We study the problem of learning high-dimensional tensor models from large-scale high-resolution spatial data. Such models can compactly describe multi-way correlations between predictive features that are spatially distributed. For example, in competitive basketball play, given the positions of all players in the court, we can predict whether a player will shoot at the basket using a high-dimensional tensor model. Previous research on individual player models has demonstrated the effectiveness of matrix latent factor models (Yue et al., 2014). For joint team behavior, we can model a basketball player’s profile by learning the latent factors among offensive team positions, defending team positions and the player’s own profile, which can be modeled by a tensor latent factor model. This paper addresses two major challenges for learning spatial tensor models. First, we study scalability, as high resolution spatial tracking data is large-scale. Computational methods usually involve fine-grained spatial discretization, leading to high-dimensional tensors that are computationally slow and expensive to train. In addition to scalability, we are also interested in learning interpretable models. Apart from accurate prediction, we also wish to learn latent factors that can provide insights for domain experts. For instance, the latent factors in basketball player models could correspond to player performance profiles, revealing hidden traits and commonalities among players. Towards scalable learning of interpretable tensor latent factor models, a key technical challenge is their non-convex nature (Yu and Liu, 2016): optimization algorithms can converge to different local optima, depending on the model initialization. Hence, a good initialization is critical for obtaining accurate and interpretable latent factors. For instance, Figure 1 shows two basketball shooting profiles with similar prediction accuracy, learned using the same optimization algorithm (gradient descent). Using a good initialization yields an interpretable solution, while a random initialization does not.
Thus we are interested in answering how we can efficiently learn both accurate and interpretable tensor models from high-resolution spatial data. In this paper, we present a novel meta-learning algorithm: multi-resolution tensor learning (MRTL), that can 1) learn a good initialization, 2) automatically control when to fine-grain, and 3) easily scale to high-dimensional tensors. More importantly, the multi-resolution learning scheme yields interpretable solutions that are spatially smooth and relevant for the prediction tasks. MRTL is based on three key insights. First, to obtain good initializations, instead of directly learning the latent factors from a random initialization, we can first learn a full-rank parameter tensor and use the factors from this parameter tensor as initialization. An analogous approach for matrix latent factor models has been shown in (Miller et al., 2014; Yue et al., 2014)
to yield interpretable factors capturing cohesive spatial semantics. Second, to avoid the curse of dimensionality, we leverage the characteristics of spatial data and learn iteratively atmultiple resolutions
. At a high-level, the multi-resolution training has two stages: in the first stage, we learn a full tensor model using multi-resolution and factorize it as initialization for the latent factor model. In the second stage, we train the latent factor model using multi-resolutions: starting at a coarse resolution and fine-graining the latent factors during training. We prove that the resulting algorithm is faster by a logarithmic factor of the Lipschitz constant of the objective function and the estimation error. Third, we investigate fine-graining criteria. One simple criterion is to measure training loss convergence. However, it requires a preset threshold, which does not easily reflect the spatial distribution at different granularities. We thus take an information theoretical approach by monitoring the entropy of the gradients distribution for every grid cell, which we refer to asspatial entropy. We found this fine-graining criterion to be more effective than loss convergence. In summary, our main contributions are:
We present MRTL: a multi-resolution meta-algorithm to learn spatial latent factor models and a number of instantiations using various fine-graining criteria.
We theoretically analyze MRTL and show that it converges faster to a similar accuracy than training at a fixed resolution.
We empirically show that using gradient statistics, (e.g. gradient entropy), is an effective transition control method across hyperparameters and can outperform loss convergence.
We empirically demonstrate orders-of-magnitude faster learning than conventional fixed-resolution training.
We show that our approach reliably and efficiently yields interpretable spatial latent factor models on real-world basketball and animal tracking data.
