1 Introduction
The problem of missing values often arises in reallife sequential data. For example, in motion tracking, trajectories often contain missing data due to targets lying out of the view, object occlusion, trajectories crossing, and the instability of camera motion (Urtasun et al., 2006). Hence, a critical task is missing data imputation which involves filling in missing values with reasonable predictions. Missing data imputation has been studied for decades. Most statistical imputation techniques such as averaging or regression (Rubin, 2004) are either reliant on strong assumptions of the data, or are limited to shortrange sequences. In this paper we study how to perform imputation on longrange sequences with arbitrary patterns of missing data.
In recent years, deep generative models and RNNs have been combined for missing data imputation. A common technique is to append a mask indicating whether or not data is missing, and then train with the masked data (Che et al., 2018). More recently, (Fedus et al., 2018; Yoon et al., 2018a; Luo et al., 2018) propose combining sequence imputation with generative adversarial networks (GAN) (Goodfellow et al., 2014)
. However, all existing imputation models are autoregressive and impute missing values conditioned on previous data. We find in our experiments that these approaches struggle on longrange sequences with longterm dynamics, as compounding error and covariate shift become catastrophic for autoregressive models. In this paper, we introduce a novel nonautoregressive sequence imputation method. Instead of conditioning only on previous values, our model learns the distribution of missing data conditioned on both the history and the future. To tackle the challenge of longterm dependencies, we exploit the multiresolution nature of spatiotemporal data, and decomposes the complex sequential dependency into simpler ones at multiple temporal resolutions. Our model, Nonautoregressive Multiresolution Imputation (
NAOMI) uses a divide and conquer strategy to recursively fill in the missing values from coarse to finegrained resolutions. We formalize the learning problem as an imitation learning task, and train the entire model using a differentiable modelbased generative adversarial imitation learning algorithm. In summary, our contributions are as follows:
We tackle the challenging task of missing value imputation for largescale spatiotemporal sequences of longrange and longterm dependency.

We propose NAOMI, a novel deep, nonautoregressive, multiresolution sequence model that recursively imputes missing values from coarse to finegrained resolutions with minimal computational overhead.

When evaluated on billiards and basketball trajectory imputation, our approach outperforms autoregressive counterparts by in accuracy and generates realistic sequences given arbitrary missing patterns.

