Individuality in the hive - Learning to embed lifetime social behaviour of honey bees
Authors: Benjamin Wild
David Dormagen
Michael L Smith
Tim Landgraf
Venue:  (2021)
Keywords: matrix factorization
honey bees
explainable
social networks
implicit bias
dataset

Abstract: Honey bees are a popular model for complex social systems, in which global behavior emerges from the actions and interactions of thousands of individuals. 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. For example, young bees can become foragers early in life, depending on the colony’s needs. 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. 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. 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.

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INDIVIDUALITY IN THE HIVE LEARNING TO EMBED LIFETIME SOCIAL BEHAVIOR OF HONEY BEES
Anonymous authors Paper under double-blind review
ABSTRACT
Honey bees popular model for complex social systems, in which global behav- ior emerges from the actions and interactions of thousands of individuals. While the average life of bee is organized age, there is substantial variation at the individual level. For example, young bees can become foragers early in life, depending on the colony's 's needs. 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 indi- viduals in the social network over their entire lives. 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. Our method provides quantitative framework for understanding behavioral heterogeneity in complex social systems applicable in fields such as behavioral biology, social sciences, neuroscience, and information science.
sequence of tasks roughly determined by
INTRODUCTION
Animals living in large groups often coordinate their behaviors, resulting in emergent properties the group level, from flocking birds to democratic elections. 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. 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; Mathis al., 2018; Graving et al., 2019; Pereira et al., 2019). These datasets have unprecedented scale and complexity, but understanding these data has emerged itself (Pinter- Wollman et al., 2014; Krause et al., 2015; Brask et al., 2020).
new and challenging problem in
A popular approach to understand high-dimensionnll data is to learn semantic embeddings (Frome et al., 2013; Asgari& & Mofrad, 2015; Hamacho-Colladoo & Pilehvar, 2018; Nelson et al., 2019). Such embeddings can be learned without supervision, interpretable, and useful for accomplishing downstream tasks. Individuals in animal societies can be described with semantic embeddings extracted from social interaction networks using matrix factorization methods. For example, in symmetric non-negative matrix factorization (SymNMF), the dot products of any two animals' factor vectors reconstruct the interaction matrix (Wang al., 2011; Shi al., 2015), see Figure I and b). If the embeddings allow us to predict relevant behavioral properties, they serve our understanding as semantic representations. However, in temporal settings where the interaction matrices change over time, there is no straightforward extension of this algorithm. 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. the prediction of relevant behavioral properties will deteriorate.
In living systems, interaction dynamics are highly variable; individuals differ when they appear in the data and how long they live. Different non-overlapping groups of individuals, e.g. from different years, may not interact with each other at all. How can we find a common semantic embedding even in these extreme cases? How do leamn embeddings that generalize to different groups and still provide insights into each individual's functional role? If animals take on roles partially determined by common factor, such age, how
learn this dependency?
Several approaches to extend NMF to temporal settings have been proposed in a variety of problem settings. Yu et al. (2016) and Mackevicius al. (2019) propose factorization method for time

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a)
b)
Interactions
SymNMF 2D Factors
Individual ID
10 20 30 40 60
Age days
Individual ID
Factor 0
2019-08-19
2019-08-31
2019-09-26
2019-10-08
Individual ID
Individual
Individual
Figure For daily snapshot of a temporal social network, symmetric NMF is able to extract meaningful factor representations of the individuals. Colors represent the interaction frequencies of all individuals (a). The age-based division of labor honey bee colony clearly reflected in the two factors same-aged individuals are likely to interact with each other (b). For long observation windows spanning several weeks, the social network changes drastically as individuals are born, die, and switch tasks (c). Here, investigate how representation of temporal networks can be extracted, such that the factors representing individuals can be meaningfully compared over time, and across datasets.
series analysis. Gauvin et al. (2014) focus on the analysis of communities that are determined by their temporal activity patterns. Jiao et al. (2017) consider the case of communities from graphs over time and enforce temporal consistency with an additional loss term. Yu et al. (2017) and Wang al. (2017) represent networks as function of time. Temporal matrix factorization can be seen as tensor decomposition problem, for which methods with many applications have been proposed in the literature, see Kolda & Bader (2009) for review. particular, time-shifted tensor decomposition methods have been used in multi-neurona spike train analysis when recordings of multiple trials from population of neurons are available (Mørup et al., 2008; Williams, 2020). However, to our knowledge, method yet considers the case of entities following common trajectory depending on an observable property (e.g.. the age of an individual), which we show to be a powerful inductive bias.
