Now, higher-order data, which is also called tensor, frequently occur in various scientific and real-world applications. Specifically, in neuroscience, functional magnetic resonance imaging (fMRI) is an example of such tensor data consisting of a series of brain scanning images. Therefore, such data can be characterized as a 3D data (or 3-mode data) with the shape of time neuron neuron. In many fields, we can encounter the problem that analyzing the relationship between the tensor variable and the scalar response for every sample . Specifically, we assume
in which is the inner product operator, is the noise, and
is the coefficient needs to be estimated through regression. Notice that in the real world, these tensor data generally have two properties which makes the coefficient difficult to be inferred perfectly: (1)Ultra-high-dimensional setting, where the number of samples is much less than the number of variables. For example, each sample of the CMU2008 dataset  is a 3D tensor with shape of , which is 71553 voxels in total. However, only 360 trials are recorded. The high-dimensional setting will make the estimated solution breaks down because we are trying to infer a large model with a limited number of observations. (2) Higher-order structure of data. The higher-order structure of data exists in many fields, such as fMRI and videos, with the shape of time pixel
pixel. Traditional machine learning methods are proposed for processing vectors or matrices, hence, dealing with high-order data might be a difficulty. In past years, many methods are introduced to address these two problems.
To reserve the spatial structure, several methods are introduced based on the CANDECOMP/PARAFAC decomposition, which approximates an M-order tensor through
Here, is defined as the CP-rank of the tensor . For instance,  propose GLTRM which first decomposes the variable tensor and then applies the generalized linear model to estimate each component vector. In addition,  propose SURF using the divide-and-conquer strategy for each component vector. Almost all the CANDECOMP/PARAFAC-decomposition-based methods, including GLTRM, suffer a problem that the CP-rank should be pre-specified. However, we always have no prior knowledge about the value of . Even if we can use techniques, such as cross-validation, to estimate from the data, the solving procedure becomes trivial and computationally expensive for large-scale data. A method called orTRR is previously proposed in  automatically obtaining a low-rank coefficient without pre-specifying . But orTRR uses norm rather than norm for recovering the sparsity of data, which makes it performance poorer than others on variable selection.
To address such problem, some methods are proposed directly making constraints on the tensor. For example,  propose Remurs exploiting commonly used norm for enforcing sparsity on the estimated coefficient tensor. In addition, a nuclear norm is attached to it to make the solution low-rank. Remurs is solved through ADMM framework . Although able to obtain an acceptable solution, Remurs has expensive time cost for large-scale data.
In this paper, we derive ideas of a scalable estimator, Elementary Estimator , and propose Sparse and Low-Rank Higher-Order Tensor Regression (SLTR), which directly enforcing sparsity and low-rankness on the tensor with norm and nuclear norm respectively. We also propose a fast algorithm making use of parallel proximal method. Notice that because the solution can be obtained in a two-layer parallel manner, the algorithm can be implemented parallelly through multi-threading computation or GPUs, thus, the solution of SLTR is able to be obtained with small time complexity. We empirically show that the parallel version of SLTR is faster and obtain no worse solution than previous methods. See details in Section 6 and Section 7. To summarize, this paper makes the following contributions:
A sparse and low-rank tensor regression model: Deriving ideas of Elementary Estimator, in Section 4, we propose a sparse and low-rank tensor regression model through norm and nuclear norm respectively.
A fast and scalable algorithm for the model: We also provide a fast solution to our model. The solution can be obtained through two-layer parallelizatio. In Section 6 we prove that SLTR has lower time complexity than counterparts.
Theoretically prove the convergence rate of sparse and low-rank tensor regression: In Section 5, we theoretically prove the sharp error bound of our model. Specifically, we provide a more exact error bound for 3D data. To our best knowledge, we are the first who prove the error bound of sparse and low-rank tensor regression with norm and nuclear norm directly applied on the tensor.
