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1 Powere by TCPDF ( This is an eletroni reprint of the original artile. This reprint may iffer from the original in pagination an typographi etail. Cihonska, Anna; Pahikkala, Tapio; Szemak, Sanor; Julkunen, Heli; Airola, Antti; Heinonen, Markus; Aittokallio, Tero; Rousu, Juho Learning with multiple pairwise kernels for rug bioativity preition Publishe in: Bioinformatis DOI: /bioinformatis/bty277 Publishe: 01/07/2018 Doument Version Publisher's PDF, also known as Version of reor Please ite the original version: Cihonska, A., Pahikkala, T., Szemak, S., Julkunen, H., Airola, A., Heinonen, M.,... Rousu, J. (2018). Learning with multiple pairwise kernels for rug bioativity preition. Bioinformatis, 34(13), i509-i518. DOI: /bioinformatis/bty277 This material is protete by opyright an other intelletual property rights, an upliation or sale of all or part of any of the repository olletions is not permitte, exept that material may be upliate by you for your researh use or euational purposes in eletroni or print form. You must obtain permission for any other use. Eletroni or print opies may not be offere, whether for sale or otherwise to anyone who is not an authorise user.

2 Bioinformatis, 34, 2018, i509 i518 oi: /bioinformatis/bty277 ISMB 2018 Learning with multiple pairwise kernels for rug bioativity preition Anna Cihonska 1,2, *, Tapio Pahikkala 3, Sanor Szemak 1, Heli Julkunen 1, Antti Airola 3, Markus Heinonen 1, Tero Aittokallio 1,2,4 an Juho Rousu 1 1 Department of Computer Siene, Helsinki Institute for Information Tehnology HIIT, Aalto University, Espoo, Finlan, 2 Institute for Moleular Meiine Finlan FIMM, University of Helsinki, Helsinki, Finlan, 3 Department of Information Tehnology an 4 Department of Mathematis an Statistis, University of Turku, Turku, Finlan *To whom orresponene shoul be aresse. Abstrat Motivation: Many inferene problems in bioinformatis, inluing rug bioativity preition, an be formulate as pairwise learning problems, in whih one is intereste in making preitions for pairs of objets, e.g. rugs an their targets. Kernel-base approahes have emerge as powerful tools for solving problems of that kin, an espeially multiple kernel learning (MKL) offers promising benefits as it enables integrating various types of omplex biomeial information soures in the form of kernels, along with learning their importane for the preition task. However, the immense size of pairwise kernel spaes remains a major bottlenek, making the existing MKL algorithms omputationally infeasible even for small number of input pairs. Results: We introue pairwisemkl, the first metho for time- an memory-effiient learning with multiple pairwise kernels. pairwisemkl first etermines the mixture weights of the input pairwise kernels, an then learns the pairwise preition funtion. Both steps are performe effiiently without expliit omputation of the massive pairwise matries, therefore making the metho appliable to solving large pairwise learning problems. We emonstrate the performane of pairwisemkl in two relate tasks of quantitative rug bioativity preition using up to bioativity measurements an 3120 pairwise kernels: (i) preition of antianer effiay of rug ompouns aross a large panel of aner ell lines; an (ii) preition of target profiles of antianer ompouns aross their kinome-wie target spaes. We show that pairwisemkl provies aurate preitions using sparse solutions in terms of selete kernels, an therefore it automatially ientifies also ata soures relevant for the preition problem. Availability an implementation: Coe is available at Contat: anna.ihonska@aalto.fi Supplementary information: Supplementary ata are available at Bioinformatis online. 1 Introution In the reent years, several high-throughput antianer rug sreening efforts have been onute (Barretina et al., 2012; Smirnov et al., 2018; Yang et al., 2012), proviing bioativity measurements that allow for the ientifiation of ompouns that show inrease effiay in speifi human aner types or iniviual ell lines, therefore guiing both the preision meiine efforts as well as rug repurposing appliations. However, hemial ompouns exeute their ation through moulating typially multiple moleules, with proteins being the most ommon moleular targets, an ultimately both the effiay an toxiity of the treatment are a onsequene of those omplex moleular interations. Hene, eluiating rug s moe of ation (MoA), inluing both on- an off-targets, is ritial for the evelopment of effetive an safe therapies. The inrease availability of rug bioativity ata for ell lines (Smirnov et al., 2018) an protein targets (Merget et al., 2017), together with the omprehensive harateristis of rug ompouns, proteins an ell lines, has enable onstrution of supervise mahine learning moels, whih offer ost-effetive means for fast, systemati an large-sale pre-sreening of hemial ompouns an their potential targets for further experimental verifiation, with the aim of aelerating an e-risking the rug isovery proess VC The Author(s) Publishe by Oxfor University Press. i509 This is an Open Aess artile istribute uner the terms of the Creative Commons Attribution Non-Commerial Liense ( whih permits non-ommerial re-use, istribution, an reproution in any meium, provie the original work is properly ite. For ommerial re-use, please ontat journals.permissions@oup.om Downloae from