Besides imputation, our framework is also effective in forward inference and outperforms stateoftheart methods without any additional modifications.
2 Related Work
Missing Value Imputation
Existing missing value imputation approaches roughly fall into two categories: statistical methods and deep learning methods. Statistical methods either use adhoc averaging with mean and/or median values
(Acuna & Rodriguez, 2004) or regression models such as ARIMA (Ansley & Kohn, 1984) and MICE (Buuren & GroothuisOudshoorn, 2010). Other popular imputation methods include the EM algorithm (Nelwamondo et al., 2007), KNN and matrix completion
(Friedman et al., 2001). Despite its rich history, missing value imputation remains elusive as traditional methods are not scalable, often requires strong assumptions on the data, and cannot capture longterm dependencies in sequential data. Recently, imputation using deep generative models has attracted considerable attention. A common practice is to append a missing indicator mask to the input sequence. For example, MRNN (Yoon et al., 2018b) proposes a new RNN architecture to account for multivariate dependencies. GRUD (Che et al., 2018) and BRITS (Cao et al., 2018) modify the RNN cell to capture the decaying influence of missing variables. GAIN (Yoon et al., 2018a) and GRUI (Luo et al., 2018) also use GAN. But GAIN does not model the sequential nature of data, and GRUI does not explicitly take the known observations into account during training. Perhaps the work that most closely related to ours is MaskGAN (Fedus et al., 2018), which makes use of adversarial training and reinforcement learning objective to fill in the missing texts for language modeling. However, all the imputation models are autoregressive, which suffers from compounding error and inconsistency between generated and actual values at observed points.
NonAutoregressive Modeling
Nonautoregressive models have been studied in the context of natural language processing where target words/sentences become independent given the latent embedding and can be predicted nonautoregressively. For instance,
Oord et al. (2018) uses Inverse Autoregressive Flow (Kingma et al., 2016) to map a sequence of independent variables to a target sequence. Gu et al. (2018) introduce a latent fertility model and treats the input word’s “fertility” as a latent variable. Other works sharing the similar idea include Lee et al. (2018) and Libovickỳ & Helcl (2018). All these works aim to parallelize sequence generation, which can be used to speed up the inference procedure in traditional autoregressive models. Our work is an innovative demonstration of nonautoregressive modeling for sequence imputation tasks. Our model utilizes the chain rule to factorize the joint distribution of sequences without imposing any additional independence assumption.
Behavioral Cloning
To explicitly model the sequential dependencies, we formulate our learning problem as an imitation learning task in the nomenclature of reinforcement learning (RL) (Syed & Schapire, 2008; Ziebart et al., 2008). Using GAN in the sequential setting has been explored in Ho & Ermon (2016); Yu et al. (2017). (Ho & Ermon, 2016) propose Generative Adversarial Imitation Learning (GAIL) by using an equivalence between maximum entropy inverse reinforcement learning (IRL) and GANs. Yu et al. (2017) further modify the reward structure and only allow a single reward when the sequence is completed. However, these models only use simple generators, which limit their ability to model longrange sequences. In order to generate longrange trajectories, Zheng et al. (2016) propose using manually defined macro goals from trajectories as weak labels to train a hierarchical RNN. Zhan et al. (2019) further extends this idea to the multiagent setting with a hierarchical variational RNN. However, while using macro goals can significantly reduce the search space, reliably obtaining the macro goals from trajectories can be very difficult. We exploit the multiresolution nature of the spatiotemporal data, and aim to learn the hierarchical structure without supervision. Our method bears affinity with other multiresolution generative models such as Progressive GAN (Karras et al., 2018)
and multiscale autoregressive density estimation
(Reed et al., 2017).3 Multiresolution Imputation
Let be a sequence of observations, where each time step . Some of the values in are missing and the goal is to replace the missing data with reasonable values. We introduce an accurate and efficient solution for missing value imputation.
3.1 Iterative Imputation
Our model, NAOMI, is a deep, nonautoregressive, multiresolution generative model. As depicted in Figure 2 and Algorithm 1, NAOMI
has three components: 1) an encoder that learns the representation of the sequence with missing values; 2) a decoder that generates values for missing observations given the hidden representations of the sequence; and 3) a discriminator to distinguish whether the generated sequence is real or fake. The decoder is multiresolutional and operates at different time scales. The encoder and decoder combined forms a generator
. NAOMI alternates between the encoder and the decoder to impute missing values iteratively. With missing values, we need iterations to impute all missing values. Denote the imputation order as , by applying the chain rule to factorize the conditional likelihoodas a product of conditional probabilities:
