Original: A D EEP L EARNING A PPROACH FOR S URVIVAL C LUSTERING WITHOUT E ND - OF - LIFE S IGNALS
Revised: A D EEP L EARNING A PPROACH FOR S URVIVAL C LUSTERING WITHOUT E ND - OF - LIFE S IGNALS

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: The goal of survival clustering is to map subjects (e.g., users in a social network, patients in a medical study) to K clusters ranging from low-risk to high-risk.
Revised: The goal of survival clustering is to map subjects (e.g., users in a social network, patients in a medical study) to K clusters ranging from low-risk to high-risk.

Original: Existing survival methods assume the presence of clear end-of-life signals or introduce them artiﬁcially using a pre-deﬁned timeout.
Revised: Existing survival methods assume the presence of clear end-of-life signals or introduce them artiﬁcially using a pre-deﬁned timeout.

Original: In this paper, we forego this assumption and introduce a loss function that differentiates between the empirical lifetime distributions of the clusters using a modiﬁed Kuiper statistic.
Revised: In this paper, we forego this assumption and introduce a loss function that differentiates between the empirical lifetime distributions of the clusters using a modiﬁed Kuiper statistic.

Original: We learn a deep neural network by optimizing this loss, that performs a soft clustering of users into survival groups.
Revised: We learn a deep neural network by optimizing this loss, that performs a soft clustering of users into survival groups.

Original: We apply our method to a social network dataset with over 1M subjects, and show signiﬁcant improvement in C-index compared to alternatives.
Revised: We apply our method to a social network dataset with over 1M subjects, and show signiﬁcant improvement in C-index compared to alternatives.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Free online subscription services (e.g., Facebook, Pandora) use survival models to predict the relationship between observed subscriber covariates (e.g. usage patterns, session duration, gender, location, etc.)
Revised: Free online subscription services (e.g., Facebook, Pandora) use survival models to predict the relationship between observed subscriber covariates (e.g. usage patterns, session duration, gender, location, etc.)

Original: and how long a subscriber remains with an active account (Kapoor et al., 2014; Ciampaglia and Taraborelli, 2015).
Revised: and how long a subscriber remains with an active account (Kapoor et al., 2014; Ciampaglia and Taraborelli, 2015).

Original: Using the same tools, healthcare providers make extensive use of survival models to predict the relationship between patient covariates (e.g. smoking, administering drug A or B) and the duration of a disease (e.g., herpes, cancer, etc.).
Revised: Using the same tools, healthcare providers make extensive use of survival models to predict the relationship between patient covariates (e.g. smoking, administering drug A or B) and the duration of a disease (e.g., herpes, cancer, etc.).

Original: In these scenarios, rarely there is an end-of-life signal: non-paying subscribers do not cancel their accounts, tests rarely declare a patient cancer-free.
Revised: In these scenarios, rarely there is an end-of-life signal: non-paying subscribers do not cancel their accounts, tests rarely declare a patient cancer-free.

Original: We want to assign subjects into K clusters, ranging from short-lived to long-lived subscribers (diseases).
Revised: We want to assign subjects into K clusters, ranging from short-lived to long-lived subscribers (diseases).

Original: Despite the recent community interest in survival models (Alaa and van der Schaar, 2017; Luck et al., 2017), existing survival analysis approaches require an unmistakable end-of-life signal (e.g., the subscriber deletes his or her account, the patient is declared disease-free), or a pre-deﬁned endof-life “timeout” (e.g., the patient is declared disease-free after 5 years, the subscriber is declared permanently inactive after 100 days of inactivity).
Revised: Despite the recent community interest in survival models (Alaa and van der Schaar, 2017; Luck et al., 2017), existing survival analysis approaches require an unmistakable end-of-life signal (e.g., the subscriber deletes his or her account, the patient is declared disease-free), or a pre-deﬁned endof-life “timeout” (e.g., the patient is declared disease-free after 5 years, the subscriber is declared permanently inactive after 100 days of inactivity).

Original: Methods that require end-of-life signals also include (Iorio et al., 2009; Bohlourihajjar and Khazaei, 2017; Bair and Tibshirani, 2004; Eleuteri et al., 2003; 2007; Ishwaran et al., 2010; Lagani and Tsamardinos, 2010; LeBlanc and Crowley, 1993;
Revised: Methods that require end-of-life signals also include (Iorio et al., 2009; Bohlourihajjar and Khazaei, 2017; Bair and Tibshirani, 2004; Eleuteri et al., 2003; 2007; Ishwaran et al., 2010; Lagani and Tsamardinos, 2010; LeBlanc and Crowley, 1993;

Original: Witten and Tibshirani, 2010b; Bøvelstad et al., 2007;
Revised: Witten and Tibshirani, 2010b; Bøvelstad et al., 2007;

Original: Hothorn et al., 2006; Shivaswamy, Chu, and
Revised: Hothorn et al., 2006; Shivaswamy, Chu, and

Original: Jansche, 2007; Shipp et al., 2002; Gaynor and Bair, 2013; Yang et al., 2010; Kapoor et al., 2014;
Revised: Jansche, 2007; Shipp et al., 2002; Gaynor and Bair, 2013; Yang et al., 2010; Kapoor et al., 2014;

Original: Aggarwal, Gates, and Yu, 2004; Basu, Banerjee, and Mooney, 2002; Basu, Bilenko, and Mooney, 2004; Nigam et al., 1998; Witten and Tibshirani, 2010a; Law, Urtasun, and Zemel, 2017).
Revised: Aggarwal, Gates, and Yu, 2004; Basu, Banerjee, and Mooney, 2002; Basu, Bilenko, and Mooney, 2004; Nigam et al., 1998; Witten and Tibshirani, 2010a; Law, Urtasun, and Zemel, 2017).

Original: In this work, we propose to address the lifetime clustering problem without end-of-life signals for the ﬁrst time, to the best of our knowledge.
Revised: In this work, we propose to address the lifetime clustering problem without end-of-life signals for the ﬁrst time, to the best of our knowledge.

Original: Deﬁnition 1 (Dataset) .
Revised: Deﬁnition 1 (Dataset) .

Original: Dataset D consists of a set of n subjects with each subject u having the following observable quantities Ψ u
Revised: Dataset D consists of a set of n subjects with each subject u having the following observable quantities Ψ u

Original: = { X u , { Y u,i } q u i =1 , S u } , where X u are the covariates of subject u , { Y u,i } q u i =1 are the observed inter-event times (disease outbreaks, website usage), q u is the number of observed events of u , and S u is the time till censoring.
Revised: = { X u , { Y u,i } q u i =1 , S u } , where X u are the covariates of subject u , { Y u,i } q u i =1 are the observed inter-event times (disease outbreaks, website usage), q u is the number of observed events of u , and S u is the time till censoring.

Original: We give two examples for constructing the datasets the way described above.
Revised: We give two examples for constructing the datasets the way described above.

Original: • Social Network Dataset : Users join the social network at different times and participate in activities deﬁned by the social network (login,
Revised: • Social Network Dataset : Users join the social network at different times and participate in activities deﬁned by the social network (login,

Original: send/receive comments).
Revised: send/receive comments).

Original: The covariates X u are the various attributes of a user like age, gender, number of friends, etc., and Y u,i is the time
Revised: The covariates X u are the various attributes of a user like age, gender, number of friends, etc., and Y u,i is the time

Original: between user’s
Revised: between user’s

Original: ( i − 1)
Revised: ( i − 1)

Original: st
Revised: st

Original: and i th activity.
Revised: and i th activity.

Original: In this case, censoring is due to a ﬁxed point of data collection that we denote t m , the time of measurement.
Revised: In this case, censoring is due to a ﬁxed point of data collection that we denote t m , the time of measurement.

Original: Thus, time till censoring of a user u is given by, S u
Revised: Thus, time till censoring of a user u is given by, S u

Original: =
Revised: =

Original: t m − (cid:80) i
Revised: t m − (cid:80) i

Original: Y u,i .
Revised: Y u,i .

Original: Lifetime of a user is deﬁned as the time from her joining till she permanently deletes her account.
Revised: Lifetime of a user is deﬁned as the time from her joining till she permanently deletes her account.

