Original: Dimension Reduction for High-dimensional Small Counts with KL Divergence
Revised: Dimension Reduction for High-dimensional Small Counts with KL Divergence

Original: Abstract
Revised: Abstract

Original: Dimension reduction for high-dimensional count data with a large proportion of zeros is an important task in various applications.
Revised: Dimension reduction for high-dimensional count data with a large proportion of zeros is an important task in various applications.

Original: As a large number of dimension reduction methods rely on the proximity measure, we develop a dissimilarity measure that is well-suited for small counts based on the Kullback-Leibler divergence.
Revised: As a large number of dimension reduction methods rely on the proximity measure, we develop a dissimilarity measure that is well-suited for small counts based on the Kullback-Leibler divergence.

Original: We compare the proposed measure with other widely used dissimilarity measures and show that the proposed one has superior discrimination ability when applied to high-dimensional count data having an excess of zeros.
Revised: We compare the proposed measure with other widely used dissimilarity measures and show that the proposed one has superior discriminative ability when applied to high-dimensional count data having an excess of zeros.

Original: Extensive empirical results, on both simulated and publicly-available real-world datasets that contain many zeros, demonstrate that the proposed dissimilarity measure can improve a wide range of dimension reduction methods.
Revised: Extensive empirical results, on both simulated and publicly-available real-world datasets that contain many zeros, demonstrate that the proposed dissimilarity measure can improve a wide range of dimension reduction methods.

Original:  INTRODUCTION
Revised:  INTRODUCTION

Original: High-dimensional count data, especially those with a large proportion of zeros, are omnipresent in various fields, such as ecology and genomics [Warton, 2018, Townes et al., 2019].
Revised: High-dimensional count data, especially those with a large proportion of zeros, are omnipresent in various fields, such as ecology and genomics [Warton, 2018, Townes et al., 2019, Svensson, 2020].

Original: Dimension reduction (DR) techniques are used to extract useful information from high-dimensional count data, by eliminating noisy/uninformative dimensions of the data.
Revised: Dimension reduction (DR) techniques are used to extract useful information from highdimensional count data, by eliminating noisy/uninformative dimensions of the data.

Original: Owing to the mean-variance dependency that is often observed in count data, it is inappropriate to apply standard DR methods that are optimal under the normality assumption, such as principal component analysis (PCA)
Revised: Owing to the mean-variance dependency that is often observed in count data, it is inappropriate to apply standard DR methods that are optimal under the normality assumption, such as principal component analysis (PCA)

Original: [Pearson, 1901, Hotelling, 1933, Tipping and Bishop, 1999] and Gaussian process latent variable model (GPLVM)
Revised: [Pearson, 1901, Hotelling, 1933, Tipping and Bishop, 1999] and Gaussian process latent variable model (GPLVM)

Original: [Lawrence, 2005].
Revised: [Lawrence, 2005], to the data.

Original: Hence, to perform DR on count data, a number of specific strategies/methods have been proposed.
Revised: Hence, to perform DR on count data, a number of specific strategies/methods have been proposed.

Original: A common strategy is to first apply a variance-stabilizing transformation (VST) to the data [Rober, 1938, Cameron and Trivedi, 2013, Bartlett, 1947, Anscombe, 1948], aiming to make the data more Gaussian-like, and then feed the transformed data into standard DR approaches.
Revised: A common strategy is to first apply a variance-stabilizing transformation (VST) to the data [Bartlett, 1947, Anscombe, 1948], aiming to make the data more Gaussian-like, and then feed the transformed data into standard DR approaches.

Original: The transformation function is specifically chosen to remove the mean-variance dependency.
Revised: The transformation function is specifically chosen to remove the mean-variance dependency.

Original: Popular transformation functions include the square root, logarithm, and inverse hyperbolic sine functions.
Revised: Popular transformation functions include the square root, logarithm, and inverse hyperbolic sine functions.

Original: Despite the widespread use of the VSTs, they can only be guaranteed to work well with large counts [Cameron and Trivedi, 2013, Bartlett, 1947, Anscombe, 1948] and cannot reasonably be expected to stabilise the variance of small counts containing a large faction of zeros [Yu, 2009, Warton, 2018].
Revised: Despite the widespread use of the VSTs, they can only be guaranteed to work well with large counts [Bartlett, 1947, Anscombe, 1948] and cannot reasonably be expected to stabilize the variance of small counts containing a large faction of zeros [Yu, 2009, Warton, 2018].

Original: Rather than focusing on making count data more normally distributed, several approaches have been developed to  model the original data directly.
Revised: Rather than focusing on making count data more normally distributed, several approaches have been developed to directly model the original data.

Original: With the assumption that count data follow the exponential family distributions, PCA variants maximise the likelihood of the observed data to get the low-dimensional representation [Collins et al., 2002, Mohamed et al., 2008, Li and Tao, 2013, Smallman et al., 2020].
Revised: With the assumption that count data follow the exponential family distributions, PCA variants maximise the likelihood of the observed data to get the low-dimensional representation [Collins et al., 2002, Mohamed et al., 2008, Li and Tao, 2013, Smallman et al., 2020].

Original: By adopting the same distributional assumption, a robust estimator of the covariance matrix is derived and the data of reduced dimension are obtained by the eigendecomposition of this estimator [Liu et al., 2018].
Revised: By adopting the same distributional assumption, a robust estimator of the covariance matrix is derived and the data of reduced dimension are obtained by the eigendecomposition of this estimator [Liu et al., 2018].

Original: Nonnegative matrix factorisation (NMF) acquires the low-dimensional representation by factorising count data matrix into two nonnegative matrices of low rank [Dhillon and Sra, 2005,Févotte and
Revised: Nonnegative matrix factorization (NMF) acquires the low-dimensional representation by factorizing count data matrix into two nonnegative matrices of low rank [Dhillon and Sra, 2005,

Original:  Idier, 2011, Cichocki et al., 2011].
Revised: Févotte and Idier, 2011, Cichocki et al., 2011].

Original: Despite the popularity of NMF and PCA variants, it isstill unclear whether they perform well on count data having an excess of zeros.
Revised: Despite the popularity of NMF and PCA variants, it is unclear whether they perform well on count data having an excess of zeros.

Original: Unlike the aforementioned works, we focus on developing measures that can reliably quantify the pairwise dissimilarity for small count data, motivated by the importance of proximity matrix in common DR frameworks.
Revised: Unlike the aforementioned works, we focus on developing measures that can reliably quantify the pairwise dissimilarity for small count data, motivated by the importance of proximity matrix in common DR frameworks.

Original: Specifically, many DR approaches seek to preserve properties of a proximity matrix of high-dimensional data when reducing the dimension of the data.
Revised: Specifically, many DR approaches seek to preserve properties of a proximity matrix of high-dimensional data when reducing the dimension of the data.

Original: Examples of such approaches include PCA with the Euclidean distance matrix and the Gram matrix [Torgerson, 1952, Van Der Maaten et al., 2009, Lawrence, 2005], GPLVM with theGram matrix
Revised: Examples of such approaches include PCA with the Euclidean distance matrix and the Gram matrix [Mardia et al., 1979, Lawrence, 2005], GPLVM with the

Original:  [Lawrence, 2005], multidimensional scaling (MDS) with an input dissimilarity matrix [Torgerson, 1952], and t-distributed stochastic neighbor embedding (tSNE) with the matrix of the Gaussian kernels [Maaten and Hinton, 2008].
Revised: Gram matrix [Lawrence, 2005], multidimensional scaling (MDS) with an input dissimilarity matrix [Torgerson, 1952], and t-distributed stochastic neighbour embedding (tSNE) with the matrix of the Gaussian kernels [Maaten and Hinton, 2008].

Original: Therefore, a proximity measure that properly quantifies the dissimilarity between small-count data points could benefit a wide range of DR methods.
Revised: Therefore, a proximity measure that properly quantifies the dissimilarity between small-count data points could benefit a wide range of DR methods.

Original: The two core contributions of this paper can be summarised as follows.
Revised: The two core contributions of this paper can be summarized as follows.

Original: First, we develop two dissimilarity measures for small count data based on the Kullback-Leibler (KL) divergence [Kullback and Leibler, 1951] and the assumption that the data follow either Poisson or negative binomial (NB) distributions.
Revised: First, we develop two dissimilarity measures for small count data based on the Kullback-Leibler (KL) divergence [Kullback and Leibler, 1951] and the assumption that the data follow either Poisson or negative binomial (NB) distributions.

Original: We take both Poisson and NB distributions into account, as it is common to model count data with these two types of distribution
Revised: We take both Poisson and NB distributions into account, as it is common to model count data with these two types of distributions [Zeileis et al., 2008, Chan and

Original: [Chan and Vasconcelos, 2009, Lindén and Mäntyniemi, 2011,Zeileis et al., 2008, Townes et al., 2019, Kim et al., 2020].
Revised: Vasconcelos, 2009, Lindén and Mäntyniemi, 2011, Townes et al., 2019, Kim et al., 2020].

Original: Furthermore, to reliably calculate the KL divergence, we propose to use empirical Bayes estimators to estimate the distributional parameters.
Revised: Furthermore, to reliably calculate the KL divergence, we propose to use empirical Bayes estimators to estimate the distributional parameters.

Original: Secondly, we propose an index to evaluate the discrimination abilities of different dissimilarity measures and show that the measure developed with the NB assumption has superior discrimination ability compared with other widely used dissimilarity measures for high-dimensional small counts, in terms of their statistical behaviours.
Revised: Secondly, we propose an index to evaluate the discriminative abilities of different dissimilarity measures and show that the measure developed with the NB assumption has superior discriminative ability compared with other widely used dissimilarity measures for high-dimensional small counts, in terms of their statistical behaviours.

