Long-term Forecasting using
We present Tensor-Train RNN (TT-RNN), a novel family of neural sequence architectures for multivariate forecasting in environments with nonlinear dynamics. Long-term forecasting in such systems is highly challenging, since there exist long-term temporal dependencies, higher-order correlations and sensitivity to error propagation. Our proposed tensor recurrent architecture addresses these issues by learning the nonlinear dynamics directly using higher order moments and high-order state transition functions. Furthermore, we decompose the higher-order structure using the tensor-train (TT) decomposition to reduce the number of parameters while preserving the model performance. We theoretically establish the approximation properties of Tensor-Train RNNs for general sequence inputs, and such guarantees are not available for usual RNNs. We also demonstrate significant long-term prediction improvements over general RNN and LSTM architectures on a range of simulated environments with nonlinear dynamics, as well on real-world climate and traffic data.
|Rose Yu 111Equal contribution. Stephan Zheng111Equal contribution. Anima Anandkumar Yisong Yue|
|Department of Computation and Mathematical Sciences, Caltech|
One of the central questions in science is forecasting: given the past history, how well can we predict the future? In many domains with complex multivariate correlation structures and nonlinear dynamics, forecasting is highly challenging since the system has long-term temporal dependencies and higher-order dynamics. Examples of such systems abound in science and engineering, from biological neural network activity, fluid turbulence, to climate and traffic systems (see Figure 1). Since current forecasting systems are unable to faithfully represent the higher-order dynamics, they have limited ability for accurate long-term forecasting.
Therefore, a key challenge is accurately modeling nonlinear dynamics and obtaining stable long-term predictions, given a dataset of realizations of the dynamics. Here, the forecasting problem can be stated as follows: how can we efficiently learn a model that, given only few initial states, can reliably predict a sequence of future states over a long horizon of time-steps?
Common approaches to forecasting involve linear time series models such as auto-regressive moving average (ARMA), state space models such as hidden Markov model (HMM), and deep neural networks. We refer readers to a survey on time series forecasting by (Box et al., 2015) and the references therein. A recurrent neural network (RNN), as well as its memory-based extensions such as the LSTM, is a class of models that have achieved good performance on sequence prediction tasks from demand forecasting (Flunkert et al., 2017) to speech recognition (Soltau et al., 2016) and video analysis (LeCun et al., 2015). Although these methods can be effective for short-term, smooth dynamics, neither analytic nor data-driven learning methods tend to generalize well to capturing long-term nonlinear dynamics and predicting them over longer time horizons.
To address this issue, we propose a novel family of tensor-train recurrent neural networks that can learn stable long-term forecasting. These models have two key features: they 1) explicitly model the higher-order dynamics, by using a longer history of previous hidden states and high-order state interactions with multiplicative memory units; and 2) they are scalable by using tensor trains, a structured low-rank tensor decomposition that greatly reduces the number of model parameters, while mostly preserving the correlation structure of the full-rank model.
In this work, we analyze Tensor-Train RNNs theoretically, and also experimentally validate them over a wide range of forecasting domains. Our contributions can be summarized as follows:
We describe how TT-RNNs encode higher-order non-Markovian dynamics and high-order state interactions. To address the memory issue, we propose a tensor-train (TT) decomposition that makes learning tractable and fast.
We provide theoretical guarantees for the representation power of TT-RNNs for nonlinear dynamics, and obtain the connection between the target dynamics and TT-RNN approximation. In contrast, no such theoretical results are known for standard recurrent networks.
We validate TT-RNNs on simulated data and two real-world environments with nonlinear dynamics (climate and traffic). Here, we show that TT-RNNs can forecast more accurately for significantly longer time horizons compared to standard RNNs and LSTMs.