2. Related Work
Modeling spatial data enjoys a long history in the data mining community (Miller and Han, 2009). Recent applications include urban air quality prediction (Hsieh et al., 2015), social media recommendation (Yin and Cui, [n. d.]), and event detection (Zhao et al., 2016), among others. In this work, we study multi-agent tracking data extracted from high-fidelity tracking cameras. Such spatial data contains high resolution information and are large-scale (cameras operate at high frequency and every frame is one data point). Latent factor models have been a popular method for spatial data modeling (Reich and Bandyopadhyay, 2010; Yue et al., 2014; Deng et al., 2016). By projecting the raw data into a low-dimensional space, latent factor models can encode patterns such as spatial clustering. However, most existing latent factor models have only focused on capturing two-way interactions (such as user-location matrix models (Koren et al., 2009)). For multi-agent behavior modeling, we extend the matrix latent factor model to capture multi-way correlations among agents, resulting in a tensor latent factor model. For instance, we can build a “player offense position defense position” tensor model for basketball plays. Tensor models have recently gained considerable attention (cf. (Jenatton et al., 2012; Takeuchi et al., 2013; Quattoni and Carreras, 2014)), but are still of limited applicability due to computational, data sparsity, and memory efficiency concerns. There are many efforts to scale up tensor computation, such as random projection (sketching) (Wang et al., 2015), parallel computing (Austin et al., 2016), or utilizing sparsity structure (Perros et al., 2017). In this work, we take a top-down approach to learn a tensor model at multiple resolutions. The multi-resolution training procedure takes advantage of the spatial characteristics of the raw data, and is generically applicable to different tensor models and optimization algorithms. Our approach bears affinity to other multi-stage meta-algorithms (e.g., (Johnson and Guestrin, 2015)) and the multi-grid method in PDE analysis. For instance, (Chow and Tsitsiklis, 1991) studies multi-grid methods in the context of dynamic programming. In the machine learning community, our method is also related to multi-resolution sparse approximation (Mallat, 1989). For example, (Kondor et al., 2014) studies multi-resolution matrix factorization based on wavelets theory. (Schifanella et al., 2014)
proposes a multi-resolution heuristic for tensor factorization. In contrast, we study multi-resolution learning fortensor models and also provide theoretical analysis for its convergence.
3. Tensor Latent Factor Models
We motivate the tensor latent factor model using the example of competitive basketball play. Suppose we have players and want to predict whether a basketball player () takes a shot at the basket (binary label ). We first discretize the court using a grid with cells (positions
) and then construct a binary feature vectorusing the positions of all other players on the court. The prediction problem can be formulated as:
where are the weight parameters for the grid cells and is the bias unit. The weights encodes how likely a basketball player will shoot from a given position on the basketball court.
Matrix Latent Factor Models.
In matrix form, (1) is equivalent to the following model:
where is the prediction function for player , and the matrix concatenates the parameter vectors . Latent factor models assume there exist low-dimensional representations of the weights. Hence the parameter matrix factorizes into two matrices and with components:
These latent factors can represent players’ behavioral profiles at different positions on the court.
Tensor Latent Factor Models.
The limitation of the matrix latent factor model in (2) is that it can only capture the correlation between the ballhandler and all other players. In order to learn rich behaviors between teams, we need to explicitly model multi-way dependencies and generalize to high-order models. In the basketball example, we separately construct a feature vector for the offense and defense team positions, respectively, and model the multi-way interactions as:
Here is an order 3 weight tensor. To encode low-dimensional structure, we assume the weight tensor factorizes:
which corresponds to a CP tensor model (Kolda and Bader, 2009). In this case, is called the tensor rank. In general, given some input and feature transformation functions and , tensor latent factor models aim to learn a function for each prediction task , such that:
depicts the model details. The dimension of the weight tensor relates to the discretization size of the spatial data. The main motivation for considering discretizations is that they allow us to learn flexible non-parametric models. In contrast, traditional parametric models such as spatial point processes(Miller et al., 2014) have strong assumptions on the form of the spatial correlations (e.g. the generating process is a distribution with multiple modes or has a cyclic factorization structure), which can be hard to learn using conventional learning methods. Tensor models explicitly capture higher-order correlations among the tasks and spatial features. The learned latent factors can also be more interpretable due to the multi-linear nature (in the spatial feature dimensions). For example, if we see the prediction function as a distribution and are 1-hot vectors that encode spatial occupancy, the columns and (for a fixed
) can be interpreted as smooth spatial probability distributions, representing players’ shooting profile. The cost of using such models, however, is that they lead to a hard non-convex optimization problem and can suffer from multiple local optima. Moreover, different initializations can lead to more or less interpretable models. They are also computationally expensive to train, making them infeasible for large-scale high resolution spatial data. Hence, the goal of our multi-resolution method is to efficiently find accurate and interpretable solutions in this setting.