Original: • Medical Dataset : Subjects join the medical study at the same time and are checked for the presence of a particular disease.
Revised: • Medical Dataset : Subjects join the medical study at the same time and are checked for the presence of a particular disease.

Original: The covariates X u are the attributes of the disease-causing cell in subject u , Y u,i is the time between ( i − 1)
Revised: The covariates X u are the attributes of the disease-causing cell in subject u , Y u,i is the time between ( i − 1)

Original: st
Revised: st

Original: and i th observations of the presence of disease.
Revised: and i th observations of the presence of disease.

Original: The time to censoring, S u , is the difference between the time of last observation when the disease was present and the time of ﬁnal observation.
Revised: The time to censoring, S u , is the difference between the time of last observation when the disease was present and the time of ﬁnal observation.

Original: If the ﬁnal observation for a subject u indicates presence of the disease, then S u is zero.
Revised: If the ﬁnal observation for a subject u indicates presence of the disease, then S u is zero.

Original: Lifetime of the disease is deﬁned as the time between the ﬁrst observation of the disease and the time until it is permanently cured.
Revised: Lifetime of the disease is deﬁned as the time between the ﬁrst observation of the disease and the time until it is permanently cured.

Original: Deﬁnition 2 (Clustering problem) .
Revised: Deﬁnition 2 (Clustering problem) .

Original: Consider a dataset of n subjects, D , constructed as in deﬁnition 1.
Revised: Consider a dataset of n subjects, D , constructed as in deﬁnition 1.

Original: Let ˆ P ( U k ) be the latent lifetime distribution of all subjects
Revised: Let ˆ P ( U k ) be the latent lifetime distribution of all subjects

Original: U k = { u } that belong to cluster k ∈ { 1 , . . .
Revised: U k = { u } that belong to cluster k ∈ { 1 , . . .

Original: , K } .
Revised: , K } .

Original: Our goal is to ﬁnd a mapping κ : X u
Revised: Our goal is to ﬁnd a mapping κ : X u

Original: → { 1 , . .
Revised: → { 1 , . .

Original: , K } , of covariates into clusters, in the set of all possible such mappings K , that maximizes the divergence d between the latent lifetime distributions of all subjects:
Revised: , K } , of covariates into clusters, in the set of all possible such mappings K , that maximizes the divergence d between the latent lifetime distributions of all subjects:

Original: [Equation]
Revised: [Equation]

Original: where U k ( κ ) is the set of users in U mapped to cluster k through κ , and d is a distribution divergence metric.
Revised: where U k ( κ ) is the set of users in U mapped to cluster k through κ , and d is a distribution divergence metric.

Original: [Equation]
Revised: [Equation]

Original: ∗ optimized in this fashion clusters the subjects into low-risk/high-risk groups.
Revised: ∗ optimized in this fashion clusters the subjects into low-risk/high-risk groups.

Original: However, because ˆ P ( U k ) are latent distributions, we cannot directly optimize Eq.(1).
Revised: However, because ˆ P ( U k ) are latent distributions, we cannot directly optimize Eq.(1).

Original: Rather, our loss function must provide an indirect way to optimize Eq.(1) without end-of-life signals.
Revised: Rather, our loss function must provide an indirect way to optimize Eq.(1) without end-of-life signals.

Original: For this, we use a deep neural network and a new loss function, with a corresponding backpropagation modiﬁcation, for clustering subjects without end-of-life signals.
Revised: For this, we use a deep neural network and a new loss function, with a corresponding backpropagation modiﬁcation, for clustering subjects without end-of-life signals.

Original: We are able to overcome the technical challenges of this problem, in part, thanks to the ability of deep neural networks to generalize while overﬁtting the training data (Zhang et al., 2017).
Revised: We are able to overcome the technical challenges of this problem, in part, thanks to the ability of deep neural networks to generalize while overﬁtting the training data (Zhang et al., 2017).

Original: The task is challenging for the following reasons:
Revised: The task is challenging for the following reasons:

Original: • The problem is fully unsupervised, as there is no pre-deﬁned end-of-life timeout.
Revised: • The problem is fully unsupervised, as there is no pre-deﬁned end-of-life timeout.

Original: While semisupervised clustering approaches exist (Aggarwal, Gates, and Yu, 2004; Basu, Banerjee, and Mooney, 2002; Basu, Bilenko, and Mooney, 2004; Nigam et al., 1998; Witten and Tibshirani, 2010a), they assume that end-of-life signals appearing before the observation time are observed; to the best of our knowledge, there are no fully unsupervised approach that can take complex input variables.
Revised: While semisupervised clustering approaches exist (Aggarwal, Gates, and Yu, 2004; Basu, Banerjee, and Mooney, 2002; Basu, Bilenko, and Mooney, 2004; Nigam et al., 1998; Witten and Tibshirani, 2010a), they assume that end-of-life signals appearing before the observation time are observed; to the best of our knowledge, there are no fully unsupervised approach that can take complex input variables.

Original: • There is no hazard function that can be used to deﬁne the “cure” rate, as we cannot determine whether the disease is cured, or whether the subscriber will never return to the website, without observing for an inﬁnitely long time.
Revised: • There is no hazard function that can be used to deﬁne the “cure” rate, as we cannot determine whether the disease is cured, or whether the subscriber will never return to the website, without observing for an inﬁnitely long time.

Original: • Cluster assignments may depend on highly complex interactions between the observed covariates and the observed events.
Revised: • Cluster assignments may depend on highly complex interactions between the observed covariates and the observed events.

Original: The unobserved lifetime distributions may not be smooth functions.
Revised: The unobserved lifetime distributions may not be smooth functions.

Original: Contributions.
Revised: Contributions.

Original: Using the ability of deep neural networks to model complex nonlinear relationships in the input data, our contribution is a loss
Revised: Using the ability of deep neural networks to model complex nonlinear relationships in the input data, our contribution is a loss

Original: function (using the p-value from a
Revised: function (using the p-value from a

Original: modiﬁed Kuiper nonparametric two-sample test (Kuiper, 1960)) and a backpropagation algorithm that can perform model-free (nonparametric) unsupervised clustering of subjects based on their latent lifetime distributions, even in the absence of end-of-life signals.
Revised: modiﬁed Kuiper nonparametric two-sample test (Kuiper, 1960)) and a backpropagation algorithm that can perform model-free (nonparametric) unsupervised clustering of subjects based on their latent lifetime distributions, even in the absence of end-of-life signals.

Original: The output of our algorithm is a trained deep neural network classiﬁer that can (soft) assign test and training data subjects into K categories, from high-risk and to low-risk individuals.
Revised: The output of our algorithm is a trained deep neural network classiﬁer that can (soft) assign test and training data subjects into K categories, from high-risk and to low-risk individuals.

Original: We apply our method to a large social network dataset and show that our approach is more robust than competing methods and obtains better clusters (higher
Revised: We apply our method to a large social network dataset and show that our approach is more robust than competing methods and obtains better clusters (higher

Original: C-index scores).
Revised: C-index scores).

Original: Why deep neural networks.
Revised: Why deep neural networks.

Original: As with any optimization method that returns a point estimate (a set of neural network weights W in our case), our approach is subject to overﬁtting the training data.
Revised: As with any optimization method that returns a point estimate (a set of neural network weights W in our case), our approach is subject to overﬁtting the training data.

Original: And because our loss function uses p -values, the optimization and overﬁtting have a rather negative name: p -hacking (Nuzzo, 2014).
Revised: And because our loss function uses p -values, the optimization and overﬁtting have a rather negative name: p -hacking (Nuzzo, 2014).

Original: That is, the optimization is looking for a W ( hypothesis ) that decreases the p -value.
Revised: That is, the optimization is looking for a W ( hypothesis ) that decreases the p -value.

Original: Deep neural networks, however, are known to both overﬁt the training data and generalize well Zhang et al.
Revised: Deep neural networks, however, are known to both overﬁt the training data and generalize well Zhang et al.

Original: That is, the hypothesis ( W ) tends to also have small p -values in the (unseen) test data, despite overﬁtting in the training data ( p -hacking).
Revised: That is, the hypothesis ( W ) tends to also have small p -values in the (unseen) test data, despite overﬁtting in the training data ( p -hacking).