2 Forecasting using Tensor-Train RNNs
Forecasting Nonlinear Dynamics
Our goal is to learn an efficient model for sequential multivariate forecasting in environments with nonlinear dynamics. Such systems are governed by dynamics that describe how a system state evolves using a set of nonlinear differential equations:
where can be an arbitrary (smooth) function of the state and its derivatives. Continuous time dynamics are usually described by differential equations while difference equations are employed for discrete time. In continuous time, a classic example is the first-order Lorenz attractor, whose realizations showcase the “butterfly-effect”, a characteristic set of double-spiral orbits. In discrete-time, a non-trivial example is the 1-dimensional Genz dynamics, whose difference equation is:
where denotes the system state at time and are the parameters. Due to the nonlinear nature of the dynamics, such systems exhibit higher-order correlations, long-term dependencies and sensitivity to error propagation, and thus form a challenging setting for learning. Given a sequence of initial states , the forecasting problem aims to learn a model
that outputs a sequence of future states . Hence, accurately approximating the dynamics is critical to learning a good forecasting model and accurately predicting for long time horizons.
First-order Markov Models
In deep learning, common approaches for modeling dynamics usually employ first-order hidden-state models, such as recurrent neural networks (RNNs). An RNN with a single RNN cell recursively computes the output from a hidden state using:
where is the state transition function, is the output function and are model parameters. An RNN therefore learns a model for the Markov process , of order 1 (only the previous time-step is considered). A common choice is to model (4) as a nonlinear activation function applied to a linear map of and as:
where is the activation function (e.g. sigmoid, ) for the state transition, and are transition weight matrices and are biases. RNNs have many different variations, including LSTMs (Hochreiter & Schmidhuber, 1997) and GRUs (Chung et al., 2014). For instance, LSTM cells use a memory-state, which mitigate the “exploding gradient” problem and allow RNNs to propagate information over longer time horizons. Although RNNs are very expressive, they compute only using the previous state and input . Such models do not explicitly model higher-order dynamics and only implicitly model long-term dependencies between all historical states , which limits their forecasting effectiveness in environments with nonlinear dynamics.
2.1 Tensorized Recurrent Neural Networks
To effectively learn nonlinear dynamics, we propose Tensor-Train RNNs, or TT-RNNs, a class of higher-order models that can be viewed as a higher-order generalization of RNNs. We developed TT-RNNs with two goals in mind: explicitly modeling 1) -order Markov processes with steps of temporal memory and 2) polynomial interactions between the hidden states and .
First, we consider longer “history”: we keep length historic states: :
where is an activation function. In principle, early work (Giles et al., 1989) has shown that with a large enough hidden state size, such recurrent structures are capable of approximating any dynamics.
Second, to learn the nonlinear dynamics efficiently, we also use higher-order moments to approximate the state transition function. We construct a higher-order transition tensor by modeling a degree polynomial interaction between hidden states. Hence, the TT-RNN with standard RNN cell is defined by:
where is a -dimensional tensor, the index the hidden states and is the polynomial degree. Here, we defined the -lag hidden state as:
We included the bias unit 1 to model all possible polynomial expansions up to order in a compact form. The TT-RNN with LSTM cell, or “TLSTM”, is defined analogously as:
where denotes the Hadamard product. Note that the bias units are again included. TT-RNN serves as a module for sequence-to-sequence (Seq2Seq) framework (Sutskever et al., 2014), which consists of an encoder-decoder pair (see Figure 3). We use tensor-train recurrent cells both the encoder and decoder. The encoder receives the initial states and the decoder predicts . For each timestep , the decoder uses its previous prediction as an input.
2.2 Tensor-train Networks
Unfortunately, due to the “curse of dimensionality”, the number of parameters in with hidden size grows exponentially as , which makes the high-order model prohibitively large to train. To overcome this difficulty, we utilize tensor networks to approximate the weight tensor . Such networks encode a structural decomposition of tensors into low-dimensional components and have been shown to provide the most general approximation to smooth tensors (Orús, 2014). The most commonly used tensor networks are linear tensor networks (LTN), also known as tensor-trains in numerical analysis or matrix-product states in quantum physics (Oseledets, 2011).
A tensor train model decomposes a -dimensional tensor into a network of sparsely connected low-dimensional tensors as:
as depicted in Figure (3). When the are called the tensor-train rank. With tensor-train, we can reduce the number of parameters of TT-RNN from to , with as the upper bound on the tensor-train rank. Thus, a major benefit of tensor-train is that they do not suffer from the curse of dimensionality, which is in sharp contrast to many classical tensor decompositions, such as the Tucker decomposition.
