuel, Liam Paninski, Vivek Venkatachalam, and Oliver Hobert. (2019) “NeuroPAL: A Neuronal Polychromatic Atlas of Landmarks for Whole-Brain Imaging in C. elegans." bioRxiv 5. Vontobel, Pascal O. "The Bethe permanent of a nonnegative matrix.” (2012) IEEE Transactions on Information Theory
p(P |L) = 1
ZLexp (hlogL,P iF )
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Problem statementFor a permutation matrix P consider the exponential family distribution
How to compute the matrix of marginals ? Requires access to , the intractable permanent.
⇢ = E(P )<latexit sha1_base64="SUpouU4oGFbObEzcqzJ53aV2v4A=">AAAB8nicbVBNSwMxEJ2tX7V+VT16CRahXspuFfQiFEXwWMHawnYp2TTbhmaTJckKZenP8OJBEa/+Gm/+G9N2D9r6YODx3gwz88KEM21c99sprKyurW8UN0tb2zu7e+X9g0ctU0Voi0guVSfEmnImaMsww2knURTHIaftcHQz9dtPVGkmxYMZJzSI8UCwiBFsrOR31VCiK3RbbZ72yhW35s6AlomXkwrkaPbKX92+JGlMhSEca+17bmKCDCvDCKeTUjfVNMFkhAfUt1TgmOogm508QSdW6aNIKlvCoJn6eyLDsdbjOLSdMTZDvehNxf88PzXRZZAxkaSGCjJfFKUcGYmm/6M+U5QYPrYEE8XsrYgMscLE2JRKNgRv8eVl8liveWe1+v15pXGdx1GEIziGKnhwAQ24gya0gICEZ3iFN8c4L8678zFvLTj5zCH8gfP5A00mj/Q=</latexit>
ZL<latexit sha1_base64="DCwlXD44JeCuCezTlVaV5R+NBUk=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioGXQxsIiovnA5Ah7m0myZG/v2N0TwpGfYGOhiK2/yM5/4ya5QhMfDDzem2FmXhALro3rfju5ldW19Y38ZmFre2d3r7h/0NBRohjWWSQi1QqoRsEl1g03AluxQhoGApvB6HrqN59QaR7JBzOO0Q/pQPI+Z9RY6f6xe9stltyyOwNZJl5GSpCh1i1+dXoRS0KUhgmqddtzY+OnVBnOBE4KnURjTNmIDrBtqaQhaj+dnTohJ1bpkX6kbElDZurviZSGWo/DwHaG1Az1ojcV//Paielf+imXcWJQsvmifiKIicj0b9LjCpkRY0soU9zeStiQKsqMTadgQ/AWX14mjUrZOytX7s5L1assjjwcwTGcggcXUIUbqEEdGAzgGV7hzRHOi/PufMxbc042cwh/4Hz+AAg2jaE=</latexit>
Variational Inference
logZL = supµ2M
hlogL, µi �A⇤(µ)<latexit sha1_base64="XJcjFCxdSMBzbC0H5GZl80ql4C0=">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</latexit>
The permanent admits a variational representation
Where is the set of doubly stochastic matrices and is the intractable dual function, or entropy. Intractable to compute.
Key fact: , the matrix achieving the maximum above.
