ML approaches to find new drugs for ATIP3 deficient Triple Negative Breast Cancer

Talk given at CBIO meeting on 11/04/2023

Transcript

1. Gwenn Guichaoua, CBIO meeting 10/04/2023 ML approaches to find new drugs for ATIP3 deficient Triple Negative Breast Cancer Supervisors : Véronique Stoven, Chloé Azencott, Olivier Collier (Modal’X Nanterre), Clara Nahmias (IGR) 1
2. Bad Subtype Luminal A HER2- positif Chemoterapy Hormonotherapy monoclonal antibodies Luminal B Triple Negatif Phenotype Prognosis Treatment ER+ PR+ ER+ PR+ HER2+ ER- PR- HER2- Good ATIP3 protein: a new marker for a category of TNBC 2 Biological sub-typing of the breast cancers Breast cancer: 1 of the 3 most common cancers worldwide
3. Bad Subtype Luminal A HER2- positif Chemoterapy Hormonotherapy monoclonal antibodies Luminal B Triple Negatif Phenotype Prognosis Treatment ER+ PR+ ER+ PR+ HER2+ ER- PR- HER2- Good ATIP3 protein: a new marker for a category of TNBC 2 Biological sub-typing of the breast cancers Breast cancer: 1 of the 3 most common cancers worldwide A candidate biomarker to de fi ne a new breast cancer subtype, identi fi ed by Clara Nahmias’s team •Low expression of ATIP3 in TNBC [Rodriguez&al, 2009] •Poorer prognosis for tumors that not express ATIP3 (called ATIP3- tumors) [Rodriguez&al, 2019] •70% of ATIP3- tumors resistance to the chemotherapy •ATIP3- resistant tumors more agressive than ATIP3+ tumors resistant Important unmet need for new therapies and therapeutic target Lack of knowledge for understanding the mechanism of ATIP3 ATIP3-
4. Roadmap of the thesis New sub-type of patients : ATIP3 de fi cient TNBC Part 1 : Find a genetic signature To predict the chemotherapy response Part 2 : Find a new treatment To increase the survival rate 70%, avoid chemotherapy 30%, chemotherapy A -Pharmacological strategy B - Transcriptomics and system biology strategy 3
5. Part 2-A : Find a new treatment For TNBC tumors, de fi cient in ATIP3 Essential cell biology Blocking points : Unknown proteins involved in biological mechanisms for ATIP3- TNBC tumors Goal : Search for proteins, speci fi c of these tumors and their corresponding molecules (ligands) 4
6. Part 2-A : Find a new treatment For TNBC tumors, de fi cient in ATIP3 Essential cell biology Blocking points : Unknown proteins involved in biological mechanisms for ATIP3- TNBC tumors Goal : Search for proteins, speci fi c of these tumors and their corresponding molecules (ligands) 4 Tools : Public Databases of interactions between proteins and molecules Public Databases of pathways (ensembles of proteins by biological way) Data : Pharmacologic screen of molecules by Clara Nahmias’s team in IGR on cells lines TNBC ATIP3- and ATIP3+ in order to fi nd 20 molecules di ff erentially active on ATIP3- TNBC cells TNBC ATIP3+ cells Sum 52 Cell Sum52 ATIP3- Cells Sum52 Ctrl ATIP3+ Exposition of one of the 100 molecules (drugs) of TOCRIS base Apoptose Unresponsive phenotype 20 di ff erentially active molecules ATIP3- vs ATIP3+
7. Overview Find known targets in protein/molecule interactions Public databases Focus known protein targets of the 20 di ff erentially active molecules Predict other protein targets of the 20 di ff erentially active molecules 5 Pathways enrichment: from single proteins to set of proteins Construction of a large protein/molecule interactions database Large scale kernel methods
8. Finding proteins in Protein/Molecule Interaction databases Bioactivity databases: BindingDB [Tiqing&al,2007], Pubchem [Kim&al, 2019], HMS LINCS Database [Fallahi-Sichani&al,2013], A Consensus Compound/Bioactivity Dataset for Data-Driven Design and Chemogenomics [Isigkeit&al(2022)] 6 + Experimentally measures of the potency of proteins/molecules interactions, including Ki, Kd, IC50 - How to de fi ne a direct interaction ? Direct binding between protein/molecule: choice of a threshold Kd, Ki, IC50 < 100 nM Quantitative measure: Dissociation constant Kd, Ki Half maximal inhibitory concentration IC50 Activity
9. Proteins in existing databases Molecule manufacturer: 1 target protein Contribution ~8 target proteins 20 molecules differentially active on ATIP3-/ATIP3+ 7
10. 204 target proteins Measures (Kd, IC50, Ki) < 100 nM Molecular interactions databases Proteins in existing databases Molecule manufacturer: 1 target protein Contribution ~8 target proteins 20 molecules differentially active on ATIP3-/ATIP3+ Tozasertib Dasatinib 7