3 Approximation results for TT-RNN
A significant benefit of using tensor-trains is that we can theoretically characterize the representation power of tensor-train neural networks for approximating high-dimensional functions. We do so by analyzing a class of functions that satisfies some regularity condition. For such functions, tensor-train decompositions preserve weak differentiability and yield a compact representation. We combine this property with neural network estimation theory to bound the approximation error for TT-RNN with one hidden layer in terms of: 1) the regularity of the target function , 2) the dimension of the input space, 3) the tensor train rank and 4) the order of the tensor.
In the context of TT-RNN, the target function , with , describes the state transitions of the system dynamics, as in (6). Let us assume that is a Sobolev function: , defined on the input space , where each is a set of vectors. The space is defined as the functions that have bounded derivatives up to some order and are -integrable:
where is the -th weak derivative of and .111A weak derivative generalizes the derivative concept for (non)-differentiable functions and is implicitly defined as: e.g. is a weak derivative of if for all smooth with : . Any Sobolev function admits a Schmidt decomposition: , where are the eigenvalues and are the associated eigenfunctions. Hence, we can decompose the target function as:
where are basis functions , satisfying . We can truncate (13) to a low dimensional subspace (), and obtain the functional tensor-train (FTT) approximation of the target function :
In practice, TT-RNN implements a polynomial expansion of the state as in (6), using powers to approximate , where is the degree of the polynomial. We can then bound the approximation error using TT-RNN, viewed as a one-layer hidden neural network:
Let the state transition function be a Hölder continuous function defined on a input domain , with bounded derivatives up to order and finite Fourier magnitude distribution . Then a single layer Tensor Train RNN can approximate with an estimation error of using with hidden units:
where , is the size of the state space, is the tensor-train rank and is the degree of high-order polynomials i.e., the order of tensor.
For the full proof, see the Appendix. From this theorem we see: 1) if the target becomes smoother, it is easier to approximate and 2) polynomial interactions are more efficient than linear ones: if the polynomial order increases, we require fewer hidden units . This result applies to the full family of TT-RNNs, including those using vanilla RNN or LSTM as the recurrent cell, as long as we are given a state transitions (e.g. the state transition function learned by the encoder).
We validated the accuracy and efficiency of TT-RNN on one synthetic and two real-world datasets, as described below; Detailed preprocessing and data statistics are deferred to the Appendix.
The Genz “product peak” (see Figure 4 \subreffig:genz) is one of the Genz functions (Genz, 1984), which are often used as a basis for high-dimensional function approximation. In particular, (Bigoni et al., 2016) used them to analyze tensor-train decompositions. We generated samples of length using (2) with and random initial points.
The traffic data (see Figure 4 \subreffig:traffic) of Los Angeles County highway network is collected from California department of transportation http://pems.dot.ca.gov/. The prediction task is to predict the speed readings for locations across LA, aggregated every minutes. After upsampling and processing the data for missing values, we obtained sequences of length .
The climate data (see Figure 4 \subreffig:climate) is collected from the U.S. Historical Climatology Network (USHCN) (http://cdiac.ornl.gov/ftp/ushcn_daily/). The prediction task is to predict the daily maximum temperature for stations. The data spans approximately years. After preprocessing, we obtained sequences of length .
4.2 Long-term Forecasting Evaluation
To validate that TT-RNNs effectively perform long-term forecasting task in (3), we experiment with a seq2seq architecture with TT-RNN using LSTM as recurrent cells (TLSTM). For all experiments, we used an initial sequence of length as input and varied the forecasting horizon . We trained all models using stochastic gradient descent on the length- sequence regression loss where are the ground truth and model prediction respectively. For more details on training and hyperparameters, see the Appendix.
We compared TT-RNN against 2 set of natural baselines: 1st-order RNN (vanilla RNN, LSTM), and matrix RNNs (vanilla MRNN, MLSTM), which use matrix products of multiple hidden states without factorization (Soltani & Jiang, 2016)). We observed that TT-RNN with RNN cells outperforms vanilla RNN and MRNN, but using LSTM cells performs best in all experiments. We also evaluated the classic ARIMA time series model and observed that it performs worse than LSTM.
