Original: Wasserstein Learning of Determinantal Point Processes
Revised: Wasserstein Learning of Determinantal Point Processes

Original: Anonymous Author(s)

Original: AfﬁliationAddress

Original: Abstract
Revised: Abstract

Original: Determinantal point processes (DPPs) have received signiﬁcant attention as anelegant probabilistic model for discrete subset selection.
Revised: Determinantal point processes (DPPs) have received signiﬁcant attention as an elegant probabilistic model for discrete subset selection.

Original: Most prior work on DPPlearning focuses on maximum likelihood estimation (MLE).
Revised: Most prior work on DPP learning focuses on maximum likelihood estimation (MLE).

Original: While efﬁcient andscalable, MLE approaches do not leverage any subset similarity information andmay fail to recover the true generative distribution of discrete data.
Revised: While efﬁcient and scalable, MLE approaches do not leverage any subset similarity information and may fail to recover the true generative distribution of discrete data.

Original: In this work,by deriving a differentiable relaxation of a DPP sampling algorithm, we present anovel approach for learning DPPs that minimizes the Wasserstein distance betweenthe model and data composed of observed subsets.
Revised: In this work, by deriving a differentiable relaxation of a DPP sampling algorithm, we present a novel approach for learning DPPs that minimizes the Wasserstein distance between the model and data composed of observed subsets.

Original: Through an evaluation ona real-world dataset, we show that our Wasserstein learning approach providessigniﬁcantly improved predictive performance on a generative task compared to
Revised: Through an evaluation on a real-world dataset, we show that our Wasserstein learning approach provides signiﬁcantly improved predictive performance on a generative task compared to

Original: DPPs trained using MLE
Revised: DPPs trained using MLE.

Original:  Introduction
Revised:  Introduction

Original: Generative models have enjoyed a great deal of success in the recent years due to their ability tocapture insights from data distributions.
Revised: Generative models have enjoyed a great deal of success in the recent years due to their ability to capture insights from data distributions.

Original: Those models have generally been applied to continuous databy training them using maximum likelihood estimation (MLE) or, more recently, using adversariallearning with the well-known Generative Adversarial Networks framework [12, 15].
Revised: Those models have generally been applied to continuous data by training them using maximum likelihood estimation (MLE) or, more recently, using adversarial learning with the well-known Generative Adversarial Networks framework [12, 15].

Original: When dealing with discrete data, generative models trained with MLE suffer from a bias due to theasymmetrical deﬁnition of MLE.
Revised: When dealing with discrete data generative models trained with MLE suffer from a bias due to the asymmetrical deﬁnition of MLE.

Original: Equivalent to minimizing a Kullback Leibler divergence, the MLEcost function pays extremely low cost for generating low-quality samples.
Revised: Equivalent to minimizing a Kullback Leibler divergence, the MLE cost function pays extremely low cost for generating low-quality samples.

Original: Consequently, a generativemodel trained by MLE tends to cover the full data distribution at the expense of covering unnecessaryregions [2, 24].
Revised: Consequently, a generative model trained by MLE tends to cover the full data distribution at the expense of covering unnecessary regions [2, 23].

Original: On the other hand, when considering adversarial learning of discrete generativemodels, one usually exploits the gradient of the discriminator’s loss when optimizing the generator.
Revised: On the other hand, when considering adversarial learning of discrete generative models, one usually exploits the gradient of the discriminator’s loss when optimizing the generator.

Original: However, since the gradient computation requires backpropagation through the generator’s output, i.e.the data, adversarial approaches are difﬁcult to apply when generating discrete data.
Revised: However, since the gradient computation requires backpropagation through the generator’s output, i.e. the data, adversarial approaches are difﬁcult to apply when generating discrete data.

Original: Depending onthe generative model and the structure of the data, there are some ways to overcome this issue.
Revised: Depending on the generative model and the structure of the data, there are some ways to overcome this issue.

Original: Forinstance, [19] was the ﬁrst to deﬁne a sampling scheme with the use of a Gumbel softmax distribution,and several generalizations of this softmax trick have been recently proposed in the literature [26, 14].
Revised: For instance, [19] was the ﬁrst to deﬁne a sampling scheme with the use of a Gumbel softmax distribution, and several generalizations of this softmax trick have been recently proposed in the literature [26, 14].

Original: In this work, we address the problem of training a determinantal point process (DPP), a probabilisticmodel for subsets drawn from a large collection of items.
Revised: In this work, we address the problem of training a determinantal point process (DPP), a probabilistic model for subsets drawn from a large collection of items.

Original: A DPP parameterizes a probabilitydistribution over the combinatorial space of subsets of elements drawn from J , which is a discretespace composed of M distinct items.
Revised: A DPP parameterizes a probability distribution over the combinatorial space of subsets of elements drawn from J , which is a discrete space composed of M distinct items.

Original: DPPs are appealing models for this setting, since they areknown to also capture interactions between elements within subsets.
Revised: DPPs are appealing models for this setting, since they are known to also capture interactions between elements within subsets.

Original: More importantly, they offerefﬁcient polynomial-time algorithms for most probabilistic inference operations over the space of2 M possible subsets, such as normalization, learning, and sampling [10, 18].
Revised: More importantly, they offer efﬁcient polynomial-time algorithms for most probabilistic inference operations over the space of 2 M possible subsets, such as normalization, learning, and sampling [10, 18].

Original: In order to move awayfrom the standard MLE learning framework [8, 23], which may suffer from the ﬂaw described above,we deﬁne a new learning scheme for DPPs based on the minimization of the Wasserstein distancebetween the samples generated by the DPP and the training data.
Revised: In order to move away from the standard MLE learning framework [8, 22], which may suffer from the ﬂaw described above, we deﬁne a new learning scheme for DPPs based on the minimization of the Wasserstein distance between the samples generated by the DPP and the training data.