3 i510 A.Cihonska et al. (Ali et al., 2017; Azuaje, 2017; Cheng et al., 2012; Cihonska et al., 2015). Uner the general framework of rug bioativity preition, two relate mahine learning tasks are ientifie: (i) preition of antianer rug responses an (ii) preition of rug protein interations, both of whih an be takle through similar mahine learning tehniques. In partiular, kernel-base approahes have emerge as powerful tools in omputational rug isovery (Cihonska et al., 2017; Marou et al., 2016; Pahikkala et al., 2015). Both the rug response in aner ell line preition an rug protein interation preition are representative examples of pairwise learning problems, where the goal is to buil preitive moel for pairs of objets. Classial kernel-base methos for pairwise learning rely merely on a single pairwise kernel. However, suh approahes are unlikely to be optimal in appliations where a growing variety of biologial an moleular ata soures are available, inluing hemial an protein strutures, pharmaophore patterns, gene expression signatures, methylation profiles as well as genomi variants foun in ell lines. In fat, the avantage of integrating ifferent ata types for the multi-level analysis has been highlighte in the reent stuies (Ebrahim et al., 2016; Elefsinioti et al., 2016). Multiple kernel learning (MKL) methos, whih searh for an optimal ombination of several kernels, hene enabling the use of ifferent information soures simultaneously an learning their importane for the preition task, have therefore reeive signifiant attention in bioinformatis (Brouar et al., 2016; Kluas et al., 2016; Shen et al., 2014), espeially in rug bioativity inferene (Amma-u-in et al., 2016; Costello et al.,2014; Nasimento et al.,2016). However, the existing MKL methos o not sale up to the massive size of pairwise kernels, in terms of both proessing an memory requirements, making the kernel weights optimization an moel training omputationally infeasible even for small numbers of input pairs, suh as rugs an ell lines or rugs an protein targets. The reently introue KronRLS-MKL algorithm for pairwise learning of rug protein interations interleaves the optimization of the pairwise preition funtion parameters with the kernel weights optimization (Nasimento et al., 2016). However, it fins two sets of kernel weights, separately for rug kernels an protein kernels instea of pairwise kernels, an therefore it oes not fully exploit the information ontaine in the pairwise spae. Here, we propose pairwisemkl, to our knowlege, the first metho for time- an memory-effiient learning with multiple pairwise kernels, implementing both effiient pairwise kernel weights optimization an pairwise moel training. In the first phase, the algorithm etermines a onvex ombination of input pairwise kernels by maximizing the entere alignment (i.e. matrix similarity measure) between the final ombine kernel an the ieal kernel erive from the label values (response kernel); in the seon phase, the pairwise preition funtion is learne. Both steps are performe without expliit onstrution of the massive pairwise matries (Fig. 1). We emonstrate the performane of pairwisemkl in two important subtasks of quantitative rug bioativity preition. In ase of rug response in aner ell line preition subtask, we use the bioativity ata from rug ell line pairs from the Genomis of Drug Sensitivity in Caner (GDSC) projet (Yang et al., 2012). We enoe similarities between the rug ompouns an ell lines using kernels onstrute base on various types of moleular fingerprints, gene expression profiles, methylation patterns, opy number ata an geneti variants, resulting in 120 pairwise kernels (10 rug kernels 12 ell line kernels). In the larger subtask of rug protein bining affinity preition, we use the reently publishe bioativities from rug protein pairs (Merget et al., 2017) an onstrute 3120 pairwise kernels (10 rug kernels 312 protein kernels) base on moleular fingerprints, protein sequenes an gene ontology annotations. We show that pairwisemkl is very well-suite for solving large pairwise learning problems, it outperforms KronRLS-MKL in terms of both memory requirements an preitive power, an, unlike KronRLS- MKL, it (i) allows for missing values in the label matrix an (ii) fins a sparse ombination of input pairwise kernels, thus enabling automati ientifiation of ata soures most relevant for the preition task. Moreover, sine pairwisemkl sales up to large number of pairwise kernels, tuning of the kernel hyperparameters an be easily inorporate into the kernel weights optimization proess. In summary, this artile makes the following ontributions. We implement a highly effiient entere kernel alignment proeure to avoi expliit omputation of multiple huge pairwise matries in the seletion of mixture weights of input pairwise kernels. To ahieve this, we propose a novel Kroneker eomposition of the entering operator for the pairwise kernel. We introue a Gaussian response kernel whih is more suitable for the kernel alignment in a regression setting than a stanar linear response kernel. We introue a metho for training a regularize least-squares moel with multiple pairwise kernels by exploiting the struture of the weighte sum of Kroneker prouts. We therefore avoi expliit onstrution of any massive pairwise matries also in the seon stage of learning pairwise preition funtion. We show how to effetively utilize the whole exome sequening ata to alulate informative real-value geneti mutation profile feature vetors for aner ell lines, instea of binary mutation status vetors ommonly use in rug response preition moels. pairwisemkl provies a general approah to MKL in pairwise spaes, an therefore it is wiely appliable also outsie the rug bioativity inferene problems. Our implementation is freely available. 2 Materials an methos This setion is organize as follows. First, Setion 2.1 explains a general approah to two-stage multiple pairwise kernel regression whih forms the basis for our pairwisemkl metho esribe in Setion 2.2. We emonstrate the performane of pairwisemkl in the two tasks of (i) antianer rug potential preition an (ii) rug protein bining affinity preition, but we selete the former as an example to explain the methoology. Finally, Setion 2.3 introues the ata an kernels we use in our experiments. 2.1 Multiple pairwise kernel regression In supervise pairwise learning of antianer rug potential, training ata appears in the form ðx ; x ; yþ,whereðx ; x Þ enotes a feature representation of a pair of input objets, rug x 2X D an aner ell line x 2X C (e.g. moleular fingerprint vetor an gene expression profile, respetively), an y 2 R is its assoiate response value (also alle the label), i.e. a measurement of sensitivity of ell line x to rug x.givenn n n training instanes, they an be represente as matries X 2 R n t ; X 2 R nt an label vetor y 2 R N,wheren enotes the number of rugs, n the number of ell lines, t an t the number of rug an ell line features, respetively. The aim is to fin a pairwise preition funtion f that moels the relationship between ðx ; X Þan y; f an later be use to preit sensitivity measurements for rug ell line pairs outsie the training spae. The assumption is that struturally similar rugs show similar effets in ell lines having ommon genomi bakgrouns. We apply kernels to Downloae from