For traffic, we forecast up to hours ahead with hours as initial inputs. For climate, we forecast up to days ahead given days of initial observations. For Genz dynamics, we forecast for steps given initial steps. All results are averages over runs.
We now present the long-term forecasting accuracy of TLSTM in nonlinear systems. Figure 5 shows the test prediction error (in RMSE) for varying forecasting horizons for different datasets. We can see that TLSTM notably outperforms all baselines on all datasets in this setting. In particular, TLSTM is more robust to long-term error propagation. We observe two salient benefits of using TT-RNNs over the unfactorized models. First, MRNN and MLSTM can suffer from overfitting as the number of weights increases. Second, on traffic, unfactorized models also show considerable instability in their long-term predictions. These results suggest that tensor-train neural networks learn more stable representations that generalize better for long-term horizons.
Visualization of Predictions
To get intuition for the learned models, we visualize the best performing TLSTM and baselines in Figure 6 for the Genz function “corner-peak” and the state-transition function. We can see that TLSTM can almost perfectly recover the original function, while LSTM and MLSTM only correctly predict the mean. These baselines cannot capture the dynamics fully, often predicting an incorrect range and phase for the dynamics.
In Figure 7 we show predictions for the real world traffic and climate dataset. We can see that the TLSTM corresponds significantly better with ground truth in long-term forecasting. As the ground truth time series is highly chaotic and noisy, LSTM often deviates from the general trend. While both MLSTM and TLSTM can correctly learn the trend, TLSTM captures more detailed curvatures due to the inherent high-order structure.
Speed Performance Trade-off
We now investigate potential trade-offs between accuracy and computation. Figure 8 displays the validation loss with respect to the number of steps, for the best performing models on long-term forecasting. We see that TT-RNNs converge significantly faster than other models, and achieve lower validation-loss. This suggests that TT-RNN has a more efficient representation of the nonlinear dynamics, and can learn much faster as a result.
The TLSTM model is equipped with a set of hyper-parameters, such as tensor-train rank and the number of lags. We perform a random grid search over these hyper-parameters and showcase the results in Table 1. In the top row, we report the prediction RMSE for the largest forecasting horizon w.r.t tensor ranks for all the datasets with lag . When the rank is too low, the model does not have enough capacity to capture non-linear dynamics. when the rank is too high, the model starts to overfit. In the bottom row, we report the effect of changing lags (degree of orders in Markovian dynamics). For each setting, the best is determined by cross-validation. For different forecasting horizon, the best lag value also varies.
Chaotic Nonlinear Dynamics
We have also evaluated TT-RNN on long-term forecasting for chaotic dynamics, such as the Lorenz dynamics (see Figure 9\subreffig:lorenz_data). Such dynamics are highly sensitive to input perturbations: two close points can move exponentially far apart under the dynamics. This makes long-term forecasting highly challenging, as small errors can lead to catastrophic long-term errors. Figure 9 shows that TT-RNN can predict up to steps into the future, but diverges quickly beyond that. We have found no state-of-the-art prediction model is stable in this setting.
5 Related Work
Classic work in time series forecasting has studied auto-regressive models, such as the ARMA or ARIMA model (Box et al., 2015), which model a process linearly, and so do not capture nonlinear dynamics. Our method contrasts with this by explicitly modeling higher-order dependencies. Using neural networks to model time series has a long history. More recently, they have been applied to room temperature prediction, weather forecasting, traffic prediction and other domains. We refer to (Schmidhuber, 2015) for a detailed overview of the relevant literature.
From a modeling perspective, (Giles et al., 1989) considers a high-order RNN to simulate a deterministic finite state machine and recognize regular grammars. This work considers a second order mapping from inputs and hidden states to the next state. However, this model only considers the most recent state and is limited to two-way interactions. (Sutskever et al., 2011) proposes multiplicative RNN that allow each hidden state to specify a different factorized hidden-to-hidden weight matrix. A similar approach also appears in (Soltani & Jiang, 2016), but without the factorization. Our method can be seen as an efficient generalization of these works. Moreover, hierarchical RNNs have been used to model sequential data at multiple resolutions, e.g. to learn both short-term and long-term human behavior (Zheng et al., 2016).