11. 204 target proteins Measures (Kd, IC50, Ki) < 100 nM Molecular interactions databases Proteins in existing databases 72 target proteins Measures (Kd, IC50, Ki) < 100 nM Molecular interactions databases 18 molecules differentially active on ATIP3-/ATIP3+ Molecule manufacturer: 1 target protein Contribution ~8 target proteins 20 molecules differentially active on ATIP3-/ATIP3+ Tozasertib Dasatinib 7
12. Overview Find known targets in protein/molecule interactions Public databases Focus known protein targets of the 20 di ff erentially active molecules Predict other protein targets of the 20 di ff erentially active molecules 8 Pathways enrichment: from single proteins to set of proteins Construction of a large protein/molecule interactions database Large scale kernel methods
13. Enrichment analysis of pathways Pathways enrichment: fi nd groups T of proteins 72 proteins of interest relevant groups of proteins Pathways enrichment Representing biological mechanism Overlapping with the 72 proteins in Q Score(Q,T) (adjusted with Benjamini-Hockberg FDR correction) hypergeometric test p-value 9
14. Hesperadin (1_3) PTK2B (Pyk2, FAK2) BRSK2 TYK2 ROS1 LTK (TYK1) PTK2 (FAK) TPCA1 (1_11) JAK2 STAT3 PP 2 (1_15) SRC HCK What about the other molecules ? Interesting proteins to test and associated molecules 3 molecules STAT3 already tested : not a target FAK 1/2 : not such a good target (migration) Blocking points for the biologists 10
15. Overview Find known targets in protein/molecule interactions Public databases Focus known protein targets of the 20 di ff erentially active molecules Predict other protein targets of the 20 di ff erentially active molecules 11 Pathways enrichment: from single proteins to set of proteins Construction of a large protein/molecule interactions database Large scale kernel methods
16. Protein prediction Goal: enriching the set of targeted proteins 12 Target proteins Molecules
17. Protein prediction Goal: enriching the set of targeted proteins 12 Target proteins Molecules
18. Protein prediction Goal: enriching the set of targeted proteins Supervised learning Input: database of interactions 12 Target proteins Molecules
19. Protein prediction Goal: enriching the set of targeted proteins Supervised learning Input: database of interactions Output: predicted interactions 12 Target proteins Molecules
20. Protein prediction Goal: enriching the set of targeted proteins Supervised learning Input: database of interactions Output: predicted interactions Method: Binary classi fi cation problem Select balanced set of negative examples [Mathieu Najm et al., 2021] 12 Target proteins Molecules
21. Protein prediction Goal: enriching the set of targeted proteins Supervised learning Input: database of interactions Output: predicted interactions Train kernel SVM classi fi er (use & ) Method: Binary classi fi cation problem Select balanced set of negative examples [Mathieu Najm et al., 2021] 12 Target proteins Molecules
22. Protein prediction Goal: enriching the set of targeted proteins Supervised learning Input: database of interactions Output: predicted interactions Train kernel SVM classi fi er (use & ) Method: Binary classi fi cation problem Select balanced set of negative examples [Mathieu Najm et al., 2021] Work in progress: New database of direct interactions Large scale kernel method 12 Target proteins Molecules
23. Overview Find known targets in protein/molecule interactions Public databases Focus known protein targets of the 20 di ff erentially active molecules Predict other protein targets of the 20 di ff erentially active molecules 13 Pathways enrichment: from single proteins to set of proteins Construction of a large protein/molecule interactions database Large scale kernel methods
24. New training database Binary interactions database: Drugbank v1.5.1 [Wishart&al,2018]: Datas Drugbank : 2670 proteins 5071 molecules 14638 interactions + + well curated + FDA-approved drugs - indirect interactions Molecule Protein
25. New training database Protein Protein Protein Protein Molecule Binary interactions database: Drugbank v1.5.1 [Wishart&al,2018]: Datas Drugbank : 2670 proteins 5071 molecules 14638 interactions + + well curated + FDA-approved drugs - indirect interactions Molecule Protein
26. New training database Protein Protein Protein Protein Molecule Binary interactions database: Drugbank v1.5.1 [Wishart&al,2018]: Datas Drugbank : 2670 proteins 5071 molecules 14638 interactions + + well curated + FDA-approved drugs - indirect interactions Molecule Protein
