Original: I NDIVIDUALITY IN THE HIVE - L EARNING TO EMBED LIFETIME SOCIAL BEHAVIOR OF HONEY BEES
Revised: I NDIVIDUALITY IN THE HIVE - L EARNING TO EMBED LIFETIME SOCIAL BEHAVIOR OF HONEY BEES

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: Honey bees are a popular model for complex social systems, in which global behavior emerges from the actions and interactions of thousands of individuals.
Revised: Honey bees are a popular model for complex social systems, in which global behavior emerges from the actions and interactions of thousands of individuals.

Original: While the average life of a bee is organized as a sequence of tasks roughly determined by age, there is substantial variation at the individual level.
Revised: While the average life of a bee is organized as a sequence of tasks roughly determined by age, there is substantial variation at the individual level.

Original: For example, young bees can become foragers early in life, depending on the colony’s needs.
Revised: For example, young bees can become foragers early in life, depending on the colony’s needs.

Original: Using a unique dataset containing lifetime trajectories of all individuals over multiple generations in two honey bee colonies, we propose a new temporal matrix factorization model that jointly learns the average developmental path and structured variations of individuals in the social network over their entire lives.
Revised: Using a unique dataset containing lifetime trajectories of all individuals over multiple generations in two honey bee colonies, we propose a new temporal matrix factorization model that jointly learns the average developmental path and structured variations of individuals in the social network over their entire lives.

Original: Our method yields inherently interpretable embeddings that are biologically plausible and consistent over time, which allow one to compare individuals regardless of when, or in which colony, they lived.
Revised: Our method yields inherently interpretable embeddings that are biologically plausible and consistent over time, which allows comparing individuals regardless of when or in which colony they lived.

Original: Our method provides a quantitative framework for understanding behavioral heterogeneity in complex social systems applicable in ﬁelds such as behavioral biology, social sciences, neuroscience, and information science.
Revised: Our method provides a quantitative framework for understanding behavioral heterogeneity in complex social systems applicable in ﬁelds such as behavioral biology, social sciences, neuroscience, and information science.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Animals living in large groups often coordinate their behaviors, resulting in emergent properties at the group level, from ﬂocking birds to democratic elections.
Revised: Animals living in large groups often coordinate their behaviors, resulting in emergent properties at the group level, from ﬂocking birds to democratic elections.

Original: In most animal groups, the role an individual plays in this process is thought to be reﬂected in the way it interacts with group members.
Revised: In most animal groups, the role an individual plays in this process is thought to be reﬂected in the way it interacts with group members.

Original: Technological advances have made it possible to track all individuals and their interactions in animal societies, ranging from social insects to primate groups (Mersch et al., 2013; Gernat et al., 2018;
Revised: Technological advances have made it possible to track all individuals and their interactions in animal societies, ranging from social insects to primate groups (Mersch et al., 2013; Gernat et al., 2018;

Original: Mathis et al., 2018; Graving et al., 2019; Pereira et al., 2019).
Revised: Mathis et al., 2018; Graving et al., 2019; Pereira et al., 2019).

Original: These datasets have unprecedented scale and complexity, but understanding these data has emerged as a new and challenging problem in itself (Pinter-Wollman et al., 2014; Krause et al., 2015; Brask et al., 2020).
Revised: These datasets have unprecedented scale and complexity, but understanding these data has emerged as a new and challenging problem in itself (Pinter-Wollman et al., 2014; Krause et al., 2015; Brask et al., 2020).

Original: A popular approach to understand high-dimensional data is to learn semantic embeddings (Frome et al., 2013; Asgari & Mofrad, 2015; Camacho-Collados & Pilehvar, 2018; Nelson et al., 2019).
Revised: A popular approach to understand high-dimensional data is to learn semantic embeddings (Frome et al., 2013; Asgari & Mofrad, 2015; Camacho-Collados & Pilehvar, 2018; Nelson et al., 2019).

Original: Such embeddings can be learned without supervision, are interpretable, and are useful for accomplishing downstream tasks.
Revised: Such embeddings can be learned without supervision, are interpretable, and are useful for accomplishing downstream tasks.

Original: Individuals in animal societies can be described with semantic embeddings extracted from social interaction networks using matrix factorization methods.
Revised: Individuals in animal societies can be described with semantic embeddings extracted from social interaction networks using matrix factorization methods.

Original: For example, in symmetric nonnegative matrix factorization (SymNMF), the dot products of any two animals’ factor vectors reconstruct the interaction matrix (Wang et al. (2011); Shi et al. , see Figure 1 a and b).
Revised: For example, in symmetric non-negative matrix factorization (SymNMF), the dot products of any two animals’ factor vectors reconstruct the interaction matrix (Wang et al., 2011; Shi et al., 2015), see Figure 1 a and b).

Original: If the embeddings allow us to predict relevant behavioral properties, they serve our understanding as semantic representations.
Revised: If the embeddings allow us to predict relevant behavioral properties, they serve our understanding as semantic representations.

Original: However, in temporal settings, i.e. when the interaction matrices change over time, there is no straightforward extension of this algorithm.
Revised: However, in temporal settings where the interaction matrices change over time, there is no straightforward extension of this algorithm.

Original: The interaction matrices at different time points can be factorized individually, but there is no guarantee that the embeddings stay semantically consistent over time, i.e. theaccuracy or our prediction of relevant behavioral properties will deteriorate.
Revised: The interaction matrices at different time points can be factorized individually, but there is no guarantee that the embeddings stay semantically consistent over time and the prediction of relevant behavioral properties will deteriorate.