This factorization allows us to model the conditional likelihood in a tractable manner and without introducing independence assumptions. We begin by describing the encoder for sequences with incomplete data.
3.2 Incomplete Sequence Encoder
To indicate which values in are missing, we introduce a masking sequence where and 1 is the indicator function. The concatenated input is then . Our encoder maps into hidden states consisting of forward hidden states and backward hidden states . The conditional distribution of the encoder can be decomposed as:
(1) 
where represents the temporal dependence from the history and encodes the dependence on the future. We can parameterize the above distributions with a forward RNN and a backward RNN . At time step t, encodes the history of observations and imputed values up to , and encodes the future and missing masks after :
(2) 
For and
, we use GRU cells with ReLU activations. Next we describe our multiresolution imputation decoder.
3.3 Multiresolution Imputation Decoder (MID)
Given the hidden representations , the decoder learns the distribution of the complete sequence . To handle longterm dependencies and avoid error propagation, our decoder uses a divide and conquer strategy and imputes values recursively from coarse to finegrained resolutions. At each iteration, the decoder first identifies two time steps with observed/imputed values as pivots, and imputes a missing value close to their midpoint. One of the pivots is then replaced by the newly imputed point and the process repeats at a higher resolution. As shown in Figure 3, a multiresolution decoder with levels is equivalent to a collection of decoders, denoted by , each of which predicts every steps. At resolution , let the subsequence be . Then can be factorized:
(3) 
The decoder finds two pivots and and chooses a time step with a missing value that is close to the midpoint: . Let be the smallest resolution that satisfies . The decoder hidden states at time step are formed by concatenating the forward states and the backward states . Decoder
then maps the hidden states to a probability distribution over the outputs. The masking
is updated to after the prediction:(4) 
If the dynamics are deterministic,
directly outputs imputed value. For stochastic environments, we reparameterize the outputs using a Gaussian distribution with diagonal covariance and predict the mean and standard deviation.
Multiresolution as a universal approximator.
We provide the theoretical intuition for our multiresolution decoder. Consider an unknown function to be approximated by its multiresolution components at levels, that is . Multiresolution approximation is defined as a sequence of functions
from a set of nested vector spaces
that satisfy:Based on the wavelet theory (Mallat, 1989), we can decompose any function using a family of functions obtained by dilating and translating a given scaling function , resulting in a discrete wavelet transformation:
where the coefficients . Here denotes the dilation and defines the translation. In NAOMI, each decoder approximates the function
, and we have the recursive formula for neural network approximation:
. Hence, the approximation error at resolution is bounded by:and becomes progressively smaller as resolution increases.
3.4 Adversarial Imitation Learning
Given the sequential nature of our problem, we cast the imputation task as an imitation learning problem. Given a policy from states to actions, where subsequences form the state space, and {} are the actions. We treat complete sequences as roll outs from an expert policy , where The generator learns to reconstruct the original sequence given the masked sequence, leading to a learner’s policy . Imitation learning aims to learn a policy that mimic the expert policy using data. To quantify the distribution mismatch between reconstructed sequences and training data, we follow the GAIL(Ho & Ermon, 2016) framework. Formally, NAOMI uses the aforementioned generator parameterized by , and a discriminator parameterized by . Our training objective function is:
(5)  
where the outputs the probability that stateaction pair comes from data rather than the generator. One way to optimize the objective in Eqn 5
is to use policy gradients, but this procedure can be expensive and tends to suffer from high variance and sample complexity
(Kakade et al., 2003). Instead, we take a modelbased approach and assume the environment dynamics are known and differentiable. Hence, we can use the “reparameterization trick” (Kingma & Welling, 2013) to differentiate the generator objective with respect to the policy parameters. Similar ideas have been shown in (Baram et al., 2017; Rhinehart & Kris, 2018) to be more stable and sampleefficient.4 Experiments
We evaluate NAOMI on the task of missing value imputation in two environments: a billiards physical simulator of a single ball with deterministic dynamics, and a realworld basketball dataset of five player trajectories with stochastic dynamics. We present quantitative and qualitative comparisons with stateoftheart methods. Lastly, we further investigate our model on the task of forward inference.
Model details.
The forward and backward encoders are both 2layer RNNs with GRU cells. The multiresolution decoder has multiple 2layer fullyconnected neural networks. For the adversarial training, we use a 1layer RNN with GRU cells as the discriminator. We train on squared loss for billiards and adversarial loss for basketball.
Baselines.
We compare NAOMI with a set of baselines that include traditional statistical imputation methods and deep neural network based approaches.