27. New training database Protein Protein Protein Protein Molecule Bioactivity database CC : A Consensus Compound/Bioactivity Dataset for Data-Driven Design and Chemogenomics [ Isigkei&al,2022] Binary interactions database: Drugbank v1.5.1 [Wishart&al,2018]: Datas Drugbank : 2670 proteins 5071 molecules 14638 interactions + + well curated + FDA-approved drugs - indirect interactions Direct binding: Kd, Ki, IC50 < 100 nM. No binding: Kd, Ki, IC50 > 10 microM Molecule Protein Molecule Protein
28. New training database Protein Protein Protein Protein Molecule Bioactivity database CC : A Consensus Compound/Bioactivity Dataset for Data-Driven Design and Chemogenomics [ Isigkei&al,2022] Extracted from 5 databases : ChEMBL, PubChem, IUPHAR/BPS, BindingDB, and Probes & Drugs Binary interactions database: Drugbank v1.5.1 [Wishart&al,2018]: Datas Drugbank : 2670 proteins 5071 molecules 14638 interactions + + well curated + FDA-approved drugs - indirect interactions Direct binding: Kd, Ki, IC50 < 100 nM. No binding: Kd, Ki, IC50 > 10 microM Molecule Protein Molecule Protein
29. New training database Protein Protein Protein Protein Molecule Bioactivity database CC : A Consensus Compound/Bioactivity Dataset for Data-Driven Design and Chemogenomics [ Isigkei&al,2022] Extracted from 5 databases : ChEMBL, PubChem, IUPHAR/BPS, BindingDB, and Probes & Drugs + True interactions + Checked datas + More datas - less proteins Datas CC : 1627 proteins 152k molecules 229k interactions + 50k interactions - Binary interactions database: Drugbank v1.5.1 [Wishart&al,2018]: Datas Drugbank : 2670 proteins 5071 molecules 14638 interactions + + well curated + FDA-approved drugs - indirect interactions Direct binding: Kd, Ki, IC50 < 100 nM. No binding: Kd, Ki, IC50 > 10 microM Molecule Protein Molecule Protein
30. Construction Preprocessing : For a (molecule,protein) pair 1. Activity check annotation : keep multiple annotated bioactivities within one log unit di ff erence kept 2. Structure check : keep molecule which same SMILES between di ff erent sources 3. Keep IC50, Ki, Kd known 4. Make binary interactions : measure = fi rst Kd, then Ki, then IC50 measure <10nM ( M): interactions + measure > 100 microM ( M) : interactions - <latexit sha1_base64="19OAeTsEV3mWXvQneo58YjqgWMc=">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</latexit> 10 7 <latexit sha1_base64="3V19iHXrJMEsQ7O1Yf/AR3qR19A=">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</latexit> 10 4
31. Construction Preprocessing : For a (molecule,protein) pair 1. Activity check annotation : keep multiple annotated bioactivities within one log unit di ff erence kept 2. Structure check : keep molecule which same SMILES between di ff erent sources 3. Keep IC50, Ki, Kd known 4. Make binary interactions : measure = fi rst Kd, then Ki, then IC50 measure <10nM ( M): interactions + measure > 100 microM ( M) : interactions - <latexit sha1_base64="19OAeTsEV3mWXvQneo58YjqgWMc=">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</latexit> 10 7 <latexit sha1_base64="3V19iHXrJMEsQ7O1Yf/AR3qR19A=">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</latexit> 10 4
32. Overview Find known targets in protein/molecule interactions Public databases Focus known protein targets of the 20 di ff erentially active molecules Predict other protein targets of the 20 di ff erentially active molecules 16 Pathways enrichment: from single proteins to set of proteins Construction of a large protein/molecule interactions database Large scale kernel methods
33. Kernel SVM 17 Database for training : a set of molecules a set of proteins a space subset of positive interactions Choice of training set Method : kernel SVM Choice of kernel Kernel Kernel Kernel : similarity between two pairs and de fi ned by a Kronecker product: Feature embedding in kernel space where trained by SVM [Vert&al, 2008] <latexit sha1_base64="d1zt1xTPGu6nzZqk8YLkjqm8Xnw=">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</latexit> (mk)k <latexit 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sha1_base64="uKlMjUATfnk5o/q0OpHSPDa/VVQ=">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</latexit> M (m, m0) <latexit sha1_base64="u0Y1LOzkasJCuZamzYIeqjadbqQ=">AAAC0XicjVHLSsNAFD3G97vq0k2wiBWkJOJrKbpxWdE+oK1lko4amscwmQhFBHHrD7jVnxL/QP/CO2MEtYhOSHLm3HvOzL3XE2GQKsd5GbKGR0bHxicmp6ZnZufmCwuLtTTJpM+rfhImsuGxlIdBzKsqUCFvCMlZ5IW87vUOdbx+xWUaJPGp6gvejthFHJwHPlNEnbV6TAjWqZTEhlhb7xSKTtkxyx4Ebg6KyFclKTyjhS4S+MgQgSOGIhyCIaWnCRcOBHFtXBMnCQUmznGDKdJmlMUpgxHbo+8F7Zo5G9Nee6ZG7dMpIb2SlDZWSZNQniSsT7NNPDPOmv3N+9p46rv16e/lXhGxCpfE/qX7zPyvTteicI49U0NANQnD6Or83CUzXdE3t79UpchBEKdxl+KSsG+Un322jSY1teveMhN/NZma1Xs/z83wpm9JA3Z/jnMQ1DbL7k55+3iruH+Qj3oCy1hBiea5i30coYIqeUs84BFP1onVt26tu49UayjXLOHbsu7fAVGrlC0=</latexit> P (p, p0) : similarity between proteins a space subset of negative interactions <latexit sha1_base64="4i5wt7ZIDXdz8pMid/F6pvKJ+9A=">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</latexit> (mi, pi)i2I <latexit sha1_base64="2m6rYrNiGIfes0iP1MgLMArFwkk=">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</latexit> f(m, p) = h'((m, p)), wi H <latexit sha1_base64="OnhZh16IirUcgMMJYVUMPOeLWaE=">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</latexit> w <latexit sha1_base64="q6QWtrWFBtRv/SZmvw1blRY1wT8=">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</latexit> ((m, p), (m0, p0)) = h'((m, p)), '((m0, p0))i H <latexit sha1_base64="8ARjBgQhZw/muUDNQthDy8PlmtE=">AAAC+nicjVHLSsNAFD3GV31XXboJFmkKpaRi1Y1QdONGqGBVUCmTOGpoHkMyEUT9EnfuxK0/4FY/QPwD/QvvjCP4QHRCknPPvefM3LmeCINMuu5zj9Xb1z8wWBgaHhkdG58oTk5tZ0me+rztJ2GS7nos42EQ87YMZMh3RcpZ5IV8x+uuqfzOKU+zIIm35JngBxE7joOjwGeSqE6xsd9lQjDHiaqiUnWiclWUKxV7xX7nOxuUiMoVE7UcofOdYsmtuXrZP0HdgBLMaiXFJ+zjEAl85IjAEUMSDsGQ0bOHOlwI4g5wTlxKKNB5jksMkzanKk4VjNgufY8p2jNsTLHyzLTap11CelNS2pgjTUJ1KWG1m63zuXZW7G/e59pTne2M/p7xioiVOCH2L91H5X91qheJIyzrHgLqSWhGdecbl1zfijq5/akrSQ6COIUPKZ8S9rXy455trcl07+pumc6/6ErFqtg3tTle1SlpwPXv4/wJtudr9cVaY3Oh1Fw1oy5gBrNwaJ5LaGIdLbTJ+wr3eMCjdWFdWzfW7Xup1WM00/iyrLs3nNGhBQ==</latexit> ((m, p), (m0, p0)) = M (m, m0)P (p, p0)) <latexit sha1_base64="OnhZh16IirUcgMMJYVUMPOeLWaE=">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</latexit> w
34. From features to kernel Optimisation problem in feature space Hinge Loss Optimisation problem in Kernel space The solution can be shown to be of the form <latexit sha1_base64="5UtP0Vp5ugbWKXurv89sXuHRTts=">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</latexit> K = XXT <latexit sha1_base64="5EtUweOQMLtDZdJ1M1lRsrxmMKE=">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</latexit> min w 1 n n X i=1 `( yi(hxi, wi)) + 2 kwk2 <latexit sha1_base64="Mipgt3qeYp8picR9l0ok1eeIk58=">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</latexit> ` <latexit sha1_base64="6pf13LL0UdhhS3zNa9wGSSp+YK4=">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</latexit> `(s) = max(0, 1 + s) with <latexit sha1_base64="MQVh7cNgGZbcnKXyeZ9ww/1s5Ow=">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</latexit> min z2RN L( diag(y)Kz) + 2 hKz, zi <latexit sha1_base64="OnhZh16IirUcgMMJYVUMPOeLWaE=">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</latexit> w
35. Drugbank vs CC 19 Drugbank: sklearn.svm.SVC (kernel=‘precomputed’) 29 k 29 k 2513 2513 Protein kernel 4813 4813 Molecule kernel Kronecker kernel for training <latexit sha1_base64="JhUk4wXzwRCmCXl9jAE1rQYQvmU=">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</latexit> K <latexit sha1_base64="ami8dpBoKqyKqi/626Rpw+6DOLE=">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</latexit> KP <latexit sha1_base64="gxgwsrlJXtjgOKDHetTN95gLgl8=">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</latexit> K((mi, pi), (mj, pj)) = KM (mi, mj) ⇥ KP (pi, pj)
36. Drugbank vs CC 19 Drugbank: sklearn.svm.SVC (kernel=‘precomputed’) 29 k 29 k 2513 2513 Protein kernel 4813 4813 Molecule kernel Kronecker kernel for training <latexit sha1_base64="JhUk4wXzwRCmCXl9jAE1rQYQvmU=">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</latexit> K <latexit sha1_base64="ami8dpBoKqyKqi/626Rpw+6DOLE=">AAACx3icjVHLSsNAFD2Nr1pfVZdugkVwVRLxtSy6UdxUsA+opSTptA3Ni8mkWIoLf8Ct/pn4B/oX3hmnoBbRCUnOnHvPmbn3ukngp8KyXnPG3PzC4lJ+ubCyura+Udzcqqdxxj1W8+Ig5k3XSVngR6wmfBGwZsKZE7oBa7jDcxlvjBhP/Ti6EeOEtUOnH/k933OEpK461UKnWLLKllrmLLA1KEGvalx8wS26iOEhQwiGCIJwAAcpPS3YsJAQ18aEOE7IV3GGexRIm1EWowyH2CF9+7RraTaivfRMldqjUwJ6OSlN7JEmpjxOWJ5mqnimnCX7m/dEecq7jenvaq+QWIEBsX/pppn/1claBHo4VTX4VFOiGFmdp10y1RV5c/NLVYIcEuIk7lKcE/aUctpnU2lSVbvsraPibypTsnLv6dwM7/KWNGD75zhnQf2gbB+Xj64PS5UzPeo8drCLfZrnCSq4QBU18h7gEU94Ni6N2BgZd5+pRk5rtvFtGQ8fLi+QMQ==</latexit> KP CC: 152 k 152 k 100 Gb Molecule kernel Issues: - Big data - Time - sklearn impractical 460 k 460 k Kronecker kernel for training <latexit sha1_base64="JhUk4wXzwRCmCXl9jAE1rQYQvmU=">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</latexit> K <latexit sha1_base64="ami8dpBoKqyKqi/626Rpw+6DOLE=">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</latexit> KP 1627 1627 Protein kernel <latexit sha1_base64="Ux6WcUgQ/myjZYS7Q1AJsdHo4vM=">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</latexit> KM <latexit sha1_base64="gxgwsrlJXtjgOKDHetTN95gLgl8=">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</latexit> K((mi, pi), (mj, pj)) = KM (mi, mj) ⇥ KP (pi, pj)