Original: In living systems, interaction dynamics are highly variable; individuals differ in when they appear in the data, and how long they live.
Revised: In living systems, interaction dynamics are highly variable; individuals differ in when they appear in the data and how long they live.

Original: We may also have access to partial observations from nonoverlapping groups, e.g., from different years with completely disjointed individuals.
Revised: Different non-overlapping groups of individuals, e.g. from different years, may not interact with each other at all.

Original: How  do we learn embeddings that generalize to these observations and still provide insights into these individuals’ functional roles?
Revised: How can we ﬁnd a common semantic embedding even in these extreme cases? How do we learn embeddings that generalize to different groups and still provide insights into each individual’s functional role?

Original: If animals take on roles  determined by a common factor, such as age, how can we learn this dependency from independent observations?
Revised: If animals take on roles partially determined by a common factor, such as age, how can we learn this dependency?

Revised: Several approaches to extend NMF to temporal settings have been proposed in a variety of problem settings.

Revised: Yu et al.

Revised: (2016) and Mackevicius et al.

Revised: (2019) propose a factorization method for time

Original: Individual ID
Revised: Individual ID

Original: I nd i
Revised: I nd i

Original: v i du a l
Revised: v i du a l

Original: I
Revised: I

Original: D
Revised: D

Original: Interactions
Revised: Interactions

Original: Factor
Revised: Factor

Original: F ac t
Revised: F ac t

Original: o r
Revised: o r

Original: SymNMF 2D Factors
Revised: SymNMF 2D Factors

Original: I nd i
Revised: I nd i

Original: v i du a l
Revised: v i du a l

Original: I
Revised: I

Original: D 2019-082019-082019-09I nd
Revised: D 2019-082019-082019-09I nd

Original: i v i du a l
Revised: i v i du a l

Original: I
Revised: I

Original: D 2019-

Revised: D 2019-

Original: Age in days
Revised: Age in days

Original: Individual ID
Revised: Individual ID

Original: [Figure]
Revised: [Figure]

Original: Several approaches to extend NMF to temporal settings have been proposed in a variety of problem settings. Yu et al. (2016) and Mackevicius et al. (2019) propose a factorization method for time series analysis.
Revised: series analysis.

Original: Gauvin et al.
Revised: Gauvin et al.

Original: (2014) focus on the analysis of communities that are determined by their temporal activity patterns.
Revised: (2014) focus on the analysis of communities that are determined by their temporal activity patterns.

Original: Jiao et al.
Revised: Jiao et al.

Original: (2017) consider the case of communities from graphs over time and enforce temporal consistency with an additional loss term.
Revised: (2017) consider the case of communities from graphs over time and enforce temporal consistency with an additional loss term.

Original: Yu et al.
Revised: Yu et al.

Original: (2017) and Wang et al.
Revised: (2017) and Wang et al.

Original: (2017) represent networks as a function of time.
Revised: (2017) represent networks as a function of time.

Original:  However, to our knowledge, no method yet considers the case of entities following a shared temporal structure starting at the time they ﬁrst appear in the dataset, or even in completely separate datasets.
Revised: Temporal matrix factorization can be seen as a tensor decomposition problem, for which methods with many applications have been proposed in the literature, see Kolda & Bader (2009) for a review. In particular, time-shifted tensor decomposition methods have been used in multi-neuronal spike train analysis when recordings of multiple trials from a population of neurons are available (Mørup et al., 2008; Williams, 2020). However, to our knowledge, no method yet considers the case of entities following a common trajectory depending on an observable property (e.g., the age of an individual), which we show to be a powerful inductive bias.

Original: We approach this question using honey bees, a popular model system for studying individual and collective behavior (Elekonich & Roberts, 2005).
Revised: We approach this question using honey bees, a popular model system for studying individual and collective behavior (Elekonich & Roberts, 2005).

Original: Bees allocate tasks across thousands of individuals without central control, using an age-based system: young bees care for brood, middle-aged bees perform within-nest labor, and old bees forage outside (Seeley, 1982; Johnson, 2010).
Revised: Bees allocate tasks across thousands of individuals without central control, using an age-based system: young bees care for brood, middle-aged bees perform within-nest labor, and old bees forage outside (Seeley, 1982; Johnson, 2010).

Original: Colonies are also organized spatially: brood is reared in the center, honey and pollen are stored at the periphery, and foragers ofﬂoad nectar near the exit.
Revised: Colonies are also organized spatially: brood is reared in the center, honey and pollen are stored at the periphery, and foragers ofﬂoad nectar near the exit.

Original: Therefore, an individual’s role can be inferred from its location, which we use to interpret the embeddings our method learns.
Revised: Therefore, an individual’s role is partially reﬂected in its location, which allows us to evaluate whether the embeddings our method learns are semantically meaningful.

Original: We propose jointly learning two meaningful representations of honey bee social behavior:  a vector representation of the individuals’ functional position in the social network on a given day, and a parametric representation of their lifetime trajectories through this functional embedding.
Revised: We propose jointly learning two meaningful representations of honey bee social behavior: 1) an individuality embedding that characterizes the lifetime behavior of each individual and 2) a daily representation of the individual’s functional position in the social network that can be derived from the individuality embeddings .

Original: We show that these representations can be learned in an unsupervised fashion, using only interaction matrices of the individuals over time.
Revised: We show that these representations can be learned in an unsupervised fashion, using only interaction matrices of the individuals over time.

Original: We analyze a dataset obtained by tracking thousands of individually marked honey bees in two colonies, at high temporal and spatial resolution over a total of 155 days, covering entire lifespans and multiple generations.
Revised: We analyze a dataset obtained by tracking thousands of individually marked honey bees in two colonies, at high temporal and spatial resolution over a total of 155 days, covering entire lifespans and multiple generations.

