DNBP: Differentiable Nonparametric Belief Propagation

1 INTRODUCTION . A significant challenge for robotic applications is the ability to estimate the pose of articulated objects in high noise environments . Nonparametric belief propagation ( NBP ) algorithms ( Sudderth et al. , 2003 ; Isard , 2003 ) have proven effective for inference in visual perception tasks such as human pose tracking ( Sigal et al. , 2004 ) and articulated object tracking in robotic perception ( Desingh et al. , 2019 ; Pavlasek et al. , 2020 ) . Moreover , these algorithms are able to account for uncertainty in their estimates when environmental noise is high and show promising computational properties in practice ( Desingh et al. , 2019 ; Ortiz et al. , 2021 ) . Their adaptability to new applications , however , is limited by the need to define hand-crafted functions that describe the distinct statistical relationships in a particular dataset . Current methods that utilize NBP rely on extensive domain knowledge to parameterize these relationships . Reducing the domain knowledge required by NBP methods would enable their use in a broader range of applications . As a form of probabilistic graphical model inference , NBP algorithms leverage domain knowledge encoded in graph-based representations , such as the Markov random field ( MRF ) . Their capacity to perform inference using arbitrary graphs sets them apart from other algorithms such as the recursive Bayes filter ( Thrun et al. , 2005 ) ( e.g . particle filter ( Godsill , 2019 ) ) and has been shown to be important in computational perception because it allows for modeling of non-causal relationships ( Sudderth et al. , 2003 ) . Data-driven approaches are an alternative for computational perception ( Xiang et al. , 2018 ; Tremblay et al. , 2018 ) . These methods generally avoid the need for extensive domain knowledge by learning from large amounts of labelled data . Data-driven approaches , however , are prone to noisy estimates and have limited capacity to represent uncertainty inherent in their estimates . In robotic applications , both of these limitations negatively impact the ability for a robot to operate effectively in unstructured environments . In this paper , we present a differentiable nonparametric belief propagation ( DNBP ) method , a hybrid approach which leverages neural networks to parameterize the NBP algorithm . Through differentiable inference , DNBP leverages the explainability and robustness of probabilistic inference techniques and capitalizes on the efficiency and generalizability of data-driven approaches . Inspired by the differentiable particle filter ( DPF ) from Jonschkowski et al . ( 2018 ) and the pull message passing for nonparametric belief propagation ( PMPNBP ) algorithm ( Desingh et al. , 2019 ) , we develop a differentiable nonparametric belief propagation algorithm . DNBP performs end-to-end learning of each probabilistic factor required for graphical model inference . The effectiveness of DNBP is demonstrated on two simulated articulated tracking tasks and on a real-world hand pose tracking tasks in challenging noisy environments . An analysis of the learned probabilistic factors and resulting tracking performance is used to validate the approach . Results show that our approach can leverage the graph structure to report uncertainty about its estimates while significantly reducing the need for prior domain knowledge required by previous NBP methods . DNBP performs competitively in comparison to traditional learning-based approaches on the tracking tasks . Collectively , these results indicate that DNBP has the potential to be successfully applied to robotic perception tasks , where a notion of uncertainty in the inference is inevitable . 2 RELATED WORK . Belief Propagation : In the context of graphical models , inference refers to the process in which information about observed variables is used to derive the posterior distribution ( s ) of unobserved variables . Belief propagation ( BP ) is a message passing algorithm for inference on graphical models . BP computes exact marginal distributions on trees ( Pearl , 1988 ) , and has demonstrated empirical success on loopy graphs ( Murphy et al. , 1999 ; Sun et al. , 2003 ; Lee et al. , 2008 ; Lan et al. , 2006 ) . In order to apply inference techniques such as BP and LBP , the parameters of a graphical model ( e.g . the probabilistic factors ) must be fully specified . Maximum likelihood estimation ( MLE ) has been shown to be an effective approach for learning the parameters of a graphical model from data ( Murphy , 2012 ; Koller & Friedman , 2009 ; Ping & Ihler , 2017 ) . In contrast , this current study focuses on parameter learning for use with inference of continuous random variables . Nonparametric Belief Propagation : For continuous spaces , such as six degrees-of-freedom object pose , exact integrals called for in BP and LBP become intractable and approximate methods for inference have been considered . Nonparametric belief propagation ( NBP ) methods ( Isard , 2003 ; Sudderth et al. , 2003 ) , have been proposed which represent the inferred marginal distributions using mixtures of Gaussians and define efficient message passing approximations for inference . Isard ( 2003 ) demonstrated the effectiveness of PAMPAS using a set of synthetic visual datasets each modeled with hand-crafted factors . Sudderth et al . ( 2003 ) applied their NBP method successfully to a visual parts-based face localization task as well as a human hand tracking task ( Sudderth et al. , 2004 ) . In both applications , NBP relied on factor models which were chosen based on task-level domain knowledge ( e.g . valid configurations of human hands ) . Sigal et al . ( 2004 ) extended these NBP methods to human pose estimation and tracking using factors which were each trained separate from the inference algorithm using independent training objectives . Ihler & McAllester ( 2009 ) described a conceptual theory of particle belief propagation , where messages being sent to inform the marginal of a particular variable could be generated using a shared proposal distribution . Following the work of Ihler and McAllester , Desingh et al . ( 2019 ) presented an efficient “ pull ” message passing algorithm ( PMPNBP ) which uses a weighted particle set to approximate messages between random variables . PMPNBP was shown to be effective on robot pose estimation tasks using hand-crafted factors . Using a similar approximation of belief propagation , Pavlasek et al . ( 2020 ) took a step toward neural network-based potential functions by introducing a pre-trained image segmentation network to the unary factors . An important limitation of these works is they assume the probabilistic factors expressed in the graph are provided as input or rely on domain knowledge to separately model and train each function . The potential for neural networks to learn the parameters used by alternative inference techniques has been demonstrated ( Do & Artières , 2010 ; Tompson et al. , 2014 ) . The recent work of Xiong & Ruozzi ( 2020 ) demonstrated promising inference performance on discrete classification and synthetic datasets using learned probabilistic factors with a Bethe free energy approximation . In contrast , this paper focuses on inference in high dimensional continuous state spaces using learned probabilistic factors with the particle-based inference process of NBP . Differentiable Bayes Filtering : In the context of robot state estimation , many approaches have recently been proposed that incorporate neural networks with recursive inference algorithms in an endto-end fashion . Haarnoja et al . ( 2016 ) introduced a differentiable Kalman filter , and Jonschkowski & Brock ( 2016 ) proposed a differentiable , histogram-based Bayes filter algorithm . Jonschkowski et al . ( 2018 ) and Karkus et al . ( 2018 ) both proposed differentiable particle filter algorithms for modeling continuous state spaces . Kloss et al . ( 2021 ) evaluate recent differentiable filtering techniques . Yi et al . ( 2021 ) propose an end-to-end learning method for inference over factor-graph models . In contrast to these methods , which model a single object body using variants of the Bayes filter , this work sets out to study the potential for NBP to be used as an algorithmic prior for modeling multi-part articulated objects . Recently , this line of research on differentiable state estimation algorithms has extended into the planning domain ( Karkus et al. , 2019 ; Wang et al. , 2020 ; Anderson et al. , 2019 ) . Exploration of embedding DNBP within a differentiable planning system is left as future work . 3 BELIEF PROPAGATION . Consider a Markov Random Field ( MRF ) defined by the undirected graph G = { V , E } , where V denotes a set of nodes and E denotes a set of edges . An example MRF model is shown in Fig . 2b . Each node in V represents an observed ( grey ) or unobserved ( white ) random variable , while each edge in E represents a pairwise relationship between two random variables in V . The joint probability distribution for G is : p ( X , Y ) = 1 Z ∏ ( s , d ) ∈E ψsd ( Xs , Xd ) ∏ d∈V φd ( Xd , Yd ) ( 1 ) where X = { Xd | d ∈ V } is the set of unobserved variables and Y = { Yd | d ∈ V } is the set of corresponding observed variables . The scalar Z is a normalizing constant . For each node , the function φd ( · ) is the unary potential , describing the compatibility of Xd with a corresponding observed variable Yd . For each edge , the function ψsd ( · ) is the pairwise potential , describing the compatibility of neighboring variables Xs and Xd . This work considers MRF models limited to pairwise clique potentials . Given the factorization of the joint distribution defined in Eq . ( 1 ) , BP provides an algorithm for inference of the marginal posterior distributions , know as the beliefs , beld ( Xd ) . BP defines a message passing scheme for calculation of the beliefs as follows : beld ( Xd ) ∝ φd ( Xd , Yd ) ∏ s∈ρ ( d ) ms→d ( Xd ) ( 2 ) where ρ ( s ) denotes the set of neighboring nodes of s. A message from node s to d is defined as : ms→d ( Xd ) = ∫ Xs φs ( Xs , Ys ) ψsd ( Xs , Xd ) × ∏ u∈ρ ( s ) \d mu→s ( Xs ) dXs ( 3 ) Performing inference of random variables in continuous space causes the integral in Eq . ( 3 ) to become intractable . This motivates the use of efficient algorithms that approximate the message passing scheme of Eq . ( 2 ) and Eq . ( 3 ) . 3.1 NONPARAMETRIC BELIEF PROPAGATION . Nonparametric belief propagation ( NBP ) ( Sudderth et al. , 2003 ) uses Gaussian mixtures to represent the beliefs and messages for continuous random variables . Later works , including Ihler & McAllester ( 2009 ) and Desingh et al . ( 2019 ) , further improve upon the tractibility of approximate nonparametric inference by representing beliefs and messages with sets of weighted particles . These particle-based NBP methods infer an approximation of the beliefs using an iterative message passing algorithm , in which beliefs and messages are updated at each iteration t. In particular , Desingh et al . ( 2019 ) avoid the expensive message generation of NBP by approximating Eq . ( 3 ) with a “ pull ” strategy . A message , mts→d , outgoing from s to d , is generated by first sampling M independent samples from belt−1d ( Xd ) then reweighting and resampling from this set .

Summary: The method (DNBP) proposed in the paper considers a non-parametric belief propagation method where the unary potential functions, the pairwise potential functions and the particle diffusion function are modeled as feed-forward neural networks. It allows them to learn the parameters of these networks using labeled data which is the main contribution of the paper.  DNBP is evaluated on three tasks (simulated double pendulum, simulated articulated spider, first-person hand action). In each application, DNBP is not able to outperform the considered baseline but it is able to provide measures of uncertainty associated with its predictions.