Tensor methods have tight connections with neural networks. For example, (Cohen et al., 2016) shows convolutional neural networks have equivalence to hierarchical tensor factorizations. (Novikov et al., 2015; Yang et al., 2017) employs tensor-train to compress large neural networks and reduce the number of weights. (Stoudenmire & Schwab, 2016) propose to parameterizes the supervised learning models with matrix-product states for image classification. This work however, to the best of our knowledge, is the first work to consider tensor networks in RNNS for sequential prediction tasks for learning in environments with nonlinear dynamics.
6 Conclusion and Discussion
In this work, we considered forecasting under nonlinear dynamics.We propose a novel class of RNNs – TT-RNN. We provide approximation guarantees for TT-RNN and characterize its representation power. We demonstrate the benefits of TT-RNN to forecast accurately for significantly longer time horizon in both synthetic and real-world multivariate time series data.
As we observed, chaotic dynamics still present a significant challenge to any sequential prediction model. Hence, it would be interesting to study how to learn robust models for chaotic dynamics. In other sequential prediction settings, such as natural language processing, there does not (or is not known to) exist a succinct analytical description of the data-generating process. It would be interesting to further investigate the effectiveness of TT-RNNs in such domains as well.
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Appendix A Appendix
a.1 Theoretical Analysis
We provide theoretical guarantees for the proposed TT-RNN model by analyzing a class of functions that satisfy some regularity condition. For such functions, tensor-train decompositions preserve weak differentiability and yield a compact representation. We combine this property with neural network estimation theory to bound the approximation error for TT-RNN with one hidden layer, in terms of: 1) the regularity of the target function , 2) the dimension of the input space, and 3) the tensor train rank.
In the context of TT-RNN, the target function with , is the system dynamics that describes state transitions, as in (6). Let us assume that is a Sobolev function: , defined on the input space , where each is a set of vectors. The space is defined as the set of functions that have bounded derivatives up to some order and are -integrable:
where is the -th weak derivative of and .222A weak derivative generalizes the derivative concept for (non)-differentiable functions and is implicitly defined as: e.g. is a weak derivative of if for all smooth with : .
Any Sobolev function admits a Schmidt decomposition: , where are the eigenvalues and are the associated eigenfunctions. Hence, we can decompose the target function as:
where are basis functions , satisfying . We can truncate Eqn 13 to a low dimensional subspace (), and obtain the functional tensor-train (FTT) approximation of the target function :
FTT approximation in Eqn 13 projects the target function to a subspace with finite basis. And the approximation error can be bounded using the following Lemma:
Lemma A.1 (FTT Approximation Bigoni et al. (2016)).
Let be a Hölder continuous function, defined on a bounded domain with exponent , the FTT approximation error can be upper bounded as
Lemma A.1 relates the approximation error to the dimension , tensor-train rank ,and the regularity of the target function . In practice, TT-RNN implements a polynomial expansion of the input states , using powers to approximate , where is the degree of the polynomial. We can further use the classic spectral approximation theory to connect the TT-RNN structure with the degree of the polynomial, i.e., the order of the tensor. Let . Given a function and its polynomial expansion , the approximation error is therefore bounded by:
Lemma A.2 (Polynomial Approximation).
Let for . Let be the approximating polynomial with degree , Then
Here is the semi-norm of the space . is the coefficient of the spectral expansion. By definition, is equipped with a norm and a semi-norm . For notation simplicity, we muted the subscript and used for .
So far, we have obtained the tensor-train approximation error with the regularity of the target function . Next we will connect the tensor-train approximation and the estimation error of neural networks with one layer hidden units. Given a neural network with one hidden layer and sigmoid activation function, following Lemma describes the classic result of describes the error between a target function and the single hidden-layer neural network that approximates it best:
Lemma A.3 (NN Approximation Barron (1993)).
Given a function with finite Fourier magnitude distribution , there exists a neural network of hidden units , such that
where with Fourier representation .