KNN: (Friedman et al., 2001) finds the nearest sequences in the training set based on known observations. Missing values are imputed using the average of these knearest neighbors.

Linear: imputes the missing values using linear interpolation between two known observations.

MaskGAN: (Fedus et al., 2018) uses a single encoder to encode the entire incomplete sequence, a decoder to impute the sequence autoregressively, and a discriminator for adversarial training.

GRUI: (Luo et al., 2018) uses GAN to model the unconditional distribution via a random vector z. Then it uses L2 loss to find the best z based on observed steps. In the original work, complete training sequences are not available and time intervals are not fixed. Here we let the discriminator see the complete training sequence, and we just use the regular GRU cell considering only fixed time intervals. Hence GRUI may not be perfectly suited for our setting.

SingleRes: has the same encoder structure as NAOMI (a forward and backward encoder), but only has a single resolution decoder to impute missing values. The model is similar to BRITS (Cao et al., 2018), but trained adversarially with a discriminator.
Training.
For deterministic settings (e.g. Billiards), we optimize the L2 loss (teacher forcing is applied during pretraining). For stochastic settings (e.g. Basketball), we first pretrain the generator using crossentropy loss for supervised, and then optimize the generator and discriminator alternatively using the training objective in Eqn 5.
4.1 Imputing Billiards Trajectories
Dataset.
We generate 4000 training and 1000 test sequences of Billiards ball trajectories in a rectangular world using the simulator as in (Fragkiadaki et al., 2016). Each ball is initialized with a random position and random velocity and rolledout for timesteps. All balls have a fixed size and uniform density, and there is no friction in the environment. We generate a masking sequence for each trajectory with 180 to 195 missing values.
Imputation accuracy.
The three defining characteristics of the dynamics in this environment are: (1) moving in straight lines; (2) maintaining unchanging speed; and (3) reflecting upon hitting a wall. To quantify the imputation accuracy of each model, we use four metrics: (1) L2 loss between imputed missing values and their groundtruth; (2) Sinuosity to measure if the generated trajectory is straight or not; (3) Average step size change to measure if the speed of the ball is unchanging; and (4) Distance between reflection point and the wall to quantify if the model learns that a ball should bounce against a wall when it collides with one. Figure 4 compares the model performance using these metrics for imputation accuracy. The average value and 5, 95 percentile values are displayed for each metric. Statistics closer to those from the groundtruth indicate better model performance. NAOMI has the best overall performance across all metrics, followed by our singleresolution baseline model. Note that by design, linear interpolation has an average step size change closest to the groundtruth.
Generated trajectories.
We visualize the imputed trajectories from NAOMI and SingleRes in Figure 5. There are 8 known timesteps (black dots), including the starting position. NAOMI can successfully recover the original trajectory whereas SingleRes deviates from the groundtruth. In particular, SingleRes fails to use knowledge of fixed positions in the future to correctly predict the first steps of the sequence; SingleRes predicts the ball to reflect off the upper wall first instead of the left wall in the groundtruth. As such, SingleRes often has to correct its course to match known observations in the future, leading to curved and unrealistic trajectories. Another deviation from the groundtruth can be seen near the bottomleft corner, where NAOMI produces trajectory paths that are parallel after two reflections, but SingleRes does not.
4.2 Imputing Basketball Plays
Models  RNN  SingleRes  NAOMI  Expert 

Sinuosity  1.054(+5.4%)  1.038(+3.8%)  1.020(+2%)  1.00 
Step Change  11.6(+629%)  9.69(+510%)  10.8(+581%)  1.59 
Reflection point dist  0.074(+338%)  0.068(305%)  0.036(+114%)  0.018 
L2 Loss  4.698  4.753  1.682  0.0 
Models  RNN  RNN + GAN  HVRNN  SingleRes  NAOMI  Expert 