37. From kernel back to features Protein kernel Factorisation of the empirical protein kernel Choleski decomposition : Singular value decomposition (SVD) : 1627 <latexit sha1_base64="UvCjxnWXbDY2a8Km+grK5fRnbAI=">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</latexit> XP = Udiag( p ) <latexit sha1_base64="PI5cHDD6zA2xt33fLSwDtLZRCiA=">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</latexit> XP <latexit sha1_base64="RuNc87O1jkuo7rrKQB2glgfO6WM=">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</latexit> ˜ XP Empirical protein kernel Empirical features in dimension : Approximation <latexit sha1_base64="sHnEc8Q0cKstTN2ChBV9z/6uvmg=">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</latexit> dP <latexit sha1_base64="9uQnW/fT/RtwQqOp/EUtYDS4QJY=">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</latexit> dP = nP <latexit sha1_base64="9uQnW/fT/RtwQqOp/EUtYDS4QJY=">AAACzXicjVHLSsNAFD2Nr1pfVZdugkVwVVLxtRGKbtwZwT6wlpKk0zo0L5KJUKpu/QG3+lviH+hfeGecglpEJyQ5c+49Z+be68Y+T4VlveaMqemZ2bn8fGFhcWl5pbi6Vk+jLPFYzYv8KGm6Tsp8HrKa4MJnzThhTuD6rOEOTmS8ccOSlEfhhRjGrB04/ZD3uOcIoi67Hds8MsOOXegUS1bZUsucBBUNStDLjoovuEIXETxkCMAQQhD24SClp4UKLMTEtTEiLiHEVZzhDgXSZpTFKMMhdkDfPu1amg1pLz1TpfboFJ/ehJQmtkgTUV5CWJ5mqnimnCX7m/dIecq7Denvaq+AWIFrYv/SjTP/q5O1CPRwqGrgVFOsGFmdp10y1RV5c/NLVYIcYuIk7lI8Iewp5bjPptKkqnbZW0fF31SmZOXe07kZ3uUtacCVn+OcBPWdcmW/vHe+W6oe61HnsYFNbNM8D1DFKWzUyDvEI57wbJwZmXFr3H+mGjmtWce3ZTx8ACXikiA=</latexit> dP = nP <latexit sha1_base64="CyI3/gq/JbzcWvbD56H43bXoG/c=">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</latexit> dP << nP <latexit sha1_base64="p6QQVWXhksQg5WjEKyJX+0SgWzE=">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</latexit> <latexit sha1_base64="PaEq8MjefSJrw1BEAic8GlEKSpo=">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</latexit> ˜ XP = U[:, : r]diag( p [: r])
38. Nyström approximation 21 Empirical molecule kernel: computing, storage and SVD impossible =152k <latexit sha1_base64="/jKOyawgGIrWU0+wjjv6zEEcbfw=">AAACyXicjVHLSsNAFD2Nr1pfVcGNm6AIrkoq+NgIpW6EIrRgH1BLSdJpHZsmMZmItbjyB9zqP/gX/oO4ca1/4Z1pCmoRnZDkzLn3nJl7r+U7PBSG8ZrQJianpmeSs6m5+YXFpfTySiX0osBmZdtzvKBmmSFzuMvKgguH1fyAmT3LYVWreyTj1SsWhNxzT0XfZ42e2XF5m9umIKpSaJ7oh3ozvWlkDLX0cZCNwWZurfTGn/LPRS/9gjO04MFGhB4YXAjCDkyE9NSRhQGfuAYGxAWEuIoz3CJF2oiyGGWYxHbp26FdPWZd2kvPUKltOsWhNyClji3SeJQXEJan6SoeKWfJ/uY9UJ7ybn36W7FXj1iBc2L/0o0y/6uTtQi0caBq4FSTrxhZnR27RKor8ub6l6oEOfjESdyieEDYVspRn3WlCVXtsremir+rTMnKvR3nRviQt6QBZ3+OcxxUdjLZvcxuiSadx3AlsY4NbNM895HDMYook/cF7vGAR62gXWrX2s0wVUvEmlV8W9rdJ4NllFI=</latexit> KM =
39. Nyström approximation 21 Empirical molecule kernel: computing, storage and SVD impossible =152k <latexit sha1_base64="/jKOyawgGIrWU0+wjjv6zEEcbfw=">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</latexit> KM = <latexit sha1_base64="2aWE8gcf1t+n5Yu9Mld9fHhm7xw=">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</latexit> C 1 M <latexit sha1_base64="oy1qLaYHYHEVZxDJSloiMHZRk7g=">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</latexit> Z> <latexit sha1_base64="9s5B754QpkqTfiHbbeIhfu1A8P4=">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</latexit> Z <latexit sha1_base64="pxCQRq2zeg+yY6O2tp0mPARQCJk=">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</latexit> CM <latexit sha1_base64="xFnqVH93OHwwcQZCasLSlO47rBI=">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</latexit> ⇡ <latexit sha1_base64="9s5B754QpkqTfiHbbeIhfu1A8P4=">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</latexit> Z <latexit sha1_base64="oy1qLaYHYHEVZxDJSloiMHZRk7g=">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</latexit> Z> Nyström approximation <latexit sha1_base64="qUbluARsbWKNRtFd+wji/TxcnTc=">AAACyXicjVHLSsNAFD2Nr1pfVZdugkVwVRLxtSy6EdxUsA9oiyTTaR2bl8lErMWVP+BWf0z8A/0L74wpqEV0QpIz595zZu69buSJRFrWa86Ymp6ZncvPFxYWl5ZXiqtr9SRMY8ZrLPTCuOk6CfdEwGtSSI83o5g7vuvxhjs4VvHGDY8TEQbnchjxju/0A9ETzJFE1dtS+Dy5KJassqWXOQnsDJSQrWpYfEEbXYRgSOGDI4Ak7MFBQk8LNixExHUwIi4mJHSc4x4F0qaUxSnDIXZA3z7tWhkb0F55JlrN6BSP3piUJrZIE1JeTFidZup4qp0V+5v3SHuquw3p72ZePrESl8T+pRtn/lenapHo4VDXIKimSDOqOpa5pLor6ubml6okOUTEKdyleEyYaeW4z6bWJLp21VtHx990pmLVnmW5Kd7VLWnA9s9xToL6TtneL++d7ZYqR9mo89jAJrZpngeo4ARV1Mj7Co94wrNxalwbt8bdZ6qRyzTr+LaMhw//0ZG/</latexit> ⇥ <latexit sha1_base64="qUbluARsbWKNRtFd+wji/TxcnTc=">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</latexit> ⇥