We approach this question using honey bees, a popular model system for studying individual and collective behavior (Elekonich & Roberts, 2005). 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). Colonies also organized spatially: brood is reared in the center, honey and pollen are stored at the periphery, and foragers offload nectar near the exit. Therefore, individual's role is partially reflected in its location, which allows us to evaluate whether the embeddings our method learss are semantically meaningful.
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) daily representation of the individual's 's functional position in the social network that can be derived from the individuality embeddings. fashion, using only interaction matrices of the individuals over time. We analyze a dataset obtained by tracking thousands of individually marked honey bees in two colonies, at high temporal and spatial resolution over total of 155 days, covering entire lifespans and multiple generations. 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.
show that these representations can be learned an unsupervised
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Individuality embeddings
Basis functions b
())
Individual
X
(i, 1)
Interactions A,
f(t, t)
X
oi, 2)
Individual j
Mcan m
f(t.5) 0
age c(t, ()
age c(t,5)
a)
b)
d)
e)
Figure 2: Overview of the method: We learn a parametric function describing the mean life trajectory mc((t, i)) and a set basis functions of individual variation b(c(t, i)), where c(t, is the age of individual i time (a). For each individual, an embedding is learned consisting of one scalar per basis function that scales the contribution the respective basis function this vector of weights makes up the individuality embedding of individual (b). The mean trajectory m(((t, i)) plus weighted sum of the basis functions b(c(t, i)) constitute the lifetime Irajectory of each individual (c). At each time point, factors can be extracted from the individual lifetime trajectories (d) reconstruct
the interaction affinity between individuals (e). Note that the lifetime trajectories are functions of the individuals' ages, while interactions can occur at any time t.
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. We provide a new perspective on how extract fundamental factors that underlie behavioral and developmental plasticity in animal groups
2
METHODS
TEMPORAL NMF ALGORITHM
SymNMF factorizes a matrix A E .R-N such that can be approximated by the product F FFT
where F E RNN and M < N:
F argmin
A FFT
f()
f(0) = F.,. f()) E RM
(I)
F22
When applied to social networks, f(i) can represent the role of an entity within the social network 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.
Here present novel temporal NMF algorithm (7NMF) which extends SymNMF temporal settings in which A N changes over time t. We assume that the entities i (0, 1, 1,..., N) follow to some extent a common trajectory depending on an observable property (for example the age of individual). We represent an entity specific point in time using a factor vector f (t, such that
(t.
À AERTANNN EL(

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In contrast to SymNMF, do not directly factorize A to find the optimal factors that reconstruct the matrices. Instead, we decompose the problem into learning an average trajectory of factors m(c(t. and structured variations from this trajectory oft, i) that depend on the observable property c(t, i):
f(t, 1) = m(((t, i)) + o(t,a)
f (t, = max(0, f(t, (t,
(3)
e: C:NTXN +N N. m m R.M
O: O:NTXN + RM
This decomposition is an inductive bias that allows the model 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. is stable. Note that in the simplest case c(t, i) = t, TNMF can be seen as a tensor decomposition model, i.c. the trajectory of all entities aligned with the temporal dimension l. of A. In our case, e(t, i) maps to the age of individual i at time I.
While many parameterizations for the function ott, i) are possible, we only consider one particular case in this work: We learn set of individuality basis functions b(e(t, i)) (shared among all entities) that define a coordinate system of possible individual variations and the individuality embeddings Φ. which capture what extent each basis function applies to entity:
K
ott, i) =
Σ di bx(c(t, i)) k=
R NKK
by N
- R
(4)
where K is the number of learned basis functions. 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.
We implement the functions m(c(t, i)) and b(c(t, with small fully connected neural networks with non-linearities and several hidden layers. The parameters of these functions and the entities' embeddings & are learned jointly using minibatch stochastic gradient descent:
0 g argmin
A-A l
(5)
8,0
Note that non 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. e T + R.P. e.g. the day and time in a intraday analysis of social networks.