4 pairwisemkl i511 Fig. 1. Shemati figure showing an overview of pairwisemkl metho for learning with multiple pairwise kernels, using the rug response in aner ell line preition as an example. First, two rug kernels an three ell line kernels are alulate from available hemial an genomi ata soures, respetively. The resulting matries assoiate all rugs an all ell lines, an therefore a kernel an be onsiere as a similarity measure. Sine we are intereste in learning bioativities of pairs of input objets, here rug ell line pairs, pairwise kernels relating all rug ell line pairs are neee, an they are alulate as Kroneker prouts () of rug kernels an ell line kernels (2 rug kernels 3 ell line kernels ¼ 6 pairwise kernels). In the first learning stage, pairwise kernel mixture weights are etermine (Setion 2.2.1), an then a weighte ombination of pairwise kernels is use for antianer rug response preition with a regularize leastsquares pairwise regression moel (Setion 2.2.2). Importantly, pairwisemkl performs those two steps effiiently by avoiing expliit onstrution of any massive pairwise matries, an therefore it is very well-suite for solving large pairwise learning problems enoe the similarities between input objets, suh as rugs or ell lines. Kernelsoffertheavantage ofinreasing the power of lassial linear learning algorithms by proviing a omputationally effiient approah for projeting input objets into a new feature spae with very high or even infinite number of imensions. A linear moel in this impliit feature spae orrespons to a non-linear moel in the original spae (Shawe-Taylor an Cristianini, 2004). Formally, a kernel is a positive semiefinite (PSD) funtion that for all x ; x 0 2X D satisfies k x ; x 0 ¼h/ ð x Þ; / x 0 i,where/ enotes a mapping from the input spae X D to a high-imensional inner prout feature spae H D, i.e. / : x 2X D! / ðx Þ 2 H D (the same hols for ell line kernel k ). It is, however, possible to avoi expliit omputation of the mapping / an efine the kernel iretly in terms of the original input features, suh as gene expression profiles, by replaing the inner prout h; i with an appropriately hosen kernel funtion (so-alle kernel trik), e.g. the Gaussian kernel (Shawe-Taylor an Cristianini, 2004). Kernels an be easily employe for pairwise learning by onstruting a pairwise kernel matrix K 2 R NN relating all rug ell line pairs. Speifially, K is alulate as a Kroneker prout of rug kernel K 2 R n n (ompute from, e.g. rug fingerprints) an ell line kernel K 2 R nn (ompute from, e.g. gene expression), forming a blok matrix with all possible prouts of entries of K an K : K ¼ K K 0 k ðx 1 ; x 1 ÞK k ðx 1 ; x 2 ÞK k x 1 ; x n k ðx 2 ; x 1 ÞK k ðx 2 ; x 2 ÞK k x 2 ; x n ¼ k x n ; x 1 K k x n ; x 2 K k x n ; x n K K K 1 : C A (1) as Then, the preition funtion for a test pair (x ; x ) is expresse f ðx ; x Þ ¼ XN a l kððx l ; x l Þ; ðx ; x ÞÞ ¼ a T k; (2) l¼1 where k is a olumn vetor with kernel values between eah training rug ell line pair (x l ; x l ) an test pair (x ; x ) for whih the preition is mae, an a ¼ ða 1 ;...; a N Þ enotes a vetor of moel parameters to be obtaine by the learning algorithm through minimizing a ertain objetive funtion. In kernel rige regression (KRR, Sauners et al., 1998), the objetive funtion is efine in terms of total square loss along with L2-norm regularizer, an the solution for a is foun by solving the following system of linear equations: ðk þ kiþa ¼ y; (3) where k iniates a regularization hyperparameter ontrolling the balane between training error an moel omplexity ðk > 0Þ, an I is the N N ientity matrix. Due to the wie availability of ifferent hemial an genomi ata soures, both rugs an ell lines an be represente with multiple kernel matries K ð1þ ;...; K ð p Þ an K ð1þ ;...; K ðpþ, therefore forming P ¼ p p pairwise kernels K ð1þ ;...; K ðpþ (Kroneker prouts of all pairs of rug kernels an ell line kernels). The goal of two-stage multiple pairwise KRR is to first fin the ombination of P pairwise kernels K l ¼ XP i¼1 l i K ðþ i ; (4) an then use K l instea of K in Equation (3) to learn the pairwise preition funtion. Downloae from

5 i512 A.Cihonska et al Centere kernel alignment The observation that a similarity between the entere input kernel K ðþ i an a linear kernel erive from the labels K y ¼ yy T (response kernel) orrelates with the performane of K ðþ i in a given preition task, has inspire a esign of the entere kernel alignment-base MKL approah (Cortes et al., 2012). Both K ðþ i an K y measure similarities between rug ell line pairs. However, K y an be onsiere as a groun-truth as it is alulate from the bioativities whih we aim to preit, an hene the ieal input kernel woul apture the same information about the similarities of rug ell line pairs as the response kernel K y. The iea is to first learn a linear mixture of entere input kernels that is maximally aligne to the response kernel, an then use the learne mixture kernel as the input kernel for learning a preition funtion. Centering a kernel K orrespons to entering its assoiate feature mapping /, an it is performe by K b ¼ CKC, where C 2 R NN h i is an iempotent (C ¼ CC) entering operator of the form I 11T N, I iniates N N ientity matrix, an 1 is a vetor of N omponents, all equal to 1 (Cortes et al., 2012). Centere kernel alignment measures the similarity between two kernels K b an K b 0 : A K; b K b 0 hk; ¼ b K b0 i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F hk; b Ki b F hk b ¼ hb K; K b0 i F 0 ; K0 b i jjkjj b F F jjk b : (5) 0 jj F Above, h; i F enotes a Frobenius inner prout, jj jj F is a Frobenius norm an A an be viewe as the osine of the angle, orrelation, efine between two matries. Kernel mixture weights l ¼ ðl 1 ;...; l P Þ are etermine by maximizing the entere alignment between the final ombine kernel