Experiments on real-world fMRI dataset: We experiment SLTR and four baseline methods on several simulated datasets and one fMRI dataset with nine projects. Results show that SLTR is more fast and stable than previous ones. See detailed explanation in Section 7.
denotes the element-wise norm, denotes the nuclear norm, denotes the norm, and denotes the spectral norm. Unfolding tensor along its -th mode is denoted as , where the columns of are the th mode vectors of . Its opposite operator is denoted as . The operation converts a vector into a tensor.
3.1 Elementary estimator for linear regression models
In the literature of linear regression (LR), some methods are proposed to address the difficulties of estimating in the high-dimensional setting, in which the number of samples less than the number of variables. For instance, Lasso attaches an
norm to the ordinary least square (OLS) problem in order to reduce the number of variables. Lasso can be iteratively solved via proximal gradient method, which takestime. On the other hand, 
further propose Dantzig selector for estimations high-dimensional settings. Dantzig selector is solved through linear programming (LP), which hastime complexity if using interior point method. These previous methods has prohibitively large time complexity when there are large number of variables. Therefore, to address this problem,  propose a fast estimator for high-dimensional linear regression named Elementary Estimator (EE). Given vector samples and scalar responses , EE aims to solve
Here, is an arbitrary regularization and is its dual norm. EE shares similarities with Dantzig selector, however, it has been shown in  that EE is more practical scalable in high-dimensional settings.
Furthermore,  propose the superposition-structured elementary estimator by combining several estimators, which has the formulation
This class of superposition-structured estimators is able to be solved through independently and parallelly solving every single estimator.
3.2 Regularized matrix regression
Although plenty of methods are proposed for LR, these can not be directly used in tasks with matrix data,such as digital images. One intuitive way is to first vectorize these two-dimensional data and then use extant LR methods. However, this will hurt the higher-order structure of data, such as spatial coherence. Therefore,  extend the ordinary liner regression to two-dimensional data and propose the regularized matrix regression. Because, for 2D-data, the true coefficient can often be well approximated by a low-rank structure, the regularized matrix regression exploits nuclear norm. Given matrix covariates and scalar responses , the regularized matrix regression solves
Here, is nuclear norm and controls the rankness of coefficient . Although effective for second-order data, this matrix regression model is infeasible for higher-order tensor data.
Because Eq. (5) is introduced only for matrix data, we need a new model for solving problems with higher-order tensor data. Therefore, deriving ideas of Eq. (5)  propose a new tensor regression method named Remurs, which combines the low-rank constraint (nuclear norm) with the sparsity constraint ( norm). Specifically, given -order tensor samples and corresponding responses , Remurs solves
Notice that the nuclear norm regularization here can not be directly utilized for higher-order tensors. Moreover, as  state, the computation of tensor rank is NP-hard, therefore, we use a convex tensor nuclear norm regularization 
as the convex relaxation of nuclear norm. Remurs exploits ADMM method for estimation, which is computationally expensive for a large problem. To address this problem, deriving ideas of EE (Eq. (3)), we propose our Sparse and Low-Rank Higher-Order Tensor Regression (SLTR) model. SLTR aims to solve
Here, are three tuning parameters, nuclear norm is in the form of Eq. (7), and denotes its dual norm. Moreover, , , and reshapes a matrix into a tensor. Notice that due to the definition of elementwise norm, we have . Integrating this with the definition of tensor nuclear norm, SLTR can be decomposed into
After estimating , the final estimation is obtained by averaging them through
The tensor nuclear norm Eq. (7) we use here is based on the unfolding tensors. Unfolding a tensor loses certain information and desires large number of computations. Directly applying low rank constraints onto the tensor rather its unfolded version could be explored in future works.
4.1 Two-layer parallel solution
In this section, we propose an algorithm to solve SLTR fast and accurately. In Eq. (9), although SLTR is decomposed into subproblems, these subproblems are not independent, thus, can not be solved separately, because each shares same elements. To overcome this drawback, we introduce several auxiliary variables into Eq. (9). Then, for , each subproblem can separately solve
Therefore, SLTR can be solved in a parallel manner.
Here, , , , and . denotes an indicator function of set as if , otherwise . To solve Eq. (12), we propose a parallel proximal method based algorithm, as summarized in Algorithm 1. Due to space limitations, the four proximal operators of functions are postponed in the appendix.