4. Multi-resolution Learning
We now describe our approach for training interpretable spatial latent factor models, such as the tensor model in (4). In this section, for simplicity we consider only fine-graining a single dimension of , although our analysis can be easily generalized.
Initialization from Factorization.
The non-convex nature of the tensor model requires a good initialization. We obtain this by factorizing a trained full-rank tensor model as in (6) and using the factors as initializations. This approach has been shown to be effective in learning predictive spatial patterns (Miller et al., 2014; Yue et al., 2014). We also have similar observations in our experiments.
It is generally not tractable in memory to learn a high-dimensional tensor latent factor model, that operates at some high resolution. To make training feasible, the main idea of our algorithm is iterative fine-graining. We will denote the resolution (i.e. size of a grid cell) as , which is inversely proportional to the number of grid cells . During the first (full-rank) phase, we start training with a full-rank model that operates at a coarse resolution and increase the spatial resolution of the data and model using a sequence of resolutions . That is, instead of learning , we learn a sequence of . When moving to a higher resolution, we use the learned weights to initialize the next weights , by scaling up along the spatial dimension. We leverage the fact that spatial data can be viewed at multiple resolutions. Lower resolution models can provide good initializations for higher resolution models with high accuracy. To determine when to transition, we study different fine-graining criterion, which we discuss in section 4.2. In the second phase, we apply a tensor factorization to at resolution as initialization. We use a similar fine-graining procedure for the latent factor model to the final resolution . The resolution is determined by considering the memory requirements of the model at different resolutions. This meta-algorithm is outlined in Algorithm 1 and depicted in Figure 3.
4.1. Computational Complexity Analysis
We analyze the computational complexity for MRTL (Algorithm 1). Intuitively, as most of the training iterations are spent on coarser resolutions with fewer number of parameters, multi-resolution training is more efficient than fixed-resolution training. In this section, we formalize this intuition and give a rigorous analysis of MRTL. The analysis is based on the multi-grid method (Stüben, 2001)
commonly used in partial differential equation analysis. We denote the label, features , and weight tensor . The prediction model is:
Denote the objective function as , MRTL (Algorithm 1) aims to solve the optimization problem:
where is the logistic loss. MRTL follows gradient descent:
We first start with the coarsest resolution (discretization size), compute an initial estimate , and keep iterating until satisfies certain fine-graining criteria. Then we replace with and use as an initialization for the next level. In general, this iterative fine-graining procedure yields a sequence of discretization sizes . Suppose for each resolution , we use the following as the fine-graining criterion:
where is the number of iterations needed at level . The algorithm terminates when the estimation error reaches . We now show that MRTL reduces the number of computation steps. This is done by first formulating a gradient descent update as a fixed point iteration operator 111Stochastic gradient descent converges to a noise ball instead of a fixed point.:
Assume the objective function is Lipschitz continuous, with a contraction constant of , meaning:
The following main theorem characterizes the speed-up gained by multi-resolution training w.r.t. the contraction factor and the terminal estimation error .
Theorem 4.1 ().
Suppose the fixed point iteration operator (gradient descent) for the optimization algorithm has a contraction factor (Lipschitz constant) of , the multi-resolution training procedure is faster than that of the fixed resolution algorithm by a factor of , with as the terminal estimation error.
We prove several useful Lemmas before proving the main Theorem 4.1. We first analyze the computational cost of the fixed-resolution algorithm.
Lemma 4.2 ().
Given a fixed point iteration operator with a contraction factor of , the computational complexity of a fixed-resolution training for a -order tensor with rank is
At a high level, we prove this by choosing a small enough resolution such that the approximation error is bounded with a fixed number of iterations. Let be the optimal estimate at level and be the estimate at step . Then we have
Choose a fixed resolution that is small enough such that
then the termination criteria gives . Initialize and iterate over times such that:
Since , , we obtain that
Note that for an order tensor with rank , the computational complexity of every iteration in MRTL is with as the discretization size. Hence, the computational complexity of the fixed resolution training is
In the multi-resolution setting, the spatial weights that are learned can be seen as a distribution that is approximated by its values at finitely many points. Given a spatial discretization , we can construct an operator that learns discretized tensor weights. The next Lemma relates the estimation error with resolution:
Lemma 4.3 ().