Original: Outline :
Revised: Outline :

Original: In section 2, we describe the traditional survival analysis concepts that assume the presence of end-of-life signals.
Revised: In section 2, we describe the traditional survival analysis concepts that assume the presence of end-of-life signals.

Original: In section 3, we deﬁne a loss function that quantiﬁes the divergence between empirical lifetime distributions of two clusters without assuming end-of-life signals.
Revised: In section 3, we deﬁne a loss function that quantiﬁes the divergence between empirical lifetime distributions of two clusters without assuming end-of-life signals.

Original: We also provide a neural network approach to optimize said loss function.
Revised: We also provide a neural network approach to optimize said loss function.

Original: We describe the dataset used in our experimentsin section 4, followed by results in section 4.
Revised: We describe the dataset used in our experiments followed by results in section 4.

Original: In section 5, we describe a few methods in literature that are related to our work.
Revised: In section 5, we describe a few methods in literature that are related to our work.

Original: Finally, we present our conclusions in section 6.
Revised: Finally, we present our conclusions in section 6.

Original:  B ACKGROUND
Revised:  B ACKGROUND

Original: In this section, we review the major concepts in survival analysis that are used in this paper.
Revised: In this section, we review the major concepts in survival analysis that are used in this paper.

Original: Let T u denote the lifetime of a subject u .
Revised: Let T u denote the lifetime of a subject u .

Original: For now, our description assumes an Oracle that provides end-oflife signals.
Revised: For now, our description assumes an Oracle that provides end-oflife signals.

Original: Thus, in addition to Ψ u , we assume for each subject u , another observable quantity, y u , that denotes whether end-of-life has been reached.
Revised: Thus, in addition to Ψ u , we assume for each subject u , another observable quantity, y u , that denotes whether end-of-life has been reached.

Original: In survival applications, y u is typically used to specify if the required event did not occur until the end of study, known as right-censoring .
Revised: In survival applications, y u is typically used to specify if the required event did not occur until the end of study, known as right-censoring .

Original: We shall forego this assumption in subsequent sections and provide a way around the lack of these signals.
Revised: We shall forego this assumption in subsequent sections and provide a way around the lack of these signals.

Original: Lifetime distribution & Hazard function (Oracle).
Revised: Lifetime distribution & Hazard function (Oracle).

Original: Lifetime (or survival) distribution is deﬁned as the probability that a subject u survives at least until time t ,
Revised: Lifetime (or survival) distribution is deﬁned as the probability that a subject u survives at least until time t ,

Original: [Equation]
Revised: [Equation]

Original: where F u ( t ) is the cumulative distribution function of T u .
Revised: where F u ( t ) is the cumulative distribution function of T u .

Original: In survival applications, it is typically convenient to deﬁne the hazard function, that represents the instantaneous rate of death of a subject given that she has survived until time t .
Revised: In survival applications, it is typically convenient to deﬁne the hazard function, that represents the instantaneous rate of death of a subject given that she has survived until time t .

Original: The hazard function of a subject u is λ u ( t ) =
Revised: The hazard function of a subject u is λ u ( t ) =

Original: dF u ( t ) S u ( t ) , where dF u is the probability density of F u .
Revised: dF u ( t ) S u ( t ) , where dF u is the probability density of F u .

Original: Kaplan-Meier Estimates and the Cox Model of Lifetime Distribution (Oracle).
Revised: Kaplan-Meier Estimates and the Cox Model of Lifetime Distribution (Oracle).

Original: Due to rightcensoring, we do not observe the true lifetimes of the subjects even in the presence of end-of-life signals, y u .
Revised: Due to rightcensoring, we do not observe the true lifetimes of the subjects even in the presence of end-of-life signals, y u .

Original: We deﬁne the observed lifetime of subject u , H u , as the difference between the time of ﬁrst event and time of last observed event, i.e.,
Revised: We deﬁne the observed lifetime of subject u , H u , as the difference between the time of ﬁrst event and time of last observed event, i.e.,

Original: [Equation]
Revised: [Equation]

Original: Kaplan and Meier (1958) provide a way to estimate the lifetime distribution for a set of subjects while incorporating the right censoring effect.
Revised: Kaplan and Meier (1958) provide a way to estimate the lifetime distribution for a set of subjects while incorporating the right censoring effect.

Original: The Kaplan-Meier estimates of lifetime distribution are given by,
Revised: The Kaplan-Meier estimates of lifetime distribution are given by,

Original: [Equation]
Revised: [Equation]

Original: where d j = (cid:80) u ∈D I
Revised: where d j = (cid:80) u ∈D I

Original: [ H u = j ] denotes the number of subjects with end-of-life at time j , and r j = (cid:80) u ∈D
Revised: [ H u = j ] denotes the number of subjects with end-of-life at time j , and r j = (cid:80) u ∈D

Original: I [ H u ≥ j ] denotes the number of subjects at risk just prior to time j .
Revised: I [ H u ≥ j ] denotes the number of subjects at risk just prior to time j .

Original: Cox regression model (Cox, 1992) is a widely used method in survival analysis to estimate the hazard function λ u ( t ) using the covariates, X u , of a subject u .
Revised: Cox regression model (Cox, 1992) is a widely used method in survival analysis to estimate the hazard function λ u ( t ) using the covariates, X u , of a subject u .

Original: The hazard function has the form, λ ( t | X u ) =
Revised: The hazard function has the form, λ ( t | X u ) =

Original: λ 0 ( t ) · e { β T X u } , where λ 0 ( t ) is a base hazard function common for all subjects, and β are the regression coefﬁcients.
Revised: λ 0 ( t ) · e { β T X u } , where λ 0 ( t ) is a base hazard function common for all subjects, and β are the regression coefﬁcients.

Original: The model assumes that the ratio of hazard functions of any two subjects is constant over time.
Revised: The model assumes that the ratio of hazard functions of any two subjects is constant over time.

Original: This assumption is violated frequently in real-world datasets (Li et al., 2015).
Revised: This assumption is violated frequently in real-world datasets (Li et al., 2015).

Original: A near-extreme case when this assumption does not hold is shown in Figure 1(c), where the survival curves of two groups of subjects cross each other.
Revised: A near-extreme case when this assumption does not hold is shown in Figure 1(c), where the survival curves of two groups of subjects cross each other.

Original: Survival Based Clustering Methods (Oracle).
Revised: Survival Based Clustering Methods (Oracle).

Original: There have been relatively fewer works that perform survival based clustering.
Revised: There have been relatively fewer works that perform survival based clustering.

Original: Bair and Tibshirani (2004) proposed a semi-supervised method for clustering survival data in which they assign Cox scores (Cox, 1992) for each feature in their dataset and considered only the features with scores above a predetermined threshold.
Revised: Bair and Tibshirani (2004) proposed a semi-supervised method for clustering survival data in which they assign Cox scores (Cox, 1992) for each feature in their dataset and considered only the features with scores above a predetermined threshold.

Original: Then, an unsupervised clustering algorithm, like k-means, is used to group the individuals using only the selected features.
Revised: Then, an unsupervised clustering algorithm, like k-means, is used to group the individuals using only the selected features.

Original: Gaynor and Bair (2013) proposed supervised sparse clustering as a modiﬁcation to the sparse clustering algorithm of Witten and Tibshirani (2010a).
Revised: Gaynor and Bair (2013) proposed supervised sparse clustering as a modiﬁcation to the sparse clustering algorithm of Witten and Tibshirani (2010a).

Original: The sparse clustering algorithm has a modiﬁed k-means score that uses distinct weights in the feature set.
Revised: The sparse clustering algorithm has a modiﬁed k-means score that uses distinct weights in the feature set.

Original: Supervised sparse clustering initializes these feature weights using Cox scores (Cox, 1992) and optimizes the same objective function.
Revised: Supervised sparse clustering initializes these feature weights using Cox scores (Cox, 1992) and optimizes the same objective function.

Original: Both these methods assume the presence of end-of-life signals.
Revised: Both these methods assume the presence of end-of-life signals.

Original: In this paper, we consider the case when end-of-life signals are not available.
Revised: In this paper, we consider the case when end-of-life signals are not available.