2.2 REGULARIZATION
Due to the inductive bias, the model performs well without additional explicit regularization in the datasets considered in this work (see Table 2). However, we find that the model's 's interpretability can be improved using additional regularization terms without significantly affecting its performance. We encourage sparsity in both the number of used factors and individuality basis functions by adding L1 penalties of the mean absolute magnitude of the factors ft, i) and basis functions b(c(t, i)) to the objective. We encourage individuals' lifetimes to be represented with a sparse embedding using an L1 penalty the learned individuality embeddings Φb
We also introduce an optional adversarial loss term to encourage the model to learn embeddings that are semantically consistent over time, i.c. 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. We jointly train a discriminative network d(():) that tries to classify the time of the first occurrence of all entities based their individuality embeddings d. 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 al., 2014). See appendix 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.
2.3 DATA
2.3.1
HONEY BEE DATA
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Two colonies of honey bees continuously recorded over a total of 155 days. Each individ- ual was manually tagged at emergence, the date of birth known for each bee. Locations and identities of all honey bees (N=9286) were extracted from the raw images and used to con-
Table
Honey bee datasets
Dataset
Days
Individuals
Interaction pairs
BN16
56
2443
43 174 748
BN19
99
6843
167 366 381
struct daily aggregated temporal interaction networks based on the counts of spatial proximity events. 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. See table 1, and appendix A.I for details. We publish the data together with this paper'..
2.3.2 SYNTHETIC DATA
Although some ground truth labels exist for the honey bee datasets (section 2.5), factorizing these data is still fundamentally an unsupervised learning problem. We generated synthetic datasets to evaluate whether the model can identify groups of individuals with common latent factors that determine their interaction frequencies. We model common lifetime trajectory and groups of individual variation of factors using smoothed Gaussian random walks. 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. We then measure how well the individuality embeddings á of fitted model correspond 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 ft. We evaluate the model on 128 different random synthetic datasets with increasing Gaussian noise levels in the interaction tensor. See appendix A. for more details on the generation process.
In both datasets, we define c(t, i) as the age in days of an individual i at time t.
2.4 BASELINE MODELS
We compute the factors that optimally reconstruct the original interaction matrices using the standard symmetric NMF algorithm (Shi et al., 2015; Kuang et al., 2015), for each day separately, using the same number of factors as in the TNMF model.
Symmetric NMF:
We consider simple extension of the standard SymNMF algorithm that aligns the factors to be more consistent over time. For each pair of subsequent days, we consider all combinatorial reorderings of the factors computed for the second day. For each reordering, we compute the mean L2 distance of all individuals that were alive on both days. We then select the reordering that minimizes those pairwise L2 distances and greedily continue with the next pair of days until all factors are aligned. Furthermore, we align the factors across colonies (where individuals cannot overlap) 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. As before, now select from all combinatorial possibilities the reordering that minimizes the L2 distance between the embeddings obtained from both datasets. See section A.5.1 for pseudo code.
Aligned symmetric NMF:
Tensor decomposition:
We also compare against a constrained non-negative tensor decomposition model with symmetric factors F R. M and temporal dynamics constrained to the diagonals, i.e. D E R TxMXM and D. (d). d, E TR..
A FD, FT
(6)
F. D argmin T
Σ A.
A.l
F.D
Temporal NMF models: We evaluate variants of the temporal symmetric matrix factorization algorithms proposed by Jiao et al. (2017) and Yu et al. (2017).
See Anonymous (2020) for dataset
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For the tensor decomposition and temporal NMF baselines, 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.
2.5 EVALUATION METRICS
Reconstruction: We measure how well the original interaction matrices A can be reconstructed from the factors. We do not require the model reconstruct the interaction matrices well possible because we only use the reconstruction as a proxy objective to learn meaningful representation. Still, a high reconstruction loss could indicate problems with the model, such as excessive regularization.
Consistency: We measure to what extent the individuality embeddings & change over time. For each model, we train multinomial logistic regression model to predict the source cohort (date of birth) and calculate the area under the ROC curve (AUCcohon) using stratified 100-fold cross-validation with scikit-learn (Pedregosa et al., 201 The baseline models do not learn an individuality embedding: therefore compute how well the model can predict the cohort using the mean factor representation of the individuals over their lives. We define consistency as - AUCoohont of this linear model. Note that 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.