The tensor nuclear norm is a norm function and it is decomposable . Specifically, given a pair of subspaces , we have for all and
(C1:sparse) The true coefficient is exactly sparse with non-zero elements.
(C2:low-rank) The true coefficient is a -rank tensor where and denotes the orthogonal rank of tensor .
Suppose that the true coefficient tensor satisfies conditions (C1:sparse) and (C2:low-rank). Furthermore, suppose we solve Eq. (8) with controlling parameters and satisfying the constraints. Then, the optimal solution satisfies the following error bound:
If the coefficient is a three-mode tensor such that and conditions (C1:sparse) and (C2:low-rank) are held for true coefficient , the optimal solution of Eq. (8) satisfies the following error bound:
where denotes the rank of the unfolding tensor and
6.1 Complexity analysis
We begin with the time complexity analysis. First of all, each needs to be vectorized once, which costs time. Then the initial approximation is computed with time complexity . Next, each subproblem of each mode is solved simultaneously using parallel proximal method. Obviously, the time complexity of parallely solving one subproblem is dominated by the largest time cost of calculating a certain proximal operator, which is the SVD procedure with time complexity. In a conclusion, by the virtue of our two-layer parallel solution, the total complexity of solving SLTR is only .
As for the memory complexity, this is the bottleneck of our approach. Because of the parallel proximal method, many auxiliary variables are needed, such as . In a total, there are memory spaces used for each in our approach, which is quite expensive for large data. We leave improvements on the memory complexity in our future works.
6.2 Relevance to previous works
Many methods have been proposed in the literature of regression tasks on higher-order tensor data. In this paper, we focus on the setting that the variables are represented by a tensor while the responses are denoted by a vector . Several models were already recently proposed to estimate the coefficient tensor for this specific, what we call, higher-order tensor regression problem.
One group of these methods is the direct extension of regularized linear regression. Naively, one way to solve this regression problem is vectorization. All the elements in the tensor are first stacked into a vector and then existing linear regression models can be applied to it. One obvious shortcoming of vectorization is that it will cause a loss of latent structural information of the data. To reserve certain potential information,  is proposed to estimate a sparse and low-rank coefficient tensor, by integrating the tensor nuclear norm and norm into the optimization problem. In Remurs, the tensor nuclear norm is approximated by the summation of ranks of several unfolded matrices. In addition,  improve Remurs by substituting the nuclear norm into Tubal nuclear norm [29, 28]
, which can be efficiently solved through discrete Fourier transform. However, these methods are computationally expensive because the non-differential regularizer,norm or nuclear norm exists in their objective function. Therefore, currently, this group of methods is not a good choice for higher-order tensor regression.
To reserve the latent structure when dealing with tensors, another prevailing group of methods [7, 31, 5, 23] are proposed based on CANDECOMP/PARAFRAC decomposition. Generally, instead of directly estimate the coefficient tensor , we aim at inferring every component vector in each sub-task. For example,  propose GLTRM using generalized linear model (GLM) to solve each sub-task. Moreover, orTRR is proposed in  enforcing sparsity and low-rankness on the estimated coefficient tensor. Instead of norm, orTRR utilize norm to obtain the sparsity. In addition, recently,  propose SURF exploiting divide-and-conquer strategy where the sub-task has a similar formulation of Elastic Net . In the paper of SURF, authors empirically show that their method can converge, but a statistical convergence rate is not proved. On the contrary, in this paper, we theoretically prove the error bound of our method. Noticeably, the main limitation of CANDECOMP/PARAFAC-decomposition-based method is that the decomposition rank should be pre-specified, however, we generally have no prior knowledge about the tensor rank in real-world applications. Although orTRR is able to automatically obtain a low-rank result, the estimated result is sub-optimal due to the fact that norm is inferior to norm in the sparse setting. Hence, these methods are not suitable for real-world applications.