Original: Compared to MLE, one of the main beneﬁts of this Wasserstein-based approach is that it allows us to deﬁne a transportation costfunction ( e.g a Jaccard distance) that induces a bias on the assumed structure of the space of subsets.
Revised: Compared to MLE, one of the main beneﬁts of this Wasserstein-based approach is that it allows us to deﬁne a transportation cost function ( e.g a Jaccard distance) that induces a bias on the assumed structure of the space of subsets.

Original: Minimizing this cost function allows the learning to take into account differences between pairsof subsets and to reduce the distance between subsets based on their similarities.
Revised: Minimizing this cost function allows the learning to take into account differences between pairs of subsets and to reduce the distance between subsets based on their similarities.

Original: We argue thatthis Wasserstein-based scheme leverages more information from the data and results in a betterapproximation of the target distribution.
Revised: We argue that this Wasserstein-based scheme leverages more information from the data and results in a better approximation of the target distribution.

Original: The contributions of this work are the following: 1.)
Revised: The contributions of this work are the following: 1.)

Original: We present a new framework when learning
Revised: We present a new framework when learning

Original: DPPs that minimizes the Wasserstein distance between the DPP and data composed of observedsubsets.
Revised: DPPs that minimizes the Wasserstein distance between the DPP and data composed of observed subsets.

Original: This framework can be applied to any generative probabilistic model for discrete sets.
Revised: This framework can be applied to any generative probabilistic model for discrete sets.

Original: Leveraging recent work on a DPP sampling algorithm with computational complexity that is sublinearin the size of the ground set [3], and stochastic softmax tricks for gradient estimation of discretedistributions [14], we present a differentiable DPP sampling algorithm that can scale to large groundsets.
Revised: Leveraging recent work on a DPP sampling algorithm with computational complexity that is sublinear in the size of the ground set [3], and stochastic softmax tricks for gradient estimation of discrete distributions [14], we present a differentiable DPP sampling algorithm that can scale to large ground sets.

Original: We evaluate our Wasserstein learning approach on a real-world dataset, and show substantialimprovements in predictive performance compared to DPPs trained using MLE.
Revised: We evaluate our Wasserstein learning approach on a real-world dataset, and show substantial improvements in predictive performance compared to DPPs trained using MLE.

Original: This experimentalevaluation is one of the ﬁrst to focus on a generative modeling task for DPPs.
Revised: This experimental evaluation is one of the ﬁrst to focus on a generative modeling task for DPPs.

Original:  Background and related work
Revised:  Background and related work

Original: Determinantal Point Processes Consider a ﬁnite set J = { 1 , 2 , . . .
Revised: Determinantal Point Processes Consider a ﬁnite set J = { 1 , 2 , . . .

Original: , M } of cardinality M , whichwe will also denote by [[ M ]] .
Revised: , M } of cardinality M , which we will also denote by [[ M ]] .

Original: A DPP deﬁnes a probability distribution over all 2 M subsets.
Revised: A DPP deﬁnes a probability distribution over all 2 M subsets.

Original: It isparameterized by a matrix L ∈ R M × M , called the kernel , such that the probability of each subset
Revised: It is parameterized by a matrix L ∈ R M × M , called the kernel , such that the probability of each subset

Original: J ⊆
Revised: J ⊆

Original: [[ M ]] is proportional to the determinant of its corresponding principal submatrix: Pr( Y ) ∝det( L J ) , where L J =
Revised: [[ M ]] is proportional to the determinant of its corresponding principal submatrix: Pr( Y ) ∝ det( L J ) , where L J =

Original: [ L ij ] i,j ∈ J is the submatrix of L indexed by J .
Revised: [ L ij ] i,j ∈ J is the submatrix of L indexed by J .

Original: The normalization constant forthis distribution can be expressed as a single M × M determinant: (cid:80) J ⊆
Revised: The normalization constant for this distribution can be expressed as a single M × M determinant: (cid:80) J ⊆

Original: [[ M ]] det( L J )
Revised: [[ M ]] det( L J )

Original: = det( L + I )[18, Theorem 2.1].
Revised: = det( L + I ) [18, Theorem 2.1].

Original: Therefore, Pr( J ) = det( L J ) / det( L + I ) .
Revised: Therefore, Pr( J ) = det( L J ) / det( L + I ) .

Original: In order to ensure that the DPP deﬁnes a probability distribution, all principal minors of L mustbe non-negative: det( L J ) ≥ 0 .
Revised: In order to ensure that the DPP deﬁnes a probability distribution, all principal minors of L must be non-negative: det( L J ) ≥ 0 .

Original: Matrices that satisfy this property are called P 0 -matrices [4,
Revised: Matrices that satisfy this property are called P 0 -matrices [4,

Original: Deﬁnition 1].
Revised: Deﬁnition 1].

Original: Several decompositions of L that partially cover the P 0 space are known.
Revised: Several decompositions of L that partially cover the P 0 space are known.

Original: One commondecomposition that covers the space of symmetric P 0 -matrices exploits the fact that L ∈ P 

Revised: One common decomposition that covers the space of symmetric P 0 -matrices exploits the fact that L ∈ P

Original: if L ispositive semideﬁnite (PSD) [29].
Revised: if L is positive semideﬁnite (PSD) [29].

Original: Any symmetric PSD matrix can be written as the Gramian matrixof some set of vectors:
Revised: Any symmetric PSD matrix can be written as the Gramian matrix of some set of vectors:

Original: L := V V (cid:62) , where V ∈ R M × K .
Revised: L := V V (cid:62) , where V ∈ R M × K .

Original: We restrict our work in this paper to suchsymmetric DPPs with this decomposition, since efﬁcient sampling algorithms, such as [3], are onlyavailable for symmetric DPPs.
Revised: We restrict our work in this paper to such symmetric DPPs with this decomposition, since efﬁcient sampling algorithms, such as [3], are only available for symmetric DPPs.