Original: Moreover, consistent with our statistical investigation, the experimental results, on both real and simulated count data, also demonstrate that the measure obtained with the NB assumption is superior to other measures when handling small counts.Additionally, representative DR methods with the proposed superior measure outperform the variants of PCA and NMF.
Revised: Moreover, consistent with our statistical investigation, the experimental results, on both real and simulated count data, also demonstrate that the measure obtained with the NB assumption is superior to other measures when handling small counts.

Original: The rest of this paper can be summarised as follows.
Revised: The rest of this paper can be summarized as follows.

Original: First, we present standard transformations for count data in Section 2.1.
Revised: First, we present standard transformations for count data in Section 2.1.

Original: We then derive two new dissimilarity measures with the KL divergence and the empirical Bayes estimators in Section 2.2.
Revised: We then derive two new dissimilarity measures with the KL divergence and the empirical Bayes estimators in Section 2.2.

Original: Secondly, we propose an index to evaluate the discriminate ability of a dissimilarity measure (Section 3.1) and compare different dissimilarity measures according to the proposed index.
Revised: Secondly, we propose an index which evaluates the discriminative ability of a dissimilarity measure (Section 3.1) and compare different dissimilarity measures according to the proposed index.

Original: It is shown that, when applied to small counts, the Euclidean distance of the transformed data exhibits better discrimination ability than the original Euclidean distance, although the corresponding VST is unable to stabilise the variances (Section 3.2).
Revised: It is shown that, when applied to small counts, the Euclidean distance of the transformed data exhibits better discriminative ability than the original Euclidean distance, although the corresponding VST is unable to stabilize the variances (Section 3.2).

Original: More importantly, the measure obtained with the NB assumption is expected to perform the best when used for separating different distributions of small count dataincluding many zeros (Section 3.3).
Revised: More importantly, the measure obtained with the NB assumption is expected to perform the best when used for separating different distributions of small count data (Section 3.3).

Original: Lastly, we present the experimental results of representative DR methods with different measures, NMF, and variants of PCA, on both real and simulated datasets (Section 4).
Revised: Lastly, we present the experimental results of representative DR methods with different measures on both real and simulated datasets (Section 4).

Original:  DISSIMILARITY MEASURES FOR COUNT DATA
Revised:  DISSIMILARITY MEASURES FOR COUNT DATA

Original: In this section, we first present widely used VSTs for count data and then derive two new dissimilarity measures with the KL divergence and the empirical Bayes estimators for small counts.
Revised: In this section, we first present widely used VSTs for count data and then derive two new dissimilarity measures with the KL divergence and the empirical Bayes estimators for small counts.

Original: 2.1 VARIANCE-STABILIZING TRANSFORMATIONS(VSTS)
Revised: 2.1 VARIANCE-STABILIZING TRANSFORMATIONS(VSTS)

Original: A VST is a data transformation that applies to data such that the variance of the transformed data is independent of their mean.
Revised: A VST is a data transformation that applies to data such that the variance of the transformed data is independent of their mean.

Original: Most VSTs for count data are developed by assuming data follow an either Poisson or NB distribution [Anscombe, 1948, Cameron and Trivedi, 2013, Bartlett, 1947].
Revised: Most VSTs for count data are developed by assuming data follow an either Poisson or NB distributions [Bartlett, 1947, Anscombe, 1948].

Original: Let y be the raw counts.
Revised: Let y be the raw counts.

Original: The square root transformation
Revised: The square root transformation

Original: [Equation]
Revised: [Equation]

Original: is a popular technique for stabilising the variance of a Poisson random variable.
Revised: is a popular technique for stabilizing the variance of a Poisson random variable.

Original: For an NB random variablewith Var( y ) = E ( y) +1 r E ( y ) 2 ,  a prevalent transformation is
Revised: For an NB random variable y which counts the number of successes and has the PMF (cid:0) y + r −y (cid:1) (1 − p ) r p y , where p is the probability of success and r represents the number of failures, a prevalent transformation is

Original: [Equation]
Revised: [Equation]

Original: where arcsinh is the inverse hyperbolic sine function.
Revised: where arcsinh is the inverse hyperbolic sine function.

Original: Since g asin ( y ) requires an approximate knowledge of r and in some cases it cannot be estimated well enough, a simpler logarithm transformation with a pseudocount 1 is preferred in practice, which is given by
Revised: Since g asin ( y ) requires an approximate knowledge of r and in some cases it cannot be estimated well enough, a simpler logarithm transformation with a pseudocount 1 is preferred in practice, which is given by

Original: [Equation]
Revised: [Equation]

Original: As mentioned before, these transformations fail to stabilise the variance of small counts.
Revised: As mentioned before, these transformations fail to stabilize the variance of small counts.

Original: Thus, there is no guarantee that the Euclidean distances of the data transformed from raw counts by these VSTs perform well on small counts.
Revised: Thus, there is no guarantee that the Euclidean distances of the data transformed from raw counts by these VSTs perform well on small counts.

Original: 2.2 TWO NEW DISSIMILARITY MEASURES DEVELOPED WITH KL DIVERGENCE
Revised: 2.2 TWO NEW DISSIMILARITY MEASURES DEVELOPED WITH KL DIVERGENCE

Original: The KL divergence is a statistical measure of how one probability distribution is different from a second, reference probability distribution [Kullback and Leibler, 1951].
Revised: The KL divergence is a statistical measure of how one probability distribution is different from a second, reference probability distribution [Kullback and Leibler, 1951].

Original: For discrete probability distributions P and Q defined on the same probability space Z , the KL divergence is defined as D KL ( P | Q ) =
Revised: For discrete probability distributions P and Q defined on the same probability space Z , the KL divergence is defined as D KL ( P | Q ) =

Original: (cid:80)
Revised: (cid:80)

Original: z ∈ Z P ( z )log P ( z ) Q ( z ) .
Revised: z ∈ Z P ( z )log P ( z ) Q ( z ) .

Original: The KL divergence for continuous random variables can be defined similarly by replacing the sum with the integral.
Revised: The KL divergence for continuous random variables can be defined similarly by replacing the sum with the integral.

Original: For a pair of univariate normal distributions, P : N ( µ x , σ 2 ) and Q : N ( µ y , σ 2 ) ,we have D KL (cid:2) N ( µ x , σ 2 ) | N ( µ y , σ 2 ) (cid:3) =
Revised: For a pair of univariate normal distributions, P : N ( µ x , σ 2 ) and Q : N ( µ y , σ 2 ) , we have D KL (cid:2) N ( µ x , σ 2 ) | N ( µ y , σ 2 ) (cid:3) =

Original: ( µ x − µ y ) 2 2 σ.
Revised: ( µ x − µ y ) 2 2 σ.

Original: The squared Euclidean distance D 2 E between two vectors x = [ x 1 , . . .
Revised: The squared Euclidean distance D 2 E between two vectors x = [ x 1 , . . .

Original: , x p ] T , y =
Revised: , x p ] T , y =

Original: [ y 1 , . . .
Revised: [ y 1 , . . .

Original: , y p ] T ∈
Revised: , y p ] T ∈

Original: R p is equivalent to the sum of the KL divergence between two univariate normal distributions across dimensions up to a constant 12 σ 2 and the mean values of the distributions are estimated by MLEs.
Revised: R p is equivalent to the sum of the KL divergence between two univariate normal distributions across dimensions up to a constant 12 σ 2 and the mean values of the distributions are estimated by the maximum likelihood estimators (MLEs).

Original: This equivalence is shown by the fol- lowing equation:
Revised: This equivalence is shown by the following equation:

Original: [Equation]
Revised: [Equation]

Original: where x i and y i are the MLEs of mean parameters of the normal distributions on the i -th dimension of x and y , respectively, when there is only one realisation observed for each distribution.
Revised: where x i and y i are the MLEs of mean parameters of the normal distributions on the i -th dimension of x and y , respectively, when there is only one realisation observed for each distribution.

Original: Stimulated by the equivalence between D 2 E and the KL divergence, we propose to quantify the pairwise dissimilarity for count data by the KL divergence.
Revised: Stimulated by the equivalence between D 2 E and the KL divergence, we propose to quantify the pairwise dissimilarity for count data with the KL divergence.

Original: To calculate the
Revised: To calculate the

Original: KL divergence, the distribution type and the corresponding parameter values are required to be specified.
Revised: KL divergence, the distribution type and the corresponding parameter values are required to be specified.

Original: For the distribution type, we assume the observed data follow either
Revised: For the distribution type, we assume the observed data follow either

Original: Poisson or NB distributions, which are commonly used for modelling count data.
Revised: Poisson or NB distributions, which are commonly used for modelling count data.

Original: Regarding the parameter estimation, a straightforward estimator is the MLE.
Revised: Regarding the parameter estimation, a straightforward estimator is the MLE.

Original: However, the  MLE incurs a numerical problem in practice.
Revised: However, the use of MLE incurs a numerical problem in practice.

Original: To clarify this problem, we  derive two dissimilarity measures with the MLEs for Poisson and NB distributions, respectively.
Revised: To clarify this problem, we first derive two dissimilarity measures with the MLEs for Poisson and NB distributions, respectively.

Original: Suppose x i and y i follow Pois( λ ix ) and Pois( λ ix ) , respectively.
Revised: Suppose x i and y i follow Pois( λ ix ) and Pois( λ iy ) , respectively.

Original: The respective MLEs of λ ix and λ iy are x i and y i .
Revised: The respective MLEs of λ ix and λ iy are x i and y i .

Original: The KL divergence between x and y with these MLEs is thus
Revised: The KL divergence between x and y with these MLEs is thus

Original: [Equation]
Revised: [Equation]

Original: Analogously, we suppose x i and y i follow NB ( r, p ix ) and
Revised: Analogously, we suppose x i and y i follow NB ( r, p ix ) and

Original: NB ( r, p iy ) , respectively, with known r .
Revised: NB ( r, p iy ) , respectively, with known r .