K l an the response kernel K y (Cortes et al., 2012): max l A K b l ; K b y hk ¼ max b l ; K y i F l jjk b ; l jj F subjet to : jjljj 2 ¼ 1; l 0: In (6), jjk b y jj F is omitte beause it oes not epen on l, an hk b l ; K b y i F ¼hK b l ; K y i F through the properties of entering. The optimization problem (6) an be solve via: (6) min v0 vt Mv 2v T a; (7) with the vetor a an the symmetri matrix M efine by: ðþ a i ¼hK b ðþ i ; K y i F ; i ¼ 1;...; P; (8) ðmþ ij ¼hK b ðþ i ; K b ðþ j i F ; i; j ¼ 1;...; P: (9) Optimal kernel weights are given by l ¼ v =jjv jj, where v is the solution to (7). Then, the ombine kernel K l is alulate with Equation (4) an use to train a kernel-base preition algorithm (Cortes et al., 2012). Suh entere kernel alignment-base strategy has prove to have a goo preitive performane (Brouar et al., 2016; Cortes et al., 2012; Kluas et al., 2016; Shen et al., 2014), but it is not appliable to most of the pairwise learning problems beause the size of pairwise kernel matries, K ð1þ ;...; K ðpþ, grows very quikly with the number of rugs an ell lines (Supplementary Fig. S1) making the mixture weights optimization proeure omputationally intratable even for small number of inputs (Table 1). For instane, given 10 kernels for 100 rugs an 10 kernels for 100 ell lines (P ¼ ¼ 100, N ¼ ¼ assuming Table 1. Memory an time neee for a naïve MKL approah expliitly omputing pairwise kernels (Setion 2.1) an pairwisemkl (Setion 2.2), epening on the number of rugs an ell lines use in the rug bioativity preition experiment Number Number of rugs of ell lines Naïve approah Memory (GB) that bioativities of all ombinations of rugs an ell lines are known), the omputation of the matrix M requires ðpþ1þp 2 ¼ evaluations of Frobenius prouts between pairwise kernels ompose of 100 million entries eah, an aitional 100 evaluations to alulate vetor a (the number of evaluations inreases when applying a ross valiation). Given 200 rugs an 200 ell lines, the size of a single pairwise kernel grows to 1.6 billion entries taking roughly 12 GB memory. For omparison, in ase of a more stanar learning problem, suh as rug response preition in a single aner ell line using rug features only, there woul be 200 rugs as inputs instea of rug ell line pairs, an the resulting kernel matrix woul be ompose of elements taking 0.32 MB memory. 2.2 pairwisemkl Stage 1: optimization of pairwise kernel weights In this work, we evise an effiient proeure for optimizing kernel weights in pairwise learning setting. Speifially, we exploit the known ientity hk K ; K 0 K0 i¼hk ; K 0 ihk ; K 0 i (10) to avoi expliit omputation of the massive Kroneker prout matries in Equations (8) an (9). The main iffiulty omes from the entering of the pairwise kernel; in partiular, the fat that one annot obtain a entere pairwise kernel b K simply by omputing the Kroneker prout of entere rug kernel b K an ell line kernel bk, i.e. b K 6¼ b K b K. In orer to aress this limitation, we introue here a new, highly effiient Kroneker eomposition of the entering operator for the pairwise kernel: C ¼ X2 Q ðqþ q¼1 Time (h) pairwisemkl Naïve approah pairwisemkl > a > b Note: A single roun of 10-fol CV was run using ifferent-size subsets of the ata on antianer rug responses (esribe in Setion 2.3.1) with 10 rug kernels an 12 ell line kernels. Regularization hyperparameter k was set to 0.1 in both methos. a Program i not omplete within 7 ays (168 h). b Program i not run given 256 GB of memory. Q ðqþ ; (11) ðq where Q Þ 2 R n n ðq an Q Þ 2 R nn are the fators of C. Exploiting the struture of C allows us to ompute the fators effiiently by solving the singular value problem for a matrix of size 2 2 only, regarless of how large N is (the etaile proeure is provie in Supplementary Material). Downloae from

6 pairwisemkl i513 Deomposition (11) allows us to greatly simplify the alulation of the matrix M an vetor a neee in the kernel mixture weights optimization by (7): ðmþ ij ¼hK b ðþ i ; K b ðþ j i F ¼ tr CK ðþ i CCK j ¼ X2 X 2 q¼1 r¼1 tr Q q ð Þ K ðþ i Q ðþ r K ðþ j ðþ C tr Q q ð Þ K ðþ i Q ðþ r K ðþ j ; (12) with trðþenoting a trae of a matrix (a full erivation is given in Supplementary Material). Hene, the inner prout in the massive pairwise spae (N N) is reue to a sum of inner prouts in the original muh smaller spaes of rugs (n n ) an ell lines (n n ). The omputation of the elements of a is simplifie to the inner prout between two vetors by first exploiting the blok struture of the Kroneker prout matrix through the ientity ða BÞveðDÞ ¼ ve BDA T : D E hk ðþ i ; K y i F ¼ K ðþ i K ðþ i ; yy T F D E ¼ y; K ðþ i K ðþ i y (13) D E ¼ y; ve K ðþ i ðþ ; an then aounting for the entering: ðþ a i ¼hK b ðþ i ; K y i F ¼hy; hi; h ¼ X2 X 2 q¼1 r¼1 ve Q ðqþ K ðþ i Q ðþ r YK i Y Q q ð Þ K ðþ i Q ðþ r ; (14) where Y 2 R nn is the label matrix (if N < n n, missing values in Y are impute with olumn (rug) averages to alulate a), an veðþis the vetorization operator whih arranges the olumns of a matrix into a vetor, veðyþ ¼ y. Gaussian response kernel. The stanar linear response kernel yy T use in Equations (13) an (14) is well-suite for measuring similarities betweenlabelsinlassifiationtasks,wherey 2f 1; þ1g, but not regression, where y 2 R. pairwisemkl therefore employs a Gaussian response kernel, a gol stanar for measuring similarities between real numbers (Shawe-Taylor an Cristianini, 2004). In partiular, we first represent eah label value y i ; i ¼ 1;...; N; with a feature vetor of length S whih is a histogram orresponing to a probability ensity funtion of all the labels y, entere at y i, an store as row vetor in the matrix W 2 R NS. Then, the Gaussian response kernel ompares the feature vetors of all pairs of labels by alulating a sum of S inner prouts: K y ¼ XS s¼1 w ðþ s w ðþt s ; (15) where w ðþ s 2 R N is a olumn vetor of W. By replaing the linear response kernel yy T in Equations (13) an (14) with the Gaussian response kernel efine in Equation (15), vetor a in regression setting is alulate as a sum of S inner prouts between two vetors: ðþ a i ¼hK b ðþ i ; K y i F ¼ XS w ¼ X2 X 2 q¼1 r¼1 ve s¼1 hw ðþ s ; wi; Q ðqþ K ðþ i Q ðþ r Z Q q ð Þ K ðþ i Q ðþ r ; (16) where Z 2 R nn, veðzþ ¼ w ðþ s [Z is analogous to Y in Equations (13) an (14)]. We use S ¼ 100 in our experiments. Taken together, pairwisemkl etermines pairwise kernel mixture weights l effiiently through (7) with the matrix M an vetor a onstrute by (12) an (16), respetively, without expliit alulation of massive pairwise matries Stage 2: pairwise moel training Given pairwise kernel weights l, Equation (3) of pairwise KRR has the following form: l 1 K ð1þ K ð1þ þ;...; þl P K ðpþ K ðpþ þ ki a ¼ y: (17) Sine the bioativities of all ombinations of rugs an ell lines might not be known, meaning that there might be missing values in the label matrix Y 2 R nn, veðyþ ¼ y, we further get Ua ¼ y; U ¼ B l 1 K ð1þ K ð1þ þ;...; þl P K ðpþ K ðpþ þ ki B T ; (18) where B is an inexing matrix enoting the orresponenes between the rows an olumns of the kernel matrix an the elements of the vetor a: B i ¼ 1 enotes that the oeffiient a i orrespons to the th row/olumn in the kernel matrix. Training the moel, i.e. fining the parameters a of the pairwise preition funtion, is equivalent to solving the above system of linear equations (18). We solve the system with the onjugate graient (CG) approah that iteratively improves the result by arrying out matrix-vetor prouts between U an a, whih in general requires a number of iterations proportional to the number of ata. However, in pratise one usually obtains as goo or even better preitive performane with only a few iterations. Restriting the number of iterations ats as an aitional regularization mehanism known in the literature as early stopping (Engl et al, 1996). We further aelerate the matrix-vetor prout Ua by taking avantage of the strutural properties of the matrix U. In(Airola an Pahikkala, 2017), we introue the generalize ve-trik algorithm that arries out matrix vetor multipliations betweena prinipal submatrix of a Kroneker prout of type B K ð1þ K ð1þ B T an a vetor a in ONn ð þ Nn Þtime, without expliit alulation of pairwise kernel matries. Here, we exten the algorithm to work with sums of multiple pairwise kernels, i.e. to solve the system of equations (18).Inpartiular, the matrix U is a sum of P submatries of type B K ð1þ K ð1þ B T,an hene eah iteration of CG is arrie out in OPNn ð þ PNn Þ time (see Supplementary Material for pseuooe an more etails). In summary, our pairwisemkl avois expliit omputation of any pairwise matries in both stages of fining pairwise kernel weights an pairwise moel training, whih makes the metho suitable for solving problems in large pairwise spaes, suh as in ase of rug bioativity preition (Table 1). 2.3 Dataset Drug bioativity ata Drug responses in aner ell lines. In orer to test our framework, we use antianer rug response ata from GDSC projet initiate by Wellome Trust Sanger Institute (release June 2014, Yang et al., 2012). Our ataset onsists of 124 rugs an 124 human aner ell lines, for whih omplete ¼ rug sensitivity measurements are available in the form of ln(ic 50 ) values in nanomolars (Amma-u-in et al., 2016). Downloae from

7 i514 A.Cihonska et al. Fig. 2. Pairwise kernel mixture weights obtaine with pairwisemkl an KronRLS-MKL (average aross 10 outer CV fols) in the task of (a) rug response in aner ell line preition an (b) rug protein bining affinity preition (note: KronRLS-MKL i not exeute with 1 TB memory); only the weights ifferent from 0 are shown. KronRLS-MKL fins separate weights for rug kernels an ell line (protein) kernels instea of pairwise kernels. Numbers at the en of kernel names iniate the kernel hyperparameter values, in partiular (i) kernel with hyperparameter in ase of Gaussian kernels (e.g. K-n-146 with r ¼ 146), an (ii) maximum sub-string length L, r 1 ontrolling for the shifting ontribution term an r 2 ontrolling for the amino ai similarity term in ase of GS kernels (e.g. Kp-GS-atp with L ¼ 5; r 1 ¼ r 2 ¼ 4, see Setion for etails). () Summary of rug, ell line an protein kernels use in this work for the two preition problems. Drug protein bining affinities. In the seon task of rug protein bining affinity preition, we use a omprehensive kinome-wie rug target interation map generate by Merget et al. (2017) from publily available ata soures, an further upate with the bioativities from version 22 of ChEMBL atabase by Sorgenfrei et al. (2017). Sine the original interation map is extremely sparse, we selete rugs with at least 1% of measure bioativity values aross the kinase panel, an also kinases with kinase omain an ATP bining poket amino ai sub-sequenes available in PROSITE (Sigrist et al., 2013), resulting in 2967 rugs, 226 protein kinases, an bining affinities between them in the form of log 10 (IC 50 ) values in molars. We ompute rug kernels, ell line kernels an protein kernels as esribe in the following setions an summarize in Figure Kernels Drug kernels. For rug ompouns, we ompute Tanimoto kernels using 10 ifferent moleular fingerprints (Fig. 2), i.e. binary vetors representing the presene or absene of ifferent substrutures in the moleule, obtaine with rk R pakage (Guha, 2007): k x ; x 0 ¼ H x ;x 0 = H x þ H x 0 H x ;x 0 ; where H x is the number of 1-bits in the rug s fingerprint x, an H x ;x 0 iniates the number of 1-bits ommon to fingerprints of two rug moleules x an x 0 uner omparison. The above Tanimoto similarity measure is a vali PSD kernel funtion (Gower, 1971). Cell line kernels. For ell lines, we alulate Gaussian kernels k x ; x 0 ¼¼exp jjx x 0 jj2 =2r 2, where x an x 0 enote feature representation of two ell lines in the form of (i) gene expression signature, (ii) methylation pattern, (iii) opy number variation or Downloae from