Original: We evaluate how well the embeddings capture the semantic differences of individual honey bee development by evaluating their predictiveness for different tasks and behaviorally relevant metrics.
Revised: We evaluate how well the embeddings capture the semantic differences of individual honey bee development by evaluating their predictiveness for different tasks and behaviorally relevant metrics.

Original: Our method can be used to study dynamic groups in which individual units change their latent properties over time, and group composition is not ﬁxed due to birth, death, or migration of individuals.

Original: e φ ( i, 0)e φ ( i, 1)e φ ( i, 2) B
Revised: φ ( i, 0) φ ( i, 1) φ ( i, 2) B

Original: a s
Revised: a s

Original: i s f un c ti on s bω k
Revised: i s f un c ti on s b k

Original: M ea n mθ
Revised: M ea n m

Original: Individual i
Revised: Individual i

Original: Individual j f ( t, j ) f ( t, i ) i
Revised: Individual j f ( t, j ) f ( t, i ) i

Original: j a) b) c) d) e)
Revised: j a) b) c) d) e)

Original: Interactions L adversarial
Revised: Interactions A t age

Original: L reconstruction c ( t, i )

Original: [Figure]
Revised: [Figure]

Original: [Figure] , i )
Revised: Our method can be used to study dynamic groups in which individual units change their latent properties over time, and group composition is not ﬁxed due to birth, death, or migration of individuals.

Original: We provide a new perspective on how to extract fundamental factors that underlie behavioral and developmental plasticity in animal groups.
Revised: We provide a new perspective on how to extract fundamental factors that underlie behavioral and developmental plasticity in animal groups.

Original:  M ETHODS 2. T EMPORAL NMF ALGORITHM
Revised:  M ETHODS 2. T EMPORAL NMF ALGORITHM

Original: SymNMF factorizes a matrix A ∈ R N × N + such that it can be approximated by the product F F T , where F ∈ R N × M + and M (cid:28) N :
Revised: SymNMF factorizes a matrix A ∈ R N × N + such that it can be approximated by the product F F T , where F ∈ R N × M + and M (cid:28) N :

Original: [Equation]
Revised: [Equation]

Revised: When applied to social networks, f ( i ) can represent the role of an entity within the social network

Revised: A (Wang et al., 2011; Shi et al., 2015) - however, in temporal settings, factorizing the matrices for different times separately will result in semantically inconsistent factors F .

Original: Here we present a novel temporal NMF algorithm ( TNMF ) which extends SymNMF to temporal settings in which A ∈ R T × N × N + changes over time t .
Revised: Here we present a novel temporal NMF algorithm ( TNMF ) which extends SymNMF to temporal settings in which A ∈ R T × N × N + changes over time t .

Original: We assume that the entities i ∈ { 0 , 1 , . . .
Revised: We assume that the entities i ∈ { 0 , 1 , . . .

Original: , N } follow to some extent a common trajectory depending on an observable property (for example the age of an individual).
Revised: , N } follow to some extent a common trajectory depending on an observable property (for example the age of an individual).

Original: We represent an entity at a speciﬁc point in time t using a factor vector f + ( t, i )∈ R M + such that
Revised: We represent an entity at a speciﬁc point in time t using a factor vector f + ( t, i ) such that

Original: [Equation]
Revised: [Equation]

Original: In contrast to SymNMF, we do not directly factorize A t to ﬁnd the optimal factors that reconstruct the matrices.
Revised: In contrast to SymNMF, we do not directly factorize A t to ﬁnd the optimal factors that reconstruct the matrices.

Original: Instead, we decompose the problem into learning a common trajectory of factors m θ and structured variations from this trajectory o φ,ω :
Revised: Instead, we decompose the problem into learning an average trajectory of factors m ( c ( t, i )) and structured variations from this trajectory o ( t, i ) that depend on the observable property c ( t, i ) :

Original: [Equation]
Revised: [Equation]

Revised: This decomposition is an inductive bias that allows the model to learn semantically consistent factors for entities, even if they do not share any data points (e.g., there is no overlap in their interaction partners), as long as the relationship between functional role and c ( t, i ) is stable.

Original: Note that in the simplest case c ( t, i ) = t ,  i.e. the trajectory of all entities is aligned with the temporal dimension t of A .
Revised: Note that in the simplest case c ( t, i ) = t , TNMF can be seen as a tensor decomposition model, i.e. the trajectory of all entities is aligned with the temporal dimension t of A .

Original: In social networks c ( t, i ) could map to the age of individual i at time t .
Revised: In our case, c ( t, i ) maps to the age of individual i at time t .

Original: We obtain o φ,ω from alearned set of individuality basis functions b ω k (shared among all entities) that deﬁne a coordinate system of possible individual variations and  individuality embeddingse φ  which capture to what extent each basis function applies to an entity:
Revised: While many parameterizations for the function o ( t, i ) are possible, we only consider one particular case in this work: We learn a set of individuality basis functions b ( c ( t, i )) (shared among all entities) that deﬁne a coordinate system of possible individual variations and the individuality embeddings φ , which capture to what extent each basis function applies to an entity:

Original: [Equation]
Revised: [Equation]

Original: where N k is the number of learned basis functions.
Revised: where K is the number of learned basis functions.

Revised: This parameterization allows us to disentangle the forms of individual variability ( individuality basis functions ) and the distribution of this variability ( individuality embeddings ) in the data.