Mortality and Rhythmicity: We evaluate how well linear regression model predict the mortality (number of days until death) and circadian rhythmicity of the movement (R² score of a sine with a period of 24 h fitted to the velocity over a daree-day window). These metries strongly correlated with an individual's 's behavior (e.g. foragers exhibit strong circadian rhythms because they can only forage during the daytime; foragers also have high mortality). We follow the procedure given in Wild et al. (2020) and report the 100-fold cross-validated R² scores for these regression tasks.
For this data, expect the factors ft and individuality embeddings ó 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 the consistency metric).
RESULTS
We implemented the model using PyTorch (Paszke al., 2019) and trained it in minibatches of 256 individuals for 200 000 iterations with the Adam optimizer (Kingma & Ba, 2015). See appendix A.2.3 for the architecture of the learned functions, precise description of the regularization losses, and further hyperparameters. The code is publicly available"
TNMF
TNMF (regularized)
SymNMF
1.00
0.75
MSE (factorrs)
10
0.50
AMI
0.25
10-1
0.00
-27
-30
-32
inf
-18
-30
-32
SNR in dB
SNR in dB
Figure 3: AMI score and mean squared error between true factors and the best permutation of learned factors for increasing noise levels. The median values over 128 trial runs are shown.
3.1 SYNTHETIC DATA
We factorize the interaction matrices of the 128 synthetic datasets with varying levels of Gaussian noise. We confirmed that our model converges in all datasets and evaluate whether we can distinguish
4open sci www.nce/eeccccccc.coo. http///t2222e//// aa04 4cf8 F 85c7 64 6d8dc4 6400

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Factor trujectories and relationship to allocated tasks
mfc(tt 8)
(t,
Brood
Honey
Dance floor
Factor strengith
0.75
Propurtion of time 00 csll callstr substrite
Jo
0.50
fo
fi
0.25
11
0
20
40
20
40
0
20
40
0
2
2
Age in days
Factor strength (f
Figure 4: Left: Mean lifetime trajectories according to mn((t, i)). The model Iearns a sparse representation of the functional position of the individuals in the social network. fo (blue) mostly corresponds to middle-aged and older bees, and fi (orange) predominantly describes young bees. Only factors with a mean magnitude of at least 0.01 are shown. Even though the model uses only these two factors, it is still expressive enough to capture individual variability, as can be seen in randomly sampled individuals' lifetime trajectories. Right: The individual factors f+ and the proportion of time the individuals spent on different nest substrates. The strong correlation indicates that the learned factors are a good representation of the individuals' roles in the colonies. Note that the factors have been divided by their standard deviation here for ease of comparability.
the individuals' ground truth group assignments. To that end, we extract the individuality embeddings 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. 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). 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 Ln penalty the learned individuality embeddings & (Nembseddings- because this regularization is required to meaningfully extract clusters), which was set to 0.1. See appendix A.1.2 for more details on the synthetic datasets.
3.2 HONEY BEES
Mean lifetime model: The model learns a sparse representation of the developmental trajectory of honey bee in the space of social interactions. Only two factors are effectively used (they exceed the threshold value of 0.01). 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).
Interpretability of factors: To understand the relationship between the factors and division of labor, calculate how the factors map the fraction of time individual spent the brood area, honey storage, or dance floor (where foragers aggregate). Time spent on these different substrates a strong indicator of individual's task. The factor fi, which peaks at young age (Figure 4), correlates with the proportion of time spent in the brood area, while a high fo indicates increased time spent on the dance floor. Therefore, the model learned to map biologically relevant processes.
Individuality basis functions and individuality embeddings: Due to the regularization of the embeddings, the model learns sparse set of individuality basis functions. As encouraged by the model, most individuals can predominantly be described by single basis function. That means that while each honey bee can collect a unique set of experiences, most can described with a few common individuality embeddings which are consistent across cohorts and colonies. In the context of honey bee division of labor, the basis functions are interpretable because the factors correspond to different task groups For example, bi2(c(t, i)) (accounting for sy 10.7% of the individuals) describes workers that occupy nursing tasks much longer than most bees. As the individuality embeddings ó only scale the magnitude of the basis functions, they can be interpreted in the same way. 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 (o). See figure 5 for examples
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of individual lifetime trajectories from workers that most strongly corresponded the common individuality basis functions.