8 pairwisemkl i515 (iv) somati mutation profile (etails given in Figure 2 an Supplementary Material); r iniates a kernel with hyperparameter. We erive real-value mutation profile feature vetors, instea of employing ommonly use binary mutation iniators. In partiular, eah element x i ; i ¼ 1;...; M, orrespons to one of M mutations. If a ell line represente by x has a negative ith mutation status, then x i ¼ 0; otherwise, x i iniates a negative logarithm of the proportion of all ell lines with positive mutation status. This way, x i is high for a mutation speifi to a ell line represente by x, giving more importane to suh geneti variant. Protein kernels. For proteins, we ompute Gaussian kernels base on real-value gene ontology (GO) annotation profiles, as well as Smith Waterman (SW) kernels an generi string (GS) kernels base on three types of amino ai sequenes: (i) full kinase sequenes, (ii) kinase omain sub-sequenes an (iii) ATP bining poket subsequenes (Fig. 2). Gaussian GO-base kernels were alulate separately for moleular funtion, biologial proess an ellular omponent omains as k p x p ; x 0 p ¼ exp jjx p x 0 p jj2 =2r 2 p, where x p an x 0 p enote GO profiles of two protein kinases. Eah element of the GO profile feature vetor, x pi ; i ¼ 1;...; G, orrespons to one of G GO terms from a given omain. If a kinase represente by x p is not annotate with term i, then x pi ¼ 0; otherwise, x pi iniates a negative logarithm of the proportion of all proteins annotate with term i. SW kernel measures similarity between amino ai sequenes x p an x 0 p using normalize SW alignment sore SW (Smith an Waterman, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1981): k p ðx p ; x 0 p Þ¼SWðx p; x 0 p Þ= SW x p ; x p SW x 0 p ; x 0 p. Although SW kernel is ommonly use in rug protein interation preition, it is not a vali PSD kernel funtion, an hene we examine all obtaine matries. The matrix orresponing to ATP bining poket subsequenes was not PSD, an therefore, we shrunk its off-iagonal entries until we reahe the PSD property (126 shrinkage iterations with the shrinkage fator of were neee for the matrix to beome PSD). There are other ways of fining the nearest PSD matrix, e.g. by setting negative eigenvalues to 0, but we selete shrinkage sine it smoothly moifies the whole spetrum of eigenvalues. Finally, GS kernel ompares eah sub-string of x p of size l L with eah sub-string of x 0 p having the same length: k p x p ; x 0 p ¼ P L P jxpj l P jx0pj l l¼1 i¼0 j¼0 exp exp ði jþ2 jjnl n 0l jj 2, where vetor n l 2r 2 1 ontains properties of l amino ais inlue in the sub-string uner omparison (Giguère et al., 2013). Eah omparison results in a sore that epens on the shifting ontribution term (ifferene in the position of two sub-strings in x p an x 0 p ) ontrolle by r 1, an the similarity of amino ais inlue in two sub-strings, ontrolle by r 2. We use BLOSUM 50 matrix as amino ai esriptors in the GS kernel an in the SW sequene alignments. We ompute eah Gaussian kernel with three ifferent values of kernel with hyperparameter, etermine by alulating pairwise istanes between all ata points, an then seleting 0.1, 0.5 an 0.9 quantiles. In ase of eah GS kernel, we selete the potential values for its three hyperparameters L ¼f5; 10; 15; 20g; r 1 ¼f0:1; 1; 2; 3; 4g an r 2 ¼f0:1; 1; 2; 3; 4g by ensuring that the resulting kernel matries have a spetrum of ifferent histograms of kernel values. 3 Results To emonstrate the effiay of pairwisemkl for learning with multiple pairwise kernels, we teste the metho in two relate 2r 2 2 regression tasks of (i) preition of antianer effiay of rug ompouns an (ii) preition of target profiles of antianer rug ompouns. In partiular, we arrie out a neste 10-fol ross valiation (CV; 10 outer fols, 3 inner fols) using rug responses in aner ell lines an rug protein bining affinities, as well as hemial an genomi information soures in the form of kernels. We onstrute a total of 120 pairwise rug ell line kernels from 10 rug kernels an 12 ell line kernels, an 3120 pairwise rug protein kernels from 10 rug kernels an 312 protein kernels (Fig. 2). We ompare the performane of pairwisemkl against the reently introue algorithm for pairwise learning with multiple kernels KronRLS-MKL (Nasimento et al., 2016). Both are regularize least-squares moels, but pairwisemkl first etermines the kernel weights, an then optimizes the moel parameters, whereas KronRLS-MKL interleaves the optimization of the moel parameters with the optimization of kernel weights. Although KronRLS- MKL was originally use for lassifiation only, it is a regression algorithm in its ore, an with few moifiations to the implementation (see Supplementary Material), we applie it here to quantitative rug bioativity preition. We use the same CV fols for both methos to ensure their fair omparison. We also onute elasti net regression with stanar feature vetors instea of kernels (see Supplementary Material). Unlike pairwisemkl, KronRLS-MKL assumes that bioativities of all ombinations of rugs an ell lines (proteins) are known, i.e. it oes not allow for missing values in the label matrix storing rug ell line (rug protein) bioativities. Therefore, in the experiments with KronRLS-MKL, we mean-impute the originally missing bioativities, as well as bioativities orresponing to rug ell line (rug protein) pairs in the test fols. We assesse the preitive power of the methos with root mean square error (RMSE), Pearson orrelation an F1 sore between original an preite bioativity values. We tune the regularization hyperparameter k of pairwisemkl an regularization hyperparameters k an r of KronRLS-MKL within the neste CV from the set f10 5 ; 10 4 ;...; 10 0 g. Instea of tuning the kernel hyperparameters in this stanar way, we onstrute several kernels with ifferent arefully selete hyperparameter values (see Setion for etails). 3.1 Drug response in aner ell line preition In the task of antianer rug response preition with 120 pairwise kernels, pairwisemkl provie aurate preitions, espeially for those rug ell line pairs with more training ata points (Fig. 3a). It outperforme KronRLS-MKL in terms of preitive power, running time an memory usage (Table 2 an Supplementary Fig. S2). In partiular, pairwisemkl was almost 6 times faster an use 68- times less memory. Even though both pairwisemkl an KronRLS- MKL ahieve high Pearson orrelation of an 0.849, respetively, the auray of preitions from KronRLS-MKL erease graually when going away from the mean value to the tails of the response istribution (Supplementary Fig. S2), as iniate by 13% inrease in RMSE an 40% erease in F1 sore when omparing to pairwisemkl (Table 2). These extreme responses are often the most important ases to preit in pratie, as they orrespon to sensitivity or resistane of aner ells to a partiular rug treatment. Importantly, pairwisemkl returne a sparse ombination of only 11 out of 120 input pairwise kernels, whereas all kernel weights from KronRLS-MKL are nearly uniformly istribute (Fig. 2a). The final moel generate by pairwisemkl is muh Downloae from