Original: We implement the functions m θ , b ω j with small fully connected neural networks with non-linearities and several hidden layers.
Revised: We implement the functions m ( c ( t, i )) and b ( c ( t, i )) with small fully connected neural networks with non-linearities and several hidden layers.

Original: The parameters  of these functions and the entities’ embeddingse φ are learned jointly using minibatch stochastic gradient descent:
Revised: The parameters θ of these functions and the entities’ embeddings φ are learned jointly using minibatch stochastic gradient descent:

Original: [Equation]
Revised: [Equation]

Original: where the regularisation term R f − is added to encourage the model to learn positive values for all possible inputs because the gradient of the max operation in equation 3 is zero for negative values. Note thatthe non-negativityconstraint is not strictly necessary, but we only consider the non-negative case in this work for consistency with prior work (Wang et al., 2011; Shi et al., 2015). 
Revised: Note that non-negativity is not strictly necessary, but we only consider the non-negative case in this work for consistency with prior work (Wang et al., 2011; Shi et al., 2015). Furthermore, instead of one common property with discrete time steps, the factors could depend on multiple continuous properties, i.e. c : R T × N → R P , e.g. the day and time in a intraday analysis of social networks.

Original: 2. R EGULARIZATION
Revised: 2. R EGULARIZATION

Revised: Due to the inductive bias, the model performs well without additional explicit regularization in the datasets considered in this work (see Table 2).

Original: We use several regularization terms to increase the  interpretability and semantic consistency of the model.
Revised: However, we ﬁnd that the model’s interpretability can be improved using additional regularization terms without signiﬁcantly affecting its performance.

Original: We encourage sparsity in both the number of used factors and individuality basis functions by adding L 1 penalties of the mean absolute magnitude of the factors f
Revised: We encourage sparsity in both the number of used factors and individuality basis functions by adding Lpenalties of the mean absolute magnitude of the factors f

Original: ( t, i ) and basis functions b ω j to the objective.
Revised: ( t, i ) and basis functions b ( c ( t, i )) to the objective.

Original: We encourage individuals’ lifetimes to be represented with a sparse embedding using an L 1 penalty of the learned individuality embeddingse φ .
Revised: We encourage individuals’ lifetimes to be represented with a sparse embedding using an L 1 penalty of the learned individuality embeddings φ .

Original: We also introduce an  adversarial loss term to encourage the model to learn embeddings that are semantically consistent over time, i.e. to only represent two entities that were present in the dataset at different times with different embeddings if this is strictly necessary to factorize the matrices A .
Revised: We also introduce an optional adversarial loss term to encourage the model to learn embeddings that are semantically consistent over time, i.e. to only represent two entities that were present in the dataset at different times with different embeddings if this is strictly necessary to factorize the matrices A .

Original: We jointly train a discriminative network dψ ( e ) that tries to classify the time of  ﬁrst occurrence of all entities based on their individuality embeddingse φ .
Revised: We jointly train a discriminative network d ( φ i ) that tries to classify the time of the ﬁrst occurrence of all entities based on their individuality embeddings φ .

Original: The  cross-entropy loss of this model is added as a regularization term to equationin a training regime similar to generative adversarial networks (Goodfellow et al., 2014).
Revised: The negative cross-entropy loss of this model is added as a regularization term to equation 5 in a training regime similar to generative adversarial networks (Goodfellow et al., 2014).

Original: We found that the model performs well without these regularizations, but we are willing to accept a higher methodological complexity for the honey bee data if this allows us to achieve a better interpretability of the results.

Original: See appendix A.2.1 for more details and A.4 for an ablation study of the effect the individual regularization terms have on the results of the model.
Revised: See appendix A.2.1 for more details and A.4 for an ablation study of the effect the individual regularization terms have on the results of the model.

Original: 2. D ATA2.3. H ONEY BEE DATA
Revised: 2. D ATA

Revised: H ONEY BEE DATA

Original: [Table]
Revised: [Table]

Original: Two colonies of honey bees were continuously recorded over a total of 155 days.
Revised: Two colonies of honey bees were continuously recorded over a total of 155 days.

Original: Each individual was manually tagged at emergence, so the date of birth is known for each bee.
Revised: Each individual was manually tagged at emergence, so the date of birth is known for each bee.

Original: Locations and identities of all honey bees (N= 9286 ) were extracted from the raw images and used to construct daily aggregated temporal interaction networks based on  spatial proximity.
Revised: Locations and identities of all honey bees (N= 9286 ) were extracted from the raw images and used to construct daily aggregated temporal interaction networks based on the counts of spatial proximity events.

Original: We embed all individuals from these two datasets into a common space of individuality embeddings e φ and factor vectors f + .

Original: The dataset also contains labels that can be used in proxy tasks (see section 2.5) to quantify if the learned embeddings and factors are semantically meaningful and temporally consistent.
Revised: The dataset also contains labels that can be used in proxy tasks (see section 2.5) to quantify if the learned embeddings and factors are semantically meaningful and temporally consistent.

Original: See Table 1, and appendix A.1 for details.
Revised: See table 1, and appendix A.1 for details.

Original: We publish the data together with this paper 1 .
Revised: We publish the data together with this paper 1 .

Original: 2.3. S YNTHETIC DATA
Revised: 2.3. S YNTHETIC DATA

Original: Although some ground truth labels exist for the honey bee datasets (section 2.5), factorizing these data is still fundamentally an unsupervised learning problem.
Revised: Although some ground truth labels exist for the honey bee datasets (section 2.5), factorizing these data is still fundamentally an unsupervised learning problem.