Path Length  1.36(+138%)  0.62(+9%)  0.67(+18%)  0.62(+9%)  0.55(3%)  0.57 
OOB Rate  29.2(+1395%)  4.33(+122%)  7.16(+266%)  3.62(+86%)  3.07(+57%)  1.95 
Step size Change  10.0(+403%)  2.20(+11%)  2.70(+36%)  2.35(+19%)  2.46(+23%)  1.99 
Path Difference  1.07(+91%)  0.41(27%)  0.59(+5%)  0.42(25%)  0.45(20%)  0.56 
Player Distance  0.450(+7%)  0.402(4%)  0.416(2%)  0.412(3%)  0.422(0%)  0.424 
Dataset.
The basketball dataset contains the trajectories of professional basketball players on offense. Each trajectory contains the (x, y)coordinates of 5 players for 50 timesteps at 6.25Hz and takes place in the left halfcourt. In total we have 107,146 training sequences and 13,845 test sequences. We generate a masking sequence for each trajectory with 40 to 49 missing values.
Imputation accuracy.
Since the environment is stochastic (basketball players on offense aim to be unpredictable), measuring L2 loss between our model output and the groundtruth is not necessarily a good indicator of realistic trajectories. Instead, we use the following 5 metrics to quantify how realistic the trajectories are: (1) Average trajectory length to measure the typical player movement in 8 seconds; (2) Average outofbound rate to measure whether the model recognizes court boundaries; (3) Average step size change to quantify the relationship between consecutive actions; (4) MaxMin path diff; and (5) Average player distance to analyze the team coordination. These metrics will serve as our proxy for evaluating imputation accuracy Figure 6 compares model performance using these metrics for imputation accuracy. The median value and 25, 75 percentile values are displayed for each metric. Statistics closer to the expert data indicate better model performance. NAOMI has the best overall performance across all metrics.
Generated trajectories.
We visualize imputed trajectories from all models in Figure 7. NAOMI produces trajectories that are the most consistent with known observations and have the most realistic player velocities and speeds. On the contrary, other baseline models commonly fail in these regards and exhibit problems similar to those observed in our Billiards experiments: KNN generates trajectories with unnatural jumps when there are too many known observations because finding neighbors becomes infeasible; Linear fails to generate trajectories with natural curvature when few observations are known; GRUI fails to generate trajectories consistent with known observations due to mode collapse in the generator when learning the unconditional distribution; and MaskGAN, which consists of a single forward encoder, fails to use known observations in the future and predicts straight lines.
Robustness to percentage of missing values.
Figure 8 compares the performance of NAOMI and SingleRes as we increase the number of missing values in the data. Generally speaking, the performance of both models degrade as more missing values are introduced, which makes intuitive sense since imputing more values is a harder task. However, at a certain percentage of missing values, the performance can improve for both models. There is an inherent tradeoff between two factors that affect model performance: Available Information and Amount of Constraints. Observed information can help models recover the original pattern of the trajectory, but it also serves as constraints that restrict the model output. As we have discussed before, generative models can easily generate trajectories that are consistent with its own historical outputs, but it is harder to be consistent with fixed observations that may deviate from its training distribution. Especially when mode collapse occurs, a single observation not captured by the model can lead to unpredictable trajectory generations.
Learned conditional distribution.
Our model learns the conditional distribution of the complete sequence given observations, which we visualize in Figure 9. For a given set of known observations, we use NAOMI to impute missing values with 50 different random seeds and overlay the generated trajectories. We can see that as the number of known observations increases, the uncertainty of the conditional distribution decreases. However, we also observe some mode collapse in our model: the trajectory of the purple player in the right image is not captured in the conditional distribution in the left image.
4.3 Imputation as Forward Inference
Imputation reduces to forward inference when all observations, except for a leading sequence, are missing. We show that NAOMI can also be trained to perform forward inference without modifying the model structure. We take a trained imputation model as initialization, and continue training for forward inference by using the masking sequence (first step is known). We evaluate forward inference performance using the same metrics. Table 2 shows the quantitative comparison of different models using our metrics. RNN statistics significantly deviate from the groundtruth, but greatly improve with adversarial training. HVRNN (Zhan et al., 2019) uses “macro goals”, and performs reasonably w.r.t macro metrics including average path length and maxmin path difference. However, the big step size changes lead to unnatural trajectories. SingleRes has similar performance as the the naive RNN + GAN model, which means our imputation model structure can be used to do forward inference. Finally, the single resolution model does better than NAOMI in terms of the average step size change, but NAOMI has the best performance across all metrics. Similarly, Table 1 compares forward inference performance in Billiards. NAOMI generates straighter lines and learns the reflection dynamics better than other baselines.
5 Conclusion
We propose a deep generative model NAOMI for imputing missing data in longrange spatiotemporal sequences. NAOMI recursively finds and predicts missing values from coarse to finegrained resolutions using a nonautoregressive approach. Leveraging multiresolution modeling and adversarial training, NAOMI is able to learn the conditional distribution given very few known observations and achieves superior performances in various experiments of both deterministic and stochastic dynamics. Future work will investigate how to infer the underlying distribution when complete training data is unavailable. The tradeoff between partial observations and external constraints is another direction for deep generative imputation models.
References