Original: Note that there are multiple definitions of the NB distribution and we use the following ones: the PMF of NB ( r, p ix ) for x i is (cid:0) x
Revised: Note that there are multiple definitions of the NB distribution and we use the following ones: the PMF of NB ( r, p ix ) for x i is (cid:0) x

Original: i + r −x
Revised: i + r −x

Original: i (cid:1) (1 − p ix ) r p x i ix and similarly for NB ( r, p iy ) , where p ix and p iy are the probabilities of success, r represents the number of failures, and x i and y
Revised: i (cid:1) (1 − p ix ) r p x i ix and similarly for NB ( r, p iy ) , where p ix and p iy are the probabilities of success, r represents the number of failures, and x i and y

Original: i count the numbers of successes.
Revised: i count the numbers of successes.

Original: The respective MLEs of p ix and p iy are x i x i + r
Revised: The respective MLEs of p ix and p iy are x i x i + r

Original: and y i y i
Revised: and y i y i

Original: + r .
Revised: + r .

Original: The KL divergence between x and y with the MLEs is given by 
Revised: The KL divergence between x and y with the MLEs is given by p (cid:88)

Original: [Equation]
Revised: [Equation]

Original: The dissimilarity measures presented in Equation (4) and
Revised: The dissimilarity measures presented in Equation (4) and

Original: Equation (5) both involve the logarithm terms, and thus zeros in count data would result in the numerical problem.
Revised: Equation (5) both involve the logarithm terms, and thus zeros in count data would result in the numerical problem.

Original: Further, since the MLEs are close to the true values of parameters only if the number of observations is sufficiently large, the MLEs calculated from one observation respectively are unreliable, so are ˆ D KL [Pois( λ ix )
Revised: Further, since the MLEs are close to the true values of parameters only if the number of observations is sufficiently large, the MLE calculated from one observation respectively are unreliable, so are ˆ D KL [Pois( λ ix )

Original: | Pois( λ iy )] and ˆ D KL [NB( r, p ix )
Revised: | Pois( λ iy )] and ˆ D KL [NB( r, p ix )

Original: | NB( r, p iy )] .
Revised: | NB( r, p iy )] .

Original: To address these issues, we propose to use the empirical
Revised: To address these issues, we propose to use the empirical

Original: Bayes estimators rather than the MLEs.
Revised: Bayes estimators rather than the MLEs.

Original: The conjugate priors of Poisson and NB distributions are employed for estimating the parameters ( λ ix , λ iy , p ix , p iy ).
Revised: The conjugate priors of Poisson and NB distributions are employed for estimating the parameters ( λ ix , λ iy , p ix , p iy ).

Original: In addition, the hyperparameters of these priors are learned from data themselves, sidestepping the difficulty of specifying proper priors to some degree.
Revised: In addition, the hyperparameters of these priors are learned from data themselves, sidestepping the difficulty of specifying proper priors to some degree.

Original: Concretely, we specify a Gamma prior distribution G ( m i , 1) , where the shape parameter m i is the mean value of the i -th dimension across all data points and the other parameter is the scale parameter, for the Poisson means ( λ ix , λ iy ).
Revised: Concretely, we specify a Gamma prior distribution G ( m i , 1) , where the shape parameter m i is the mean value of the i -th dimension across all data points and the other parameter is the scale parameter, for the Poisson means ( λ ix , λ iy ).

Original: For the probability parameters of NB distributions ( p ix , p iy ), we specify a Beta prior distribution B ( m i , r ) for them.
Revised: For the probability parameters of NB distributions ( p ix , p iy ), we specify a Beta prior distribution B ( m i , r ) for them.

Original: Note that the mean value can be thought of an additional observation.
Revised: Note that the mean value can be thought of an additional observation.

Original: With the priors, we obtain the posterior distribution of λ ix is G ( m i + x i , 12 ) and that of λ iy is G ( m i
Revised: With the priors, we obtain the posterior distribution of λ ix is G ( m i + x i , 12 ) and that of λ iy is G ( m i

Original: + y i , 12 ) .
Revised: + y i , 12 ) .

Original: The posterior means, which are m i + x iand m i + y i, respectively, are used as the estimated distributional parameters.
Revised: The posterior means, which are m i + x iand m i + y i, respectively, are used as the estimated distributional parameters.

Original: Analogously, we obtain the posterior mean m i + x i m i
Revised: Analogously, we obtain the posterior mean m i + x i m i

Original: + x i +

Revised: + x i +

Original: r from the posterior distribution B ( m i + x i , 2 r ) and m 
Revised: r from the posterior distribution B ( m i + x i , 2 r ) and m i

Original: [Equation]
Revised: + y

Original:  m i + x i +2{++
+} r from B ( m i + y i , 2 r ) , as the estimated probability parameters.
Revised: i m i + y i {++
+} r from B ( m i + y i , 2 r ) , as the estimated distributional parameters for NB distributions.

Original: Now we obtain the KL divergence between x and y with the Bayes estimators (posterior means):
Revised: Now we obtain the KL divergence between x and y with the Bayes estimators (posterior means):

Original: [Equation]
Revised: [Equation]

Original: The logarithm terms in Equation (6) are well defined for m i > 0 , which is easily satisfied in practice as only meaningless features in the form of all zeros have m i
Revised: The logarithm terms in Equation (6) are well defined for m i > 0 , which is easily satisfied in practice as only meaningless features in the form of all zeros have m i

Original: = 0 .
Revised: = 0 .

Original: Owing to the asymmetry of the KL divergence, we propose to use
Revised: Owing to the asymmetry of the KL divergence, we propose to use

Original: [Equation]
Revised: [Equation]

Original: to measure the pairwise dissimilarity for count data.
Revised: to measure the pairwise dissimilarity for count data.

Original: Tablelists the dissimilarity measures that we take into account in this paper.
Revised: Tablelists the dissimilarity measures that we take into account in this paper.

Original: Note that for D P and D NB we ignore their multiplicative constant 12 for conciseness.
Revised: Note that for D P and D NB we ignore their multiplicative constant 12 for conciseness.

Original: [Table]
Revised: [Table]

Original:  COMPARISON OF DISSIMILARITY MEASURES FOR HIGH-DIMENSIONAL SMALL COUNTS
Revised:  COMPARISON OF DISSIMILARITY MEASURES FOR HIGH-DIMENSIONAL SMALL COUNTS

Original: In this section
Revised: In this section

Original: [Table]
Revised: [Table]

Original: we compare different measures listed in 
Revised: we compare different measures listed in Table

Original: Table 1 according to their abilities to distinguish distributions that tend to produce small counts.
Revised: [Table] according to their abilities to distinguish distributions that tend to produce small counts.

Original: First
Revised: First

Original: [Table]
Revised: [Table]

Original: we propose an index to quantify the discrimination abilities of different dissimilarity measures.
Revised: we propose an index to quantify the discriminative abilities of different dissimilarity measures.

Original: Then,  we investigate and compare the statistical behaviours of different measuresbased on the proposed index when the dimension is high and the count data consist of many zeros.
Revised: Then, based on the proposed index [Table] we investigate and compare the statistical behaviours of different measures when the dimension is high and the count data consist of many zeros.

Original: 3.1 EVALUATION INDEX
Revised: 3.1 EVALUATION INDEX

Original: The main  goal of DR is to eliminate noisy or uninformative dimensions of high-dimensional data andto assist  classification/clustering algorithms to uncover meaningful classes in the data.
Revised: The main practical goal of DR is to eliminate noisy or uninformative dimensions of high-dimensional data and assist downstream classification/clustering algorithms in uncovering meaningful classes/clusters in the data.

Original: Different classes of count data can be characterised by different distributions, and thus a dissimilarity measure that benefits DR techniques should distinguish these distributions well.
Revised: Different classes/clusters of count data can be characterized by different distributions [Table] and thus a dissimilarity measure that distinguishes these distributions well could benefit the downstream analysis of the data when integrated into standard DR approaches.

Original: In this subsection, we propose an index to evaluate how well a dissimilarity measure separates  data points generated from different distributions and groups those from the same distribution.
Revised: In this subsection, we propose an index to evaluate how well a dissimilarity measure separates those data points generated from different distributions and groups those from the same distribution.

Original: The definition of the proposed index is given in Definition 1.
Revised: The definition of the proposed index is given in Definition 1.

Original: Definition 1 Suppose there are two count data distributions, denoted by F x and F y , respectively.
Revised: Definition 1 Suppose there are two count data distributions, denoted by F x and F y , respectively.

Original: Let S X
Revised: Let S X

Original: = { x 1 , . . .
Revised: = { x 1 , . . .

Original: , x n x } be the set of samples generated from F x and S Y = (cid:8) y 1 , . . .
Revised: , x n x } be the set of samples generated from F x and S Y = (cid:8) y 1 , . . .

Original: , y n y (cid:9) the set of samples from F y .
Revised: , y n y (cid:9) the set of samples from F y .

Original: For a given dissimilarity measure D ( · , · ) , the proposed index
Revised: For a given dissimilarity measure D ( · , · ) , the proposed index

Original: R ( F x , F y ) is defined asfollows:
Revised: R ( F x , F y ) is defined as

Original: [Equation]
Revised: [Equation]

Original: The proposed index R ( F x , F y ) assesses the variation between data points from the same distribution and that from different distributions simultaneously, which are measured by D ( · , · ) .

Original: In particular, R ( F x , F y ) > 1 implies that using the corresponding dissimilarity measure makes the separation between F x and F x greater than the within-distribution variation.