Original: We generated synthetic datasets to evaluate whether the model can identify groups of individuals with common latent factors that determine their interaction frequencies.
Revised: We generated synthetic datasets to evaluate whether the model can identify groups of individuals with common latent factors that determine their interaction frequencies.

Original: We model a common lifetime trajectory and groups of individual variation  using smoothed Gaussian random walks.
Revised: We model a common lifetime trajectory and groups of individual variation of factors using smoothed Gaussian random walks.

Original: We randomly assign individuals to a group with random times of emergence and disappearance from the data and compute interaction matrices by calculating the dot product of the factors of 1024 individuals for 100 simulated days.
Revised: We randomly assign individuals to a group with random times of emergence and disappearance from the data and compute interaction matrices by calculating the dot product of the factors of 1024 individuals for 100 simulated days.

Original: We then measure how well the individuality embeddingse φ of a ﬁtted model correspond to the truth groups using the adjusted mutual information score (Vinh et al., 2009).
Revised: We then measure how well the individuality embeddings φ of a ﬁtted model correspond to the truth groups using the adjusted mutual information score (Vinh et al., 2009), and the mean squared error between the ground truth factors and the best permutation of the factors f + .

Original: We evaluate the model on 128 different random synthetic datasets with increasing Gaussian noise levels in the interaction tensor.
Revised: We evaluate the model ondifferent random synthetic datasets with increasing Gaussian noise levels in the interaction tensor.

Original: See appendix A.1. for more details on the generation process.
Revised: See appendix A.1.2 for more details on the generation process.

Original: In both datasets, we deﬁne c ( t, i ) as the time since the date of emergence of an individual i at time t (i.e., for the honey bees c ( t, i ) is an individuals’ age).
Revised: In both datasets, we deﬁne c ( t, i ) as the age in days of an individual i at time t .

Original: This inductive bias allows the model to learn semantically consistent embeddings for individuals, even if their interaction partners are entirely disjointed.

Original: 2. B ASELINE MODELS
Revised: 2. B ASELINE MODELS

Original: Symmetric NMF: We compute the factors that optimally reconstruct the original interaction matrices using the standard symmetric NMF algorithm
Revised: Symmetric NMF: We compute the factors that optimally reconstruct the original interaction matrices using the standard symmetric NMF algorithm

Original: (Shi et al., 2015; Kuang et al., 2015), for each day separately, using the same number of factors as in the TNMF model.
Revised: (Shi et al., 2015; Kuang et al., 2015), for each day separately, using the same number of factors as in the TNMF model.

Original: Aligned symmetric NMF: We consider a simple extension of the standard SymNMF algorithm that aligns the factors to be more consistent over time.
Revised: Aligned symmetric NMF: We consider a simple extension of the standard SymNMF algorithm that aligns the factors to be more consistent over time.

Original: For each pair of subsequent days, we consider all combinatorial reorderings of the factors computed for the second day.
Revised: For each pair of subsequent days, we consider all combinatorial reorderings of the factors computed for the second day.

Original: For each reordering, we compute the mean L 2 distance of all individuals that were alive on both days.
Revised: For each reordering, we compute the mean L 2 distance of all individuals that were alive on both days.

Original: We then select the reordering that minimizes those pairwise L 2 distances and greedily continue with the next pair of days until all factors are aligned.
Revised: We then select the reordering that minimizes those pairwise L 2 distances and greedily continue with the next pair of days until all factors are aligned.

Original: Furthermore, we align the factors across colonies (where individuals cannot overlap) as follows: we run this algorithm for both datasets separately and align the resulting factors by ﬁrst computing the mean embedding for all individuals grouped by their ages.
Revised: Furthermore, we align the factors across colonies (where individuals cannot overlap) as follows: we run this algorithm for both datasets separately and align the resulting factors by ﬁrst computing the mean embedding for all individuals grouped by their ages.

Original: As before, we now select from all combinatorial possibilities the reordering that minimizes the L 2 distance between the embeddings obtained from both datasets.
Revised: As before, we now select from all combinatorial possibilities the reordering that minimizes the L 2 distance between the embeddings obtained from both datasets.

Original: See section A.5.1 for pseudo code.
Revised: See section A.5.1 for pseudo code.

Revised: Tensor decomposition: We also compare against a constrained non-negative tensor decomposition model with symmetric factors F ∈ R N × M + and temporal dynamics constrained to the diagonals, i.e.

Revised: D ∈ R T × M × M + and D t = diag ( d t ) , d t ∈ R M + .

Original:  Temporal NMF models: We evaluate variants of the temporal symmetric matrix factorization algorithms proposed by Jiao et al.
Revised: [Equation] [Equation] Temporal NMF models: We evaluate variants of the temporal symmetric matrix factorization algorithms proposed by Jiao et al.

Original: (2017) and Yu et al.
Revised: (2017) and Yu et al.

Original: Because they did not consider nonoverlapping datasets, we follow the procedure given above for the Aligned symmetric NMF to ﬁnd the optimal reordering to align the factors obtained by applying models to the two datasets separately.
Revised: For the tensor decomposition and temporal NMF baselines, we follow the procedure given above for the Aligned symmetric NMF to ﬁnd the optimal reordering to align the factors obtained by applying models to the two datasets separately.

Original: 2. E VALUATION METRICS
Revised: 2. E VALUATION METRICS

Original: Reconstruction: We measure how well the original interaction matrices
Revised: Reconstruction: We measure how well the original interaction matrices

Original: A can be reconstructed from the factors.
Revised: A can be reconstructed from the factors.

Original: We do not require the model to reconstruct the interaction matrices as well as possible because we only use the reconstruction as a proxy objective to learn a meaningful representation.
Revised: We do not require the model to reconstruct the interaction matrices as well as possible because we only use the reconstruction as a proxy objective to learn a meaningful representation.