Original: We provide a loss function that quantiﬁes the divergence between survival distributions of the clusters, and we minimize said loss function using a neural network in order to obtain the optimal clusters.
Revised: We provide a loss function that quantiﬁes the divergence between survival distributions of the clusters, and we minimize said loss function using a neural network in order to obtain the optimal clusters.

Original:  M ETHODOLOGY
Revised:  M ETHODOLOGY

Original: Our goal is to cluster the subjects into K clusters ranging from low-risk to high-risk by keeping the empirical lifetime distributions of these K groups as different as they can be, while ensuring that the observed difference is statistically signiﬁcant.
Revised: Our goal is to cluster the subjects into K clusters ranging from low-risk to high-risk by keeping the empirical lifetime distributions of these K groups as different as they can be, while ensuring that the observed difference is statistically signiﬁcant.

Original: In this section, we assume there are no end-of-life signals.
Revised: In this section, we assume there are no end-of-life signals.

Original: 3. L OSS F UNCTION
Revised: 3. L OSS F UNCTION

Original: We introduce a loss function that is based on a divergence measure between empirical lifetime distributions of two groups, and at the same time takes into account the uncertainty regarding the end-of-life of the subjects.
Revised: We introduce a loss function that is based on a divergence measure between empirical lifetime distributions of two groups, and at the same time takes into account the uncertainty regarding the end-of-life of the subjects.

Original: Instead of a clear end-of-life signal, we specify a probability for each subject u that represents how likely her last observed activity coincides with her end-of-life .
Revised: Instead of a clear end-of-life signal, we specify a probability for each subject u that represents how likely her last observed activity coincides with her end-of-life .

Original: Deﬁnition 3 (Probability of end-of-life ) .
Revised: Deﬁnition 3 (Probability of end-of-life ) .

Original: Given a dataset D (Deﬁnition 1), we deﬁne a function, by an abuse of notation, p ( X u , S u ) → [0 , 1] that gives a probability of end-of-life of each subject u .
Revised: Given a dataset D (Deﬁnition 1), we deﬁne a function, by an abuse of notation, p ( X u , S u ) → [0 , 1] that gives a probability of end-of-life of each subject u .

Original: Divergence measures like Kullback-Leibler divergence and Earth-Mover’s distance that do not incorporate the empirical nature of the given probability distributions are not appropriate for our task as they do not discourage highly imbalanced groups (Figure 1b).
Revised: Divergence measures like Kullback-Leibler divergence and Earth-Mover’s distance that do not incorporate the empirical nature of the given probability distributions are not appropriate for our task as they do not discourage highly imbalanced groups (Figure 1b).

Original: This motivates the use of twosample tests that allow for the probability distributions to be empirical.
Revised: This motivates the use of twosample tests that allow for the probability distributions to be empirical.

Original: Logrank test (Mantel, 1966;
Revised: Logrank test (Mantel, 1966;

Original: Peto and Peto, 1972; Bland and Altman, 2004) is commonly used to compare groups of subjects based on their hazard rates.
Revised: Peto and Peto, 1972; Bland and Altman, 2004) is commonly used to compare groups of subjects based on their hazard rates.

Original: However, the test assumes proportional hazards (section 2) and will not be able to ﬁnd groups whose hazard rates are not proportional to each other (Figure 1a).
Revised: However, the test assumes proportional hazards (section 2) and will not be able to ﬁnd groups whose hazard rates are not proportional to each other (Figure 1a).

Original: Fleming 
Revised: Fleming et al.

Revised: (1980) introduced Modiﬁed Kolmogorov-Smirnov (MKS) statistic that works for arbitrarily

Original: [Figure]
Revised: [Figure]

Original: et al. (1980) introduced Modiﬁed Kolmogorov-Smirnov (MKS) statistic that works for arbitrarily right-censored data anddoes not assume hazards proportionality.
Revised: right data and not assume hazards proportionality.

Original: ButMKS suffers from thesame drawback as the standard Kolmogorov-Smirnov statistic, namely that it is not sensitive to the differences in the tails of the distributions.
Revised: But suffers from the drawback as the standard Kolmogorov-Smirnov statistic, namely that it is not sensitive to the differ ences in the tails of the distributions.

Original: In this paper, we use p-value from the Kuiper statistic (Kuiper, 1960) which extends the Kolmogorov-Smirnov statistic to increase the statistical power of distinguishing distribution tails(Tygert, 2010).
Revised: In this paper, we use p-value from the Kuiper statistic (Kuiper, 1960) which extends the Kolmogorov-Smirnov statistic to increase the statistical power of distin- distribution tails

Original: Deﬁnition 4 (Optimization of Kuiper loss) .
Revised: Deﬁnition 4 (Optimization of Kuiper loss) .

Original: Given a dataset D (Deﬁnition 1), we deﬁne a loss
Revised: Given a dataset D (Deﬁnition 1), we deﬁne a loss

Original: L ( κ, p ) where, by an abuse of notation, κ ( X u ) →
Revised: L ( κ, p ) where, by an abuse of notation, κ ( X u ) →

Original: [0 , 1] is a mapping that performs soft clustering of subjects into two clusters 0 & 1 by outputting a probability of a subject belonging in cluster 0, and p ( X u , S u ) → [0 , 1] is a function that gives a probability of end-of-life of a subject in D .
Revised: [0 , 1] is a mapping that performs soft clustering of subjects into two clusters 0 & 1 by outputting a probability of a subject belonging in cluster 0, and p ( X u , S u ) → [0 , 1] is a function that gives a probability of end-of-life of a subject in D .

Original: Our goal is to obtain
Revised: Our goal is to obtain

Original: [Equation]
Revised: [Equation]

Original: where the loss function
Revised: where the loss function

Original: [Equation]
Revised: [Equation]

Original: returns the logarithm of a p -value from the Kuiper statistic (Press et al., 1996), with
Revised: returns the logarithm of a p -value from the Kuiper statistic (Press et al., 1996), with

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation] (cid:33) · (cid:0) D + ( κ, p ) + D − ( κ, p ) (cid:1) , and D + ( κ, p ) =
Revised: D + ( κ, p ) + D − ( κ, p ) (cid:1) , and D + ( κ, p ) =

Original: sup t { S 0 ( t ; κ, p ) − S 1 ( t ; κ, p ) } , D − ( κ, p )
Revised: sup t { S 0 ( t ; κ, p ) − S 1 ( t ; κ, p ) } , D − ( κ, p )

Original: =
Revised: =

Original: sup t { S 1 ( t ; κ, p ) − S 0 ( t ; κ, p ) } , n ( κ ) = n 0 ( κ ) · n 1 ( κ ) n 0 ( κ )+ n 1 ( κ ) , and for k = 0 , 1 ,
Revised: sup t { S 1 ( t ; κ, p ) − S 0 ( t ; κ, p ) } , n ( κ ) = n 0 ( κ ) · n 1 ( κ ) n 0 ( κ )+ n 1 ( κ ) , and for k = 0 , 1 ,

Original: [Equation]
Revised: [Equation]

Original: where d j ( κ, p ) =
Revised: where d j ( κ, p ) =

Original: (cid:80) u ∈D I
Revised: (cid:80) u ∈D I

Original: [ H u
Revised: [ H u

Original: = j ] · p ( X u , S u ) · α u,k ( κ ) , r j ( κ ) =
Revised: = j ] · p ( X u , S u ) · α u,k ( κ ) , r j ( κ ) =

Original: (cid:80) u ∈D I
Revised: (cid:80) u ∈D I

Original: [ H u ≥ j ] · α u,k ( κ ) , with H u computed from { Y u,i } q u i =1 in eq.
Revised: [ H u ≥ j ] · α u,k ( κ ) , with H u computed from { Y u,i } q u i =1 in eq.

Original: (3) , and n k ( κ ) =
Revised: (3) , and n k ( κ ) =

Original: (cid:80) u α u,k ( κ ) , α u,k ( κ ) = κ ( X u ) 1 − k · (1 − κ ( X u ))
Revised: (cid:80) u α u,k ( κ ) , α u,k ( κ ) = κ ( X u ) 1 − k · (1 − κ ( X u ))

Original: k .
Revised: k .

Original: The following theorem states a few properties of our loss function.
Revised: The following theorem states a few properties of our loss function.