A⇤(µ)<latexit sha1_base64="P5MK/3KaSyr6Dt0el1YwfHJZsXo=">AAAB8HicbVBNSwMxEJ31s9avqkcvwSJUD2W3CnqsevFYwX5Iu5Zsmm1Dk+ySZIWy9Fd48aCIV3+ON/+NabsHbX0w8Hhvhpl5QcyZNq777Swtr6yurec28ptb2zu7hb39ho4SRWidRDxSrQBrypmkdcMMp61YUSwCTpvB8GbiN5+o0iyS92YUU1/gvmQhI9hY6QFdPZ6WOiI56RaKbtmdAi0SLyNFyFDrFr46vYgkgkpDONa67bmx8VOsDCOcjvOdRNMYkyHu07alEguq/XR68BgdW6WHwkjZkgZN1d8TKRZaj0RgOwU2Az3vTcT/vHZiwks/ZTJODJVktihMODIRmnyPekxRYvjIEkwUs7ciMsAKE2MzytsQvPmXF0mjUvbOypW782L1OosjB4dwBCXw4AKqcAs1qAMBAc/wCm+Ocl6cd+dj1rrkZDMH8AfO5w8f+o9Q</latexit>
M<latexit sha1_base64="O0ZuVSc93kx5NZ9IzTaY5ls9v1g=">AAAB8nicbVDLSgMxFL1TX7W+qi7dBIvgqsxUQZdFN26ECvYB06Fk0kwbmkmGJCOUoZ/hxoUibv0ad/6NmXYW2nogcDjnXnLuCRPOtHHdb6e0tr6xuVXeruzs7u0fVA+POlqmitA2kVyqXog15UzQtmGG016iKI5DTrvh5Db3u09UaSbFo5kmNIjxSLCIEWys5PdjbMYE8+x+NqjW3Lo7B1olXkFqUKA1qH71h5KkMRWGcKy177mJCTKsDCOczir9VNMEkwkeUd9SgWOqg2weeYbOrDJEkVT2CYPm6u+NDMdaT+PQTuYR9bKXi/95fmqi6yBjIkkNFWTxUZRyZCTK70dDpigxfGoJJorZrIiMscLE2JYqtgRv+eRV0mnUvYt64+Gy1rwp6ijDCZzCOXhwBU24gxa0gYCEZ3iFN8c4L86787EYLTnFzjH8gfP5A4PYkWc=</latexit>
⇢ = µopt<latexit sha1_base64="VG2+kI4hzRuaaeRA8jVXgXHEW74=">AAAB9XicbVBNSwMxEM3Wr1q/qh69BIvgqexWQS9C0YvHCvYDumvJptk2NJssyaxSlv4PLx4U8ep/8ea/MW33oK0PBh7vzTAzL0wEN+C6305hZXVtfaO4Wdra3tndK+8ftIxKNWVNqoTSnZAYJrhkTeAgWCfRjMShYO1wdDP1249MG67kPYwTFsRkIHnEKQErPfh6qK78OO1lKoFJr1xxq+4MeJl4OamgHI1e+cvvK5rGTAIVxJiu5yYQZEQDp4JNSn5qWELoiAxY11JJYmaCbHb1BJ9YpY8jpW1JwDP190RGYmPGcWg7YwJDs+hNxf+8bgrRZZBxmaTAJJ0vilKBQeFpBLjPNaMgxpYQqrm9FdMh0YSCDapkQ/AWX14mrVrVO6vW7s4r9es8jiI6QsfoFHnoAtXRLWqgJqJIo2f0it6cJ+fFeXc+5q0FJ585RH/gfP4A/I2S1A==</latexit>
Our Approximation: solve instead
The solution is realized through Sinkhorn algorithm (Mena et al., 2018)
S(L) := µopt<latexit sha1_base64="i8FUVUAKyuAUETGVg5P3D/IkCXM=">AAAB+HicbVDLSgNBEJz1GeMjqx69DAYhXsJuFBRBCHrx4CGieUCyLLOTSTJkZmeZhxCXfIkXD4p49VO8+TdOkj1oYkFDUdVNd1eUMKq05307S8srq2vruY385tb2TsHd3WsoYSQmdSyYkK0IKcJoTOqaakZaiSSIR4w0o+H1xG8+EqmoiB/0KCEBR/2Y9ihG2kqhW7gv3R5fXHa4CVOR6HHoFr2yNwVcJH5GiiBDLXS/Ol2BDSexxgwp1fa9RAcpkppiRsb5jlEkQXiI+qRtaYw4UUE6PXwMj6zShT0hbcUaTtXfEyniSo14ZDs50gM1703E/7y20b3zIKVxYjSJ8WxRzzCoBZykALtUEqzZyBKEJbW3QjxAEmFts8rbEPz5lxdJo1L2T8qVu9Ni9SqLIwcOwCEoAR+cgSq4ATVQBxgY8AxewZvz5Lw4787HrHXJyWb2wR84nz/aK5KU</latexit>
Input OutputIterate until convergence
Normalize each row of Normalize each column of L
<latexit sha1_base64="QE9oDXOSeRcavWmZZHNQHB2S4vk=">AAAB6HicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioGXQxsIiAfMByRH2NnPJmr29Y3dPCCG/wMZCEVt/kp3/xk1yhSY+GHi8N8PMvCARXBvX/XZya+sbm1v57cLO7t7+QfHwqKnjVDFssFjEqh1QjYJLbBhuBLYThTQKBLaC0e3Mbz2h0jyWD2acoB/RgeQhZ9RYqX7fK5bcsjsHWSVeRkqQodYrfnX7MUsjlIYJqnXHcxPjT6gynAmcFrqpxoSyER1gx1JJI9T+ZH7olJxZpU/CWNmShszV3xMTGmk9jgLbGVEz1MveTPzP66QmvPYnXCapQckWi8JUEBOT2dekzxUyI8aWUKa4vZWwIVWUGZtNwYbgLb+8SpqVsndRrtQvS9WbLI48nMApnIMHV1CFO6hBAxggPMMrvDmPzovz7nwsWnNONnMMf+B8/gCk14zU</latexit>