Original: Still, a high reconstruction loss could indicate problems with the model, such as excessive regularization.
Revised: Still, a high reconstruction loss could indicate problems with the model, such as excessive regularization.

Original: Consistency: We measure to what extent the adversarial model can constrain the information in the lifetime embeddingse φ ( i, k ) .
Revised: Consistency: We measure to what extent the individuality embeddings φ change over time.

Original: For each model, we train a multinomial logistic regression model to predict the source cohort (date of birth) and calculate the area under the ROC curve (AUC cohort ) using a stratiﬁed 100-fold cross-validation with scikit-learn (Pedregosa et al., 2011).
Revised: For each model, we train a multinomial logistic regression model to predict the source cohort (date of birth) and calculate the area under the ROC curve (AUC cohort ) using a stratiﬁed 100-fold cross-validation with scikit-learn (Pedregosa et al., 2011).

Original: Because the baseline models do not learn a lifetime embedding, weinstead compute how well the model can predict the cohort using the mean factor representation of the individuals over their lives.
Revised: The baseline models do not learn an individuality embedding; therefore we compute how well the model can predict the cohort using the mean factor representation of the individuals over their lives.

Original: We deﬁne consistency as 1 − AUC cohort of this linear model.
Revised: We deﬁne consistency as 1 − AUC cohort of this linear model.

Revised: Note that a very low temporal consistency would indicate that the development of individual bees changes strongly between cohorts and colonies, which we know not to be true.

Original: Mortality and Rhythmicity: We evaluate how well a linear regression model can predict the mortality (number of days until death) and circadian rhythmicity of the movement ( R 2 score of a sine with a period of 24 h ﬁtted to the velocity over a three day window).
Revised: Mortality and Rhythmicity: We evaluate how well a linear regression model can predict the mortality (number of days until death) and circadian rhythmicity of the movement ( R 2 score of a sine with a period of 24 h ﬁtted to the velocity over a three-day window).

Original: These metrics are strongly correlated with an individual’s behavior (e.g. foragers exhibit strong circadian rhythms because they can only forage during the daytime; foragers also have a highrisk of mortality).
Revised: These metrics are strongly correlated with an individual’s behavior (e.g. foragers exhibit strong circadian rhythms because they can only forage during the daytime; foragers also have a high mortality).

Original: We follow the procedure given in Wild et al.
Revised: We follow the procedure given in Wild et al.

Original: and report the 100-fold cross-validated R 2 scores for these regression tasks.
Revised: and report the 100-fold cross-validated R 2 scores for these regression tasks.

Original: For this data, we expect the factors f + and lifetime embeddingse φ( i, k ) to be semantically meaningful and temporally consistent if they reﬂect an individual’s behavioral metrics (mortality and rhythmicity) and if they do not change strongly over time (measured in the consistency metric).
Revised: For this data, we expect the factors f + and individuality embeddings φ to be semantically meaningful and temporally consistent if they reﬂect an individual’s behavioral metrics (mortality and rhythmicity) and if they do not change strongly over time (measured in the consistency metric).

Original:  R ESULTS
Revised:  R ESULTS

Original: We implemented the model using PyTorch (Paszke et al., 2019) and trained it in minibatches ofindividuals for 100 000 iterations with the Adam optimizer (Kingma & Ba, 2015).
Revised: We implemented the model using PyTorch (Paszke et al., 2019) and trained it in minibatches ofindividuals for 200 000 iterations with the Adam optimizer (Kingma & Ba, 2015).

Original: See appendix A.2.
Revised: See appendix A.2.

Original: for the architecture of the learned functions, a precise description of the regularization losses and further hyperparameters.
Revised: for the architecture of the learned functions, a precise description of the regularization losses, and further hyperparameters.

Original: The code is publicly available 2 .
Revised: The code is publicly available 2 .

Original: 3. S YNTHETIC DATA

Original: [Figure]
Revised: [Figure]

Original: We factorize the interaction matrices of the 128 synthetic datasets with varying levels of Gaussian noise using the TNMF model.
Revised: We factorize the interaction matrices of the 128 synthetic datasets with varying levels of Gaussian noise.

Original: We conﬁrmed that our model converges in all datasets and evaluate whether we can distinguish the ground truth group assignments of the individuals.{+F ac t o r s t r e ng t h f 0 f 20 
 Brood+}
Revised: We conﬁrmed that our model converges in all datasets and evaluate whether we can distinguish {+F ac t o r s t r e ng t h f 0 f 20 
 Brood+}

Original: To that end, we extract the individuality embeddings e φ from the models and measure how well they correspond to ground truth data using the adjusted mutual information (AMI) score.
Revised: Honey f Dance ﬂoor 0 1 2  0 1 2 3 0 1 2f Factor trajectories and relationship to allocated tasks Age in days m ( c ( t,i ))

Original: We ﬁnd that for low levels of noise, our model can identify the truth group assignments with high accuracy
Revised: f + ( t,i ) Factor strength ( f + ) in σ P r opo r ti on o f

Original: [Figure]
Revised: ti m e s p e n t on ce ll s ub s t r a t e the individuals’ ground truth group assignments. To that end, we extract the individuality embeddings φ from the models and measure how well they correspond to ground truth data using the adjusted mutual information (AMI) score. Furthermore, we measure the mean squared error between the best permutation of learned factors f + and the ground truth factors.