Most frequent individuality basis functions
a)
by (=:20.756
b1n (*11..9%
bu2 (s=10.7%)
bo (=28.6%)
by (=7.55%
(1,5) bulett, 8))
b)
individual 5476
individual 1103
individual 2690
individual 4516
individual 5346
25
0
25
25
25
Age in days
fo
ft
Figure 5; a) Magnitude of factor offsets for the five most common individuality basis functions over age (c(t,, i)). The percentage of individuals that most strongly correspond to the individual basis functions is shown in the column titles. More than 60% of the individuals strongly correspond to one of the five basis functions shown here. b) Because the basis functions describe individuality offisets 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 corresponds to that basis function (bottom row). Note that individuals can die or disappear at any time (solid lines). The mean lifetime trajectories are shown as dotted lines in the background.
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. We compare variants of our model with different adversarial loss scaling factors and factor Ln regularizations, the baseline models, and the individuals' ages. We expect a good model to be temporally consistent and semantically meaningful. All variants of 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. A very strong adversarial regularization (see row with Aud, representation of the data. See Table 2 for an overview of the results. We also evaluate the tradeoff between the different metrics using grid search over the hyperparameters (see appendix A.3).
in Table 2) prevents the model from learning good
The functions mee(t, i)) and b(e(t, i)) are learned neural networks with non-linearities. The objective is non-convex and we learn the model parameters using stochastic gradient descent. Optimization is therefore slower than the standard NMF algorithms that can be fitted using algorithms such as Alternating Least Squares (Kim et al., 2014). We found that the model converges faster the reconstruction loss of the age based model m(c(t. i)) is additionally minimized with the main objective in equation Due to the minibatch training regime, our method should scale well in larger datasets. Small neural networks were sufficient to learn the functions m(c(t, i)) and b(e(t, i)) in our experiments. Most of the runtime during training is spent on the matrix multiplication I+ (1, (L,,)) and the corresponding backwards pass.
Scalability:
4 CONCLUSION
Temporal NMF factorizes temporal matrices with overlapping and even disjointed communities by learning common embedding of the lifetime development of the individual entities. In the context of honey bees, this embedding is biologically meaningful, consistent over time, and shows that interaction patterns follow common lifetime trajectory. Differences from the mean are described in coordinate system of individual variability. 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. 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 method honey bees exemplary system
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Table 2: Evaluation metrics
Model
Method
Variant
A A
Consistency
Mortality
Rhythmicity 1
Age
0.02
0.20
SymNMF
Vanilla
0.9
0.18
0.01
0.02
SymNMF
Aligned
0.9
0.12
0.09
0.35
Tensor decomp.
1.36
0.03
0.06
0.09
Jiao et al. (2017)
0.1
0.9
0.19
0.02
0.05
Jiao et al. (2017)
1.15
0.15
0.01
0.04
at (2017)
0.01,
1.59
0.03
0.17
0.06
TNMF
No regularization Aud
1.21
0.17
0.30
0.48
TNMF
0, A 0.01
1.26
0.18
0.10
0.40
TNMF
Aud 0.1, A = 0.01
1.28
0.35
0.20
0.42
TNMF
And 1, Ar 0.01
1.88
0.5
0.03
0.25
TNMF
Aud O, Ar = 0.1
1.31
0.19
0.09
0.38
TNMF
Aud 0.1, 0.1
1.33
0.37
0.10
0.42
Table 3: The evaluation metrics for TNMF and the baseline models described in section 2.5. See appendix A.2.3 and A.5 for descriptions of the hyperparameters used. Note that the SymNMF model reconstruction loss can be seen as a lower bound for the matrix factorization models considered here, and imposing temporal structure regularization causes all models explain less variance in the data. However, for all models except TNMF this does not result in a significant increase of the other metries. The underlined model is used in plots in the results section.
Individuality embeddings ø
40
20
TSNE d2
20
-40
-50
-50 50
TSNE d1
TSNE
Individuality embedding dimension
Hierarchical clustering of individuality embeddings: Most individuals strongly correspond to single individuality basis function, making it easy to cluster their lifetime social behavior (i.e. each individual has a high value a single dimension for their individuality embedding). Because each cluster is strongly associated with specific individuality basis function, and because each basis function is interpretable (Figure 4), these blueprints lifetime development can also be intuitively understood and compared. Right: TSNE plots of the individuality embeddings colored by cluster (left) and the maximum circadian rhythmicity of an individual during her lifetime (right), indicating that the embeddings semantically meaningful.