Original: Note that here we consider only two distributions for simplicity, and to facilitate the following analysis we use the squared dissimilarity function.
Revised: Note that here we consider only two distributions for simplicity, and to facilitate the following analysis we use the squared dissimilarity function.

Original: In the following, the subscript ∗ of R ∗ ( F x , F y ) will be that of the corresponding dissimilarity measure.
Revised: In the following, the subscript ∗ of R ∗ ( F x , F y ) will be that of the corresponding dissimilarity measure.

Original: It is straightforward to see that the higher the value of  R ( F x , F y ) , the more powerful the discrimination ability of the  dissimilarity measure.
Revised: The mathematical objectives of many DR approaches are to preserve the global or local proximity of high-dimensional data, and the proposed index suits them in that R ( F x , F y ) assesses simultaneously the variation between data points from the same distribution (local proximity) and the separation between data points from different distributions (global proximity). R ( F x , F y ) > 1 implies that using the corresponding dissimilarity measure D ( · , · ) makes the separation between F x and F y greater than the within-distribution variation.

Original: Particularly, a higher value of R ( F x , F y ) indicates that the corresponding measureis more robust in the presence of noisy dimensions, which possibly reduce the inter-distribution separationdegree and increase the within-distribution variation.
Revised: By construction, a higher value of R ( F x , F y ) would tend to indicate more powerful and robust discriminative ability of the corresponding measure in the presence of noisy dimensions, which possibly reduce the between-distribution separation and increase the withindistribution variation.

Original: Before we dive into the comparison of measures using R ( F x , F y ) , the statistical behaviour of R ( F x , F y ) in the high-dimensional space should be clarified.
Revised: Before we dive into the comparison of measures using R ( F x , F y ) , the statistical behaviour of R ( F x , F y ) in the high-dimensional space should be clarified.

Original: Proposition 1 presents the behaviour of R ( F x , F y ) for the dissimilarity functions in a generic form: D 2 ( x, y ) =
Revised: Proposition 1 presents the behaviour of R ( F x , F y ) for the dissimilarity functions in a generic form: D 2 ( x , y ) =

Original: [ f ( x ) − f ( y )]
Revised: (cid:80) [Equation]

Original: [ g ( x ) − g ( y )] , which covers all the measures presented in Table 1.
Revised: which covers all the measures presented in Table 1.

Original: Proposition 1 shows that R ( F x , F y ) moves towards a constant as dimension d grows, irrespective of the number of samples from distributions.
Revised: Proposition 1 shows that R ( F x , F y ) moves toward a constant as dimension p grows, irrespective of the number of samples from distributions.

Original: The covariance of two increasing functions ( f , g ) of a random variable is positive [Schmidt, 2003], and thus the constant which R ( F x , F y ) converges to would be greater than 12 iff [E f ( x ) − E f ( y )]
Revised: The covariance of two increasing functions ( f , g ) of a random variable is positive [Schmidt, 2003], and thus the constant which R ( F x , F y ) converges to would be greater than 12 iff [E f ( x ) − E f ( y )]

Original: [E g ( x ) − E g ( y )] > 0 , which is readily satisfied in practice.
Revised: [E g ( x ) − E g ( y )] > 0 , which is readily satisfied in practice.

Original: The convergencewill still hold under some mild conditions, such as with dependent dimensions and non-identical distributions.
Revised: The convergence still holds under some mild conditions, such as with dependent dimensions and non-identical distributions.

Original: Proposition 1 Suppose points in S X ∪ S Y are independent, and each coordinate of points in S X and S Y are independently drawn from 1 -dimensional non-degenerate data distributions F x and F y , respectively.
Revised: Proposition 1 Suppose points in S X ∪ S Y are independent, and each coordinate of x , y in S X and S Y are independently drawn from 1 -dimensional non-degenerate data distributions F x and F y , respectively.

Original: For D 2 ( x, y ) =
Revised: For D 2 ( x , y ) =

Original: [ f ( x ) − f ( y )]
Revised: (cid:80) p i =1 D 2 ( x i , y i ) = (cid:80) p i

Original: {+=
 [ f ( x i ) − f ( y i )]+} [ g ( x  ) − g ( y  )] with x  ∼ F x , y  ∼ F y , where f ( · ) and g ( · ) are predetermined functions, if E [ D 2 ( x, y )] , 
Revised: {+=
 [ f ( x i ) − f ( y i )]+} [ g ( x i ) − g ( y i )] with x i ∼ F x , y i ∼ F y , where f ( · ) and g ( · ) are predetermined functions, if E [ D 2 ( x, y )] , E [ D 2 ( x, ˜ x )] and

Original: E [ D 2 ( x 1 , x 2 )] and E [ D 2 ( y1 , y 2 )] exist for independent samples x 1 , x 2 , x ∼ F x , y 1 , y 2 , y ∼ F y , we haveR ( F x , F y )
Revised: E [ D 2 ( y, ˜ y )] exist for independent samples ˜ x, x ∼ F x , ˜ y, y ∼ F y , we have

Original: [Equation]
Revised: [Equation]

Revised: where prob → denotes the convergence in probability as the dimension p goes to infinity.

Original: 3.2 COMPARE THE EUCLIDEAN DISTANCES W/O VSTS
Revised: 3.2 COMPARE THE EUCLIDEAN DISTANCES W/O VSTS

Original: In the following, we compare D E of original data with the
Revised: In the following, we compare D E of original data with the

Original: Euclidean distances of the transformed data, according to their behaviours when dealing with small counts in the space of high dimension; that is, we compare them in terms of the respective constants that their R ( F x , F y ) ’s converge to as  dimension diverges to infinity.
Revised: Euclidean distances of the transformed data, according to their behaviours when dealing with small counts in the highdimensional space; that is, we compare them in terms of the respective constants that their R ( F x , F y ) ’s converge to as the dimension diverges to infinity.

Original: We first examine the discrimination ability of D E when count data are small.
Revised: We first examine the discriminative ability of D E when count data are small.

Original: Corollary 1 provides the sufficient and necessary condition for R E ( F x , F y ) p → c E > 1 .
Revised: Corollary 1 provides the sufficient and necessary condition for R E ( F x , F y ) p → c E > 1 .

Original: This condition suggests that for any pairs of Poisson distributions that generate small counts

Original: 1.0 Poisson mean 0.

Original: c g g ( y )=

Original: y g r g log g asin (a)     6 Poisson mean      c g g ( y )= y g r g log g asin (b) with mean values less than 1 , we obtain c E < 1 ; that is, D E cannot distinguish the two distributions well.
Revised: This condition suggests that for any pairs of Poisson distributions that generate small counts with mean values less than 1 , we obtain c E < 1 ; that is, D E cannot distinguish the two distributions well.

Original: Therefore, D E is expected to perform poorly when handling small counts.
Revised: Therefore, D E is expected to perform poorly when handling small counts.

Original: An example showing D E is unable to distinguish two Poisson distributions with different patterns of small counts is provided in Section B of Appendix.
Revised: An example showing D E is unable to distinguish two Poisson distributions with different patterns of small counts is provided in Section S.2 of Supplementary Material.

Original: Corollary 1 With the same assumptions and notation as those in Proposition 1, for D ( · , · )
Revised: Corollary 1 With the same assumptions and notation as those in Proposition 1, for D ( · , · )

Original: =
Revised: =

Original: D E ( · , · ) , we have1.
Revised: D E ( · , · ) , we have

Original: R E [Equation] F x , F y ) p → c E ≥ 12 for some constant c E> 0 .
Revised: R E ( F x , F y ) prob → c E ≥ 12 for some constant c E .

Original: The equality holds iff E ( x ) =
Revised: The equality holds iff E ( x ) =

Original: E ( y ) for x ∼ F x , y ∼ F y .
Revised: E ( y ) for x ∼ F x , y ∼ F y .

Original: [Equation]
Revised: c E > 1 iff [ E ( x ) − E ( y )] 2 > Var ( x )+Var ( y ) .

Original: We then investigate the behaviours of R ( F x , F y ) ’s of the Euclidean measures based onthe VSTs when either F x or F y generates small counts.
Revised: We then investigate the behaviours of R ( F x , F y ) ’s of the Euclidean distances based on VSTs when either F x or F y generates small counts.

Original: Without loss of generality, we assume F x outputs small counts and F y is an arbitrary distribution.
Revised: Without loss of generality, we assume F x produces small counts and F y is an arbitrary distribution.

Original: Suppose there is a VST characterised by an increasing transformation function g ( · ) , such that g ( y ) ≥ 0 for y ≥ 0 .
Revised: Suppose there is a VST characterized by an increasing transformation function g ( · ) , such that g ( y ) ≥ 0 for y ≥ 0 .

Original: Note that g ( · ) covers g r ( y ) , g asin ( y ) , and g log ( y ) .
Revised: Note that g ( · ) covers g r ( y ) , g asin ( y ) , and g log ( y ) .

Original: Let D E of the data transformed from raw counts by g ( · ) be D g and the corresponding index R g ( F x , F y ) .
Revised: Let D E of the data transformed from raw counts by g ( · ) be D g and the corresponding index R g ( F x , F y ) .

Original: Corollary 2 provides the difference between c g and c E when the proportion of zeros of each data point in S X moves towards 1 .
Revised: Corollary 2 provides the difference between c g and c E when the proportion of zeros of each data point in S X moves toward 1 .

Original: It shows that D g is better suited for distinguishing data points as µ x approaches 0 in comparison with D E iff [ g (0) − E( y )]
Revised: It shows that, as µ x approaches 0 , D g is better suited for distinguishing data points than D E iff [ g (0) − E g ( y )]

Original:  Var[ g ( y )] − E 2 ( y )
Revised: 2 Var[ g ( y )] − E 2 ( y )

Original: Var( y ) > 0 .
Revised: Var( y ) > 0 .

Revised: Corollary 2 Suppose that x and y are non-negative random variables.