Original: Other decompositions of L that partially cover the nonsymmetric
Revised: There are decompositions of L that partially cover the nonsymmetric

Original: P 0space have been studied [9]; we leave an investigation of Wasserstein learning of nonsymmetric DPPs for future work.
Revised: P 0 [9]; we leave an investigation of Wasserstein learning of nonsymmetric DPPs for future work.

Original: In this work we use the DPP-VFX sampling algorithm [3], which has computational complexitysublinear in M , and is therefore one of the most efﬁcient exact sampling methods for DPPs.
Revised: In this work we use the DPP-VFX sampling algorithm [3], which has computational complexity sublinear in M , and is therefore one of the most efﬁcient exact sampling methods for DPPs.

Original: DPP-VFXrelies on a connection between ridge leverage scores [1] and DPPs to implement a distortion-freeintermediate sampling method that enables this sublinear time complexity.
Revised: DPP-VFX relies on a connection between ridge leverage scores [1] and DPPs to implement a distortion-free intermediate sampling method that enables this sublinear time complexity.

Original: Since DPP-VFX requires abase DPP sampling algorithm, we propose to use the Cholesky-based DPP sampling approach [20, 28].
Revised: Since DPP-VFX requires a base DPP sampling algorithm, we propose to use the Cholesky-based DPP sampling approach [20, 28].

Original: Estimating gradients in discrete settings The Wasserstein learning approach requires computinggradients over discrete subset samples drawn from a DPP.
Revised: Estimating gradients in discrete settings The Wasserstein learning approach requires computing gradients over discrete subset samples drawn from a DPP.

Original: Two families of approaches for discrete gradient estimation are score function estimators, such as REINFORCE [30], and continuous relaxationsof discrete distributions, most of which are based on the Gumbel-Max trick [21].
Revised: Two families of approaches for discrete gradient estimation are score function estimators, such as REINFORCE [30], and continuous relaxations of discrete distributions, most of which are based on the Gumbel-Max trick [21].

Original: REINFORCE hasthe drawback of high variance, making it impractical in many cases.
Revised: REINFORCE has the drawback of high variance, making it impractical in many cases.

Original: While techniques for variancereduction exist [25], they often involve highly engineered control variates.
Revised: While techniques for variance reduction exist [24], they often involve highly engineered control variates.

Original: Relaxed gradient estimators incorporate bias in order to reduce variance, and are often easier to implement [26].
Revised: Relaxed gradient estimators incorporate bias in order to reduce variance, and are often easier to implement [26].

Original: We choosethe relaxation approach, and leverage recent work on stochastic softmax tricks [19, 26], which is auniﬁed framework for structured relaxations of discrete combinatorial distributions.
Revised: We choose the relaxation approach, and leverage recent work on stochastic softmax tricks [19, 26], which is a uniﬁed framework for structured relaxations of discrete combinatorial distributions.

Original: In particular, weuse stochastic softmax tricks to develop a differentiable version of the DPP-VFX sampling algorithm,with a differentiable version of the Choleksy-based approach as the base DPP sampling algorithm.
Revised: In particular, we use stochastic softmax tricks to develop a differentiable version of the DPP-VFX sampling algorithm, with a differentiable version of the Choleksy-based approach as the base DPP sampling algorithm.

Original: As far as we are aware, this is the ﬁrst instance of a differentiable DPP sampling algorithm, whichmay be of independent interest.
Revised: As far as we are aware, this is the ﬁrst instance of a differentiable DPP sampling algorithm.

Original:  Learning DPPs via Wasserstein minimization
Revised:  Learning DPPs via Wasserstein minimization

Original: The classical approach for learning a DPP kernel given a collection of subsets is to maximize thelikelihood of data samples drawn from the same distribution as the one used for obtaining training examples [8, 9].
Revised: The classical approach for learning a DPP kernel given a collection of subsets is to maximize the likelihood of data samples drawn from the same distribution as the one used for obtaining training examples [8, 9].

Original: One advantage of optimizing the (log) likelihood is that the likelihood of samples hasa closed form expression with respect to the model parameters.
Revised: One advantage of optimizing the (log) likelihood is that the likelihood of samples has a closed form expression with respect to the model parameters.

Original: Since that expression is continuouslydifferentiable, a gradient ascent algorithm is a natural solution for solving the problem.
Revised: Since that expression is continuously differentiable, a gradient ascent algorithm is a natural solution for solving the problem.

Original: Insteadof likelihood maximization, we propose a DPP learning approach that minimizes the Wassersteindistance between the training data and samples generated by the model.
Revised: Instead of likelihood maximization, we propose a DPP learning approach that minimizes the Wasserstein distance between the training data and samples generated by the model.

Original: This optimization schemeseeks to improve the approximation of the generative distribution of the data.
Revised: This optimization scheme seeks to improve the approximation of the generative distribution of the data.

Original: The Wasserstein distance is a distance between probability distributions deﬁned on a given metricspace.
Revised: The Wasserstein distance is a distance between probability distributions deﬁned on a given metric space.

Original: We let X n = { x 1 , · · · , x n } denote the training dataset of size n with empirical distributionµ = (cid:80) ni
Revised: We let X n = { x 1 , · · · , x n } denote the training dataset of size n with empirical distribution µ = (cid:80) ni

Original: =1 a i δ x i , where δ refers to the Dirac distribution, and Y n = { y 1 , · · · , y n } be the collectionof n sets sampled from the DPP model with distribution ν = (cid:80) mi =1 b i
Revised: =1 a i δ x i , where δ refers to the Dirac distribution, and Y n = { y 1 , · · · , y n } be the collection of n sets sampled from the DPP model with distribution ν = (cid:80) mi =1 b i

Original: δ y i , where the a i and b ifollow a uniform distribution.
Revised: δ y i , where the a i and b i follow a uniform distribution.