40. Nyström approximation 21 Empirical molecule kernel: computing, storage and SVD impossible =152k <latexit sha1_base64="/jKOyawgGIrWU0+wjjv6zEEcbfw=">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</latexit> KM = <latexit sha1_base64="2aWE8gcf1t+n5Yu9Mld9fHhm7xw=">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</latexit> C 1 M <latexit sha1_base64="oy1qLaYHYHEVZxDJSloiMHZRk7g=">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</latexit> Z> <latexit sha1_base64="9s5B754QpkqTfiHbbeIhfu1A8P4=">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</latexit> Z <latexit sha1_base64="pxCQRq2zeg+yY6O2tp0mPARQCJk=">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</latexit> CM <latexit sha1_base64="xFnqVH93OHwwcQZCasLSlO47rBI=">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</latexit> ⇡ <latexit sha1_base64="9s5B754QpkqTfiHbbeIhfu1A8P4=">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</latexit> Z <latexit sha1_base64="oy1qLaYHYHEVZxDJSloiMHZRk7g=">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</latexit> Z> Nyström approximation <latexit sha1_base64="qUbluARsbWKNRtFd+wji/TxcnTc=">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</latexit> ⇥ <latexit sha1_base64="qUbluARsbWKNRtFd+wji/TxcnTc=">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</latexit> ⇥ Singular value decomposition (SVD) <latexit sha1_base64="/DREvQiqqMOAfLRTI95g2P5V/2U=">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</latexit> XM ⇡ ˜ XM where ˜ XM 2 RnM ⇥rM <latexit sha1_base64="p6QQVWXhksQg5WjEKyJX+0SgWzE=">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</latexit> <latexit sha1_base64="wlQqN/dnLCLVaDWz4G0wQMTCMug=">AAADAHicjVHLSuRAFD3Gd3y1ztJNYSO4atLiC9yIbtwIjtjaYLRJ0qVddF5UKj4IvfFP3LmT2c4PzHZciX/g/MXcKiP4QLRCklPn3nOq7r1+GopMOc5jn9U/MDg0PDJqj41PTE5VpmcOsiSXAW8ESZjIpu9lPBQxbyihQt5MJfciP+SHfndLxw/PucxEEu+rq5QfR95ZLE5F4CmiWpV1V/FLVVx0uOSs5yoRtnnRbO30mCti5kae6vh+sdc7KeLWDqN4xDPWprht261K1ak5ZrGPoF6CKsq1m1Qe4KKNBAFyROCIoQiH8JDRc4Q6HKTEHaMgThISJs7Rg03anLI4ZXjEdul7Rrujko1prz0zow7olJBeSUqGedIklCcJ69OYiefGWbOfeRfGU9/tiv5+6RURq9Ah9ivdS+Z3dboWhVOsmRoE1ZQaRlcXlC656Yq+OXtVlSKHlDiN2xSXhAOjfOkzM5rM1K5765n4k8nUrN4HZW6Of/qWNOD6+3F+BAeLtfpKbfnnUnVjsxz1CGYxhwWa5yo2sI1dNMj7Bn/wF/fWtXVr3Vm/nlOtvlLzA2+W9fs/gz6nJQ==</latexit> where ˜ XM 2 RnM ⇥dM <latexit sha1_base64="rX074yA6SbMo0biw2e29sIegcnA=">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</latexit> KM ⇡ XM X> M where <latexit sha1_base64="09suPseAwaTYU9j9m5Z8Qt7bwJE=">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</latexit> XM = Z Udiag(1/ p ) <latexit sha1_base64="sxJ6BP26liNUw96MaJf90Qqfn9w=">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</latexit> CM = Udiag( )U>
41. Fast Kronecker feature map 22 Pair features <latexit sha1_base64="N/jWUraLGjzi7eD0iNLDrJ+eAOc=">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</latexit> mi 152k Molecule features <latexit sha1_base64="wlQqN/dnLCLVaDWz4G0wQMTCMug=">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</latexit> where ˜ XM 2 RnM ⇥dM <latexit sha1_base64="O3Uf4vnd+aqzIufIv9BDDQSn1eg=">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</latexit> pi 1627 Protein features <latexit sha1_base64="cr9PrbFrLZbhZvMjskS2Od5QDqI=">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</latexit> ˜ XP 2 RnP ⇥dP
42. Fast Kronecker feature map 22 Pair features <latexit sha1_base64="N/jWUraLGjzi7eD0iNLDrJ+eAOc=">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</latexit> mi 152k Molecule features <latexit sha1_base64="wlQqN/dnLCLVaDWz4G0wQMTCMug=">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</latexit> where ˜ XM 2 RnM ⇥dM <latexit sha1_base64="O3Uf4vnd+aqzIufIv9BDDQSn1eg=">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</latexit> pi 1627 Protein features <latexit sha1_base64="cr9PrbFrLZbhZvMjskS2Od5QDqI=">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</latexit> ˜ XP 2 RnP ⇥dP
43. Fast Kronecker feature map 22 Tensor product Pair features <latexit sha1_base64="N/jWUraLGjzi7eD0iNLDrJ+eAOc=">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</latexit> mi 152k Molecule features <latexit sha1_base64="wlQqN/dnLCLVaDWz4G0wQMTCMug=">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</latexit> where ˜ XM 2 RnM ⇥dM <latexit sha1_base64="O3Uf4vnd+aqzIufIv9BDDQSn1eg=">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</latexit> pi 1627 Protein features <latexit sha1_base64="cr9PrbFrLZbhZvMjskS2Od5QDqI=">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</latexit> ˜ XP 2 RnP ⇥dP