Revised: Let the expectation of F x be µ x .

Revised: With the same assumptions and notation as those in Proposition 1, we have

Original:  To illustrate the advantages of applying VSTs to small counts, we obtain values of c g =
Revised: [Equation] where c g and c E are the constants that R g ( F x , F y ) and R E ( F x , F y ) approach, respectively, as the dimension p goes to infinity. To illustrate the advantages of applying VSTs to small counts, we obtain values of lim

Original: [ g (0) − E( y )]
Revised: [Equation]

Original: 2 Var[ g ( y )] for Poisson distributions  with different mean values and transformation functions by numerical computation.
Revised: for Poisson distributions F y ’s with different mean values and transformation functions by numerical computation.

Original: Figure 1 supplies the numerical results and shows that g r ,  g log , and g asin always result in a c g that is no less than c E .
Revised: Figure 1 supplies the numerical results and shows that g r , [Figure] g log , and g asin always result in a c g that is no less than c E .

Original: In particular, it is observed from Figure 1(a) that c g ’s exceed 1 when the Poisson mean is higher than 0 . 8 , indicating that the corresponding measures distinguish better between data points with large proportions of zeros compared with D E .
Revised: In particular, it is observed from Figure 1(a) that c g ’s exceed 1 when the Poisson mean is higher than 0 . 8 , indicating that the corresponding measures distinguish better between data points with large proportions of zeros compared with D E .

Original: Note that we assign a large value to r in c g asin ( r = 1000 ) since an arbitrary NB ( r, p ) approximates a Poisson distribution when r approaches infinity.
Revised: Note that we assign a large value to r in c g asin ( r = 1000 ) since an arbitrary NB ( r, p ) approximates a Poisson distribution when r approaches infinity.

Original: The above analysis suggests that although the VSTs are unable to stabilise variances when count data are small, they improve the discrimination ability over D E .
Revised: The above analysis suggests that, although the VSTs are unable to stabilize the variances of small count data, they improve the discriminative ability over D E .

Original: Proofs of Proposition 1 and Corollary 2 are provided in Section A of Appendix.
Revised: Proofs of Proposition 1 and Corollary 2 are provided in Section S.1 of Supplementary Material.

Original: Corollary 2 Suppose that x and y are non-negative random variables.

Original: Let the expectation of F x be µ x .

Original: With the same assumptions and notation as those in Proposition 1, we have

Original: [Equation]

Original: where c g is the constant that R g ( F x , F y ) approaches.

Original: 3.3 COMPARE THE TWO PROPOSED MEASURES WITH OTHER DISSIMILARITY MEASURES
Revised: 3.3 COMPARE THE TWO PROPOSED MEASURES WITH OTHER DISSIMILARITY MEASURES

Original: In the following, we will compare the proposed measures ( D P , D NB ) with the other dissimilarity measures in terms of the proposed index R ( F x , F y ) computed in the highdimensional space.
Revised: In the following, we will compare the proposed measures ( D P , D NB ) with the other dissimilarity measures in terms of the proposed index R ( F x , F y ) computed in the highdimensional space.

Original: Note that the estimate ˆ R ( F x , F y ) would be close enough to the constant that R ( F x , F y ) approaches as long as the dimension is high enough.
Revised: Note that the estimate ˆ R ( F x , F y ) would be close enough to the constant that R ( F x , F y ) approaches as long as the dimension is high enough.

Original: Given a pair of distributions ( F x , F y ) and that of measures ( D NB , D E ) , we believe D NB is superior to D E for distinguishing between F x and F y if ˆ R NB ( F x , F y ) >
Revised: Take a pair of distributions ( F x , F y ) and a pair of measures ( D NB , D E ) for example, we believe D NB is superior to D E for distinguishing between F x and F y if ˆ R NB ( F x , F y ) >

Original: ˆ R E ( F x , F y ) .
Revised: ˆ R E ( F x , F y ) .

Original: Further, to thoroughly evaluate their discrimination abilities for a specific distribution type, we compare their performances in terms of the fraction that ˆ R NB ( F x , F y ) >
Revised: Further, to thoroughly evaluate their discriminative abilities for a specific distribution type, we compare their performances in terms of the fraction that ˆ R NB ( F x , F y ) >

Original: ˆ R E ( F x , F y ) for different configurations of parameters.
Revised: ˆ R E ( F x , F y ) for different configurations of parameters.

Original: The fraction greater than [-
-]
Revised: The fraction greater than [-
-]0 .

Original: .suggests D NB is better suited for distinguishing this distribution type than D E and vice versa.
Revised: 5 suggests D NB is better suited for distinguishing this distribution type than D E and vice versa.

Original: The distributions types ( F x , F y ) taken into account are the broadly used Poisson and NB distributions.
Revised: The distributions types ( F x , F y ) taken into account are the broadly used Poisson and NB distributions.

Original: F x and F y are of the same distribution type but with different parameter configurations.
Revised: F x and F y are of the same distribution type but with different parameter configurations.

Revised: It is worth mentioning that R P ( F x , F y ) and R NB ( F x , F y ) are the same when data are Poissondistributed, because D NB approaches D P when the dispersion parameter r goes to infinity.

Revised: We thus exclude D NB from the comparison for Poisson-distributed data.

Original: More  details on the simulations and the calculation of different measures are presented in Section C of Appendix.
Revised: More [Table] [Table] details on the simulations and the calculation of different measures are presented in Section S.3 of Supplementary

Revised: Material.

Original: After simulations, we get ˆ R ( F x , F y ) ’s for a wide scope of parameter configurations and different distribution types.
Revised: After simulations [Table] we get ˆ R ( F x , F y ) ’s for a wide scope of parameter configurations and different distribution types.

Original: As mentioned before, the measures are compared in terms of ˆ R ( F x , F y ) ’
Revised: As mentioned before, the measures are compared in terms of ˆ R ( F x , F y ) ’

Original: s. Table 2 and Table 3 supply the comparison of the measures when data follow Poisson and NB distributions, respectively.
Revised: s. Table 2 and Table 3 supply the comparisons of the measures when data follow Poisson and NB distributions, respectively.

Original: For small count data following Poisson distributions, D log achieves the best performance according to its highest average fraction value, followed by D NB . Although D log  is superior to D NB for Poisson-distributed count data, D NB achieves a much higher value than D log when count data are negative-binomially distributed.In addition, for small count data following NB distributions, D P exhibits the second best discrimination ability.
Revised: For small count data following Poisson distributions, D P / D NB performs as well as D log . Furthermore, D NB is superior to the other measures when count data are negative-binomially distributed.

Original: D E , as anticipated, performs much worse than the other measures.
Revised: D E , as anticipated, performs much worse than the other measures.

Original: The simulation results show that D NB distinguishes both the equidispersed and overdispersed distributions well, and thus we expect that D NB outperforms the others when integrated into standard DR methods.
Revised: The simulation results show that D NB is better than the other measures when distinguishing Poisson and NB distributions, and thus we expect that D NB outperforms the others when integrated into standard DR methods.

Original: [Table]
Revised: Although it is shown that D NB is superior to the other measures and D P is the second-best measure, we find that the calculation of m i affects their discriminative performance. Specifically, if the value of m i is closer to the average of the expected values of the two distributions ( F x , F y ), D P and D NB would have a better discriminative ability.

Original: [Table]
Revised: We reason that the mean of the expected values can be regarded as a typical value informative to the parameter estimation when used in the priors and thus is beneficial for the calculation of the KL divergence.

Original:  EXPERIMENTAL RESULTS
Revised:  EXPERIMENTAL RESULTS

Original: In this section, we present experimental results of representative DR methods with different dissimilarity measures,
Revised: In this section [Table] we present experimental results of representative DR methods with different dissimilarity mea- [Table]

Original: PCA variants, and NMF on both real and simulated highdimensional count datasets with large fractions of zeros.
Revised: datasets used in this paper. [Figure] [Table] [Figure]

Revised: [Table]

Revised: [Figure]

Revised: [Table]

Revised: [Figure]

Revised: [Table]

Revised: [Figure]

Revised: [Table]

Revised: [Figure]

Revised: [Table]

Revised: [Figure]

Revised: [Table]

Revised: [Figure]

Revised: sures on both real and simulated high-dimensional count datasets with large fractions of zeros.

Revised: In addition, we compare the generalised PCAs (GPCAs)

Revised: [Collins et al., 2002] and NMF [Dhillon and Sra, 2005] with the proposed measure D NB in Section S.6 of Supplementary Material.

Original: 4.1 DATASETS
Revised: 4.1 DATASETS

Original: The high-dimensional count data considered in this paper is the single cell RNA sequencing (scRNA-seq) data with unique molecular identifiers (UMI).
Revised: The high-dimensional count data considered in this paper is the single cell RNA sequencing (scRNA-seq) data with unique molecular identifiers (UMI).

Original: scRNA-seq data offer a unique opportunity to investigate the stochastic heterogeneity of complex issues at a near-genome-wide scale [Saliba et al., 2014, Shapiro et al., 2013, Kolodziejczyk et al., 2015].
Revised: scRNA-seq data offer a unique opportunity to investigate the stochastic heterogeneity of complex issues at a near-genome-wide scale [Saliba et al., 2014, Shapiro et al., 2013, Kolodziejczyk et al., 2015].

Original: scRNA-seq data with UMI are often modelled by NB or
Revised: scRNA-seq data with UMI are often modelled by NB or

Original: Poisson distributions [Townes et al., 2019, Kim et al., 2020,
Revised: Poisson distributions [Townes et al., 2019, Kim et al., 2020,

Original: Svensson, 2020] and exhibit large proportions of zero counts.
Revised: Svensson, 2020] and exhibit large proportions of zero counts.