Figure Left:
with many individuals that exhibit entangled, non-overlapping social structure, our method be applied to any setting in which some interaction structure follows a general pattern over an observable (such as time) detect structured deviations at the individual level.
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12


Reviews:
[{"review_id": "jWeFKV3Cjx7", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "\nThis paper proposes a NMF formulation ||A-FF^T||^2 where A and F are different types of information extracted from social datasets. In the honeybee example the authors highlight, A represents the spatial relationship between bees, and F encodes the age of the bees. The authors setup F to be decomposable into two types of embeddings, one which characterizes the group activity and the other which characterizes the individual activity. \n\nPros:\nThe authors highlight a very interesting application. Honey bees perform tasks according to their age (information in F) and the tasks are performed in different nest substrates (information in A).  \n\n\nCons:\n It is not clear what is the form of the embeddings e_\\phi.\nIt is not clear what is the baseline for the interpretable individual embeddings. Why should these individual embeddings capture reasonable offsets and why they capture the information in c_{i,t}? This is only tested indirectly, through the AMI score in the case of the synthetic dataset. Is there a way to setup this framework to check that true F and learned F in the synthetic dataset match more directly? In the current version of the work I cannot tell how well F captures the information encoded in c_{t,i}, this makes it hard to consider why/if ||A-FF^T||  could be meaningful. \nWhat is the factor strength in Fig 4?\nThe variance of AMI score is high even when the SNR = inf. Do the authors consider the model results robust given this high variance in the synthetic dataset? \n\nComments:\nIs there a mismatch in the type of data of Y in the synthetic and in the true datasets?  In the synthetic dataset is Y the spatial distance like in the honeybee dataset? \nThe notation in Eq 3 is not clear, c: \\mathbb{N}^{TxN} -> \\mathbb{N}, but o_{\\phi, \\omega} \\mathbb{N}^{Nx T} -> \\mathbb{R}^M? Why are the inputs in \\o_{} NxT? \nThe authors use the terms functional position, functional role, and functional embedding. The way the paper is written it seems like these terms could be used interchangeably, but isn’t this is what the authors are trying to show in the paper? It would be easier to understand this work if the authors explained in more detail the different variables/mappings.", "extra_scores": {}, "extra_text": {}}, {"review_id": "iY-zvNjADA", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "# Summary\n\nThe authors introduce a novel method for non-negative matrix factorization for timeseries and apply it to longitudinal honey bee interaction data.  The model leverages consistency of individuals over time by forcing the factors (or rather, the residuals of the factors with respect to a global trajectory) to be linear combinations of a small set of temporal basis functions.  These temporal basis functions are functions of the bee’s age.  In other words, the factor embedding of each bee is a vector of linear combinations of 16 learned basis functions of time.  Since all bees use the same 16 temporal basis functions, given these basis functions the lifetime embedding of each bee is encapsulated by a small matrix of numbers, namely the coefficients for the temporal basis functions for each factor (and in practice only two factors were significant, so each bee’s life is embedded in 32-dimensional space).  There are a number of regularizations on the temporal basis functions and the embedding coefficients.\n\nThe authors show that the two significant factors have values that vary somewhat consistently as a function of a bee’s age.  The authors also offer some interpretability of the temporal basis functions as corresponding to different social roles in the bee colony.  The authors claim that this interpretable representation of individual bees is valuable for understanding social dynamics in the colony and may be useful for understanding how various perturbations affect the social order of the colony.\n\n# Pros\n\n* The paper is clear and well-written.\n* The authors are open-sourcing both their model code and their dataset (which involved a substantial undertaking to collect).\n* The authors’ method is novel as far as I know, and it is elegant (albeit has quite a lot of regularization terms).\n* The authors are pretty rigorous in their analysis, ablations, and comparison to baselines.\n\n# Cons\n\nI have a few primary concerns, listed here roughly in order of significance:\n* I am concerned that this paper might have too narrow a scope in too niche a field for it to be suitable for ICLR.  The paper is entirely dedicated to exploring honey bee social behavior, and it is not clear whether their method would be useful/interesting for other areas of research.  While the authors do suggest that it might be useful for analyzing social behavior in other animals (and make vague references to potential broader applications), they provide no concrete evidence for that.  Without concrete ideas for applications aside from animal social behavior and without strong implications for the usefulness of the representations (see following point), I am concerned that this paper will fail to capture the interest of the vast majority of ICLR attendees.  