44. Fast Kronecker feature map 22 Tensor product Pair features <latexit sha1_base64="N/jWUraLGjzi7eD0iNLDrJ+eAOc=">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</latexit> mi 152k Molecule features <latexit sha1_base64="wlQqN/dnLCLVaDWz4G0wQMTCMug=">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</latexit> where ˜ XM 2 RnM ⇥dM <latexit sha1_base64="O3Uf4vnd+aqzIufIv9BDDQSn1eg=">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</latexit> pi 1627 Protein features <latexit sha1_base64="cr9PrbFrLZbhZvMjskS2Od5QDqI=">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</latexit> ˜ XP 2 RnP ⇥dP <latexit sha1_base64="gJpHNGWX/unbZfOtW5K9inCWHZo=">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</latexit> dM <latexit sha1_base64="AIy9pngVdFN19ZDra1bb3fvvkt8=">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</latexit> dP <latexit sha1_base64="p1fsGV7BwU+Jsq+eJvCgCJqIEQo=">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</latexit> (mi, pi) <latexit sha1_base64="SoWanmzFvO02T8zjXv5MX2U+EGM=">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</latexit> N = 460k <latexit sha1_base64="3ALvQsFnUBc/gcMp0/a1E9EqME4=">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</latexit> dP ⇥ dM <latexit sha1_base64="CxFrKJQ1G0ZwAm4pomVX1fHd0JM=">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</latexit> ˜ X 2 RN⇥(dP ⇥dM )
45. Fast Kronecker feature map 22 Tensor product Pair features <latexit sha1_base64="N/jWUraLGjzi7eD0iNLDrJ+eAOc=">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</latexit> mi 152k Molecule features <latexit sha1_base64="wlQqN/dnLCLVaDWz4G0wQMTCMug=">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</latexit> where ˜ XM 2 RnM ⇥dM <latexit sha1_base64="O3Uf4vnd+aqzIufIv9BDDQSn1eg=">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</latexit> pi 1627 Protein features <latexit sha1_base64="cr9PrbFrLZbhZvMjskS2Od5QDqI=">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</latexit> ˜ XP 2 RnP ⇥dP <latexit sha1_base64="gJpHNGWX/unbZfOtW5K9inCWHZo=">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</latexit> dM <latexit sha1_base64="AIy9pngVdFN19ZDra1bb3fvvkt8=">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</latexit> dP <latexit sha1_base64="p1fsGV7BwU+Jsq+eJvCgCJqIEQo=">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</latexit> (mi, pi) <latexit sha1_base64="SoWanmzFvO02T8zjXv5MX2U+EGM=">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</latexit> N = 460k <latexit sha1_base64="3ALvQsFnUBc/gcMp0/a1E9EqME4=">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</latexit> dP ⇥ dM <latexit sha1_base64="CxFrKJQ1G0ZwAm4pomVX1fHd0JM=">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</latexit> ˜ X 2 RN⇥(dP ⇥dM ) Solving: <latexit sha1_base64="vLqJeyVlwp2MQqnrimW2L/+v2NM=">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</latexit> ˜ Xw Compute <latexit sha1_base64="zCMZvt1Helt/DSvKJ7JP32arcww=">AAADKHicjVFNb9QwEJ2Gj5YF2qUcuVhUSFuhrrIrQTmuyqUHkBbEtivVbeQ43q1V50OOw7Jyo/4S/gBH/gU3blWvnODUK9wYu6kEVAgcJXl+M+/ZMxMXSpYmDM8WgmvXb9xcXLrVun3n7vJK+97qTplXmosRz1WuxzErhZKZGBlplBgXWrA0VmI3Pnru4rtvhS5lnr0x80Lsp2yayYnkzCAVtTlNZRbZGaEyIzRl5jCO7ev6wCbRkFAjU1GSJHpZ1+RFZ4Ma8c7YRLJp3ZmvY1Qlwo5rMlsnjwmdaMYtVXh2wmrbrwk9ntHjg34raq+F3dAvchX0GrA2GJycv/+4sTzM26dAIYEcOFSQgoAMDGIFDEp89qAHIRTI7YNFTiOSPi6ghhZqK8wSmMGQPcLvFHd7DZvh3nmWXs3xFIWvRiWBR6jJMU8jdqcRH6+8s2P/5m29p7vbHP9x45Uia+AQ2X/pLjP/V+dqMTCBZ74GiTUVnnHV8cal8l1xNye/VGXQoUDO4QTjGjH3yss+E68pfe2ut8zHv/pMx7o9b3Ir+OZuiQPu/TnOq2Cn3+097T55hZPegou1BA/gIXRwnpswgG0Ywgi9P8E5fIcfwYfgc3AanF2kBguN5j78toIvPwHCLrqO</latexit> min w2RdP ⇥dM L( diag(y) ˜ Xw) + 2 kwk2
46. Fast Kronecker feature map 22 Tensor product Pair features <latexit sha1_base64="N/jWUraLGjzi7eD0iNLDrJ+eAOc=">AAACx3icjVHLSsNAFD2Nr1pfVZdugkVwVVLxtSy60V0F+wAtJUmn7dC8mEyKpbjwB9zqn4l/oH/hnXEKahGdkOTMufecmXuvlwQ8lY7zmrPm5hcWl/LLhZXVtfWN4uZWI40z4bO6HwexaHluygIesbrkMmCtRDA39ALW9IbnKt4cMZHyOLqW44S1Q7cf8R73XamosMMLnWLJKTt62bOgYkAJZtXi4gtu0UUMHxlCMESQhAO4SOm5QQUOEuLamBAnCHEdZ7hHgbQZZTHKcIkd0rdPuxvDRrRXnqlW+3RKQK8gpY090sSUJwir02wdz7SzYn/znmhPdbcx/T3jFRIrMSD2L9008786VYtED6e6Bk41JZpR1fnGJdNdUTe3v1QlySEhTuEuxQVhXyunfba1JtW1q966Ov6mMxWr9r7JzfCubkkDrvwc5yxoHJQrx+Wjq8NS9cyMOo8d7GKf5nmCKi5QQ528B3jEE56tSyu2RtbdZ6qVM5ptfFvWwwe6zpBs</latexit> mi 152k Molecule features <latexit sha1_base64="wlQqN/dnLCLVaDWz4G0wQMTCMug=">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</latexit> where ˜ XM 2 RnM ⇥dM <latexit sha1_base64="O3Uf4vnd+aqzIufIv9BDDQSn1eg=">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</latexit> pi 1627 Protein features <latexit