(Nash, 2000) For each resolution level , there exists a constant and , such that the fixed point iteration with the discretization size has an estimation error:
See (Nash, 2000) for the details. ∎We have obtained the discretization error for the fixed point operation at any resolution. Next we analyze the number of iterations needed at each resolution before fine-graining.
Lemma 4.4 ().
For every resolution , there exists a constant , such that the number of iterations before fine-graining satisfies:
We know that . At resolution , by using the estimate from the last level as initialization, we have:
According to the fixed point iteration definition, we have:
Thus the number of iterations satisfies for all resolutions. ∎.
Proof of Theorem 4.1.
By combining Lemmas 4.4 and the computational cost per iteration, we can compute the total computational cost for our MRTL algorithm, which is proportional to the total number of iterations for all resolutions:
4.2. Fine-Graining Criteria
Given the structure of MRTL (Algorithm 1), the key technical question is which fine-graining criterion to use at each stage. In Theorem 4.1, we showed the improved convergence speed when using loss-convergence as a transition criterion. We now propose several fine-graining criteria to instantiate MRTL, that empirically outperform loss-convergence.
Intuitively, one simple fine-graining condition is when training at the present resolution converges (which mimics the termination condition for iterative training of sparse models (Johnson and Guestrin, 2015)). That is, at training step , we check whether:
where is the historical mean loss and is a threshold. However, since the model at each resolution is used to initialize the training for the next resolution, the converged model at the current resolution might not be the best initialization for the next resolution (i.e., training might be overfitting to the coarser resolution).
Using Gradient Statistics.
Given the spatial nature of the model weights (6), we can use more finegrained information during training by analyzing the gradient distribution during training. Intuitively, spatial resolution is too coarse when the data prefers much more fine-grained curvature, as evidenced by substantial disagreement in the gradients between resolution and
. For example, consider learning a linear binary classifieras in (1):
where and we consider two resolutions: cells and cells. Suppose we optimize using
where is the learning-rate, is a (nonlinear) optimizer with parameters and the true gradient is
Consider the situation that 1) during training at resolution the model has converged to and 2) the ground truth model at resolution is . In this case, at the coarse resolution the distribution of the point-wise gradient has mean
(and some variance). However, if we fine-grain the model, the (distribution of) gradients for the higher-resolutions weights and have non-zero mean: the mean gradient will be negative and positive. This is illustrated in Figure 4. More generally, we can quantify disagreement between gradients at multiple resolutions via the statistics and entropy of the (empirical) gradient distribution of all weights at resolutions :
Intuitively, when the entropy is high, the gradients of the current discretization are increasingly disagreeing with each other and training can benefit from finegraining. More formally, we can express this via the information gain:
If , the gradient will have lower entropy at the higher resolution and finegraining could be beneficial.222 This is analogous to the use of information gain for decision tree regularization.
This is analogous to the use of information gain for decision tree regularization.However, in practice it is infeasible to compute information gain, as this requires gradient statistics at a higher resolution. As a proxy to (27) we can instead use a simpler condition: finegrain when for of the weights in the entropy exceeds a margin :
where and are tunable hyperparameters.
We can define other criteria based on gradient statistics: finegrain if for at least of the weights:
where , are the gradient mean and variance at step and s are tunable hyperparameters. -thresholding is a more coarse statistic compared to the entropy : if the gradient distribution is non-Gaussian or multimodal can capture higher-order statistics as well. -thresholding also tracks whether training has converged () according to the gradients.
|Gradient buffer window||10 - 1000|
|Criterion check frequency|
5. Benchmark Experiments
To validate our approach, we demonstrate that our multi-resolution method converges faster than fixed-resolution training on two real-world datasets: basketball shots and fruit-fly behavior prediction. Furthermore, we show that using spatial gradient statistics, such as its entropy, outperforms using loss convergence and provide a sensitivity analysis of the various transition criteria. Finally, in Section 6, we show that our learned models are interpretable. For both datasets, we learn a tensor model for multi-task prediction, where we instantiate the model (6) as a sum of low-rank and sparse tensors:
where indexes the tasks. A key motivation for this decomposition is that can capture semantically meaningful profiles (spatially dense and smooth, or sparse and peaky).
Empirical Gradient Distribution.