Original:  and are still signiﬁcantly better than random assignments even at very high levels of noise (see ﬁgure 3).
Revised: We ﬁnd that for low levels of noise, our model can identify the truth group assignments with high accuracy, and are still signiﬁcantly better than random assignments even at very high levels of noise (see ﬁgure 3).

Revised: Note that for this experiment, we evaluated a model with the same hyperparameters as used in all plots in the results section (see Table 2) and a variant without explicit regularization except the L 1 penalty of the learned individuality embeddings φ ( λ embeddings , because this regularization is required to meaningfully extract clusters), which was set to 0 .

Original: See appendix A.1.2 for more details on the synthetic datasets
Revised: See appendix A.1.2 for more details on the synthetic datasets.

Original: [Figure] [Figure] For example f 1 , is associated with young bees (see Figure 4) with time spent in the brood area. biologically consistent because young tend to care brood the factors divided by their
Revised: 3. H ONEY BEES

Original: Average lifetime model:
Revised: Mean lifetime model:

Original: The model learns a sparse representation of the developmental trajectory of a honey bee in the space of social interactions.
Revised: The model learns a sparse representation of the developmental trajectory of a honey bee in the space of social interactions.

Original: Only two factors are effectively used (they exceed the threshold value of 0 . 01 ).
Revised: Only two factors are effectively used (they exceed the threshold value of 0 . 01 ).

Original: These factors show a clear trend over the life of a bee, indicating that the model captures the temporal aspects of the honey bee division of labor (See Figure 4).
Revised: These factors show a clear trend over the life of a bee, indicating that the model captures the temporal aspects of the honey bee division of labor (See Figure 4).

Original: Interpretability of factors: To understand the relationship between the factors and division of labor, we calculate how the factors map to the fraction of time an individual spent on the brood area, honey storage, or dance ﬂoor (where foragers aggregate).
Revised: Interpretability of factors: To understand the relationship between the factors and division of labor, we calculate how the factors map to the fraction of time an individual spent on the brood area, honey storage, or dance ﬂoor (where foragers aggregate).

Original: Time spent on these different substrates is a strong indicator of an individual’s task.
Revised: Time spent on these different substrates is a strong indicator of an individual’s task.

Original: The factor f 1 , which peaks at young age (Figure 4), correlates with the proportion of time spent in the brood area, while a high f 3 indicates increased time spent on the dance ﬂoor.
Revised: The factor f 1 , which peaks at young age (Figure 4), correlates with the proportion of time spent in the brood area, while a high f 0 indicates increased time spent on the dance ﬂoor.

Original: Therefore, the model learned to map biologically relevant processes.
Revised: Therefore, the model learned to map biologically relevant processes.

Original: Individuality basis functions and lifetime embeddings: Due to the regularization of the embeddings, the model learns a sparse set of individuality basis functions.
Revised: Individuality basis functions and individuality embeddings: Due to the regularization of the embeddings, the model learns a sparse set of individuality basis functions .

Original: As encouraged by the model, most individuals can predominantly be described by a single basis function.
Revised: As encouraged by the model, most individuals can predominantly be described by a single basis function.

Original: That means that while each honey bee can collect a unique set of experiences, most can be described with a few  individuality blueprints which are consistent across cohorts and colonies.
Revised: That means that while each honey bee can collect a unique set of experiences, most can be described with a few common individuality embeddings which are consistent across cohorts and colonies.

Original: In the context of honey bee division of labor, the basis functions are interpretable because the factors correspond to different task groups.
Revised: In the context of honey bee division of labor, the basis functions are interpretable because the factors correspond to different task groups.

Original: For example, ω 11 (accounting for ≈ 24 .
Revised: For example, b 12 ( c ( t, i )) (accounting for ≈ 10 .

Original: 5% of the individuals) describes workers that occupy nursing tasks over their lives, whereas ω 6 (accounting for ≈ 9 .

Original: 9% of the individuals) describes workers that associate with in-hive duties such as honey storage later, but also stronger than average.
Revised: 7% of the individuals) describes workers that occupy nursing tasks much longer than most bees.

Original: As the lifetime embeddingse φ only scale the magnitude of the basis functions, they can be interpreted in the same way.
Revised: As the individuality embeddings φ only scale the magnitude of the basis functions, they can be interpreted in the same way.

Original: Individual lifetime trajectories in the factor space can be computed based on the mean lifetime trajectories ( mθ ), individuality basis functions ( b ω ) and lifetime embeddings (e φ ).
Revised: Individual lifetime trajectories in the factor space can be computed based on the mean lifetime trajectories ( m ), individuality basis functions ( b ( c ( t, i )) ) and individuality embeddings ( φ ).

Original: See Figure 5 for examples of individual lifetime trajectories from workers that most strongly corresponded to the common individuality basis functions.
Revised: See ﬁgure 5 for examples of individual lifetime trajectories from workers that most strongly corresponded to the common individuality basis functions .

Original: [Figure]
Revised: [Figure]

Original: that most strongly correspond to the individual basis functions are shown in the column titles. More than 59% of the individuals strongly correspond to one of the ﬁve basis functions shown here. b) Because the basis functions describe individuality offsets from the mean lifetime trajectory, it may be easier to interpret them by visualizing individual examples. For each of the basis functions (top row), we show a lifetime trajectory of an individual that to
Revised: [Figure]

Original: Evaluation: We verify that the learned representations of the individuals are meaningful (i.e., they relate to other properties of the individuals, not just their interaction matrices) and semantically consistent over time and across datasets using the metrics described in the section Evaluation metrics .
Revised: Evaluation: We verify that the learned representations of the individuals are meaningful (i.e., they relate to other properties of the individuals, not just their interaction matrices) and semantically consistent over time and across datasets using the metrics described in the section Evaluation metrics .