Original: Theorem 1 (Kuiper loss properties) .
Revised: Theorem 1 (Kuiper loss properties) .

Original: From Deﬁnition 4, consider two clusters with true lifetime distributions ˆ P ( U 1 ) and ˆ P ( U 2 ) .
Revised: From Deﬁnition 4, consider two clusters with true lifetime distributions ˆ P ( U 1 ) and ˆ P ( U 2 ) .

Original: Assume an inﬁnite number of samples/subjects.
Revised: Assume an inﬁnite number of samples/subjects.

Original: Then, the loss function deﬁned in equation (5) has the following properties:
Revised: Then, the loss function deﬁned in equation (5) has the following properties:

Original: (a) If the two clusters have distinct lifetime distributions,
Revised: (a) If the two clusters have distinct lifetime distributions,

Original: i
Revised: i

Original: e. ˆ P ( U 1 ) (cid:54) =
Revised: e. ˆ P ( U 1 ) (cid:54) =

Original: ˆ P ( U 2 ) then, either ∃ ˆ κ, ˆ p such that L (ˆ κ, ˆ p ) →
Revised: ˆ P ( U 2 ) then, either ∃ ˆ κ, ˆ p such that L (ˆ κ, ˆ p ) →

Original: −∞ , or ∀ κ, p , L ( κ, p ) → 
Revised: −∞ , or ∀ κ, p , L ( κ, p ) → 0 .

Original: (b) If the two clusters have the same stochastic process Ψ u (Deﬁnition 1), Ψ u
Revised: (b) If the two clusters have the same stochastic process Ψ u (Deﬁnition 1), Ψ u

Original: = Ψ v , for any two subjects u and v , regardless of cluster assignments, then ∀ κ, p , L ( κ, p ) → 0 .
Revised: = Ψ v , for any two subjects u and v , regardless of cluster assignments, then ∀ κ, p , L ( κ, p ) → 0 .

Original: We prove Theorem 1 in Appendix 7.1 by deﬁning the activity process of the subjects using shifted
Revised: We prove Theorem 1 in Appendix 7.1 by deﬁning the activity process of the subjects using shifted

Original: Random Marked Point Processes.
Revised: Random Marked Point Processes.

Original: The loss deﬁned above solves all the aforementioned issues; a) does not need clear end-of-life signals, b) use of a p-value forces sufﬁcient number of examples in both groups, c) does not assume proportionality of hazards and works even for crossing survival curves, and d) accounts for differences at the tails.
Revised: The loss deﬁned above solves all the aforementioned issues; a) does not need clear end-of-life signals, b) use of a p-value forces sufﬁcient number of examples in both groups, c) does not assume proportionality of hazards and works even for crossing survival curves, and d) accounts for differences at the tails.

Original: 3. N EURAL N ETWORK A PPROACH TO O PTIMIZE K UIPER L OSS
Revised: 3. N EURAL N ETWORK A PPROACH TO O PTIMIZE K UIPER L OSS

Original: In this section, we describe the functions κ ( · ) and p ( · ) in deﬁnition 4 κ ( · ) gives the probability of a subject u being in cluster 0, and we deﬁne it using a neural network as follows,
Revised: In this section, we describe the functions κ ( · ) and p ( · ) in deﬁnition 4 κ ( · ) gives the probability of a subject u being in cluster 0, and we deﬁne it using a neural network as follows,

Original: [Equation]
Revised: [Equation]

Original: where { W i , b i } Li =1 are the weights and the biases of a neural network with L − 1 hidden layers, X u are the covariates of subject u , φ is an activation function (tanh or relU in our experiments), and σ is the softmax function.
Revised: where { W i , b i } Li =1 are the weights and the biases of a neural network with L − 1 hidden layers, X u are the covariates of subject u , φ is an activation function (tanh or relU in our experiments), and σ is the softmax function.

Original: An example of a feedforward neural network that optimizes Kuiper loss is shown in ﬁgure 2b.
Revised: An example of a feedforward neural network that optimizes Kuiper loss is shown in ﬁgure 2b.

Original: Next, we describe the form of end-of-life probability function, p ( · ) .
Revised: Next, we describe the form of end-of-life probability function, p ( · ) .

Original: We make the reasonable assumption that p ( · ) is an increasing function of S u .
Revised: We make the reasonable assumption that p ( · ) is an increasing function of S u .

Original: For example, consider two subjects a and b , with last activities one year and one week before their respective time of censoring.
Revised: For example, consider two subjects a and b , with last activities one year and one week before their respective time of censoring.

Original: Clearly, it is more likely that subject
Revised: Clearly, it is more likely that subject

Original: a ’s activity is her last one than that b ’s activity is her last one.
Revised: a ’s activity is her last one than that b ’s activity is her last one.

Original: In our experiments, we also assume that p ( · ) only depends on S u , and not on the covariates X u .
Revised: In our experiments, we also assume that p ( · ) only depends on S u , and not on the covariates X u .

Original: Survival tasks commonly use the following naive technique to identify the end-of-life signal.
Revised: Survival tasks commonly use the following naive technique to identify the end-of-life signal.

Original: They deﬁne p ( · ) using a step function, p ( X u , S u ) = 

Revised: They deﬁne p ( · ) using a step function, p ( X u , S u ) =

Original: [ S u > W ] , where W is the size of an arbitrary window from the time of censoring (see Figure 2a).
Revised: [ S u > W ] , where W is the size of an arbitrary window from the time of censoring (see Figure 2a).

Original: However, this approach does not allow learning of the window size parameter W , and hence, the analysis can be highly coupled with the choice of W .
Revised: However, this approach does not allow learning of the window size parameter W , and hence, the analysis can be highly coupled with the choice of W .

Original: We remedy this by choosing p ( · ) to be a smooth non-decreasing function of S u , parameters of which can be learnt by minimizing the loss function L ( κ, p ) .
Revised: We remedy this by choosing p ( · ) to be a smooth non-decreasing function of S u , parameters of which can be learnt by minimizing the loss function L ( κ, p ) .

Original: We use the cumulative distribution function of an exponential distribution in our experiments, i.e, p ( X u , S u ) =
Revised: We use the cumulative distribution function of an exponential distribution in our experiments, i.e, p ( X u , S u ) =

Original: 1 − e − β · S u (Figure 2a).
Revised: 1 − e − β · S u (Figure 2a).

Original: The rate parameter, β , is learnt using gradient descent along with the weights of the neural network.
Revised: The rate parameter, β , is learnt using gradient descent along with the weights of the neural network.

Original: Extension to K Clusters Until now, we dealt with clustering the subjects into two groups.
Revised: Extension to K Clusters Until now, we dealt with clustering the subjects into two groups.

Original: However, it is not hard to extend the framework for K clusters.
Revised: However, it is not hard to extend the framework for K clusters.

Original: We increase the number of units in the output layer from 2 to K .
Revised: We increase the number of units in the output layer from 2 to K .

Original: As before, a softmax function applied at the output layer gives probabilities that deﬁne a soft clustering of the subjects into K groups.
Revised: As before, a softmax function applied at the output layer gives probabilities that deﬁne a soft clustering of the subjects into K groups.

Original: These probabilities can be used to obtain the loss, L A,B , between any two groups, D A and D B .
Revised: These probabilities can be used to obtain the loss, L A,B , between any two groups, D A and D B .

Original: We deﬁne the total loss for K groups as the average of all the pairwise losses between individual groups and get the geometric mean of the pairwise p-values, i.e.,
Revised: We deﬁne the total loss for K groups as the average of all the pairwise losses between individual groups and get the geometric mean of the pairwise p-values, i.e.,

Original: [Equation]
Revised: [Equation]

Original: In other words, the loss L 1 ...K is minimized only if each of the individual p-values are low indicating that each group’s lifetime distribution is different (in divergence) from every other group’s lifetime distribution.
Revised: In other words, the loss L 1 ...K is minimized only if each of the individual p-values are low indicating that each group’s lifetime distribution is different (in divergence) from every other group’s lifetime distribution.