Perhaps it is better suited for an animal behavior or computational biology journal/conference.\n* While the paper focuses on interpretable representations of honey bee behavior, it lacks evidence/support for the usefulness of these representations.  After reading the paper I am left wondering why being able to extract these representations is valuable.  First, in order to extract the representations one must have a longitudinal dataset of tracked honeybee positions.  This I think is only possible to obtain in controlled laboratory settings (though the authors redact the exact method they use), hence the method does not seem broadly applicable.  Second, even if the data were easy to collect, what do the representations indicate that cannot be obtained directly from the data?  If the temporal basis functions merely reflect what hive locations a bee frequents, can that not be extracted much more easily from the tracking data that the model relies on for training?  Third, the authors allude to potential usefulness of the representations for understanding the influence of pesticides or predatory pressure on the colony, but I think the paper really needs a concrete demonstration of (or at least very strong justification for) the usefulness of the representations.\n* I definitely appreciate the sweeps over the regularization hyperparameters in table 2, but I am worried about whether these hyperparameters will transfer to other dataset, since the model has quite a number of hyperparameters.  So from a user perspective it would be good to know how robust the authors’ hyperparameters are across datasets.  One way to inform this would be to evaluate the hyperparameter sweeps for the synthetic datasets and see whether optimal hyperparameters (e.g. for the AMI score) are consistent across synthetic datasets.\n* Please include the baseline models in Figure 3 (if it is possible to compute AMI for the baselines).  Without baselines, it’s difficult for a reader to determine how good the TNMF curve is.\n\nFinally, I have a few very minor points:\n* Please put a legend in the leftmost panel of Figure 4.\n* In Figure 5-b, I’m guessing the “f_2” label in the legend supposed to be “f_3”.  If that is true, please change it.  If not, please justify why you are using f_2 in Figure 5 instead of f_3 when it is previously stated that only two factors (f_1 and f_3) are significant.\n* The caption for Table 2 reads “Table 3” --- please change that to read “Table 2.”\n\n\n# Conclusions\n\nI do not recommend accepting this paper.  My main concerns are that the usefulness of the method is not demonstrated or well-justified, and the paper as a whole seems to have limited scope in a field of animal behavior that I suspect is pretty far from most ICLR attendees’ research interests.  However, the AC’s are in a better position than I am to judge the latter point, so I am not certain about my recommendation and could be convinced otherwise.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "R5Ic7OmdCDW", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors present a matrix factorization model to jointly characterize the lifetime interactions of thousands of bees over generations. The problem is fascinating as both a technical and scientific question and the modeling framework appears novel. Although not directly addressed, the authors appear to be trying to solve a *tensor* factorization problem, not just the special case of a matrix (which is of course a 2-d tensor). It would have been interesting to see results comparing their method with a non-negative variant of, say, PARAFAC/CANDECOMP or generalizations thereof. It would have at least have been appropriate to explain why or why not existing tensor methods are not appropriate.\n\nWhile I'm excited about this work and would love to see the potential applications in other domains, I'm troubled by the lack of clarity and sparsity of essential details in the main text. I am also concerned about how the authors have prioritized the regularization. I detail my concerns below:\n\nSome erroneous (or perhaps just awkward notational issues) make reading difficult and omitted material makes it hard to evaluate the results.\n* The authors do not explain in either the main text or in the supplement what $\\omega$ is (first appears in equation (3)) and what about it is being learned (as indicated in equation (5)). \n* Equation (4) uses $k$ as both an index for the terms in the sum and as a subscript for the number of learned basis functions (although it is not clear why that number has its own index). What did the authors \n* How did the authors select the hyperparameters not listed in Table 2?\n\nIt seems that the \"interpretability\" metrics reported in Table 2 are precisely the same criteria as are being regularized for. It therefor should not be surprising that TNMF outperforms SymNMF on these metrics. What is somewhat concerning is how big a difference there is in the reconstruction loss and how much TNMF seems to fail in this regard. This must mean that the features that the authors wish to represent explain very little of the variance in the interaction networks. Could the authors 1) comment on this a bit and discuss why it would be the case and, 2) give us a sense of what the scale of the errors are relative to the total variance of $\\mathbf{A}$, 3) comment on whether or not they could be missing high-variance features that they did not anticipate.