Original: We compare variants of our model with different adversarial loss scaling factors and factor Lregularizations, the baseline models, as well as the individuals’ ages.
Revised: We compare variants of our model with different adversarial loss scaling factors and factor L 1 regularizations, the baseline models, and the individuals’ ages.

Original: We expect a good model to be temporally consistent and semantically meaningful.
Revised: We expect a good model to be temporally consistent and semantically meaningful.

Original: All variants of our model outperform the baselines in terms of the semantic metrics Mortality and Rhythmicity , and the  adversarial loss term further increases the Consistency metric without negatively affecting the other metrics.
Revised: All variants of our model outperform the baselines in terms of the semantic metrics Mortality and Rhythmicity , except for the Yu et al. (2017) model, which performs comparably well in the Mortality metric. The adversarial loss term further increases the Consistency metric without negatively affecting the other metrics.

Original: A very strong adversarial regularization (see row with λ adv = 1 in Table 2) prevents the model from learning a good representation of the data.
Revised: A very strong adversarial regularization (see row with λ adv = 1 in Table 2) prevents the model from learning a good representation of the data.

Original: See Table 2 for an overview of the results.
Revised: See Table 2 for an overview of the results.

Original: We also evaluate the tradeoff between the different metrics using a grid search over the hyperparameters (see appendix A.3).
Revised: We also evaluate the tradeoff between the different metrics using a grid search over the hyperparameters (see appendix A.3).

Original: Scalability: The functions m θ and b ω j are learned neural networks with non-linearities.
Revised: Scalability: The functions m ( c ( t, i )) and b ( c ( t, i )) are learned neural networks with non-linearities.

Original: The objective is non-convex and we learn the model parameters using stochastic gradient descent.
Revised: The objective is non-convex and we learn the model parameters using stochastic gradient descent.

Original: Optimization is therefore slower than the standard NMF algorithms that can be ﬁtted using algorithms such as Alternating Least Squares (Kim et al., 2014).
Revised: Optimization is therefore slower than the standard NMF algorithms that can be ﬁtted using algorithms such as Alternating Least Squares (Kim et al., 2014).

Original: We found that the model converges faster if the reconstruction loss of the age based model m θ is additionally minimized with the main objective in equation 5.
Revised: We found that the model converges faster if the reconstruction loss of the age based model m ( c ( t, i )) is additionally minimized with the main objective in equation 5.

Original: Due to the minibatch training regime, our method should scale well in larger datasets.
Revised: Due to the minibatch training regime, our method should scale well in larger datasets.

Original: For datasets with longer time spans, minibatch sampling could also be extended over the temporal dimension.

Original: Small neural networks are sufﬁcient to learn the functions m θ and b ω j .
Revised: Small neural networks were sufﬁcient to learn the functions m ( c ( t, i )) and b ( c ( t, i )) in our experiments.

Original: Most of the runtime during training is spent on the matrix multiplication f
Revised: Most of the runtime during training is spent on the matrix multiplication f

Original: + ( t, i ) · f + ( t, j ) T and the corresponding backwards pass.[Table] [Table] [Figure]
Revised: + ( t, i ) · f + ( t, j ) T and the corresponding backwards pass.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: Temporal NMF factorizes temporal matrices with overlapping and even disjointed communities by learning a common embedding of the lifetime development of the individual entities.
Revised: Temporal NMF factorizes temporal matrices with overlapping and even disjointed communities by learning a common embedding of the lifetime development of the individual entities.

Original: In the context of honey bees, this embedding is biologically meaningful, consistent over time, and shows that interaction patterns follow a common lifetime trajectory.
Revised: In the context of honey bees, this embedding is biologically meaningful, consistent over time, and shows that interaction patterns follow a common lifetime trajectory.

Original: Differences from the mean are described in a coordinate system of individual variability.
Revised: Differences from the mean are described in a coordinate system of individual variability.

Original: Honey bee colonies are known to exhibit suites of correlated traits (Wray et al., 2011), but this method provides a perspective at the individual-level, and opens the possibility for understanding how colony-level traits respond to biotic and abiotic pressures.
Revised: Honey bee colonies are known to exhibit suites of correlated traits (Wray et al., 2011), but this method provides a perspective at the individual level and opens the possibility for understanding how colony-level traits respond to biotic and abiotic pressures.

Original: The basis functions are interpretable with respect to the division of labor within colonies, and offer a valuable tool to understand and quantify the inﬂuence of experimental manipulations (e.g. pesticides or increased predatory pressure) on an individual, even between experimental trials on completely different sets of individuals. 
Revised: The basis functions are interpretable with respect to the division of labor within colonies and offer a valuable tool to understand and quantify the inﬂuence of experimental manipulations (e.g. pesticides or increased predatory pressure) on an individual, even between experimental trials on completely different sets of individuals. While we applied our method to honey bees as an exemplary system [Table] temporal structure or regularization causes all models to explain less variance in the data.

Revised: However, for all models except TNMF this does not result in a signiﬁcant increase of the other metrics.

Original: While we applied our method to honey bees as an exemplary system with many individuals that exhibit an entangled, non-overlapping social structure, our method can be applied to any setting in which some interaction structure follows a general pattern over an observable (such as time) to detect structured deviations at the individual level.
Revised: The underlined model is used in all plots in the results section. [Figure] cluster is strongly associated with a , and interpretable ( with many individuals that exhibit an entangled, non-overlapping social structure, our method can be applied to any setting in which some interaction structure follows a general pattern over an observable (such as time) to detect structured deviations at the individual level.