Original: Given a transportation cost d deﬁned on 2 M × 2 M , the Wassersteindistance between µ and ν seeks an optimal coupling P deﬁned on [1 , n ] 2 that minimizes the costof transporting mass from µ to ν [27].
Revised: Given a transportation cost d deﬁned on 2 M × 2 M , the Wasserstein distance between µ and ν seeks an optimal coupling P deﬁned on [1 , n ] 2 that minimizes the cost of transporting mass from µ to ν [27].

Original: When dealing with discrete sets of items, we argue that theuse of the Jaccard distance [17] as as transportation cost function is a good choice.
Revised: When dealing with discrete sets of items, we argue that the use of the Jaccard distance [17] as as transportation cost function is a good choice.

Original: The Jaccarddistance between two sets takes into account both the difference in length and in the items chosen:d J ( X, Y ) =
Revised: The Jaccard distance between two sets takes into account both the difference in length and in the items chosen: d J ( X, Y ) =

Original: ( | X ∪ Y | − | X ∩ Y | ) /
Revised: ( | X ∪ Y | − | X ∩ Y | ) /

Original: | X ∪ Y | , where X, Y ∈ 2 M .
Revised: | X ∪ Y | , where X, Y ∈ 2 M .

Original: Since the cost function needs tobe differentiable, we use a differentiable proxy for Jaccard distance.
Revised: Since the cost function needs to be differentiable, we use a differentiable proxy for Jaccard distance.

Original: For x ∈ X n and y ∈ Y n , thedifferentiable Jaccard distance d S is deﬁned as follows:
Revised: For x ∈ X n and y ∈ Y n , the differentiable Jaccard distance d S is deﬁned as follows:

Original: [Equation]
Revised: [Equation]

Original: where x , ∈ { 0 , 1 } M is a binary indicator vector and y ∈ [0 , 1] M is a continuous relaxation of a binaryvector, with y k , k ∈ [1 , M ] being the inclusion probability of item k in the sample.
Revised: where x , ∈ { 0 , 1 } M is a binary indicator vector and y ∈ [0 , 1] M is a continuous relaxation of a binary vector, with y k , k ∈ [1 , M ] being the inclusion probability of item k in the sample.

Original: By combining thedeﬁnition of the Wasserstein distance with the chosen cost function in (1), we deﬁne the following
Revised: By combining the deﬁnition of the Wasserstein distance with the chosen cost function in (1), we deﬁne the following

Original: Wasserstein optimization problem for DPPs:argmin
Revised: Wasserstein optimization problem for DPPs: [Equation] P

Original: [Equation]
Revised: [Equation]

Original: where { x 1 , · · · , x n } is the training data, { y 1 , · · · , y n } is a collection of n subsets drawn from the
Revised: where { x 1 , · · · , x n } is the training data, { y 1 , · · · , y n } is a collection of n subsets drawn from the

Original: DPP, and α ≥ 0 is a tunable hyperparameter for regularization.
Revised: DPP, and α ≥ 0 is a tunable hyperparameter for regularization.

Original: Recall that we use the decomposition
Revised: Recall that we use the decomposition

Original: L = V V (cid:62) for the DPP kernel.
Revised: L = V V (cid:62) for the DPP kernel.

Original: Algorithm 1 Wasserstein learning
Revised: Algorithm 1 Wasserstein learning

Original: Input: training data, V ∈ R M × K , maxIter for maxIter steps do
Revised: Input: training data, V ∈ R M × K , maxIter for maxIter steps do

Original: Sample subsets from training data.
Revised: Sample subsets from training data.

Original: Sample subsets from DPP.
Revised: Sample subsets from DPP.

Original: Compute P (cid:63) in Eq. 2 with [6].
Revised: Compute P (cid:63) in Eq. 2 with [6].

Original: Update V using Eq. 2. end for
Revised: Update V using Eq.end for

Original: For the sake of completeness, the algorithm usedto solve this optimization scheme is described in
Revised: For the sake of completeness, the algorithm used to solve this optimization scheme is described in

Original: Algorithm 1.
Revised: Algorithm 1.

Original: Solving the optimization problemdeﬁned in Eq.
Revised: Solving the optimization problem deﬁned in Eq.

Original: 2 with backpropagation requirescomputing the gradient on minibatches with respect to the parameters V , and thus a differentiable sampling algorithm is needed.
Revised: 2 with backpropagation requires computing the gradient on minibatches with respect to the parameters V , and thus a differentiable sampling algorithm is needed.

Original: We use anestimation of the Wasserstein distance on a minibatch [5] by computing a Earth Mover distance,and consider a new differentiable formulation ofthe DPP-VFX sampling algorithm [3], shown in Algorithm 2 in Appendix A.
Revised: We use an estimation of the Wasserstein distance on a minibatch [5] by computing a Earth Mover distance, and consider a new differentiable formulation of the DPP-VFX sampling algorithm [3], shown in Algorithm 2 in Appendix A.

Original: We apply the Gumbelsoftmax trick to the base Poisson and multinomial sampling steps in DPP-VFX.
Revised: We apply the Gumbel softmax trick to the base Poisson, and multinomial, and Bernoulli sampling steps in DPP-VFX.

Original: Combined withdifferentiable Bernoulli and DPP Cholesky-based sampling procedures (Algorithm 6), this samplergenerates continuous relaxations of binary indicator vectors for subsets; see Appendix A for details.
Revised: Combined with a differentiable DPP Cholesky-based sampler (Algorithm 6), this sampler generates continuous relaxations of binary indicator vectors for subsets; see Appendix A for details.

Original:  Experiments
Revised:  Experiments

Original: We perform experiments on the Amazon Baby Registries dataset.
Revised: We perform experiments on the Amazon Baby Registries dataset.

Original: This dataset consists of registriesor "baskets" of baby products, and has been used in prior work on DPP learning [7, 9, 11, 22].
Revised: This dataset consists of registries or "baskets" of baby products, and has been used in prior work on DPP learning [7, 9, 11, 22].