Original: We run experiments on both real and simulated scRNA-seq datasets.
Revised: We run experiments on both real and simulated scRNA-seq datasets.

Original: These datasets contain large proportions of zeros, ranging from 0 .
Revised: These datasets contain large proportions of zeros, ranging from 0 .

Original: 6 to 0 .
Revised: 6 to 0 .

Original: The characteristics of the real scRNA-seq datasets used in this paper are summarised in
Revised: The characteristics of the real scRNA-seq datasets used in this paper are summarized in

Original: Table 4.
Revised: Table 4.

Revised: Cluster labels provided by the real scRNA-seq datasets correspond to different cell types: labels for the datasets obtained from Tian et al.

Revised: [2019] are assigned in terms of cancer cell lines and those for the Zheng8eq dataset based on the types of purified peripheral blood mononuclear cells.

Revised: All the cluster labels reported in these datasets are defined independently of gene expression profiles and can be used as the ground-truth labels.

Original: We simulate three additional scRNA-seq datasets by using the R’s Splatter package [Zappia et al., 2017] with most of the parameters learned from real datasets except for differential expression factors, which determine the difference between groups of cells and the number of clusters.
Revised: We simulate three additional scRNA-seq datasets by using the R’s Splatter package [Zappia et al., 2017] with most of the parameters learned from real datasets except for differential expression factors, which determine the difference between groups of cells and the number of clusters.

Original: The information of the simulated datasets and the corresponding real datasets used for simulations is summarised in Table 5.Note that different clusters presented by the datasets represent different cell types. [Table] [Table]
Revised: The information of the simulated datasets and the corresponding real datasets used for simulations are summarized in Table 5.

Original: 4.2 EVALUATION
Revised: 4.2 EVALUATION

Original:  We comparethe proposed measures with two different types of methods that deal with count data: the VSTs and  the DR methods designed specifically for original non-negative data.
Revised: Representative DR methods. We compare different measures with three representative DR methods: PCA, GPLVM, and tSNE. GPLVM and PCA seek to retain the global struc-

Original: The proposed measures and D E ’s ofthe datatransformed by  the VSTs are compared based on their performances when integrated into representative DR approaches, including PCA, tSNE, and GPLVM.
Revised: [Figure] [Figure] ture of data by preserving the pairwise proximity for all pairs of data points, while tSNE predominantly preserves the local structure with the pairwise proximity amongst neighbouring data points.

Original: In addition, the generalised
Revised: Therefore, DR results presented by

Original: PCAs (GPCAs) [Collins et al., 2002] and NMF [Dhillon and Sra, 2005], which handle original non-negative data, are included in comparison. GPCAs assume observed data follow either Poisson or NB distributions, denoted by GPCA-P and GPCA-NB, respectively.
Revised: PCA/GPLVM and tSNE, respectively, are complementary. The proposed measures and D E ’s of the data transformed by the VSTs are compared based on their performance when integrated into the DR approaches.

Original: Note that r in D NB and D asin is set to the common NB dispersion parameter estimated by the R’s edgeR package [Robinson et al., 2009].
Revised: Note that r in D NB and D asin is set to the common NB dispersion parameter estimated by the R’s edgeR package [Robinson et al., 2009].

Revised: As we treat mean values of features as pseudo observations when deriving the proposed measures, we replace each value x in the data matrix with x + m 2 , where m is the mean value of the corresponding feature column, when estimating r .

Original: We evaluate theperformance of DR from two perspectives: visualisation and clustering. We project the highdimensional dataset onto a two-dimensional (2D) space with a DR method and assess the 2D visualisation by visual inspection.
Revised: Visualization. As visualization is an important application of DR, we evaluate the DR methods with different dissimilarity measures by visually inspecting their DR results in a two-dimensional (2D) space.

Original: A good visualisation should exhibit wellseparated groups of data.
Revised: A good visualization should exhibit well-separated groups of data.

Revised: Clustering.

Revised: Apart from visualization in a 2D space, DR techniques can also be used for improving clustering of highdimensional data in real-world applications.

Revised: For instance, high-dimensional scRNA-seq data are often projected into a low-dimensional space whose dimension could be greater than 2 , and clustering methods, such as k -means and hierarchical clustering, are performed on the dimension-reduced data to improve clustering [Sun et al., 2019, Petegrosso et al., 2020].

Revised: Furthermore, it has been shown that applying k -means clustering in a PCA subspace can significantly improve clustering accuracy [Ding and He, 2004].

Revised: As clustering is an important downstream task to DR, we also evaluate the DR approaches with different measures based on the clustering performance in the dimension-reduced space.

Original: The k -means algorithm
Revised: The k -means algorithm

Original: [Lloyd, 1982, MacQueen, 1967] is used for inferring the cluster labels of data in the space of reduced dimension, and the number of clusters  is set to be the ground truth.
Revised: [MacQueen, 1967, Lloyd, 1982] is used for inferring the cluster labels of data in the space of reduced dimension. The number of clusters in the k -means algorithm is set to be the ground truth.

Original: The clustering performance is assessed in terms of the adjusted rand index (ARI)  between the cluster labels from the original publication/simulation and the inferred ones.
Revised: The clustering performance is assessed in terms of the adjusted rand index (ARI) [Hubert and Arabie, 1985] between the cluster labels from the original publication/simulation and the inferred ones.

Original: The higher the ARI, the better the performance.
Revised: The higher the ARI, the better the performance.

Original: Normally, tSNE and GPLVM map high-dimensional data to a 2D space, and thus we consider only 2D projections when evaluating their clustering performance.
Revised: Normally, tSNE and GPLVM map high-dimensional data to a 2D space, and thus we consider only 2D projections when evaluating their clustering performance.

Original: Since the results of the tSNE algorithm, GPCA-P, and GPCA-NB could be variable, we replicate the procedure, of first performing tSNE/GPCA-P/GPCA-NB and then applying k -means, for 10 times on the datasets for a more reliable comparison.We evaluate the clustering [Figure]
Revised: Since the results of the tSNE algorithm could be variable, we replicate the procedure, of first performing tSNE and then applying k -means, for 10 times on the datasets for a more reliable comparison.

Original: [Figure]

Original: performances of GPCAs and NMF in the spaces of different dimensions for a thorough comparison, ranging from 2 to 7 .

Original: More  details are provided in Section D of Appendix.
Revised: More experimental details are provided in Section S.4 of Supplementary Material.

Original: 4.3 VISUALISATION
Revised: 4.3 VISUALIZATION

Original: In this subsection [Figure] we examine whether the use of D NB can produce better visualisation.
Revised: In this subsection, we examine whether the application of D NB can produce better visualization.

Original: First, we visualise the dimension-reduced data obtained by  different dissimilarity measures. Figure 2 and Figures 612 of Appendix provide the visualisation results obtained by GPLVM with different measures.
Revised: First, we visualize the dimension-reduced data obtained by GPLVMs with different measures in Figure 2 and Figures S2-S8 of Supplementary Material.

Original: It is observed that
Revised: It is observed that

Original: GPLVMs with D NB , D r , D asin , and D log perform equally well on most datasets except for the sim-manno-ESCsdataset andthe sim-manno-vm dataset.
Revised: GPLVMs with D NB , D r , D asin , and D log perform equally well on most datasets except for the sim-manno-ESCs and sim-manno-vm datasets.

Original: For the sim-mannoESCs dataset (Figure 12 of Appendix), GPLVMs with D P and D NB display well-grouped data in the 2D space while those with the other measures fail to do so.
Revised: For the sim-manno-ESCs dataset (Figure S8 of Supplementary Material), GPLVMs with D P and D NB display well-grouped data in the 2D space while those with the other measures fail to do so.

Original: Furthermore, only the GPLVM with D NB can distinguish different groups of data points on the sim-manno-vm dataset (Figure 2).
Revised: Furthermore, only the GPLVM with D NB can distinguish different groups of data points on the sim-manno-vm dataset (Figure 2).

Original: Figures 13-19 of Appendix supply the visualisation obtained by tSNE with different measures.
Revised: Secondly, we compare the visualization results obtained by tSNE with different measures in Figures S9-S16 of Supplementary Material.

Original: tSNEs with D NB ,D r ,
Revised: The tSNE algorithms with D NB ,

Original: D  asin , and D log produce well-distinguished groups of data on most datasets except  the sim-manno-ESCs dataset (Figure 18 of Appendix) and the sim-manno-vm dataset (Figure 19 of Appendix), where all the measures  fail to recognise the groups of data.
Revised: D r , D asin , and D log produce well-distinguished groups of data on most datasets except for the Zheng8eq dataset (Figure S13), the sim-manno-ESCs dataset (Figure S15) and the sim-manno-vm dataset (Figure S16), where all the measures [Figure] [Figure] fail to recognize the groups of data.

Original: By comparing PCA results with different measures in the 2D space (Figure 3 and Figures 20-26 of Appendix), we found that PCAs with D NB , D asin , D log produce more distinguished groups of data on the sc-CELseq2-5cl-p1 dataset (Figure 3) and the sc-CEL-seq2-5cl-p dataset (Figure 20 of Appendix).
Revised: Thirdly [Figure] by comparing PCA results with different measures in the 2D space (Figure 3 and Figures S17-S23 of Supplementary Material), we find that the PCA algorithms with D NB , D asin , D log produce more distinguished groups of data on the sc-CEL-seq2-5cl-p1 dataset (Figure 3) and the sc-CEL-seq2-5cl-p2 dataset (Figure S17).

Original: For the sc-CEL-seq2-5clp3 dataset (Figure 21 of Appendix), the PCA with D log presents 3 distinguished groups while those with the other measures display only 2 groups.
Revised: For the scCEL-seq2-5cl-p3 dataset (Figure S18), the PCA with D log presents three distinguished groups while those with the other measures display only two groups.