Acuna & Rodriguez (2004)
Acuna, E. and Rodriguez, C.
The treatment of missing values and its effect on classifier accuracy.
In Classification, clustering, and data mining applications, pp. 639–647. Springer, 2004.  Ansley & Kohn (1984) Ansley, C. F. and Kohn, R. On the estimation of arima models with missing values. In Time series analysis of irregularly observed data, pp. 9–37. Springer, 1984.
 Baram et al. (2017) Baram, N., Anschel, O., Caspi, I., and Mannor, S. Endtoend differentiable adversarial imitation learning. In International Conference on Machine Learning, pp. 390–399, 2017.
 Buuren & GroothuisOudshoorn (2010) Buuren, S. v. and GroothuisOudshoorn, K. mice: Multivariate imputation by chained equations in r. Journal of statistical software, pp. 1–68, 2010.
 Cao et al. (2018) Cao, W., Wang, D., Li, J., Zhou, H., Li, L., and Li, Y. Brits: Bidirectional recurrent imputation for time series. In Advances in Neural Information Processing Systems 31, pp. 6776–6786, 2018.
 Che et al. (2018) Che, Z., Purushotham, S., Cho, K., Sontag, D., and Liu, Y. Recurrent neural networks for multivariate time series with missing values. Scientific reports, 8(1):6085, 2018.

Fedus et al. (2018)
Fedus, W., Goodfellow, I., and Dai, A.
Maskgan: Better text generation via filling in the (blank).
In International Conference on Learning Representations (ICLR), 2018.  Fragkiadaki et al. (2016) Fragkiadaki, K., Agrawal, P., Levine, S., and Malik, J. Learning visual predictive models of physics for playing billiards. In International Conference on Learning Representations (ICLR), 2016.
 Friedman et al. (2001) Friedman, J., Hastie, T., and Tibshirani, R. The elements of statistical learning, volume 1. Springer series in statistics New York, NY, USA:, 2001.
 Goodfellow et al. (2014) Goodfellow, I., PougetAbadie, J., Mirza, M., Xu, B., WardeFarley, D., Ozair, S., Courville, A., and Bengio, Y. Generative adversarial nets. In Advances in neural information processing systems, pp. 2672–2680, 2014.

Gu et al. (2018)
Gu, J., Bradbury, J., Xiong, C., Li, V. O., and Socher, R.
Nonautoregressive neural machine translation.
In International Conference on Learning Representations (ICLR), 2018.  Ho & Ermon (2016) Ho, J. and Ermon, S. Generative adversarial imitation learning. In Advances in Neural Information Processing Systems, pp. 4565–4573, 2016.
 Kakade et al. (2003) Kakade, S. M. et al. On the sample complexity of reinforcement learning. PhD thesis, University of London London, England, 2003.
 Karras et al. (2018) Karras, T., Aila, T., Laine, S., and Lehtinen, J. Progressive growing of gans for improved quality, stability, and variation. In International Conference on Learning Representations (ICLR), 2018.
 Kingma & Welling (2013) Kingma, D. P. and Welling, M. Autoencoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.
 Kingma et al. (2016) Kingma, D. P., Salimans, T., Jozefowicz, R., Chen, X., Sutskever, I., and Welling, M. Improved variational inference with inverse autoregressive flow. In Advances in neural information processing systems, pp. 4743–4751, 2016.
 Lee et al. (2018) Lee, J., Mansimov, E., and Cho, K. Deterministic nonautoregressive neural sequence modeling by iterative refinement. In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, 2018.
 Libovickỳ & Helcl (2018) Libovickỳ, J. and Helcl, J. Endtoend nonautoregressive neural machine translation with connectionist temporal classification. In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, 2018.
 Luo et al. (2018) Luo, Y., Cai, X., Zhang, Y., Xu, J., et al. Multivariate time series imputation with generative adversarial networks. In Advances in Neural Information Processing Systems, pp. 1603–1614, 2018.
 Mallat (1989) Mallat, S. G. A theory for multiresolution signal decomposition: the wavelet representation. IEEE transactions on pattern analysis and machine intelligence, 11(7):674–693, 1989.