Original: Theregistries contain items from 15 different categories, such as “apparel”, with a catalog of up to100 items per category.
Revised: The [Figure] [Table]

Original: We evaluate on the most popular apparel category, which contains 14,970registries, as well as the popular diaper and feeding categories.
Revised: [Figure] [Table] [Figure]

Original: We leave experiments on largerdatasets for future work.
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: [Table] 
Revised: [Table] [Figure] [Table]

Original: [Figure]
Revised: [Figure]

Revised: registries contain items from 15 different categories, such as “apparel”, with a catalog of up to 100 items per category.

Revised: We evaluate on the most popular apparel category, which contains 14, registries, as well as the popular diaper and feeding categories.

Original: 4.1 Setup and evaluation metrics
Revised: 4.1 Setup and evaluation metrics

Original: A small set consisting of 300 randomly-selected baskets is kept for validation and a further randomselection of 2000 baskets is used for testing.
Revised: A small set consisting of 300 randomly-selected baskets is kept for validation, and a further random selection of 2000 baskets is used for testing.

Revised: We implement our models using PyTorch [25];

Original: Adam [16] is used for optimization, in conjunction withthe solver from the POT package [6].
Revised: Adam [16] is used for optimization, in conjunction with the solver from the POT package [6].

Original: We use the low-rank symmetric DPP (SDPP) [8] and the low-rank nonsymmetric DPP (NDPP) [9],both trained using MLE, as baseline models for all experiments.
Revised: We use the low-rank symmetric DPP (SDPP) [8] and the low-rank nonsymmetric DPP (NDPP) [9], both trained using MLE, as baseline models for all experiments.

Original: We evaluate these baselines andour Wasserstein DPP model (WDPP) model on a subset generation task, where we estimate the
Revised: We evaluate these baselines and our Wasserstein DPP model (WDPP) model on a subset generation task, where we estimate the

Original: Wasserstein distance (WD) between subsets sampled from the model and subsets in the test set bycomputing the Earth Mover’s distance between these two subset collections using POT [6].
Revised: Wasserstein distance (WD) between subsets sampled from the model and subsets in the test set by computing the Earth Mover’s distance between these two subset collections using POT [6].

Original: 4.2 Results
Revised: 4.2 Results

Original: Consistent with prior work, we see that the MLE NDPP outperforms the MLE SDPP on the testlog-likelihood metric.
Revised: Consistent with prior work, we see that the MLE NDPP outperforms the MLE SDPP on the test log-likelihood metric.

Original: However, we also observe that MLE is not directly connected to the generativetask, and higher performance on test log-likelihood does not result in higher performance on the
Revised: However, we also observe that MLE is not directly connected to the generative task, and higher performance on test log-likelihood does not result in higher performance on the

Original: WD metric.
Revised: WD metric.

Original: As expected, since the Wasserstein learning approach directly optimizes a proxy for thegenerative task, the WDPP model signiﬁcantly outperforms the baseline models in terms of WD.
Revised: As expected, since the Wasserstein learning approach directly optimizes a proxy for the generative task, the WDPP model signiﬁcantly outperforms the baseline models in terms of WD.

Original: Toprovide some evidence of the connection between the WD metric and the quality of generated subsets, Fig.
Revised: To provide some evidence of the connection between the WD metric and the quality of generated subsets, Fig.

Original: 1 shows the percentage of generated subsets from each model that have a Jaccard distance ofat most (cid:15) with at least one subset in the test set, for (cid:15) ∈ (0 , 1] .
Revised: 1 shows the percentage of generated subsets from each model that have a Jaccard distance of at most (cid:15) with at least one subset in the test set, for (cid:15) ∈ (0 , 1] .

Original: For any given (cid:15) , we see that WDPPoutperforms MLE models.
Revised: For any given (cid:15) , we see that WDPP outperforms MLE models.

Original: This highlights that the WDPP, by being able to take the Jaccard distanceinto account, exploits the underlying structure of the combinatorial space 2 M , while the MLE-trainedmodels do not and thus treat all subsets as completely different.
Revised: This highlights that the WDPP, by being able to take the Jaccard distance into account, exploits the underlying structure of the combinatorial space 2 M , while the MLE-trained models do not and thus treat all subsets as completely different.

Original: We present additional experimentalresults in Appendix C.
Revised: We present additional experimental results in Appendix C.

Original: These results provide further evidence that, compared to DPPs learned by
Revised: These results provide further evidence that, compared to MLE

Original: MLE, the WDPP model recovers signiﬁcantly more structure, and is able to generate subsets that aresubstantially closer to observed data.
Revised: DPPs, the WDPP model recovers signiﬁcantly more structure, and is able to generate subsets that are substantially closer to observed data.

Original:  Conclusion
Revised:  Conclusion

Original: We have presented a new Wasserstein learning approach for DPPs.
Revised: We have presented a new Wasserstein learning approach for DPPs.

Original: Unlike conventional MLElearning approaches for DPPs, this learning approach optimizes a proxy for discrete subset generation.
Revised: Unlike conventional MLE learning approaches for DPPs, this learning approach optimizes a proxy for discrete subset generation.

Original: Empirical results indicate that the proposed approach leads to substantially improved generativeperformance compared to MLE.
Revised: Empirical results indicate that the proposed approach leads to substantially improved generative performance compared to MLE.

Original: This approach is fully general, and can be readily applied to otherfamilies of models for discrete subsets.
Revised: This approach is fully general, and can be readily applied to other families of models for discrete subsets.

Original: We leave such an investigation for future work.
Revised: We leave such an investigation for future work.