Original: Both PCAs withD NB and
Revised: The PCA with

Original: D  asin  separate the groups in the sc-CEL-seq2 dataset (Figure 22 of Appendix).
Revised: D NB and D asin can separate the groups in the sc-CEL-seq dataset (Figure S19).

Original: PCAs with the proposed measures and VSTs perform equally well on the Zheng8eq dataset (Figure 23 of Appendix) and the sim-Zheng8eq dataset (Figure 24 of Appendix).
Revised: The PCA algorithms with the proposed measures and VSTs perform equally well on the Zheng8eq dataset (Figure S20) and the sim-Zheng8eq dataset (Figure S21).

Original: For the sim-manno-ESCs dataset (Figure 25 of Appendix), the uses of D r , D P , and D NB lead to more distinguished groups in the data.
Revised: For the sim-manno-ESCs dataset (Figure S22), the applications of D r , D P , and D NB lead to more distinguished groups in the data.

Original: Furthermore, it is observed in Figure 26 of Appendix that only the PCA with D NB can separate groups of data to some degree on the sim-manno-vm dataset while those with the other measures fail to do so.
Revised: Furthermore, it is observed in Figure S23 that only the PCA with D NB can separate groups of data to some degree on the sim-manno-vm dataset while those with the other measures fail to do so.

Original: Secondly, we assess the visualisation produced by GPCAs and NMF (Figure 27 and Figures 28-34 of Appendix) and compare them with those obtained by the PCA with D NB (Figure 3).

Original: It is observed that the PCA with D NB display more well-grouped data than GPCAs and NMF on most datasets except the Zheng8eq dataset (Figure 31 of Appendix) and sim-Zheng8eq dataset (Figure 32 ofAppendix). For these two datasets, GPCAs distinguish the groups existing in the data more clearly compared with the PCA with
Revised: It is found that the GPLVM/PCA with D NB often presents distinguished groups of data in the 2D space while the tSNE with D NB fails to do so on some datasets. This difference may be due to the characteristics of the DR methods, but not the dissimilarity measures themselves.

Original: D NB .
Revised: As mentioned before,

Original: To sum up, the representative DR approaches with D NB outperform those with the other measures, and the use of D NB often results in visualisation better than those produced by GPCAs and NMF.
Revised: GPLVM/PCA aims to preserve the global structure of data and tSNE predominantly preserves the local structure.

Original: We first compare the proposed dissimilarity measures with those based on the VSTs in terms of the clustering performances in the reduced spaces.
Revised: Thus, tSNE may fail to preserve the global structure (inter-cluster proximity) due to its preference for the local proximity. In such cases, visualizing clusters with tSNE would not work well.

Original: The k -means clustering results with GPLVM and different dissimilarity measures are shown in Figure 35 of Appendix.
Revised: Although Kobak and Berens [2019] suggest using informative initialization or multi-scale similarities to improve the preservation of the global structure, but we find that these strategies do not result in better-distinguished clusters in the 2D space for the Zheng8eq, sim-manno-ESCs and sim-manno-vm datasets.

Original: It is observed that the GPLVM  with D NB performs consistently well on most datasets except the sc-CEL-seq2-5cl-p2, where its ARI value is slightly lower than those obtained by the VSTs. Further, the GPLVM with D NB obtains much higher values than
Revised: To sum up, the GPLVM and PCA with D NB can produce better visualization results than those with the other measures, while the tSNE with D NB may not distinguish clusters well due to its preference for the local structure of data.

Original: 7 Number of dimensions 0.
Revised: [Figure] [Figure]

Original: AR I
Revised: A

Original: ( K - m e a n s ) D P D NB
Revised: [Figure]

Original: GPCA-PGPCA-NB NMF (a)     7 Number of dimensions 0. A R I( K - m e a n s )
Revised: I

Original: D P D NB 
Revised: D P D NB D r

Original: GPCA-PGPCA-NB NMF (b)     7 Number
Revised: D log D asinY D Y (a)

Original: [Table]
Revised: [Figure]

Original:  dimensions0.
Revised: Number of dimensions

Original: [Table] [Table] A R I
Revised: [Figure] [Figure]

Original: ( K - m e a n s )
Revised: [Figure]

Original: D P D NB GPCA-PGPCA-NB NMF (c)     7 Number of dimensions 0.
Revised: [Figure]

Original: A R I ( K - m e a n s ) D P D NB GPCA-PGPCA-NB
Revised: [Figure]

Original: NMF (d) [Table] of dimensions A R I
Revised: [Figure]

Original: ( K - m e a n s ) D P D NB GPCA-PGPCA-NB
Revised: [Figure]

Original: NMF (e)     7 Number of dimensions0.
Revised: I (b)     7 Number of dimensions

Original: A R I
Revised: [Figure]

Original: ( K - m e a n s )
Revised: [Figure]

Original: D P D NB
Revised: [Figure] [Figure] [Figure] [Figure]

Original: GPCA-
Revised: I (c)

Original: [Table]
Revised: [Figure]

Original: NMF (f)
Revised: Number

Original: [Table]
Revised: [Figure]

Original: [Table] [Table] [Table]
Revised: (d)

Original: [Table]
Revised: [Figure]

Original: A R I
Revised: (h)

Original: ( K - m e a n s )
Revised: h) sim-manno-vm.

Original: D P D NB
Revised: 4.4 CLUSTERING RESULTS

Original: GPCA-PGPCA-NB NMF (g)     7 Number of dimensions 0.
Revised: In the following, we compare different measures in terms of their clustering performance in the dimension-reduced spaces.

Original: A R I
Revised: The k -means clustering results with GPLVMs and different dissimilarity measures are shown in Figure 4.

Original: ( K - m e a n s ) D P D NB GPCA-PGPCA-NB
Revised: It is observed that the GPLVM with D NB performs consistently well on most datasets except the sc-CEL-seq2-5cl-p2.

Original: NMF (h) the other measures  on the sim-manno-ESCs dataset and the sim-manno-vm dataset.
Revised: Furthermore, the GPLVM with D NB obtains much higher values than the other measures based on VSTs on the simmanno-ESCs dataset and the sim-manno-vm dataset.

Original: To sum up, D NB outperforms the other measures when integrated into GPLVM.
Revised: To sum up, D NB outperforms the other measures when integrated into GPLVM.

Original: Figure 36 of Appendix presents the k -means clustering results with tSNE and different measures.
Revised: Figure 5 presents the k -means clustering results with tSNE and different measures.

Original: The clustering performances obtained by tSNEs on the datasets differ greatly from those by GPLVMs. For the sc-CEL-seq2-5cl-p dataset, the sc-CEL-seq2-5cl-p2 dataset, and the sc-CELseq2 dataset, tSNEs withthe VSTs and the proposed measures perform equally well according to their average ARI values. For the sc-CEL-seq2-5cl-p3 dataset, D NB and D log achieve more stable and higher ARI values than the others.
Revised: The tSNE with D NB performs comparably well on some datasets except for the Zheng8eq, sim-manno-ESCs, and sim-manno-vm datasets.

Original: Unlike GPLVM, the tSNE with D E achieves the highest
Revised: The clustering performance of the tSNE with D NB are consistent with its visualization results.

Original: ARI value onboth the Zheng8eq dataset and  the sim-mannoESCs dataset, as well as a value approximately the same as those with D P and D NB on the sim-Zheng8eq dataset. All measures perform poorly on the sim-manno-vm dataset.
Revised: As we discussed before, its relatively weak performance on the Zheng8eq, sim-mannoESCs, and sim-manno-vm datasets may be due to the tSNE’s preference for the preservation of the local structure of data.

Original: There is no measure that maintains the superiority over the other measures across the datasets when input into tSNE. It is observed in Figure 37 of Appendix, D NB is superior to the other measures when combined with PCA, irrespective of the number of dimensions, on most datasets except for the sc-CEL-seq2-5cl-p3 dataset and the Zheng8eq dataset. Furthermore,D NB outperforms most measures when the dimension is greater than 5 on the the sc-CEL-seq2-5cl-p dataset and the Zheng8eq dataset. Secondly, we compare the proposed measures, which are input into PCA, with GPCAs and NMF according to the clustering results provided in Figure 4.
Revised: Furthermore, the clustering performance of tSNE suggests that clustering on the outputs of DR techniques must be done with caution.

Original: On the sc-CEL-seq dataset and the sim-manno-vm dataset, the PCA with D NB not only outperforms GPCAs and NMF by a large margin when dimension is 2 but also achieves the highest ARI value with the lowest dimension. The value of ARI obtained by D NB is much higher than those by GPCAs and NMF when dimension equals 2 on the sc-CEL-seq2-5cl-p1 dataset. For the sim-manno-ESCs dataset, the ARI values of GPCAs dramatically decrease as the dimension increases fromto 7 , while those obtained by PCAs with D NB and D P are higher and more stable.
Revised: DR approaches, such as non-linear tSNE, could be unsuccessful to preserve clusters and thus adversely affect the cluster analysis.

Original: Although the ARI values of

Original:  D NB do not rise as the dimension grows from 2 to4 on the sc-CEL-seq2-5cl-p1 dataset and the sc-CEL-seq2-5cl-p dataset, PCA with D NB outperforms GPCAs and NMF by a wide margin in the 2D space. For the Zheng8eq dataset andthe sim-Zheng8eq dataset, GPCA-P accomplishes the highest value with the lowest dimension. It is found that the results of GPCAs could be highly variable.
Revised: It is observed in Figure 6, D NB is superior to the other measures when combined with PCA, irrespective of the number of dimensions, on most datasets except for the scCEL-seq2-5cl-p3, Zheng8eq, and sim-Zheng8eq datasets. On these three datasets, D NB outperforms most measures when the dimension is greater than five.