During training, we estimate the gradient distribution by recording the empirical minibatch gradients and their statistics over a rolling window of training steps. While collecting gradients, the weights are typically updated too, introducing a bias in the gradient statistics, as gradients are computed for different models at each step. However, we found empirically that using (28) remains effective.
First, we compared MRTL with fixed-resolution training. Secondly, we compared the various fine-graining criteria (entropy divergence, loss convergence, -threshold, -threshold) as described in section 4.2. We instantiated MRTL by adaptively upscaling the model and features and as in Algorithm 1, checking transition criteria each steps for and various . All models were trained using Adam using cross-entropy loss and () regularization for the dense (sparse) factors. We selected optimal hyperparameters for all models using a grid search over hyperparameters (see Table 1). For the sensitivity analysis, we used the s as in Table 1.
Basketball tracking data.
We evaluated our method on basketball shot prediction, where the tasks correspond to unique basketball players, indexed by , and our multi-task model predicts whether player will shoot at the basket immediately after frame ( indexes the dataset). We used a large player tracking dataset BBShot (Yue et al., 2014; Zheng et al., 2016) that includes hundreds of players and covers millions of game frames captured during competitive basketball gameplay. For each frame , we have binary labels : whether player will (not) shoot at the basket in frame . The features are 1-hot vectors for the location of the ballhandler and for the defenders in a grid around the ballhandler. At full resolution, the court is discretized using a grid of 2000 cells of size . For lower resolutions, we used and cells.
The Caltech Fly-vs-Fly behavior dataset Fly-vs-Fly (Eyjolfsdottir et al., 2014) consists of several million video-frames of 2 fruit-flies, labeled by neuro-biologists with 10 behavioral fly actions: touch, wing threat, charge, lunge, hold, tussle, wing extension, circle, copulation attempt, copulation. The goal is to predict whether either fly performs any of these 10 actions in video frame . Here, the tasks are the actions indexed by and the binary labels correspond to whether an action is present in frame or not. The dataset includes 12 pose and spatial features, including the velocity and the wing angles of a single fly, and pairwise features, such as the distance and angle between the two flies (see Figure 5).
|Criterion||Minimal time (s)||Mean||Minimal||Mean|
5.2. Accuracy and Convergence Speed
We compare MRTL with fixed-resolution learning. Moreover, we compare a number of fine-graining criteria: fine-graining when the loss has converged versus spatial entropy control.
For the latter, we used condition (28) to determine when to fine-grain.
Figure 6 and Table (2) show our results using models with total latent dimension 20.333Performance trend is stable for a wide range of latent dimensions, omitted for brevity.
The left plot shows the results for the full-rank model, which is usually the more computationally intensive part.
We see that our multi-stage approach dramatically outperforms (by multiple orders-of-magnitude) a naive approach which only uses the finest resolution.444Note that for more complex models, the naive approach would not even fit in memory.
Moreover, spatial entropy control outperforms, by an order-of-magnitude, using the loss as a termination criterion.
The right plot shows the performance of MRTL on the latent factor model after initializing using a factorized model of the previous stage. We see that the learning objective continues to decrease as learning enters what is essentially a fine-tuning phase.555 Note the absolute objective of the un-factorized model is lower than the latent-factor model because the former has more degrees of freedom and is overfitting.
Note the absolute objective of the un-factorized model is lower than the latent-factor model because the former has more degrees of freedom and is overfitting.
5.3. Sensitivity Analysis
To evaluate the efficiency and transition behavior using the transition criteria from Section 4.2, we evaluated their sensitivity to the threshold parameters . For this, we used BBShot-200k with 200k BBShot examples and evaluated Algorithm 1 using a random search over . The results are in Table 2 and Figure 7. We see that using gradient statistics ((28), (29)) consistently outperforms loss convergence in the short-term (-threshold) and long-term (entropy-threshold). We empirically find that -threshold is harder to stabilize than other criteria, although it can outperform other methods.
6. Analysis of Interpretable Solutions
The latent factors are interpretable: they capture characteristic basketball shooting profiles and spatial fly behavior.
Smooth and sparse latent factors correspond to various types of basketball behavior and spatial configurations of fruit flies.
6.1. Basketball shot prediction
Ball handler shooting profiles.