Original: A Differentiable DPP-VFX Sampling Algorithm
Revised: A Differentiable DPP-VFX Sampling Algorithm

Original: As indicated in Section 2, we have leveraged stochastic softmax tricks (SST)
Revised: As indicated in Section 2, we have leveraged stochastic softmax tricks (SST)

Original: [26] to develop adifferentiable version of the DPP-VFX sampling algorithm [3].
Revised: [26] to develop a differentiable version of the DPP-VFX sampling algorithm [3].

Original: Compared to other DPP samplingalgorithms, DFF-VFX can be substantially faster, since it has time complexity sublinear in M .
Revised: Compared to other DPP sampling algorithms, DFF-VFX can be substantially faster, since it has time complexity sublinear in M .

Original: DPPVFX uses a connection between ridge leverage scores [1] and DPPs to implement a distortion-freeintermediate sampling method that enables this sublinear time complexity.
Revised: DPPVFX uses a connection between ridge leverage scores [1] and DPPs to implement a distortion-free intermediate sampling method that enables this sublinear time complexity.

Original: The ﬁrst step of thesampling algorithm downsamples the items in [[ M ]] i.i.d.
Revised: The ﬁrst step of the sampling algorithm downsamples the items in [[ M ]] i.i.d.

Original: with probability proportional to the ridgeleverage score of each item, and then runs a conventional DPP sampling algorithm on this thinnedor downsampled set of items, whose cardinality is much smaller than M .
Revised: with probability proportional to the ridge leverage score of each item, and then runs a conventional DPP sampling algorithm on this thinned or downsampled set of items, whose cardinality is much smaller than M .

Original: We use a differentiableversion of the Cholesky-based DPP sampling algorithm [20, 28] on this thinned set of items.
Revised: We use a differentiable version of the Cholesky-based DPP sampling algorithm [20, 28] on this thinned set of items.

Original: Whendownsampling the items, the ﬁrst step is to select the number of items that will be kept using a Poissonsampling step, followed by a multinomial sampling step that selects the items that will be included inthe downsampled set.
Revised: When downsampling the items, the ﬁrst step is to select the number of items that will be kept using a Poisson sampling step, followed by a multinomial sampling step that selects the items that will be included in the downsampled set.

Original: Finally, a Bernoulli sampling step is used to perform rejection sampling, inorder to ensure that the ﬁnal exact DPP sample will be contained within the downsampled set.
Revised: Finally, a Bernoulli sampling step is used to perform rejection sampling, in order to ensure that the ﬁnal exact DPP sample will be contained within the downsampled set.

Original: Our differentiable DPP-VFX sampling algorithm is presented in Algorithm 2, where β is a kernelrescaling parameter that ensures that the Poisson parameter ( s ∗ e s/q ) is equal to the catalog size M .
Revised: Our differentiable DPP-VFX sampling algorithm is presented in Algorithm 2, where β is a kernel rescaling parameter that ensures that the Poisson parameter ( s ∗ e s/q ) is equal to the catalog size M .

Original: Differentiable versions of the Poisson, multinomial, Bernoulli, and Cholesky-based samplers invokedby Algorithm 2 are shown in Algorithm 3, Algorithm 4, Algorithm 5, and Algorithm 6, respectively.
Revised: Differentiable versions of the Poisson, multinomial, Bernoulli, and Cholesky-based samplers invoked by Algorithm 2 are shown in Algorithm 3, Algorithm 4, Algorithm 5, and Algorithm 6, respectively.

Original: Input: L ∈ R M × M , β >
Revised: Input: L ∈ R M × M , β >

Original: Initialization: L ← β ∗ L , K ← I − ( L + I ) −l i ← K ii ≈ Pr
Revised: Initialization: L ← β ∗ L , K ← I − ( L + I ) −l i ← K ii ≈ Pr

Original: ( i ∈ S )
Revised: ( i ∈ S )

Original: , s ← (cid:80) i l
Revised: , s ← (cid:80) i l

Original: i if s > 1 then q ← s 2 else q ← s (cid:101)
Revised: i if s > 1 then q ← s 2 else q ← s (cid:101)

Original: L ← sq [ 1 √ l i lj L i,j ] i,j
Revised: L ← sq [ 1 √ l i lj L i,j ] i,j

Original: Downsampling:
Revised: Downsampling:

Original: Acc ← False while not Acc do t ∼ SST-Poisson ( s ∗ e s/q ) (Algorithm 3) σ 1 , ..., σ t i.i.d ∼ SST-Multinomial (cid:0) l 1 s , ..., l n s (cid:1) (Algorithm 4)
Revised: Acc ← False while not Acc do t ∼ SST-Poisson ( s ∗ e s/q ) (Algorithm 3) σ 1 , ..., σ t i.i.d ∼ SST-Multinomial (cid:0) l 1 s , ..., l n s (cid:1) (Algorithm 4)

Original: Acc ∼ Differentiable-Bernoulli (cid:16) e s det( I + (cid:101) L σ ) e ts/q det( I + (cid:101) L ) (cid:17) (Algorithm 5) end while { Sample from thinned item catalog: } (cid:101) S ∼ DPP ( (cid:101)
Revised: Acc ∼ SST-Bernoulli (cid:16) e s det( I + (cid:101) L σ ) e ts/q det( I + (cid:101) L ) (cid:17) (Algorithm 5) end while { Sample from thinned item catalog: } (cid:101) S ∼ DPP ( (cid:101)

Original: L σ ) (Algorithm 6) return S
Revised: L σ ) (Algorithm 6) return S

Original: = { σ i : i ∈ (cid:101) S }
Revised: = { σ i : i ∈ (cid:101) S }

Original: Algorithm3 SST-Poisson sampling
Revised: Algorithm

Original: Input: λ , temperature τ
Revised: Input: λ , temperature τ

Original: STEP 1: {Truncate the total support and compute the probabilities of the integers from 1 to 2 ∗ λ } massProb ←  Poisson λ ( i ) for i in [0 , ..., 2 ∗ λ ]
Revised: STEP 1: {Truncate the total support and compute the probabilities of the integers from 1 to 2 ∗ λ } massLogProb ← log( Poisson λ ( i )) for i in [0 , ..., 2 ∗ λ ]

