arXiv:1809.10680v2 [cs.LG] 9 Oct 2018 n hnw ple tt C otlt ikpeito and prediction risk mortality supe meth ICU conventional this other to methods. of over NMF it superiority effectiveness applied its us the demonstrated we We verify procedure. then an to optimization framework and data alternating NMF simulation an the the with into the it integrating function solved by loss One algorithm pro regression this SANMF data. predicti with logistic supervised poor deal patient a To its nature. proposed the is unsupervised we its method analyze t to SANMF due adopted to original ability provide we the backbone to Therefore, of the data limitation well. series features as the time approach interpret to to applied (SANMF) path successfully recentl Factorization been but Matrix interpretability, has Nonnegative the Augmented of problems.graph ca lack accuracy prediction methods and feature mortalit interpretability existing good effective the the hand, prevent to to the other actions correspond data timely the identify temporal take on clinicians complex to help easily; the difficult them to is interpret due hand, it one collected, On task. portant rsin uevsdlann,Rpeetto,IUmorta ICU Representation, learning, Supervised gression, enegSho fMedicine of School Feinberg hsmkspeitv oeigo h otlt ikan risk mortality the of decision modeling death. correct problem. and predictive important patient makes timely is makes usually make ICU data to This ICU difficult in the clinicians event of the continuou nature key the complex One (especially the However, data). types complicated variable series and in diverse time abundant generated the are clinicians data to (ICU) help due The Unit and Care decisions. Intensive data accurate di the the and can in timely models hidden make These patterns institutions. latent clinical dat cover possible and medical available hospitals becomes increasingly in the modeling with combining predictive when automated mining, enegSho fMedicine of School Feinberg [email protected] [email protected] ne Terms Index Abstract ihtefs eeomn fmcielann n data and learning machine of development fast the With otwsenUniversity Northwestern otwsenUniversity Northwestern uevsdNneaieMti atrzto to Factorization Matrix Nonnegative Supervised hcg,U.S Chicago, uqn Chao Guoqing hcg,U.S Chicago, unLuo Yuan IUmraiyrs rdcini og e im- yet tough a is prediction risk mortality —ICU Nneaiemti atrzto,Lgsi re- Logistic factorization, matrix —Nonnegative .I I. NTRODUCTION rdc C otlt Risk Mortality ICU Predict [email protected] enegSho fMedicine of School Feinberg otwsenUniversity Northwestern hnsegMao Chengsheng hcg,U.S Chicago, .These y. iyrisk lity Sub- y and s rvised blem, Most od, his on ed s- s. n d a a s iegne hc samc oegnrlls ihboth with loss general more much a Bregman by is case measured the which the is to loss NMF divergence, discussed approximation further extended also [8] matrix [6] the They [7]. when rows. in NMF and of co-clustering [5]. variants columns perform various purpose matrix can clustering both that approach on data t-NMF for a developed used between NMF equivalence claimed be thus some and can is clustering there K-means/spectral that and NMF showed [4] [3]. they paper aspects. their and in solution, algorithm the stationary multiplicative of a convergence a the obtain developed prove they to and NMF algorithm problem constraints, updates convex nonnegativity a from the image not of face is the Because reconstruct part bases. to ha with were matrix those basis we other the image if coefficients and of one faces, the set that human the found of as representation they based interpreted matrices, such vectors be two factorizing can intensity of matrix After product pixel the images. of nonnegative into facial a matrix set human form the from they where stretched concatenating [2], by become work matrix has the problem since The prod- popular matrices. the nonnegative as several matrix of nonnegative uct a approximating on problems eis[]so infiatpromneipoeet from improvements performance baselines. significant multiple patien show for [1] simultaneousl features physiologica series effective and patient on form Applications patients, prediction. to outcome for trends types temporal group feature dimensiona different reduces identification Moreov of representation. group graph latent fe into supervised higher-order higher-order data of converting the mining by and the tures atomic automate We predicti of jointly. exploration performs features the that by algorithm modeling (NMF) factorization trix ic hnpol aebe okn nNFfo various from NMF on working been have people then Since of set the to refers (NMF) Factorization Matrix Nonnegative nti ae,w ecieasprie ongtv ma- nonnegative supervised a describe we paper, this In [email protected] el onl Medicine Cornell Weill onl University Cornell e ok U.S York, New e Wang Fei I R II. LTDWORK ELATED [email protected] enegSho fMedicine of School Feinberg otwsenUniversity Northwestern hcg,U.S Chicago, unZhao Yuan time l lity er, ve a- y s t Frobenius norm and KL divergence that was introduced in [3] III. METHODS as its special cases. On the solution procedure aspect, multi- A. NMF and Logistic Regression plicative updates has been recognized for its slow convergence and poor quality. [9] reviewed the general algorithms before NMF [2] is a popular representation learning technique. n×m 2007, three classes of algorithms are categorized. The first Assume that we have data matrix X ∈ R and its corre- n class is multiplicative updates, the second class is gradient sponding label y ∈ R . NMF aims to learn two nonnegative n×r r×m based methods such as [10], [11], [12], the third class is the matrices U ∈ R and V ∈ R to approximate a n×m alternating least squares (ALS) algorithm [13], [14], [15]. Also nonnegative matrix X ∈ R , each row of which contains in 2007, [16] proposed an efficient projected gradient approach m features of one of the n samples. Then NMF can be for NMF, which adopted Taylor expansion strategy to reduce formulated in matrix form X = UV , and at the same time the cost. [17] also proposed an active set type of method U and V are constrained to be nonnegative. Therefore, NMF called principal block pivoting to solve the NMF problem. is formulated as the following optimization problem. [18] adopted the alternating direction method of multipliers 2 minkX − UV kF (ADMM) to solve the NMF with beta-divergence. Hsieh and U,V (1) Dhillon [19] designed a fast coordinate descent method to s.t.U 0, V 0, accelerate the FastHals [20]. 2 where k·kF indicates squared frobenius norm and U 0, V U V Apart from basic NMF method introduced above, there 0 mean that and are both entry-wise nonnegative. r rows V U are also many variants. They can be grouped into three of are considered as new bases while r columns of are groups. The first group enforced some constraints into basic viewed as the coefficients with which the original samples can NMF to obtain certain desirable charateristics, such as sparse be represented with a linear combination of bases. U can be X NMF [10], [21], [22], orthogonal NMF [23], [24], [25], considered as a low-dimensional representation of since discrininant NMF [26], [27] and manifold NMF [28], [29]. The generally r
Supervised Nonnegative Matrix Factorization to Predict ICU
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