Non-approximate Inference for Collective Graphical Models on Path Graphs via Discrete Difference of Convex Algorithm

1 Introduction . In recent years , the importance of aggregated count data , which is calculated from the data of multiple individuals , has been increasing [ 21 , 27 ] . Although technologies for acquiring individual data such as sensors and GPS have greatly advanced , it is still very difficult to handle individual data due to privacy concerns and the difficulty of tracking individuals . However , there are many situations where data aggregated from multiple individuals can be obtained and utilized easily . For example , Mobile Spatial Statistics [ 22 ] , which is the hourly population data of fixed-size square grids calculated from 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . cell phone network data in Japan , are available for purchase ; such data is being used for disaster prevention and urban planning [ 20 ] . In traffic networks , traffic volume data at each point can be obtained more easily by sensors or cameras than the trajectories of individual cars , and the data is useful for managing traffic congestion [ 9 , 26 ] . Collective Graphical Model ( CGM ) [ 17 ] is a probabilistic model to describe aggregated statistics of a sample drawn from a graphical model . CGM makes it possible to conduct various practical tasks on aggregated count data , such as estimating movements from population snapshots , parameter learning of the underlying graphical model , and interpolation and denoising of count tables . Particularly , the case where the underlying graph is a path graph is important because CGMs on path graphs can treat time series data in which the states of interest follow Markov chains . In fact , most of the real-world applications of CGMs utilize CGMs on path graphs to represent the collective movement of humans and animals [ 2 , 4 , 19 ] . Detailed analyses of time series of collective people movements from limited observations would be useful for controlling people flow to avoid congestion and to maintain social distancing in urban spaces . One of the most important operations in CGM is maximum a posteriori ( MAP ) inference . MAP inference is the discrete ( combinatorial ) optimization problem of finding an assignment of unobserved variables that maximizes the posterior probability under given observations . MAP inference makes it possible to interpolate missing values of aggregated data and to estimate more detailed information that lies behind the observations . Unfortunately , MAP inference for general CGMs has been shown to be NP-hard [ 16 ] and thus is difficult to solve exactly and efficiently . Therefore , an alternative approach that solves an approximate problem , which is derived by applying Stirling ’ s approximation and continuous relaxation , has been proposed [ 16 ] . Subsequent studies have focused on solving this approximate problem efficiently [ 12 , 18 , 19 , 23 ] . However , there are inherent problems with this approach of solving the approximate problem . First , this approach tends to output a solution with a low posterior probability when the values in the count tables are small , because Stirling ’ s approximation , log x ! ≈ x log x− x , is inaccurate when x is small . Such a situation frequently occurs when the number of values that each variable in the graphical model takes is large , or when the sample size is small . Second , since continuous relaxation is applied , the integrality constraints of count table values are violated in the output . As a result , values that should be integers ( e.g. , the number of people ) are no longer integers , which not only reduces interpretability , but also makes the output less sparse , resulting in high memory consumption to maintain the output . It is possible to obtain integer-valued results by rounding the output , but this rounding process destroys the sum constraints among the estimated counts , e.g. , the sum of the count table values at each node may not match the sample size . To resolve these issues , in this paper , we propose a new method for MAP inference for CGMs on path graphs . We first show that the objective function of the