Original: STEP 2: {Differentiable sampling using the Gumbel Softmax trick over the massProb mass probability distribution} oneHotSample ∼ GumbelSoftmax τ ( massProb )
Revised: STEP 2: {Differentiable sampling using the Gumbel Softmax trick over the massLogProb log mass probability distribution} oneHotSample ∼ GumbelSoftmax τ ( massLogProb )

Original: STEP 3: {Rearrange the one-hot-vector sample into the desired output format using matrix operations} sstPoissonSample ← I · oneHotSample return sstPoissonSample
Revised: STEP 3: {Rearrange the one-hot-vector sample into the desired output format using matrix operations} sstPoissonSample ← I · oneHotSample return sstPoissonSample

Original: A.1 Gumbel-Softmax trick
Revised: A.1 Gumbel-Softmax trick

Original: Our differentiable DPP sampling approach relies on the Gumbel-Softmax reparameterizationtrick [13], which is an efﬁcient gradient estimator that replaces the non-differentiable sample from adiscrete distribution with a differentiable sample from a Gumbel-Softmax distribution.
Revised: Our differentiable DPP sampling approach relies on the Gumbel-Softmax reparameterization trick [13], which is an efﬁcient gradient estimator that replaces the non-differentiable sample from a discrete distribution with a differentiable sample from a Gumbel-Softmax distribution.

Original: The GumbelMax trick provides a simple and efﬁcient way to draw samples z from a discrete distribution with 
Revised: The GumbelMax trick provides a simple and efﬁcient way to draw samples z from a discrete distribution with Algorithm

Original: Input: massprob, nbsample, upperbound, temperature τ
Revised: Input: masslogprob, nbsample, upperbound, temperature τ

Original: STEP  {Sample upperbound differentiable multinomial samples} allSamples ← [GumbelSoftmax 
Revised: STEP 1: {Sample upperbound differentiable multinomial samples} allSamples ← [GumbelSoftmax τ ( masslogprob ) ]

Original: ( massprob ) ] for i in [0,...,upperbound]
Revised: for i in [0,...,upperbound]

Original: STEP  nbsample unique  allSamples} uniqueMultinomialSamples ← unique(allSamples) return uniqueMultinomialSamples
Revised: STEP 2: {Select nbsample unique samples from allSamples} uniqueMultinomialSamples ← unique(allSamples) return uniqueMultinomialSamples

Original: Algorithm
Revised: Algorithm

Original: Input: value
Revised: Input: value, temperature τ

Original: STEP distribution} randValue ← uniform(0, 1) STEP 2: {Compute
Revised: STEP 1: Sample a uniform value and build a massLogProb out of the two values randValue ← uniform(0, 1) massLogProb ← [log( value ) , log( randValue )] , {massLogProb ∈ R 2 }

Original: Bernoulli sample} bernoulliSample ←  ( < randValue) return bernoulliSample
Revised: STEP 2: {Differentiable sampling using the Gumbel Softmax trick over the massLogProb log mass probability distribution} oneHotSample ← GumbelSoftmax τ ( massLogProb ) return oneHotSample[0] Algorithm

Original: Input: L ∈ R M × M , temperature τ
Revised: Input: L ∈ R M × M , temperature τ

Original: K ← I − ( L + I ) −
Revised: K ← I − ( L + I ) −

Original: S ← [] for each item i in catalog do itemValue ∼ Differentiable-Bernoulli
Revised: S ← [] for each item i in catalog do itemValue ∼ Differentiable-Bernoulli

Original: ( K i,i ) (Algorithm 5) {Add 0 or soft-value to S:}
Revised: ( K i,i ) (Algorithm 5) {Add 0 or soft-value to S:}

Original: S ← S + binary ( itemValue ) ∗ sigmoid ( itemValue /τ ) {Update the kernel according to the item sample:}
Revised: S ← S + binary ( itemValue ) ∗ sigmoid ( itemValue /τ ) {Update the kernel according to the item sample:}

Original: K i,i ← K i,i − (1 − binary ( itemValue ))
Revised: K i,i ← K i,i − (1 − binary ( itemValue ))

Original: K [i+1:M],i ← K [i+1:M],i /K
Revised: K [i+1:M],i ← K [i+1:M],i /K

Original: i,i K
Revised: i,i K

Original: [i+1:M],[i+1:M] ← K
Revised: [i+1:M],[i+1:M] ← K

Original: [i+1:M],[i+1:M] − K
Revised: [i+1:M],[i+1:M] − K

Original: [i+1:M],i ⊗ K i,[i+1:
Revised: [i+1:M],i ⊗ K i,[i+1:

Original: M] end for return S class probabilities
Revised: M] end for return S class probabilities

Original: π i : [Equation] = oneHot (cid:18) argmax 
Revised: π i : z = oneHot (cid:18) argmax i

Original: [Equation] [ g [Equation] + log( π
Revised: [ g i

Original: [Equation]
Revised: [Equation]

Original: )] (cid:19) (3) where g 1 ...g k are i.i.d samples drawn from Gumbel(0, 1).
Revised: where g 1 ...g k are i.i.d samples drawn from Gumbel(0, 1).

Original: The softmax is used as a continuous,differentiable approximation to argmax , and generates k -dimensional sample vectors y ∈ ∆ k − 1 ,where each component y i is:
Revised: The softmax is used as a continuous, differentiable approximation to argmax , and generates k -dimensional sample vectors y ∈ ∆ k − 1 , where each component y i is:

Original: [Equation]
Revised: [Equation]

Original: where τ is the temperature hyperparameter.
Revised: where τ is the temperature hyperparameter.