9 i516 A.Cihonska et al. Fig. 3. Preition performane of pairwisemkl in the tasks of (a) rug response in aner ell line preition an (b) rug protein bining affinity preition. Satter plots between original an preite bioativity values aross (a) rug ell line pairs an (b) rug protein pairs. Performane measures were average over 10 outer CV fols. F1 sore was alulate using the threshol of (a) ln(ic 50 ) ¼ 5 nm, (b) log 10 (IC 50 ) ¼ 7 M, both orresponing to low rug onentration of roughly 100 nm, i.e. relatively stringent poteny threshol (re otte lines). Color oing iniates the number of training ata points, i.e. rug ell line (respetively rug protein) pairs inluing the same rug or ell line (rug or protein) as the test ata point. whereas pairwisemkl require just a fration of that memory (2.21 GB). pairwisemkl assigne the highest weights to two pairwise kernels buil upon amino ai sub-sequenes of ATP bining pokets, together with either shortest path fingerprints or extene onnetivity fingerprints. A relatively high weight was given also to the pairwise kernel onstrute from Klekota-Roth fingerprints an subsequenes of kinase omains. In fat, kinase omain sequenes inlue short sequenes of ATP bining pokets, an apture also their neighboring ontext. In all of the above selete pairwise kernels, protein sequenes were ompare using GS kernel. Our results therefore suggest that ATP bining pokets are more informative than full amino ai sequenes, an that GS kernel is more powerful in apturing similarities between amino ai sequenes than a ommonly use protein kernel base on SW amino ai sequene alignments, at least for the preition of rug interations with protein kinases investigate. Finally, gene ontology profiles of proteins, in partiular those from biologial proess an ellular omponent omains, provie also a moest ontribution to the optimal pairwise kernel use for rug protein bining affinity preition (Fig. 2b). Table 2. Preition performane, memory usage an running time of pairwisemkl an KronRLS-MKL methos in the task of rug response in aner ell line preition. Antianer rug response preition RMSE r Pearson F1 sore Memory (GB) Time (h) pairwisemkl KronRLS-MKL Performane measures were average over 10 outer CV fols. F1 sore was alulate using the threshol of ln(ic 50 ) ¼ 5 nm. simpler ue to fewer ative kernels, an an therefore inform about the preitive power of ifferent hemial an genomi ata soures. Figure 2a shows that the pairwise kernel alulate using gene expression in aner ell lines an moleular fingerprint efine by Klekota an Roth (2008) arries the greatest weight in the moel. Klekota-Roth fingerprint is the longest among the onsiere ones, onsisting of 4860 bits representing ifferent substrutures in the hemial ompoun, an gene expression profiles have previously been reporte as most preitive of antianer rug effiay (Costello et al., 2014). Other fingerprints ientifie by pairwisemkl as relevant for rug response inferene inlue information on Estate substrutures (Hall an Kier, 1995), onnetivity an shortest paths between ompoun s atoms. Among the ell line kernels, the ones alulate using geneti mutation an methylation patterns, in aition to gene expression, paire with the abovementione fingerprint-base rug kernels, were selete for the onstrution of the optimal pairwise kernel. Those information soures an therefore be onsiere as omplementing eah other. The opy number variation ata i not prove effetive in the preition of antianer rug responses. 3.2 Drug protein bining affinity preition In the larger experiment of preition of target profiles of almost 3000 antianer rug ompouns, pairwisemkl ahieve again high preitive performane (Pearson orrelation of 0.883, RMSE of 0.364, F1 sore of 0.713; Fig. 3b). Notably, our metho onstrute the final moel using only 8 out of 3120 pairwise kernels (Fig. 2b). KronRLS-MKL i not exeute given 1 TB memory, 3.3 Kernel hyperparameters tuning Our results emonstrate that pairwisemkl provies also a useful tool for tuning the kernel hyperparameters. In partiular, we onstrute eah kernel with ifferent hyperparameter values from a arefully hosen range (see Setion for etails), an the algorithm then selete the optimal hyperparameters by assigning nonzero mixture weights to orresponing kernels (Fig. 2). Notably, pairwisemkl always pike a single value for the Gaussian kernel with hyperparameter, r in ase of ell line kernels an r p in ase of protein kernels. This is well-represente in Figure 2a where, for eah ell line ata soure, the weights are ifferent from zero only in one of the three rows of the heatmap. Furthermore, pairwisemkl selete also only a single out of 100 ombinations of three values of the hyperparameters (L,r 1,r 2 ) for the GS kernel (Fig. 2b). 4 Disussion The enormous size of the hemial universe, estimate to onsist of up to moleules isplaying goo pharmaologial properties (Reymon an Awale, 2012), makes the experimental bioativity profiling of the full rug-like ompoun spae infeasible in pratie, an therefore alls for effiient in silio approahes that oul ai various stages of rug evelopment proess an ientifiation of optimal therapeuti strategies (Azuaje, 2017; Cheng et al., 2012; Cihonska et al., 2015). Espeially kernel-base methos have prove goo performane in many appliations, inluing inferene of rug responses in aner ell lines (Costello et al., 2014) an eluiation of rug MoA through rug protein bining affinity preitions (Cihonska et al., 2017). Pairwise learning is a natural approah for solving suh problems involving pairs of objets, an the benefits from integrating multiple hemial an genomi information soures into linially ationable preition moels are wellreporte in the reent literature (Cheng an Zhao, 2014; Costello et al., 2014; Ebrahim et al., 2016). To takle the omputational limitations of the urrent MKL approahes, we introue here pairwisemkl, a new framework for time- an memory-effiient learning with multiple pairwise kernels. pairwisemkl is well-suite for massive pairwise spaes, owing to our novel, highly effiient formulation of Kroneker eomposition of the entering operator for the pairwise kernel, an a fast Downloae from