In competitive basketball play, shots at the basket happen throughout the court and peak at certain hot-spot locations at short range (close to the basket), medium range (between the basket and 3-point line) and long range (at the 3-point line or beyond). Inspecting the smooth shooting profiles in Figure 8, we see that all profiles are spatially cohesive and have small peaks around different subsets of hot-spots. Profiles 1, 2 and 3 capture medium range and long range hot-spots, while profile 4 covers the short-range zone. In contrast, the sparse shooting profiles are more spatially concentrated and activate only on specific hot-spots. For example, profile 5 covers 2 hot-spots to the far left and right side at the back of the court and profiles 7 and 8 only activate on the short range zone directly around the basket. This leads to an attractive semantic interpretation: smooth shooting profiles describe inconsistent players that tend to shoot from many locations throughout the basketball court, whereas sparse shooting profiles capture consistent players that shoot from only specific hot-spots. Figure 9 show the latent factor activations and of 6 players that can be grouped as such: players 1, 2, and 3 are consistent; players 4, 5, and 6 are more inconsistent.
Defender influence profiles.
The bottom row of Figure 8 depicts four dense defender profiles and four sparse defender profiles . The first two dense profiles capture defender suppression in front of the ballhandler across the entire court, since the companion shooting profiles (1 and 2) are diffused throughout. In contrast, the first two sparse defender profiles (5 and 6) describe defender influence specifically at long range, which has a more peaked behavior. The last two sparse profiles (7 and 8) describe ballhandlers that are prone to shoot with a defender close by. This is likely due to confusing correlation with causation, since players close to the basket tend to shoot, even though typically a defender is close by.
6.2. Fruit fly behavior
Figure 11 depicts a sample dense and a sparse learned profile that are interpretable. In the dense profile, the fly extends at least one wing which defines wing extension, a sign of courtship. The sparse profile similarly captures more concentrated spatial characteristics, when the flies are close. Such wing configurations are important in both aggressive and courtship behavior, when the two flies interact closely to each other.
Actions and behaviorial types.
Figure 10 provides a more holistic view across all actions in the Fly-vs-Fly dataset. For example, we see that lunge and wing threat have similar activations. This is natural: the former is active physical aggression, while the latter can be interpreted as a fly showing a preview of physical aggression to intimidate the other fly. We observe that both dense and sparse profiles are useful for modeling a wide range of actions/tasks.
Our adaptive multi-resolution approach learns models at a fixed global resolution at each stage. An open problem remains how to learn at different resolutions locally, i.e. by finegraining individual cells rather than all. Such methods could be used to learn the dynamics in systems that intrinsically behave differently at different resolution, such as many dynamical systems. Local gradient statistics could be used in this case to decide when and which cells to finegrain to optimize model performance.
- Austin et al. (2016) Woody Austin, Grey Ballard, and Tamara G Kolda. 2016. Parallel tensor compression for large-scale scientific data. In Parallel and Distributed Processing Symposium, 2016 IEEE International. IEEE, 912–922.
- Chow and Tsitsiklis (1991) C-S Chow and John N Tsitsiklis. 1991. An optimal one-way multigrid algorithm for discrete-time stochastic control. IEEE transactions on automatic control 36, 8 (1991), 898–914.
- Deng et al. (2016) Dingxiong Deng, Cyrus Shahabi, Ugur Demiryurek, Linhong Zhu, Rose Yu, and Yan Liu. 2016. Latent space model for road networks to predict time-varying traffic. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM, 1525–1534.
- Eyjolfsdottir et al. (2014) Eyrun Eyjolfsdottir, Steve Branson, Burgos-A, Xavier P Rtizzu, Eric D Hoopfer, Jonathan Schor, David J Anderson, and Pietro Perona. 2014. Detecting Social Actions of Fruit Flies. Computer Vision–ECCV 2014 (2014), 772–787.
- Hsieh et al. (2015) Hsun-Ping Hsieh, Shou-De Lin, and Yu Zheng. 2015. Inferring air quality for station location recommendation based on urban big data. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM, 437–446.
- Jenatton et al. (2012) Rodolphe Jenatton, Antoine Bordes, Nicolas Le Roux, and Guillaume Obozinski. 2012. A Latent Factor Model for Highly Multi-relational Data. Advances in Neural Information Processing Systems 25 (2012), 3167–3175.
- Johnson and Guestrin (2015) Tyler B Johnson and Carlos Guestrin. 2015. BLITZ: A Principled Meta-Algorithm for Scaling Sparse Optimization. In International Conference on Machine Learning (ICML).