Nelwamondo et al. (2007)
Nelwamondo, F. V., Mohamed, S., and Marwala, T.
Missing data: A comparison of neural network and expectation maximization techniques.
Current Science, pp. 1514–1521, 2007.  Oord et al. (2018) Oord, A. v. d., Li, Y., Babuschkin, I., Simonyan, K., Vinyals, O., Kavukcuoglu, K., Driessche, G. v. d., Lockhart, E., Cobo, L. C., Stimberg, F., et al. Parallel wavenet: Fast highfidelity speech synthesis. In International Conference on Machine Learning, 2018.
 Reed et al. (2017) Reed, S., Oord, A. v. d., Kalchbrenner, N., Colmenarejo, S. G., Wang, Z., Belov, D., and de Freitas, N. Parallel multiscale autoregressive density estimation. In International Conference on Machine Learning, 2017.

Rhinehart & Kris (2018)
Rhinehart, N. and Kris, M.
R2p2: A reparameterized pushforward policy for diverse, precise
generative path forecasting.
In
Proceedings of the European Conference on Computer Vision (ECCV)
, pp. 772–788, 2018.  Rubin (2004) Rubin, D. B. Multiple imputation for nonresponse in surveys, volume 81. John Wiley & Sons, 2004.
 Syed & Schapire (2008) Syed, U. and Schapire, R. E. A gametheoretic approach to apprenticeship learning. In Advances in neural information processing systems, pp. 1449–1456, 2008.

Urtasun et al. (2006)
Urtasun, R., Fleet, D. J., and Fua, P.
3d people tracking with gaussian process dynamical models.
In
Computer Vision and Pattern Recognition, 2006 IEEE Computer Society Conference on
, volume 1, pp. 238–245. IEEE, 2006.  Yoon et al. (2018a) Yoon, J., Jordon, J., and van der Schaar, M. Gain: Missing data imputation using generative adversarial nets. In International Conference on Machine Learning, 2018a.
 Yoon et al. (2018b) Yoon, J., Zame, W. R., and van der Schaar, M. Estimating missing data in temporal data streams using multidirectional recurrent neural networks. IEEE Transactions on Biomedical Engineering, 2018b.
 Yu et al. (2017) Yu, L., Zhang, W., Wang, J., and Yu, Y. Seqgan: Sequence generative adversarial nets with policy gradient. In AAAI, pp. 2852–2858, 2017.
 Zhan et al. (2019) Zhan, E., Zheng, S., Yue, Y., and Lucey, P. Generating multiagent trajectories using programmatic weak supervision. In International Conference on Learning Representations (ICLR), 2019.
 Zheng et al. (2016) Zheng, S., Yue, Y., and Hobbs, J. Generating longterm trajectories using deep hierarchical networks. In Advances in Neural Information Processing Systems, pp. 1543–1551, 2016.
 Ziebart et al. (2008) Ziebart, B. D., Maas, A. L., Bagnell, J. A., and Dey, A. K. Maximum entropy inverse reinforcement learning. In AAAI, volume 8, pp. 1433–1438. Chicago, IL, USA, 2008.
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