10 pairwisemkl i517 metho for training a regularize least-squares moel with a weighte ombination of multiple kernels. To illustrate our approah, we applie pairwisemkl to two important problems in omputational rug isovery: (i) the inferene of antianer potential of rug ompouns an (ii) the inferene of rug protein interations using up to bioativities an 3120 kernels. pairwisemkl integrates heterogeneous ata types into a single moel whih is a sparse ombination of input kernels, thus allowing to haraterize the preitive power of ifferent information soures an ata representations by analyzing learne kernel mixture weights. For instane, our results emonstrate that among the genomi views, gene expression, followe by geneti mutation an methylation patterns ontribute mostly to the final pairwise kernel aopte for antianer rug response preition (Fig. 2a). Although methylation plays an essential role in the regulation of gene expression, typially repressing the gene transription, the assoiation between these two proesses remains inompletely unerstoo (Wagner et al., 2014). Therefore, it an be hypothesize that these genomi an epigeneti information levels are inee omplementing eah other in the task of rug response moeling. In ase of preition of target profiles of antianer rugs, we observe the highest ontribution to the final pairwise moel from Tanimoto rug kernels, ouple with GS protein kernels applie to ATP bining pokets (Fig. 2b). This oul be explaine by the fat that majority of antianer rugs, inluing those onsiere in this work, are kinase inhibitors esigne to bin to ATP bining pokets of protein kinases, an therefore onstruting kernels from short sequenes of these pokets is more meaningful in the ontext of rug-kinase bining affinity preition ompare to using full protein sequenes. Moreover, GS kernel is more avane than the ommonly use SW kernel as it ompares protein sequenes inluing properties of amino ais. GS kernel also enables to math short sub-sequenes of two proteins even if their positions in the input sequenes iffer notably. In both preition problems, pairwisemkl was able to tune kernel hyperparameters by seleting a single kernel out of several kernels with ifferent hyperparameter values (Fig. 2). It has been note by Cortes et al. (2012) in the ontext of ALIGNF algorithm that sparse kernel weight vetor is a onsequene of the onstraint l 0 in the kernel alignment maximization (Equation 6). This has been observe empirially in other works as well (e.g. Brouar et al., 2016; Shen et al., 2014). In partiular, it appears that pairwisemkl an ALIGNF, given a set of losely relate kernels, suh as those alulate using the same ata soure an kernel funtion but ifferent hyperparameters, ten to selet a representative kernel for the group to the optimize kernel mixture. We ompare the performane of pairwisemkl to reently introue metho for pairwise learning with multiple kernels KronRLS-MKL. pairwisemkl outperforme KronRLS-MKL in terms of preitive power, running time an memory usage (Table 2 an Supplementary Fig. S2). Unlike pairwisemkl, KronRLS-MKL oes not onsier optimizing pairwise kernel weights, i.e. it fins separate weights for rug kernels an ell line (protein) kernels (Fig. 2a), an therefore it oes not fully exploit the information ontaine in the pairwise spae. The reue preitive performane of KronRLS-MKL an be also attribute to the fat that it oes not allow for any missing values in the label matrix storing bioativities between rugs an ell lines (proteins), whih nee to be impute as a pre-proessing step an inlue in the moel training. KronRLS- MKL has two regularization hyperparameters that nee to be tune, hene lengthening the training time. Furthermore, etermining the parameters of the pairwise preition funtion involves a omputation of large matries, whih requires signifiant amount of memory that grows quikly with the number of rugs an ell lines (proteins). Finally, KronRLS-MKL applies L2 regularization on the kernel weights, thus not enforing sparse kernel seletion. In fat, KronRLS-MKL returne a nearly uniform ombination of input kernels, not allowing for the interpretation of the preitive power of ifferent ata soures. We teste pairwisemkl using CV on the level of rug ell line (rug protein) pairs whih orrespons to evaluating the performane of the metho in the task of filling experimental gaps in bioativity profiling stuies. However, pairwisemkl oul also be applie, e.g. to the inferene of antianer potential of a new aniate rug ompoun or preition of sensitivity of a new ell line to a set of rugs. We plan to takle these important problems in the future work. In this work, we put an emphasis on the regression task, sine rug bioativity measurements have a real-value nature, but we also implemente an analogous metho for solving the lassifiation problems with support vetor mahine. Other potential appliations of our effiient Kroneker eomposition of the entering operator for the pairwise kernel inlue methos whih involve kernel entering in the pairwise spae, suh as pairwise kernel PCA. Finally, even though we fouse here on the problems of antianer rug response preition an rug target interation preition, pairwisemkl has wie appliations outsie this fiel, suh as in the inferene of protein protein interations, bining affinities between proteins an pepties or mrnas an mirnas. Aknowlegements We aknowlege the omputational resoures provie by the Aalto Siene- IT projet an CSC - IT Center for Siene, Finlan. Funing This work was supporte by the Aaemy of Finlan [ to A.A.; an to J.R.; to M.H.; an to T.P.; , an to T.A.]. Conflit of Interest: none elare. 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(2014) Mahine learning-base preition of rug-rug interations by integrating rug phenotypi, therapeuti, hemial, an genomi properties. J. Am. Me. Inform. Asso., 21, e278 e286. Downloae from