Original: Implementation
Revised: Implementation

Original: We implement a feedforward neural network in Theano (Theano Development
Revised: We implement a feedforward neural network in Theano (Theano Development

Original: Team, 2016) and use ADAM (Kingma and Ba, 2014) to optimize the loss L 1 ...K deﬁned in equation 8.
Revised: Team, 2016) and use ADAM (Kingma and Ba, 2014) to optimize the loss L 1 ...K deﬁned in equation 8.

Original: Each iteration of the optimization takes as input a batch of subjects (full batch or a minibatch), generates a value for the loss, calculates the gradients, and updates the parameters ( β, { W i , b i } L i =1 ).
Revised: Each iteration of the optimization takes as input a batch of subjects (full batch or a minibatch), generates a value for the loss, calculates the gradients, and updates the parameters ( β, { W i , b i } L i =1 ).

Original: This is done repeatedly until there is no improvement in the validation loss.
Revised: This is done repeatedly until there is no improvement in the validation loss.

Original: We use L2 regularization over the weights and experiment with different values for the regularization parameter.
Revised: We use L2 regularization over the weights and experiment with different values for the regularization parameter.

Original: We also experiment with different neural network sizes (number of hidden layers, number of hidden units), activation functions for the hidden layers, and weight initialization techniques.
Revised: We also experiment with different neural network sizes (number of hidden layers, number of hidden units), activation functions for the hidden layers, and weight initialization techniques.

Original: We applied different deep learning techniques like batch normalization (Ioffe and Szegedy, 2015) and dropout (Srivastava et al., 2014) to better learn the neural network.
Revised: We applied different deep learning techniques like batch normalization (Ioffe and Szegedy, 2015) and dropout (Srivastava et al., 2014) to better learn the neural network.

Original: [Figure]
Revised: [Figure]

Original:  R ESULTS
Revised:  R ESULTS

Original: Dataset In this paper, we analyze a large-scale social network dataset collected from Friendster.
Revised: Dataset In this paper, we analyze a large-scale social network dataset collected from Friendster.

Original: After processing 30TB of data, originally collected by the Internet Archive in June 2011, the resulting network has around 15 million users with 335 million friendship links.
Revised: After processing 30TB of data, originally collected by the Internet Archive in June 2011, the resulting network has around 15 million users with 335 million friendship links.

Original: Each user has proﬁle information such as age, gender, marital status, occupation, and interests.
Revised: Each user has proﬁle information such as age, gender, marital status, occupation, and interests.

Original: Additionally, there are user comments on each other’s proﬁle pages with timestamps that indicate activity in the site.
Revised: Additionally, there are user comments on each other’s proﬁle pages with timestamps that indicate activity in the site.

Original: In our experiments, we only use the data up to March 2008 as Friendster’s monthly active users have been signiﬁcantly affected with the introduction of “new Facebook wall” (Ribeiro and Faloutsos, 2015).
Revised: In our experiments, we only use the data up to March 2008 as Friendster’s monthly active users have been signiﬁcantly affected with the introduction of “new Facebook wall” (Ribeiro and Faloutsos, 2015).

Original: From this, we only consider a subset of 1.1 million users who had participated in atleast one comment, and had speciﬁed their basic proﬁle information like age and gender.
Revised: From this, we only consider a subset of 1.1 million users who had participated in atleast one comment, and had speciﬁed their basic proﬁle information like age and gender.

Original: We make our processed data available to the public at location (anonymized).
Revised: We make our processed data available to the public at location (anonymized).

Original: We build the dataset D : { X u , { Y u,i } q u i =1 , S u } (Deﬁnition 1) for our clustering task as follows
Revised: We build the dataset D : { X u , { Y u,i } q u i =1 , S u } (Deﬁnition 1) for our clustering task as follows.

Original: X u : We use each user’s proﬁle information (like age, gender, relationship status, occupation and location) as features.
Revised: X u : We use each user’s proﬁle information (like age, gender, relationship status, occupation and location) as features.

Original: To help improve the performance of competing methods, we also construct additional features using the user’s activity over the initial 5 months (like number of comments sent and received, number of individuals interacted with, etc.).
Revised: To help improve the performance of competing methods, we also construct additional features using the user’s activity over the initial 5 months (like number of comments sent and received, number of individuals interacted with, etc.).

Original: Y u,i : We deﬁne Y u,i as the time between u ’s i th comment (sent or received) and ( i − 1) st comment (sent or received).
Revised: Y u,i : We deﬁne Y u,i as the time between u ’s i th comment (sent or received) and ( i − 1) st comment (sent or received).

Original: q ( u ) is then deﬁned as the total number of comments the user participated in.
Revised: q ( u ) is then deﬁned as the total number of comments the user participated in.

Original: S u : We calculate S u as the time between u ’s last activity and the chosen time of measurement (March 2008).
Revised: S u : We calculate S u as the time between u ’s last activity and the chosen time of measurement (March 2008).

Original: Model Training & Evaluation
Revised: Model Training & Evaluation

Original: We perform 10-fold cross validation as follows.
Revised: We perform 10-fold cross validation as follows.

Original: We split the dataset D randomly into 10 folds, sample 100,000 users without replacement from i th fold for testing and sample 100,000 users similarly from the remaining 9 folds for training.
Revised: We split the dataset D randomly into 10 folds, sample 100,000 users without replacement from i th fold for testing and sample 100,000 users similarly from the remaining 9 folds for training.

Original: We use 20% of the training data as validation  for early stopping in our neural network model.
Revised: We use 20% of the training data as validation data for early stopping in our neural network training.

Original: We repeat this for i ranging from 1 to 10.
Revised: We repeat this for i ranging from 1 to 10.

Original: We compare our clustering approach with the only two survival-based clustering methods in literature; a) Semi-supervised clustering (Bair and Tibshirani, 2004) and b) Supervised sparse clustering (Gaynor and Bair, 2013).
Revised: We compare our clustering approach with the only two survival-based clustering methods in literature; a) Semi-supervised clustering (Bair and Tibshirani, 2004) and b) Supervised sparse clustering (Gaynor and Bair, 2013).

Original: Since both these methods require clear end-of-life signals, we use an arbitrary window of 10 months (i.e., a pre-deﬁned “timeout”) prior to the time of measurement in order to obtain these signals (Figure 2a).
Revised: Since both these methods require clear end-of-life signals, we use an arbitrary window of 10 months (i.e., a pre-deﬁned “timeout”) prior to the time of measurement in order to obtain these signals (Figure 2a).

Original: We also try window sizes of 0 months (only the users with activity at t m are censored) and 5 months, and obtain similar results (not reported here).
Revised: We also try window sizes of 0 months (only the users with activity at t m are censored) and 5 months, and obtain similar results (not reported here).

Original: We test our approach in two cases – in the presence and lack of end-of-life signals.
Revised: We test our approach in two cases – in the presence and lack of end-of-life signals.

Original: In the former case, we optimize the loss function in equation (8) keeping p ( · ) ﬁxed to the end-of-life signals obtained from using a window of 10 months.
Revised: In the former case, we optimize the loss function in equation (8) keeping p ( · ) ﬁxed to the end-of-life signals obtained from using a window of 10 months.

Original: In the latter case, our approach learns the latent end-of-life signals.
Revised: In the latter case, our approach learns the latent end-of-life signals.

Original: We report performance of both the approaches.
Revised: We report performance of both the approaches.

Original: We evaluate the clusters obtained from each of the methods using concordance index.
Revised: We evaluate the clusters obtained from each of these methods using concordance index.

Original: [Table] Concordance Index Concordance index or C-index (Harrell et al., 1982) is a commonly used metric in survival applications (Alaa and van der Schaar, 2017; Luck et al., 2017) to quantify a model’s ability to discriminate between subjects with different lifetimes.
Revised: Concordance Index Concordance index or C-index (Harrell et al., 1982) is a commonly used metric in survival applications (Alaa and van der Schaar, 2017; Luck et al., 2017) to quantify a model’s ability to discriminate between subjects with different lifetimes.

Original: It calculates the fraction of pairs of subjects for which the model predicts the correct order of survival while also incorporating  censoring.
Revised: It calculates the fraction of pairs of subjects for which the model predicts the correct order of survival while also incorporating [Table] censoring.

Original: We use the end-of-life signalswe calculated using a pre-deﬁned “timeout” of 10 months.
Revised: We use the end-of-life signals calculated using a pre-deﬁned “timeout” of 10 months.