- Kolda and Bader (2009) Tamara G. Kolda and Brett W. Bader. 2009. Tensor Decompositions and Applications. SIAM Rev. 51, 3 (2009), 455–500. https://doi.org/10.1137/07070111X
- Kondor et al. (2014) Risi Kondor, Nedelina Teneva, and Vikas Garg. 2014. Multiresolution matrix factorization. In Proceedings of the 31st International Conference on Machine Learning (ICML-14). 1620–1628.
- Koren et al. (2009) Yehuda Koren, Robert Bell, and Chris Volinsky. 2009. Matrix factorization techniques for recommender systems. Computer 8 (2009), 30–37.
- Mallat (1989) Stephane G Mallat. 1989. Multiresolution approximations and wavelet orthonormal bases of (). Transactions of the American mathematical society 315, 1 (1989), 69–87.
- Miller et al. (2014) Andrew Miller, Luke Bornn, Ryan Adams, and Kirk Goldsberry. 2014. Factorized Point Process Intensities: A Spatial Analysis of Professional Basketball. In International Conference on Machine Learning (ICML).
- Miller and Han (2009) Harvey J Miller and Jiawei Han. 2009. Geographic data mining and knowledge discovery. CRC Press.
- Nash (2000) Stephen G Nash. 2000. A multigrid approach to discretized optimization problems. Optimization Methods and Software 14, 1-2 (2000), 99–116.
- Perros et al. (2017) Ioakeim Perros, Evangelos E Papalexakis, Fei Wang, Richard Vuduc, Elizabeth Searles, Michael Thompson, and Jimeng Sun. 2017. SPARTan: Scalable PARAFAC2 for Large & Sparse Data. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM, 375–384.
- Quattoni and Carreras (2014) Ariadna Quattoni and Xavier Carreras. 2014. Spectral Regularization for Max-Margin Sequence Tagging. Proceedings of The 31st International Conference on Machine Learning 32 (2014).
- Reich and Bandyopadhyay (2010) Brian J Reich and Dipankar Bandyopadhyay. 2010. A latent factor model for spatial data with informative missingness. The annals of applied statistics 4, 1 (2010), 439.
- Schifanella et al. (2014) Claudio Schifanella, K Selçuk Candan, and Maria Luisa Sapino. 2014. Multiresolution tensor decompositions with mode hierarchies. ACM Transactions on Knowledge Discovery from Data (TKDD) 8, 2 (2014), 10.
- Stüben (2001) Klaus Stüben. 2001. A review of algebraic multigrid. In Partial Differential Equations. Elsevier, 281–309.
- Takeuchi et al. (2013) Koh Takeuchi, Ryota Tomioka, Katsuhiko Ishiguro, Akisato Kimura, and Hiroshi Sawada. 2013. Non-negative multiple tensor factorization. In IEEE International Conference on Data Mining (ICDM).
- Wang et al. (2015) Yining Wang, Hsiao-Yu Tung, Alexander J Smola, and Anima Anandkumar. 2015. Fast and guaranteed tensor decomposition via sketching. In Advances in Neural Information Processing Systems. 991–999.
- Yin and Cui ([n. d.]) Hongzhi Yin and Bin Cui. [n. d.]. Spatio-temporal recommendation in social media. Springer.
- Yu and Liu (2016) Rose Yu and Yan Liu. 2016. Learning from multiway data: Simple and efficient tensor regression. In International Conference on Machine Learning. 373–381.
- Yue et al. (2014) Yisong Yue, Patrick Lucey, Peter Carr, Alina Bialkowski, and Iain Matthews. 2014. Learning Fine-Grained Spatial Models for Dynamic Sports Play Prediction. In IEEE International Conference on Data Mining (ICDM).
- Zhao et al. (2016) Liang Zhao, Jieping Ye, Feng Chen, Chang-Tien Lu, and Naren Ramakrishnan. 2016. Hierarchical incomplete multi-source feature learning for spatiotemporal event forecasting. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM, 2085–2094.
- Zheng et al. (2016) Stephan Zheng, Yisong Yue, and Jennifer Hobbs. 2016. Generating long-term trajectories using deep hierarchical networks. In Advances in Neural Information Processing Systems. 1543–1551.