Original: Overall, the clustering performances obtained by the representative DR methods with D NB are superior to those with the other measures.Moreover, the PCA with D NB outperforms GPCAs and NMF.
Revised: In summary, consistent with the visualization results, the clustering performance obtained by the representative GPLVM/PCA with D NB are superior to those with the other measures.

Original:  CONCLUSION
Revised:  CONCLUSION

Original: This paper investigates how to perform DR for highdimensional small count data.
Revised: This paper investigates how to perform DR for highdimensional small count data.

Original: We propose a dissimilarity measure D NB that is well-suited for count data with many zeros.
Revised: We propose a dissimilarity measure D NB that is well-suited for count data with many zeros.

Original: The statistical behaviours of different dissimilarity measures when the dimension is high enough are investigated in terms of a proposed index.
Revised: The statistical behaviours of different dissimilarity measures when the dimension is high enough are investigated in terms of a proposed index.

Original: It is found that the proposed measure D NB is superior to the other measures in the sense that it distinguishes data points from different distributions better.
Revised: It is found that the proposed measure D NB is superior to the other measures in the sense that it distinguishes data points from different distributions better.

Original: Consistent with the statistical comparison, the experimental results demonstrate that D NB enhances a variety of standard DR approaches.A PROOFS
Revised: Consistent with the statistical comparison, the experimental results demonstrate that D NB enhances a variety of standard DR approaches.

Original: In this section, we provide the proofs of Proposition 1 and

Original: Corollary 2 in the main paper.

Original: A.1 PROOF OF PROPOSITION

Original: Proof 1 First note that

Original: R ( F x , F y ) converges to E [ D 2 ( x,y )]

Original: E [ D 2 ( x 1 ,x 2 )]

Original: + E

Original: [ D 2 ( y 1 ,y 2 )] in probability, as d → ∞ , based on the weak law of large numbers and the Slutsky’s theorem.

Original: Now we have [Equation] and [Equation]
Revised: Acknowledgements

Original: Combining results from Equation (9) and Equation (10), we obtain

Original: [Equation]

Original: which completes the proof.

Original: A.2 PROOF OF COROLLARY

Original: Proof 2 First note that lim µ x → 0 p ( x = 0)

Original: = 1 for non-negative random variable x and lim µ x → 0 Var ( x ) = 0 . Further, we obtain lim µ x → 0 E [ g ( x )] = g (0) and lim µ x → 0 Var [ g ( x )]
Revised: The authors thank the anonymous reviewers for helpful discussions and suggestions.

Original: = 0 .

Original: Now consider the limiting difference between c g and c E : lim µ x → 0 c g − c E = lim

Original: µ x →(cid:34) [E g ( x

Original: [Equation]

Original: g

Original: [Equation]

Original: y )]

Original: Var [ g ( x )]

Original: + Var [ g ( y )] −

Original: [Equation]

Original: [Equation]

Original: Var

Original: [Equation]

Original: B AN EXAMPLE SHOWING THE EUCLIDEAN DISTANCE PERFORMS BADLY ON SMALL COUNTS

Original: Suppose we have two Poisson distributions with mean values 0 .

Original: 01 and 0 .

Original: 96 , respectively.

Original: Although both distributions

Original: [Figure]

Original: outputs small counts, the PMFs of these two distributions, provided in Figure 5, display disparate patterns: the mass of one distribution concentrates in 0 , while the frequent values of the other one lie in { 0 , 1 , 2 } .

Original: It is highly likely that

Original: R E ( F x , F y ) < 1 in the space of high dimension according to Corollary 1 due to the mean-variance dependency, indicating that D E cannot distinguish these two distributions well.

Original: C SIMULATION DETAILS

Original: To obtain ˆ R ( F x , F y ) , we first generate high-dimensional samples in S X and S Y with each coordinate i.i.d., and then calculate ˆ R ( F x , F y ) in Equation (8) with different measures.

Original: As we compared D E with the Euclidean distances of the transformed data, we let F x be a count distribution that frequently generates zeros.

Original: Specifically, F x is either Pois (0 . 05) or NB ( r, 0 . 05) that has the same r as that of F y .

Original: Note that the probability of getting a zero from Pois (0 . 05) is around 0 .

Original: 951 and that from NB ( r, 0 . 05) is 0 .

Original: 95 r .

Original: The parameter spaces for F y are provided in Table 6.

Original: Note that the parameters for simulating NB and Poisson distributions are selected in an evenly spaced way from the intervals provided in Table 6.

Original: With the mean value increasing, the proportion of zeros produced from a Poisson distribution decreases to 0 .

Original: We hence bound the mean value from above by 1 to generate small counts and the expected fractions of zeros in the Poissondistributed count data lie in the range [0 .

Original: When r approaches infinity, the NB distribution approximates an equi-dispersed Poisson distribution.

Original: Thus, the upper bound of r for simulating NB distributions should be small in order to generate overdispersed distributions.

Original: When simulating data from NB distributions, the values of r for both F x and

Original: F y are the same, and we use the true value of r for calculating D NB and D asin .

Original: Since r < 34 results in the numerical problem when calculating D asin , we only take values that is larger than 3 4 for r .

Original: Again, we restrict the range of p in

Original: NB distributions with the aim of simulating small counts.

Original: The proportion of zeros in the simulated NB data ranges from 0 .

Original: 363 to 0 . 963 .

Original: For Poisson distributions, we set r to 1000 in both D NB and D asin due to the approximation of

Original: NB distributions to Poisson distributions when r is large.

Original: Distributions

Original: [Table]

Original: Poisson means: [0 .

Original: #samples:variances: the same as the space of means

Original: NB r (# failures): ( 34 , 5] ; #samples:p (probability of success): [0 . 05 , 0 . 2] ; #samples:

Original: The dimension for each data point is set to be 5000 which is high enough to obtain a reliable estimation of the constant that R ( F x , F y ) converges to.

Original: Although the convergence of

Original: R ( F x , F y ) holds regardless of the number of samples in each distribution, small values of n x and n y possibly result in the meaningless coordinates, in the form of vectors of zeros.

Original: Therefore, we set n x and n y to 200 to avoid such cases.

Original: D EVALUATION DETAILS

Original: When applying tSNE with the proposed measures to the datasets, we replace the Euclidean distance with the proposed ones to characterise the dissimilarities between data points for the construction of conditional probabilities.

Original: When performing PCA with the proposed measures, we first get a pseudo Gram matrix by double-centring the corresponding pairwise dissimilarity matrix and then get the low-dimensional components by eigen-decomposing the pseudo Gram matrix.

Original: We empirically found that making the pseudo Gram matrix positive semi-definite would produce better results when combined with GPLVM.

Original: Thus, we first eigen-decompose the pseudo Gram matrix and keep all the eigenvectors with non-negative eigenvalues.

Original: We then feed the modified pseudo Gram matrix Q Λ Q T into the GPLVM, where Q is the matrix whose columns are the kept eigenvectors and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues.

Original: E MORE VISUALISATION RESULTS E.1 VISUALISATION OF GPLVM RESULTS

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: E.2 VISUALISATION OF TSNE RESULTS

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: D P

Original: [Figure]

Original: [Figure]

Original: D NB

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: E.4 VISUALISATION OF GPCAS AND NMF

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: D E

Original: D r

Original: D asin

Original: D log

Original: D P

Original: D NB 0.0 0.

Original: A R I

Original: ( K - m e a n s ) (a)

Original: D E

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Original: D NB 0.0 0.

Original: A R I

Original: ( K - m e a n s ) (b)

Original: D E

Original: D r

Original: D asin

Original: D log

Original: D P

Original: D NB 0.0 0.

Original: A R I

Original: ( K - m e a n s ) (

Original: [Figure]

Original: E

Original: [Figure]

Original: r

Original: [Figure]

Original: asin

Original: D log

Original: D P

Original: D NB

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: A R I (

Original: [Figure]

Original: e a n s ) (

Original: [Figure]

Original: D E

Original: D r

Original: D asin

Original: D log

Original: D P

Original: D NB 0.

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: A R I

Original: ( K

Original: [Figure]

Original: e a n s ) (e)

Original: D E

Original: D r

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Original: D log

Original: [Figure]

Original: P

Original: [Figure]

Original: NB

Original: [Figure]

Original: n

Original: [Figure]

Original: [Figure]

Original: E

Original: D r

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Original: D log

Original: D P

Original: D NB 0.

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: A R I

Original: ( K - m e a n s ) (

Original: [Figure]

Original: D E

Original: D r

Original: D asin

Original: D log

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Original: D NB 0.0 0.

Original: A R I

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Original: A R I

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Original: D E

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Original: D NB 0.0 0.

Original: A R I

Original: ( K - m e a n s ) (c)

Original: D E

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Original: D NB 0.0 0.

Original: A R I

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Original: D E

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Original: D NB 0.0 0.

Original: A R I

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Original: D E

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Original: D NB 0.0 0.

Original: A R I

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Original: D E

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Original: D log

Original: D P

Original: D NB 0.0 0.

Original: A R I

Original: ( K - m e a n s ) (g)

Original: D E

Original: D r

Original: D asin

Original: D log

Original: D P

Original: D NB 0.0 0.

Original: [Figure]

Original: [Table]

Original: A R I

Original: ( K - m e a n s )

Original: D P D NB

Original: [Table]

Original: D log

Original: [Table]

Original: Y

Original: [Table]

Original: a)

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: a n

Original: [Table]

Original: a n s )

Original: [Table]

Original: NB

Original: [Table]

Original: [Table]

Original: A R I

Original: ( K - m e a n s )

Original: D P D NB D r

Original: D log D asinY D Y (g)     7 Number of dimensions 0.

Original: A R I

Original: ( K - m e a n s )

Original: D P D NB

Original: D r D log

Original: D asinY D Y (h)