Original: Rather than populating all possible pairs of users, we sample 10000 random pairs to calculate the
Revised: Rather than populating all possible pairs of users, we sample 10000 random pairs to calculate the

Original: C-index.
Revised: C-index.

Original: Table 1 shows the C-index values for the baselines and the proposed method.
Revised: Table 1 shows the C-index values for the baselines and the proposed method.

Original:  The proposed neural network approach(with and without end-of-life signals) performs better on average than the two competing methods.
Revised: Discussion The proposed neural network approach performs better on average than the two competing methods.

Original: Even with no end-of-life signals, the proposed approach achieves comparable scores for K = 3 , 4 and the best C-index score for K = 2 .
Revised: Even without end-of-life signals, the proposed approach achieves comparable scores for K = 3 , 4 and the best C-index score for K = 2 .

Original: We use the end-of-life signals obtained using a window of 10 months to calculate the empirical lifetime distributions of the clusters identiﬁed by the neural network (Figure 3).
Revised: We use the end-of-life signals obtained using a window of 10 months to calculate the empirical lifetime distributions of the clusters identiﬁed by the neural network (Figure 3).

Original: The empirical lifetime distributions of clusters seem distinct from each other at K = 2 but not at K = 3 , 4 .
Revised: The empirical lifetime distributions of clusters seem distinct from each other at K = 2 but not at K = 3 , 4 .

Original: In addition, there is not signiﬁcant gain in the C-index values as we increase the number of clusters from K = 2 to K = 4 .
Revised: In addition, there is not signiﬁcant gain in the C-index values as we increase the number of clusters from K = 2 to K = 4 .

Original: Hence, we can conclude that there are only two types of users in the Friendster dataset – long-lived and short-lived .
Revised: Hence, we can conclude that there are only two types of users in the Friendster dataset – long-lived and short-lived .

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: Majority of the work in survival analysis has dealt with the task of predicting the survival outcome especially when the number of features is much higher than the number of subjects (Witten and
Revised: Majority of the work in survival analysis has dealt with the task of predicting the survival outcome especially when the number of features is much higher than the number of subjects (Witten and

Original: Tibshirani, 2010b; Bøvelstad et al., 2007; Hothorn et al., 2006; Shivaswamy, Chu, and Jansche, 2007).
Revised: Tibshirani, 2010b; Bøvelstad et al., 2007; Hothorn et al., 2006; Shivaswamy, Chu, and Jansche, 2007).

Original: A number of approaches have also been proposed to perform feature selection in survival data (Ishwaran et al., 2010; Lagani and Tsamardinos, 2010).
Revised: A number of approaches have also been proposed to perform feature selection in survival data (Ishwaran et al., 2010; Lagani and Tsamardinos, 2010).

Original: In the social network scenario, Sun et al.
Revised: In the social network scenario, Sun et al.

Original: (2012) tried to predict the relationship building time, that is, the time until a particular link is formed in the network.
Revised: (2012) tried to predict the relationship building time, that is, the time until a particular link is formed in the network.

Original: Many unsupervised approaches have been proposed to identify cancer subtypes in gene expression data without considering the survival outcome (Eisen et al., 1998; Alizadeh et al., 2000; Bhattacharjee et al., 2001).
Revised: Many unsupervised approaches have been proposed to identify cancer subtypes in gene expression data without considering the survival outcome (Eisen et al., 1998; Alizadeh et al., 2000; Bhattacharjee et al., 2001).

Original: Traditional semi-supervised clustering methods (Aggarwal, Gates, and Yu, 2004;
Revised: Traditional semi-supervised clustering methods (Aggarwal, Gates, and Yu, 2004;

Original: Basu, Banerjee, and Mooney, 2002; Basu, Bilenko, and Mooney, 2004; Nigam et al., 1998) do not perform well in this scenario since they do not provide a way to handle the issues with right censoring.
Revised: Basu, Banerjee, and Mooney, 2002; Basu, Bilenko, and Mooney, 2004; Nigam et al., 1998) do not perform well in this scenario since they do not provide a way to handle the issues with right censoring.

Original: Semi-supervised clustering (Bair and Tibshirani, 2004) and supervised sparse clustering Witten and Tibshirani (2010a) use Cox scores (Cox, 1992) to identify features associated with survival.
Revised: Semi-supervised clustering (Bair and Tibshirani, 2004) and supervised sparse clustering Witten and Tibshirani (2010a) use Cox scores (Cox, 1992) to identify features associated with survival.

Original: They treat these features differently in order to perform clustering based on the survival outcome.
Revised: They treat these features differently in order to perform clustering based on the survival outcome.

Original: Although there are quite a few works on using neural networks to predict the hazard rates of individuals (Eleuteri et al., 2003; 2007), to the best of our knowledge, there hasn’t been a work on using neural networks for a survival-based clustering task.
Revised: Although there are quite a few works on using neural networks to predict the hazard rates of individuals (Eleuteri et al., 2003; 2007), to the best of our knowledge, there hasn’t been a work on using neural networks for a survival-based clustering task.

Original: Recently, Alaa and van der Schaar (2017) proposed a nonparametric Bayesian approach for survival analysis in the case of more than one competing events (multiple diseases).
Revised: Recently, Alaa and van der Schaar (2017) proposed a nonparametric Bayesian approach for survival analysis in the case of more than one competing events (multiple diseases).

Original: They not only assume the presence of end-of-life signals but also the type of event that caused the end-of-life .
Revised: They not only assume the presence of end-of-life signals but also the type of event that caused the end-of-life .

Original: Luck et al.
Revised: Luck et al.

Original: (2017) optimize a loss based on Cox’s partial likelihood along with a penalty using a deep neural network to predict the probability of survival at a point in time.
Revised: (2017) optimize a loss based on Cox’s partial likelihood along with a penalty using a deep neural network to predict the probability of survival at a point in time.

Original: Here we considered the task of clustering subjects into low-risk/high-risk groups without observing any end-of-life signals.
Revised: Here we considered the task of clustering subjects into low-risk/high-risk groups without observing any end-of-life signals.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this work we introduced a Kuiper-based nonparametric loss function, and a corresponding backpropagation procedure (which backpropagates the loss over clusters rather than the loss per training example).
Revised: In this work we introduced a Kuiper-based nonparametric loss function, and a corresponding backpropagation procedure (which backpropagates the loss over clusters rather than the loss per training example).

Original: These procedures are then used to train a feedforward neural network to inductively assign observed subject covariates into K survival-based clusters, from high-risk to low-risk subjects, without requiring an end-of-life signal
Revised: These procedures are then used to train a feedforward neural network to inductively assign observed subject covariates into K survival-based clusters, from high-risk to low-risk subjects, without requiring an end-of-life signal.

Original:  A PPENDIX 7. P ROOF OF T HEOREM
Revised:  A PPENDIX 7. P ROOF OF T HEOREM

Original: We restate the theorem for convenience.
Revised: We restate the theorem for convenience.

Original: Theorem 1 (Kuiper loss properties) .
Revised: Theorem 1 (Kuiper loss properties) .

Original: From Deﬁnition 4, consider two clusters with true lifetime distributions ˆ P ( U 1 ) and ˆ P ( U 2 ) .
Revised: From Deﬁnition 4, consider two clusters with true lifetime distributions ˆ P ( U 1 ) and ˆ P ( U 2 ) .

Original: Assume an inﬁnite number of samples/subjects.
Revised: Assume an inﬁnite number of samples/subjects.

Original: Then, the loss function deﬁned in equation (5) has the following properties:
Revised: Then, the loss function deﬁned in equation (5) has the following properties:

Original: (a) If the two clusters have distinct lifetime distributions,i.e. ˆ P ( U 1 ) (cid:54)
Revised: (a) If the two clusters have distinct lifetime distributions,

Original: =
Revised: i

Original:  ˆ P ( U 2 ) then, either ∃ ˆ κ, ˆ p such that L (ˆ κ, ˆ p ) →
Revised: e. ˆ P ( U 1 ) (cid:54) = ˆ P ( U 2 ) then, either ∃ ˆ κ, ˆ p such that L (ˆ κ, ˆ p ) →

Original: −∞ , or ∀ κ, p , L ( κ, p ) → 0 .
Revised: −∞ , or ∀ κ, p , L ( κ, p ) → 0 .