\n\nLastly, there is a fundamental problem I have with the authors' regularization approach to interpretability. It seems as though they have enforced a handful of individual features for their model in order to achieve interpretability. However, these features are ostensibly things they already know about the individuals in the population. The point of an unsupervised method is to allow yourself the opportunity to discover things that you didn't already know from observing the features of individuals. Moreover, \"interpretability\" is somewhat subjective. Therefore, I'm concerned that the authors are using these regularization terms a little too haphazardly and have failed to make a strong argument for their use vis-a-vis their effect on reconstruction error.\n\nI pains me to give this a mediocre score because I do find the work fascinating.  The paper requires more principled motivation for the choices the authors made as well as cleaning up the notation.", "extra_scores": {}, "extra_text": {}}, {"review_id": "PAM2GlryvfR", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors propose a factorization method for exploratory analysis in temporal graphs / social networks. If I understand the model correctly, the new idea is to allow individual-specific differences in temporal trajectories (to account, e.g., for difference in age). Some of these ideas might be under the surface in time-shifted tensor decomposition models (linked below), but I do think that the authors' work is novel:\n\nhttps://www.biorxiv.org/content/10.1101/2020.03.02.974014v2\nhttps://doi.org/10.1016/j.neuroimage.2008.05.062\n\nThough I think the core idea is interesting, I have mixed feelings about the method and some of the analyses.\n\n(1) The method seems potentially more complex than is necessary. Honestly I found its presentation in the paper confusing and difficult to read. Perhaps section 2.1 is a bit too short, and more visualizations like Fig 2 would be helpful.\n\nI think the biggest problem is that the authors' motivation for various choices is not well-described. For example, the adversarial loss is not well-explained, and it is referred to as \"L_adversarial\" in Fig 2, but \"R_adv\" in the Appendix (or are these two different things?).\n\nPerhaps a bigger question is why the authors chose to decompose their factors $f_i = m(t_i) + o_i$ (see eq. 3). I think this is an interesting choice, but it seems like there could be many others like $f_i = \\sum_j \\alpha_{ij} m_j(t_i)$ or $f_i = m(t_i)$. If I understood the paper correctly, the second option is a simpler extension of Yu, Aggarwal, & Wang (2017), which seems like natural to explore before trying the more complicated model proposed in this work.\n\nSmoothness over time seems to be an important aspect of the model (as judged from the schematic figures), but it is not emphasized by the authors how this is achieved in practice. Rather than having a neural network output the mean m, why not use a gaussian process, or a linear layer with a smooth spline basis / RBFs?\n\n(2) If I understand the model correctly, it seems very similar to a classic tensor factorization, so it would seem to me that a better baseline to include would be $\\hat{A}_t = F D_t F^T$ where each $D_t$ for $t = 1, ..., T$ is constrained to be diagonal. There are very fast and well-established algorithms for this (see Kolda & Bader, 2009). Also see the time-shifted model variants linked above.\n\n(3) The evaluation metrics are a bit strange. The model performs worse than the baselines on reconstruction, though this does not seem to be the primary goal of the paper. Rather, the idea is to pull out interpretable factors. I had a hard time understanding whether the consistency / mortality / rhythmicity metrics were actually a good proxy for model interpretability. My suggestion would be to focus on qualitative / visual comparisons across the different models rather than these numeric comparisons.\n\n(4) It would improve the paper to show the method working on a second dataset (e.g. a different social network or temporal graph of interactions).\n\nIn Summary: I like some of the ideas presented in this work, but I found it difficult to read. I think the biggest problem is that the core idea (individual-specific time courses) seems to get buried and lost with other modeling innovations like adversarial training and new terms like \"individuality basis functions.\" Some of these bells and whistles may not even be needed, so simplifying the model and streamlining the text would go a long way for me.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The authors present a matrix factorization for the social behavior of honey bees in a hive. All the reviewers appreciated the interesting application.  However, substantial concerns were raised about the model motivation and the interpretation of the learned factors. To quote one reviewer, "Some of these bells and whistles may not even be needed, so simplifying the model and streamlining the text would go a long way for me." Another said, "The paper requires more principled motivation for the choices the authors made as well as cleaning up the notation."  The authors did address some of these concerns in discussion, but there are too many lingering concerns to recommend acceptance.  Given the unique application of this paper, the authors might also consider a journal that specializes in computational biology instead. 

Decision: Reject