S(L)<latexit sha1_base64="6bvUsijW6qvVE+rVddgGwxJWgsk=">AAAB63icbVA9SwNBEJ2LXzF+RS1tFoMQm3AXBS2DNhYWEY0JJEfY2+wlS3b3jt09IRz5CzYWitj6h+z8N+4lV2jig4HHezPMzAtizrRx3W+nsLK6tr5R3Cxtbe/s7pX3Dx51lChCWyTikeoEWFPOJG0ZZjjtxIpiEXDaDsbXmd9+okqzSD6YSUx9gYeShYxgk0n31dvTfrni1twZ0DLxclKBHM1++as3iEgiqDSEY627nhsbP8XKMMLptNRLNI0xGeMh7VoqsaDaT2e3TtGJVQYojJQtadBM/T2RYqH1RAS2U2Az0oteJv7ndRMTXvopk3FiqCTzRWHCkYlQ9jgaMEWJ4RNLMFHM3orICCtMjI2nZEPwFl9eJo/1mndWq9+dVxpXeRxFOIJjqIIHF9CAG2hCCwiM4Ble4c0Rzovz7nzMWwtOPnMIf+B8/gANzI2W</latexit>
L <latexit sha1_base64="44cWxetFoT515T0PlXPctkmgXds=">AAAB8nicbVA9SwNBEN3zM8avqKXNYRCswl0UtAzaWFhEMB9wOcLeZi5Zsrd77M4pIeRn2FgoYuuvsfPfuEmu0MQHA4/3ZpiZF6WCG/S8b2dldW19Y7OwVdze2d3bLx0cNo3KNIMGU0LpdkQNCC6hgRwFtFMNNIkEtKLhzdRvPYI2XMkHHKUQJrQvecwZRSsFdx0BMVKt1VO3VPYq3gzuMvFzUiY56t3SV6enWJaARCaoMYHvpRiOqUbOBEyKncxAStmQ9iGwVNIETDienTxxT63Sc2OlbUl0Z+rviTFNjBklke1MKA7MojcV//OCDOOrcMxlmiFINl8UZ8JF5U7/d3tcA0MxsoQyze2tLhtQTRnalIo2BH/x5WXSrFb880r1/qJcu87jKJBjckLOiE8uSY3ckjppEEYUeSav5M1B58V5dz7mrStOPnNE/sD5/AGHhZFq</latexit>
L <latexit sha1_base64="44cWxetFoT515T0PlXPctkmgXds=">AAAB8nicbVA9SwNBEN3zM8avqKXNYRCswl0UtAzaWFhEMB9wOcLeZi5Zsrd77M4pIeRn2FgoYuuvsfPfuEmu0MQHA4/3ZpiZF6WCG/S8b2dldW19Y7OwVdze2d3bLx0cNo3KNIMGU0LpdkQNCC6hgRwFtFMNNIkEtKLhzdRvPYI2XMkHHKUQJrQvecwZRSsFdx0BMVKt1VO3VPYq3gzuMvFzUiY56t3SV6enWJaARCaoMYHvpRiOqUbOBEyKncxAStmQ9iGwVNIETDienTxxT63Sc2OlbUl0Z+rviTFNjBklke1MKA7MojcV//OCDOOrcMxlmiFINl8UZ8JF5U7/d3tcA0MxsoQyze2tLhtQTRnalIo2BH/x5WXSrFb880r1/qJcu87jKJBjckLOiE8uSY3ckjppEEYUeSav5M1B58V5dz7mrStOPnNE/sD5/AGHhZFq</latexit>
L<latexit sha1_base64="QE9oDXOSeRcavWmZZHNQHB2S4vk=">AAAB6HicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioGXQxsIiAfMByRH2NnPJmr29Y3dPCCG/wMZCEVt/kp3/xk1yhSY+GHi8N8PMvCARXBvX/XZya+sbm1v57cLO7t7+QfHwqKnjVDFssFjEqh1QjYJLbBhuBLYThTQKBLaC0e3Mbz2h0jyWD2acoB/RgeQhZ9RYqX7fK5bcsjsHWSVeRkqQodYrfnX7MUsjlIYJqnXHcxPjT6gynAmcFrqpxoSyER1gx1JJI9T+ZH7olJxZpU/CWNmShszV3xMTGmk9jgLbGVEz1MveTPzP66QmvPYnXCapQckWi8JUEBOT2dekzxUyI8aWUKa4vZWwIVWUGZtNwYbgLb+8SpqVsndRrtQvS9WbLI48nMApnIMHV1CFO6hBAxggPMMrvDmPzovz7nwsWnNONnMMf+B8/gCk14zU</latexit>
L<latexit sha1_base64="QE9oDXOSeRcavWmZZHNQHB2S4vk=">AAAB6HicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioGXQxsIiAfMByRH2NnPJmr29Y3dPCCG/wMZCEVt/kp3/xk1yhSY+GHi8N8PMvCARXBvX/XZya+sbm1v57cLO7t7+QfHwqKnjVDFssFjEqh1QjYJLbBhuBLYThTQKBLaC0e3Mbz2h0jyWD2acoB/RgeQhZ9RYqX7fK5bcsjsHWSVeRkqQodYrfnX7MUsjlIYJqnXHcxPjT6gynAmcFrqpxoSyER1gx1JJI9T+ZH7olJxZpU/CWNmShszV3xMTGmk9jgLbGVEz1MveTPzP66QmvPYnXCapQckWi8JUEBOT2dekzxUyI8aWUKa4vZWwIVWUGZtNwYbgLb+8SpqVsndRrtQvS9WbLI48nMApnIMHV1CFO6hBAxggPMMrvDmPzovz7nwsWnNONnMMf+B8/gCk14zU</latexit>