sha1_base64="cr9PrbFrLZbhZvMjskS2Od5QDqI=">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</latexit> ˜ XP 2 RnP ⇥dP <latexit sha1_base64="gJpHNGWX/unbZfOtW5K9inCWHZo=">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</latexit> dM <latexit sha1_base64="AIy9pngVdFN19ZDra1bb3fvvkt8=">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</latexit> dP <latexit sha1_base64="p1fsGV7BwU+Jsq+eJvCgCJqIEQo=">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</latexit> (mi, pi) <latexit sha1_base64="SoWanmzFvO02T8zjXv5MX2U+EGM=">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</latexit> N = 460k <latexit sha1_base64="3ALvQsFnUBc/gcMp0/a1E9EqME4=">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</latexit> dP ⇥ dM <latexit sha1_base64="CxFrKJQ1G0ZwAm4pomVX1fHd0JM=">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</latexit> ˜ X 2 RN⇥(dP ⇥dM ) Solving: <latexit sha1_base64="vLqJeyVlwp2MQqnrimW2L/+v2NM=">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</latexit> ˜ Xw Compute <latexit sha1_base64="zCMZvt1Helt/DSvKJ7JP32arcww=">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</latexit> min w2RdP ⇥dM L( diag(y) ˜ Xw) + 2 kwk2 <latexit sha1_base64="F7mqqx/DEERYCXY3OdqD8uJBfn8=">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</latexit> ˜ XP Implicit Time: Space: computation computation Explicit and <latexit sha1_base64="gxwnlu3SjkMfdpf44lgDTDo+5uk=">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</latexit> ˜ XM <latexit sha1_base64="cu3X3368bZOnEZtmpc5niiMd/JM=">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</latexit> N ⇥ dM <latexit sha1_base64="xK3UK5JksgHBpV+cSy8GY9x1UHo=">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</latexit> nP ⇥ dP + nM ⇥ dM <latexit sha1_base64="l+AUOZ8HKWX1Jx1QvP+UFto6UdM=">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</latexit> ˜ X <latexit sha1_base64="0lD8nje0x6Mx2P0FV1mERUaJ+EI=">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</latexit> ! and using using <latexit sha1_base64="YLZJWNIWUFPTrQjYtuH6LQhDaPo=">AAAC4XicjVHLSsNAFD2NrxpfVZe6CBahbkoqvpZFN26UCvYBKiWZjjo0L5KJUIobd+7ErT/gVn9G/AP9C++MKahFdEKSM+fec2buvW7kiUTa9mvOGBkdG5/IT5pT0zOzc4X5hUYSpjHjdRZ6YdxynYR7IuB1KaTHW1HMHd/1eNPt7ql484rHiQiDY9mL+JnvXATiXDBHEtUuLB9ap1L4PLFKnXZtgDvtgzXTNNuFol229bKGQSUDRWSrFhZecIoOQjCk8MERQBL24CCh5wQV2IiIO0OfuJiQ0HGOa5ikTSmLU4ZDbJe+F7Q7ydiA9soz0WpGp3j0xqS0sEqakPJiwuo0S8dT7azY37z72lPdrUd/N/PyiZW4JPYv3SDzvzpVi8Q5dnQNgmqKNKOqY5lLqruibm59qUqSQ0Scwh2Kx4SZVg76bGlNomtXvXV0/E1nKlbtWZab4l3dkgZc+TnOYdBYL1e2yptHG8XqbjbqPJawghLNcxtV7KOGOnnf4BFPeDaYcWvcGfefqUYu0yzi2zIePgA7sJiq</latexit> N ⇥ (dP ⇥ dM ) <latexit sha1_base64="YLZJWNIWUFPTrQjYtuH6LQhDaPo=">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</latexit> N ⇥ (dP ⇥ dM )
47. Results 23 Distribution of True labels in function of the predicted score Cross validation with CC
48. Results 23 Distribution of True labels in function of the predicted score Cross validation with CC Results on Drugbank [Playe&al, 2018] Performance : bias ?
49. Results 23 Distribution of True labels in function of the predicted score Cross validation with CC Results on Drugbank [Playe&al, 2018] Performance : bias ? Number of CC molecules which are similar Back to the 20 molecules How hard is the prediction ? Similarity
50. Conclusion 24 Initial problem: understanding biological mechanisms associated to a set of diverse proteins Chemogenomics: enlarge and consolidate this set of proteins Contributions: A large new molecule/protein interactions dataset Fast large scale kernel method Perspectives: Analysis of the target proteins predicted for the 20 di ff erentially active molecules Use transcriptomic datas to fi nd transcription factors and pathways activities
51. Perspective New sub-type of patients : ATIP3 de f i
52. Transcriptomics strategy 26 6 samples with 2 conditions : Counts matrix (Cj,k ) ∈ N^(J×K) where K = 6 and J = 25472. From a certain gene j, Cj,k is proportional to the abundance of its mRNA in the cell. Have more robust data in public data base of TNBC cell lines Experimental dataset
53. Transcriptomics strategy 26 6 samples with 2 conditions : Counts matrix (Cj,k ) ∈ N^(J×K) where K = 6 and J = 25472. From a certain gene j, Cj,k is proportional to the abundance of its mRNA in the cell. To do : Make a network of the proteins involved these pathways Find the di ff erentially active pathways (GSEA) Search in network the proteins known and predicted (A) Have more robust data in public data base of TNBC cell lines Experimental dataset