Original: (b) If the two clusters have the same stochastic process Ψ u (Deﬁnition 1), Ψ u
Revised: (b) If the two clusters have the same stochastic process Ψ u (Deﬁnition 1), Ψ u

Original: = Ψ v , for any two subjects u and v , regardless of cluster assigments, then ∀ κ, p , L ( κ, p ) → 0 .
Revised: = Ψ v , for any two subjects u and v , regardless of cluster assigments, then ∀ κ, p , L ( κ, p ) → 0 .

Original: Both parts (a) and (b) of our proof need the following general deﬁnition that translates into the observed data D u for subject u into a stochastic process.
Revised: Both parts (a) and (b) of our proof need the following general deﬁnition that translates into the observed data D u for subject u into a stochastic process.

Original: The deﬁnition is general enough to avoid making any signiﬁcant assumption about the data.
Revised: The deﬁnition is general enough to avoid making any signiﬁcant assumption about the data.

Original: Deﬁnition 5 (Observable shifted RMPP cluster process) .
Revised: Deﬁnition 5 (Observable shifted RMPP cluster process) .

Original: We will deﬁne the activity process of subjects in cluster k as a Random Marked Point Processes (RMPP).
Revised: We will deﬁne the activity process of subjects in cluster k as a Random Marked Point Processes (RMPP).

Original: The RMPP is Φ k = { X k , { ( A k,i , Y k,i ) } i ∈
Revised: The RMPP is Φ k = { X k , { ( A k,i , Y k,i ) } i ∈

Original: Z ∗ , S k } , where Y k,i is the inter-event time between the ( i − 1) -st and the ith activities, X k are the random variable representing the covariates of subjects in cluster k , S k is the time from last event until censoring at cluster k , and A k,i = 0 indicates an activity with an end-of-life signal,
Revised: Z ∗ , S k } , where Y k,i is the inter-event time between the ( i − 1) -st and the ith activities, X k are the random variable representing the covariates of subjects in cluster k , S k is the time from last event until censoring at cluster k , and A k,i = 0 indicates an activity with an end-of-life signal,

Original: otherwise A k,i = 1 .
Revised: otherwise A k,i = 1 .

Original: This deﬁnition is model-free, i.e., we will not prescribe a model for Φ k .
Revised: This deﬁnition is model-free, i.e., we will not prescribe a model for Φ k .

Original: Note that, theoretically, Φ k continues to evolve beyond the end-of-life signal, but this evolution will be ignored as it is irrelevant to us.
Revised: Note that, theoretically, Φ k continues to evolve beyond the end-of-life signal, but this evolution will be ignored as it is irrelevant to us.

Original: The relative time of the i -th activity of a subject of cluster k , since the subject’s ﬁrst activity, is (cid:80)i (cid:48) ≤ i Y k,i (cid:48) , as long as we haven’t seen an end-of-life signal, i.e., (cid:81) i (cid:48) < i A k,i = 1 .
Revised: The relative time of the i -th activity of a subject of cluster k , since the subject’s ﬁrst activity, is (cid:80)

Original: Deﬁnition 6 (RMPP Lifetime) .
Revised: Deﬁnition 6 (RMPP Lifetime) .

Original: The random variable that deﬁnes the lifetime of a subject of cluster k is
Revised: The random variable that deﬁnes the lifetime of a subject of cluster k is

Original: [Equation]
Revised: [Equation]

Original: We now deﬁne H u using Φ k .
Revised: We now deﬁne H u using Φ k .

Original: Deﬁnition 7 (RMPP Censored Lifetimes) .
Revised: Deﬁnition 7 (RMPP Censored Lifetimes) .

Original: The random variable that deﬁnes the last observed action time of a subject u of cluster k is
Revised: The random variable that deﬁnes the last observed action time of a subject u of cluster k is

Original: [Equation]
Revised: [Equation]

Original: where δ k,i = 1 { (cid:80) i (cid:48) ≤
Revised: where δ k,i = 1 { (cid:80) i (cid:48) ≤

Original: i Y k,i (cid:48)
Revised: i Y k,i (cid:48)

Original: ≤ S k } .
Revised: ≤ S k } .

Original: Let i (cid:63) ( S k ) be a random variable that denotes the number of events until the censoring time S k .
Revised: Let i (cid:63) ( S k ) be a random variable that denotes the number of events until the censoring time S k .

Original: The main challenge is not knowing when H k = T k , because we are unable to observed the end-of-life signal A k,i (cid:63) ( S k )
Revised: The main challenge is not knowing when H k = T k , because we are unable to observed the end-of-life signal A k,i (cid:63) ( S k )

Original: = 0 .
Revised: = 0 .

Original: Clearly, this affects the decision of which subjects have been censored and which have not.
Revised: Clearly, this affects the decision of which subjects have been censored and which have not.

Original: The end-of-life signal probability p : ( X u , D u ) → [0 , 1] only affects S k ( t ; κ, p ) of equation (6).
Revised: The end-of-life signal probability p : ( X u , D u ) → [0 , 1] only affects S k ( t ; κ, p ) of equation (6).

Original: Proof of (a) : If the two clusters have distinct lifetime distributions, it means that the distributions of T 0 and T 1 in eq.
Revised: Proof of (a) : If the two clusters have distinct lifetime distributions, it means that the distributions of T 0 and T 1 in eq.

Original: (9) are different.
Revised: (9) are different.

Original: Then, either the right-censoring δ in eq.
Revised: Then, either the right-censoring δ in eq.

Original: (10) does not allow us to see the difference between T 0 and T 1 , and then there is no mappings ˆ p and ˆ κ that can get the distribution of S 0 ( t ; ˆ κ, ˆ p ) and S 1 ( t ; ˆ κ, ˆ p ) to be distinct, implying an L ( κ, p ) → 0 , as n → ∞ as the observations come from the same distribution, making the Kuiper score asymptotically equal to one; or δ does allow us to see the difference and then, clearly ˆ p ≡ 0 with a mapping ˆ κ that assigns more than half of the subjects to their correct clusters, which would allow us to see the difference in Hand H 1 , would give Kuiper score asymptotically equal to zero.
Revised: (10) does not allow us to see the difference between T 0 and T 1 , and then there is no mappings ˆ p and ˆ κ that can get the distribution of S 0 ( t ; ˆ κ, ˆ p ) and S 1 ( t ; ˆ κ, ˆ p ) to be distinct, implying an L ( κ, p ) → 0 , as n → ∞ as the observations come from the same distribution, making the Kuiper score asymptotically equal to one; or δ does allow us to see the difference and then, clearly ˆ p ≡ 0 with a mapping ˆ κ that assigns more than half of the subjects to their correct clusters, which would allow us to see the difference in Hand H 1 , would give Kuiper score asymptotically equal to zero.

Original: Thus, L ( κ, p ) →
Revised: Thus, L ( κ, p ) →

Original: −∞ , as n → ∞ .
Revised: −∞ , as n → ∞ .

Original: Proof of (b) : Because κ only take the subject covariates as input, and there are no dependencies between the subject covariates and the subject lifetime in eq.
Revised: Proof of (b) : Because κ only take the subject covariates as input, and there are no dependencies between the subject covariates and the subject lifetime in eq.

Original: (9), any clustering based on the covariates will be a random assignment of users into clusters.
Revised: (9), any clustering based on the covariates will be a random assignment of users into clusters.

Original: Moreover, from eq.
Revised: Moreover, from eq.

Original: (10), the censoring time of subject u , S u , has the same distribution for both clusters because the RMPPs are the same.
Revised: (10), the censoring time of subject u , S u , has the same distribution for both clusters because the RMPPs are the same.

Original: Thus,
Revised: Thus,

Original: H 

Revised: H

Original: d = H 1 , i.e., H 0 and H 1 have the same distributions, and the Kuiper p -value test returns zero,
Revised: d = H 1 , i.e., H 0 and H 1 have the same distributions, and the Kuiper p -value test returns zero,

Original: L ( κ, p ) → 0 , as n → ∞ .
Revised: L ( κ, p ) → 0 , as n → ∞ .