problem can be expressed as the sum of univariate discrete convex functions and discrete concave functions . Based on this expression , we utilize the idea of the Difference of Convex Algorithm ( DCA ) [ 6 ] . DCA is a framework to minimize a function expressed as the sum of a convex function and a concave function . In DCA , a solution is obtained by repeatedly minimizing a surrogate function that upper-bounds the objective function , and the objective function value decreases monotonically in each iteration . In addition , the algorithm terminates in a finite number of iterations in our case since the variables are discrete not continuous . The key to make the DCA-based algorithm efficient is a fast minimization algorithm for the surrogate function . Because the feasible region of our problem is limited to integer lattice points , continuous optimization methods such as the gradient descent , which are usually used in DCAs , can not be applied to minimize our surrogate function . Instead , we utilize the special structure of path graphs ; it enables us to formulate the minimization problem of the surrogate function as a combinatorial optimization problem called the minimum convex cost flow problem . Fast algorithms for the minimum convex cost flow problem are known and we can minimize the surrogate function efficiently by using these algorithms . The proposed method has several practical advantages . First , since the proposed method does not use Stirling ’ s approximation , it offers an accurate inference even when the values in the count tables are small . This makes it possible to output solutions with much higher posterior probability than the approximation-based approach . Second , because the proposed method does not apply continuous relaxation , the obtained solution is guaranteed to be integer-valued , which results in sparse and interpretable outputs . In Section 5 , we show results gained from synthetic and real-world datasets ; they indicate that the proposed method outputs higher quality solutions than the existing approach . We show that the superiority of the proposed method is much greater when the sample size is not very large or the number of states on nodes in the graphical model is large . 2 Collective Graphical Models ( CGMs ) . Collective Graphical Model ( CGM ) is a probabilistic generative model that describes the distributions of aggregated statistics of a sample drawn from a certain graphical model [ 17 ] . Let G = ( V , E ) be an undirected tree graph ( i.e. , a connected graph with no cycles ) . We consider a pairwise graphical model over discrete random variable X : = ( Xu ) u∈V defined by Pr ( X = x ) = 1Z ∏ ( u , v ) ∈E ϕuv ( xu , xv ) , where ϕuv ( xu , xv ) is a local potential function on edge ( u , v ) and Z : = ∑ x ∏ ( u , v ) ∈E ϕuv ( xu , xv ) is the partition function for normalization . In this paper , we assume that xu takes values on the set [ R ] for all u ∈ V , where [ k ] denotes the set { 1 , 2 , . . . , k } for a positive integer k. We draw an ordered sample ( X ( 1 ) , . . . , X ( M ) ) independently from the graphical model , where M is the sample size . Let nu : = ( nu ( i ) ) i∈ [ R ] and nuv : = ( nuv ( i , j ) ) i , j∈ [ R ] , where nu ( i ) : = | { m | X ( m ) u = i } | and nuv ( i , j ) : = | { m | X ( m ) u = i , X ( m ) v = j } | . Each entry of nu and nuv is the number of occurrences of a particular variable setting . We call ( nu ) u∈V node contingency table and ( nuv ) ( u , v ) ∈E edge contingency table , and denote n : = ( ( nu ) u∈V , ( nuv ) ( u , v ) ∈E ) . We assume that observations y : = ( yu ) u∈V are generated by adding noise to the node contingency table ( nu ) u∈V , and the distribution of y is given by Pr ( y|n ) = ∏ u∈V ∏ i∈ [ R ] pui ( yu ( i ) |nu ( i ) ) , where pui is the noise distribution . An additional assumption is described below . Assumption 1 . For u ∈ V and i ∈ [ R ] , log pui ( y|n ) is a concave function in n. Assumption 1 is a quite common assumption in CGM studies [ 16 , 19 ] . Commonly used noise distributions such as Gaussian distribution pui ( yu ( i ) |nu ( i ) ) = 1√2πσ2 exp ( − ( yu ( i ) −nu ( i ) ) 2 2σ2 ) and Poisson distribution pui ( yu ( i ) |nu ( i ) ) = nu ( i ) yu ( i ) /yu ( i ) ! · exp ( −nu ( i ) ) satisfy Assumption 1 . The MAP inference problem for CGM is to find n that maximizes the posterior probability Pr ( n|y ) . Since Pr ( n|y ) = Pr ( n , y ) /Pr ( y ) from Bayes ’ rule , it suffices to maximize the joint probability Pr ( n , y ) = Pr ( n ) · Pr ( y|n ) . Pr ( n ) is called CGM distribution and calculated as follows [ 19 ] : Pr ( n ) = F ( n ) · I ( n ∈ LZM ) , ( 1 ) F ( n ) : = M ! ZM · ∏ u∈V ∏ i∈ [ R ] ( nu ( i ) ! ) νu−1∏ ( u , v ) ∈E ∏ i , j∈ [ R ] nuv ( i , j ) ! · ∏ ( u , v ) ∈E ∏ i , j∈ [ R ] ϕuv ( i , j ) nuv ( i , j ) , ( 2 ) LZM : = { n ∈ Z|V |R+|E|R 2 ≥0 ∣∣∣ M = ∑ i∈ [ R ] nu ( i ) ( u ∈ V ) , nu ( i ) = ∑ j∈ [ R ] nuv ( i , j ) ( ( u , v ) ∈ E , i ∈ [ R ] ) } . ( 3 ) Here , I ( · ) is the indicator function , νu is the degree of node u in G , and LZM is the set of possible contingency tables . Using the above notations , the MAP inference problem can be written as min n∈LZM − logF ( n ) − log Pr ( y|n ) . ( 4 ) 3 CGMs on Path Graphs . Hereafter , we focus on CGMs on path graphs , which is the main topic of this paper . Path graph Pn is an undirected graph whose vertex set is V = [ N ] and edge set is E = { ( t , t+ 1 ) | t ∈ [ N − 1 ] } . A graphical model ( not CGM ) on path graph is the most basic graphical model that represents a time series generated by a Markov model ; that is , the current state depends only on the previous state . A CGM on a path graph represents the distribution of aggregated statistics when there are many individuals whose state transition is determined by a Markov model . In the rest of this paper , we use the notation nti : = nt ( i ) , ntij : = nt , t+1 ( i , j ) , and ϕtij : = ϕt , t+1 ( i , j ) for simplicity . From ( 1 ) – ( 4 ) , the MAP inference problem for CGMs on path graphs can be written as follows : min n N−1∑ t=1 ∑ i , j∈ [ R ] ftij ( ntij ) + N−1∑ t=2 ∑ i∈ [ R ] g ( nti ) + N∑ t=1 ∑ i∈ [ R ] hti ( nti ) , s.t . ∑ i∈ [ R ] nti = M ( t ∈ [ N ] ) , ∑ j∈ [ R ] ntij = nti ( t ∈ [ N − 1 ] , i ∈ [ R ] ) , ∑ i∈ [ R ] ntij = nt+1 , j ( t ∈ [ N − 1 ] , j ∈ [ R ] ) , ntij , nti ∈ Z≥0 , ( 5 ) where ftij ( z ) : = log z ! − z · log ϕtij , g ( z ) : = − log z ! , hti ( z ) : = − log pti ( yti|z ) . For the details of the derivation , please see Appendix . We give an example of a CGM on a path graph which models human mobility . Consider that a space is divided into R distinct areas and that M people are moving around in the space . The random variable X ( m ) t represents the area to which person m belongs at time step t , and the time series X ( m ) = ( X ( m ) 1 , . . . , X ( m ) N ) is determined by the graphical model p ( x ) = 1Z ∏N−1 t=1 ϕtxtxt+1 . Here , ϕtij is the affinity between two areas i and j at time step t → t + 1. nti represents the number of people in area i at time step t , and ntij represents the number of people who moved from area i to j at time step t→ t+ 1 . We have noisy observations yti for t ∈ [ N ] and i ∈ [ R ] , which are generated by adding noise to nti . The MAP inference problem we want to solve is to find the true number of people of each area at each time step , ( nti ) t∈ [ N ] , i∈ [ R ] , and the true number of people moving between each two areas , ( ntij ) t∈ [ N−1 ] , i , j∈ [ R ] , with the highest posterior probability given the observation ( yti ) t∈ [ N ] , i∈ [ R ] .

Summary: The authors consider the problem of MAP estimate of count data given noisy count data. For example $X_1(t), X_2(t), \ldots, X_M(t)$ describe the location of an agent at time $t$, $N_i(t)$ is the number of agents at location $i$ at time $t$, and $Y_i(t)$ is a noisy estimate of $N_i(t)$. The goal is to estimate $N$ given $Y$. The noise distribution is assumed to be log-convex.  The authors develop an approach based on the difference of convex algorithm, which is an extension of the convex-concave procedure. They show that this procedure can monotonically improve a solution by repeatedly solving the minimum convex flow problem, which is known to be efficiently solvable.  The authors demonstrate that their approach improves over non-linear belief propagation with a continuous approximation of the discrete time problem.