Original: As the softmax temperature τ approaches 0, samples fromthe Gumbel-Softmax distribution become one-hot and the Gumbel-Softmax distribution becomesidentical to the categorical distribution p ( z ) .
Revised: As the softmax temperature τ approaches 0, samples from the Gumbel-Softmax distribution become one-hot and the Gumbel-Softmax distribution becomes identical to the categorical distribution p ( z ) .

Original: B Hyperparameters for experiments in Table1
Revised: B Hyperparameters for experiments in Table

Original: We perform a grid search using a held-out validation set to select the best performing hyperparametersfor each model and dataset.
Revised: We perform a grid search using a held-out validation set to select the best performing hyperparameters for each model and dataset.

Original: The hyperparameter settings used for each model and dataset aredescribed below.
Revised: The hyperparameter settings used for each model and dataset are described below.

Original: Baseline MLE SDPP [7].
Revised: Baseline MLE SDPP [7].

Original: For this model, we use K for the number of item feature dimensionsfor the symmetric component V , and α for the regularization hyperparameter for V .
Revised: For this model, we use K for the number of item feature dimensions for the symmetric component V , and α for the regularization hyperparameter for V .

Original: We use thefollowing hyperparameter settings:
Revised: We use the following hyperparameter settings:

Original: • All datasets: K = 30 , α = 0 , batch-size = 200 .
Revised: • All datasets: K = 30 , α = 0 , batch-size = 200 .

Original: Baseline MLE NDPP [9].
Revised: Baseline MLE NDPP [9].

Original: For this model, to ensure consistency with the notation used in [9], weuse D to denote the number of item feature dimensions for the symmetric component V , and D (cid:48)to denote the number of item feature dimensions for the nonsymmetric components, B and C .
Revised: For this model, to ensure consistency with the notation used in [9], we use D to denote the number of item feature dimensions for the symmetric component V , and D (cid:48) to denote the number of item feature dimensions for the nonsymmetric components, B and C .

Original: Asdescribed in [9], α is the regularization hyperparameter for the V , while β and γ are the regularizationhyperparameters for B and C , respectively.
Revised: As described in [9], α is the regularization hyperparameter for the V , while β and γ are the regularization hyperparameters for B and C , respectively.

Original: We use the following hyperparameter settings:
Revised: We use the following hyperparameter settings:

Original: • All datasets: D = D (cid:48)
Revised: • All datasets: D = D (cid:48)

Original: = 30 , α = β
Revised: = 30 , α = β

Original: = γ = 0 , batch-size = 200 .
Revised: = γ = 0 , batch-size = 200 .

Original: WDPP (ours) 
Revised: WDPP (ours) .

Original: We use K to denote the number of item feature dimensions for V .
Revised: We use K to denote the number of item feature dimensions for V .

Original: α is the regular- ization hyperparameter.
Revised: α is the regularization hyperparameter.

Original: τ C , τ P ,and τ M  are the temperature hyperparameters for Cholesky-based DPP sampling, stochastic softmax trick (SST)
Revised: τ C , τ P , τ M and τ B are the temperature hyperparameters for Choleskybased DPP sampling, stochastic softmax trick (SST)

Original: Poisson sampling,and SST multinomial sampling,respectively.
Revised: Poisson sampling, SST multinomial sampling, and the SST Bernoulli sampling, respectively.

Original: We use the following hyperparameter settings:
Revised: We use the following hyperparameter settings:

Original: • All datasets: K = 30 , α = 0 .
Revised: • All datasets: K = 30 , α = 0 .

Original: 01 , τ C
Revised: 01 , τ C

Original: =
Revised: =

Original: τ P = 0 .
Revised: τ P = 0 .

Original: 1 , τ M = 1 ,  batch-size = 400 .
Revised: 1 , τ M = 1 , τ B = 10 − 8 , batch-size = 400 .

Original: During WDPP training, we anneal both the learning rate and α .
Revised: During WDPP training, we anneal both the learning rate and α .

Original: C Additional Experimental Results
Revised: C Additional Experimental Results

Original: Fig.
Revised: Fig.

Original: 2 shows a plot of the kernels learned by the MLE SDPP and WDPP models for the Amazonfeeding dataset.
Revised: 2 shows a plot of the kernels learned by the MLE SDPP and WDPP models for the Amazon feeding dataset.

Original: We see more apparent structure in the WDPP kernel, suggesting that our Wassersteinlearning approach allows the DPP to capture more structure from the data than when trained using
Revised: We see more apparent structure in the WDPP kernel, suggesting that our Wasserstein learning approach allows the DPP to capture more structure from the data than when trained using

Original: MLE.
Revised: MLE.

Original: In Fig.
Revised: In Fig.

Original: 3 we compare a portion of the empirical marginal item distribution with the marginalscaptured by the MLE SDPP and WDPP models when trained on the Amazon diaper dataset.
Revised: 3 we compare a portion of the empirical marginal item distribution with the marginals captured by the MLE SDPP and WDPP models when trained on the Amazon diaper dataset.

Original: Wesee that WDPP appears to learn a better approximation of the true marginal distribution of the itemsin the data.
Revised: We see that WDPP appears to learn a better approximation of the true marginal distribution of the items in the data.

Original: Finally, Table 2 shows a collection of some of the most common non-singleton subsets(modes) from the test set, and samples generated by the WDPP, SDPP, and NDPP models, for the
Revised: Finally, Table 2 shows a collection of some of the most common non-singleton subsets (modes) from the test set, and samples generated by the WDPP, SDPP, and NDPP models, for the

Original: Amazon apparel dataset.
Revised: Amazon apparel dataset.

Original: Compared to the DPPs trained by MLE, we see that our WDPP modelgenerates subsets that are much closer to subsets found in the empirical test set.
Revised: Compared to the DPPs trained by MLE, we see that our WDPP model generates subsets that are much closer to subsets found in the empirical test set.

Original: [Table]
Revised: [Table]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]