S(L) L<latexit sha1_base64="zkSZ6QQ25ok6w/xBUzMoqCTLW8Y=">AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBahXkpSBT0WvXjooaL9gDaUzXbTLt1swu6mUkL/iRcPinj1n3jz37htc9DWBwOP92aYmefHnCntON9Wbm19Y3Mrv13Y2d3bP7APj5oqSiShDRLxSLZ9rChngjY005y2Y0lx6HPa8ke3M781plKxSDzqSUy9EA8ECxjB2kg9234o1c67nAYaSxk9oVrPLjplZw60StyMFCFDvWd/dfsRSUIqNOFYqY7rxNpLsdSMcDotdBNFY0xGeEA7hgocUuWl88un6MwofRRE0pTQaK7+nkhxqNQk9E1niPVQLXsz8T+vk+jg2kuZiBNNBVksChKOdIRmMaA+k5RoPjEEE8nMrYgMscREm7AKJgR3+eVV0qyU3Yty5f6yWL3J4sjDCZxCCVy4gircQR0aQGAMz/AKb1ZqvVjv1seiNWdlM8fwB9bnD2cTkt0=</latexit>
logSZ = supµ2M
hlogL, µi �X
i,j
µi,j logµi,j
<latexit sha1_base64="R+Aq1HhMpfNxmSsHIXhPz7q8C+c=">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</latexit>
Application to neural identification in C. Elegans• The worm C.elegans is a unique species since its nervous system is stereotypical, neurons (~300) and
connections remain the same from animal to animal. • Now it is possible to image worm’s whole brains to better understand mind (Fig. 1)• However, before that, a technical problem has to be solved: given colored volumetric images of the worm we
need to identify the neurons, that is, assign a canonical label (a name) to each of them. • Further, we prefer to have probabilistic identities to acknowledge the possibility of errors.
Fig 1. NeuroPAL technology (Yemini et al., 2019) enables the deterministic coloring of C.elegans neurons to facilitate neural identificationw1 = f1(z1)
w2 = f2(z2)
z1
z2
• Given labeled worms we considered a gaussian to represent the variability in color and position of each of neurons.
• We also inferred afine mappings to a canonical space to account for rotations, stretching, etc (Fig. 2)
• For an unlabeled worm, the the above model induces a posterior distribution over assignments between observed neurons and canonical identities. Is the likelihood of observed neuron under the model of neuron . The resulting is the likelihood that observed neuron has identity
p(z) =NY
i=1
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Fig 2. Observations are mapped to canonical space
Results
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• We used to efficiently guide annotation. • We iteratively look for neurons where model is most uncertain, and ask the
human to annotate manually. After, is updated. • This strategy is efficient, accuracy grows fast (Fig 3) • The so-called Bethe (Vontobel, 2014) approximation does similarly, but it is
~100 times slower
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Fig 3. Identification accuracy as a function of human-labeled neurons
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