We can now generalize Barron’s approximation lemma A.3 to TT-RNN. The target function we are approximating is the state transition function . We can express the function using FTT, followed by the polynomial expansion of the states concatenation . The approximation error of TT-RNN, viewed as one layer hidden
Where is the order of tensor and is the tensor-train rank. As the rank of the tensor-train and the polynomial order increase, the required size of the hidden units become smaller, up to a constant that depends on the regularity of the underlying dynamics .
a.2 Training and Hyperparameter Search
We trained all models using the RMS-prop optimizer and employed a learning rate decay of schedule. We performed an exhaustive search over the hyper-parameters for validation. Table 2 reports the hyper-parameter search range used in this work.
|Hyper-parameter search range|
|learning rate||hidden state size|
|tensor-train rank||number of lags|
|number of orders||number of layers|
For all datasets, we used a train-validation-test split and train for a maximum of steps. We compute the moving average of the validation loss and use it as an early stopping criteria. We also did not employ scheduled sampling, as we found training became highly unstable under a range of annealing schedules.
a.3 Dataset Details
Genz functions are often used as basis for evaluating high-dimensional function approximation. In particular, they have been used to analyze tensor-train decompositions (Bigoni et al., 2016). There are in total different Genz functions. (1) , (2) , (3) , (4) (5) (6) . For each function, we generated a dataset with samples using (2) with and and random initial points draw from a range of .
We use the traffic data of Los Angeles County highway network collected from California department of transportation http://pems.dot.ca.gov/. The dataset consists of month speed readings aggregated every minutes . Due to large number of missing values () in the raw data, we impute the missing values using the average values of non-missing entries from other sensors at the same time. In total, after processing, the dataset covers time-series. We treat each sequence as daily traffic of time stamps. We up-sample the dataset every minutes, which results in a dataset of sequences of daily measurements. We select sensors as a joint forecasting tasks.
We use the daily maximum temperature data from the U.S. Historical Climatology Network (USHCN) daily (http://cdiac.ornl.gov/ftp/ushcn_daily/) contains daily measurements for climate variables for approximately years. The records were collected across more than locations and span over days. We analyze the area in California which contains stations. We removed the first years of day, most of which has no observations. We treat the temperature reading per year as one sequence and impute the missing observations using other non-missing entries from other stations across years. We augment the datasets by rotating the sequence every days, which results in a data set of sequences.
We also perform a DickeyâFuller test in order to test the null hypothesis of whether a unit root is present in an autoregressive model. The test statistics of the traffic and climate data is shown in Table 3, which demonstrate the non-stationarity of the time series.
|p-value||0.96||0.96||1.12 e-13||2.52 e-7|
|Number Lags Used||2||7||0||1|
|Critical Value (1%)||-3.49||-3.51||-3.63||2.7|
|Critical Value (5%)||-2.89||-2.90||-2.91||-3.70|
|Critical Value (10%)||-2.58||-2.59||-2.60||-2.63|
a.4 Prediction Visualizations
Genz functions are basis functions for multi-dimensional Figure 10 visualizes different Genz functions, realizations of dynamics and predictions from TLSTM and baselines. We can see for “oscillatory”, “product peak” and “Gaussian ”, TLSTM can better capture the complex dynamics, leading to more accurate predictions.
a.5 More Chaotic Dynamics Results
Chaotic dynamics such as Lorenz attractor is notoriously different to lean in non-linear dynamics. In such systems, the dynamics are highly sensitive to perturbations in the input state: two close points can move exponentially far apart under the dynamics. We also evaluated tensor-train neural networks on long-term forecasting for Lorenz attractor and report the results as follows:
The Lorenz attractor system describes a two-dimensional flow of fluids (see Figure 9):
This system has chaotic solutions (for certain parameter values) that revolve around the so-called Lorenz attractor. We simulated trajectories with the discretized time interval length . We sample from each trajectory every units in Euclidean distance. The dynamics is generated using , . The initial condition of each trajectory is sampled uniformly random from the interval of .
Figure 11 shows steps ahead predictions for all models. HORNN is the full tensor TT-RNN using vanilla RNN unit without the tensor-train decomposition. We can see all the tensor models perform better than vanilla RNN or MRNN. TT-RNN shows slight improvement at the beginning state.