Some other models were introduced previously for other problem settings. Recently, [6, 9, 26] propose models for non-parametric estimation by assuming that the response , making use of either additive model or Gaussian process. Apart from the above-mentioned ones, many models [21, 19, 27, 32, 14] were put forward to estimate the relationship between the variable tensor and a response tensor . Another line of statistical models involving tensor data is tensor decomposition [11, 22, 1, 12, 16]. Tensor decomposition can be considered as an unsupervised problem which aims at approximating the tensor with lower-order data. On the contrary, our SLTR is a supervised method estimating the latent relationship between variables and responses. Because these methods have different objectives from our method, we pay little attention to them and exclude them from experiments. In section 7, we compare SLTR with several introduced higher-order tensor regression methods, including Lasso, Elastic Net, Remurs, GLTRM, and SURF.
7.1 Experiment Setups
Baselines and metrics:
To compare with our method, we use four previous methods as baselines. 1) Linear regression, specifically, Lasso and Elastic Net (with trade-off ratio between and norm being 0.5), 2) Remurs, 3) SURF, and 4) GLTRM. Furthermore, we use three metrics to evaluate performances of our method and baselines as: 1) time cost, 2) coefficient error, which is defined as , and 3) mean squared error (MSE).
We compare SLTR with baselines on simulated datasets and a real-world fMRI dataset. Specifically, our simulated datasets are generated through several steps. First, generate and
with each element drawn from the normal distribution. Then, randomly set elements of to be . Compute the response with respect to . Here, controls the ratio of noises and is also generated from normal distribution. In addition to simulated datasets, we further experiment SLTR on CMU2008 fMRI datasets .
In the experiments, parameters of all the methods are selected through 5-fold cross-validation procedures which take the average MSE on validation datasets as the selecting criteria. Detailed descriptions about ranges of tuning parameters are shown in the appendix. As for the experiment environment, all the experiments are implemented on a Linux server with 2 Intel Xeon Silver 4216 CPUs and 256 GB RAMs. Moreover, our SLTR is implemented in MATLAB. Out of fairness, we set the maximal number of iterations to be for all the methods and let them terminate when . We run every single experiment 20 times and report the averaged value of metrics over these 20 trials.
7.2 Experiments on simulated data
First, we experiment SLTR on both 2D and 3D simulated datasets. The shape of data varies from , , , , for 2D data and , , , , , , and for 3D data. We fix the sparsity level and noise coefficient . The number of samples are determined through and for 2D and 3D data respectively. The time costs of all the methods are shown in Figure (1). Apparently, we can see that the not only the sequential version of SLTR has lower time cost, but also the parallel implementation of SLTR achieves more obvious improvements on the time cost, compared with baselines. Notice that the code of SURF is inapplicable for 2D data and the GLTRM is infeasible for 3D data, we omit these two methods in subfigures correspondingly. Ideally, if a better environment is provided, the speedup of SLTR can be larger. Then, we report the MSE of all the methods on these simulated datasets, in Table (1). The bold number denotes the best MSE value and the underlined value represents the second-best result. The result indicates that in most cases SLTR obtains the best solution, while in other cases an estimation is computed by SLTR with only a little difference with the best one. Because of the infeasibility of SURF and GLTRM, we also omit these two methods in the table under the corresponding conditions. The experiment results in Figure (1) and Table (1) indicate that SLTR is able to spend less time on obtaining a solution no worse than solutions of other methods.
|1-7[0.8pt/2pt] 2D Data – 50% samples|
|1-7[0.8pt/2pt] 3D Data – 8% samples|
|10 10 5||0.8532||0.8538||0.8472||Infeasible||1.7993||1.799|
|15 15 5||0.9892||0.9867||0.9994||2.3151||2.3132|
|20 20 5||0.9383||0.9378||1.0049||2.5469||2.5193|
|25 25 5||0.9275||0.9398||0.9391||2.0149||2.0149|
|30 30 5||0.9186||0.919||0.9289||1.9381||1.9377|
|35 35 5||0.9336||0.9370||0.9527||2.0147||2.0147|
|40 40 5||0.9073||0.9072||1.0006||2.1059||2.1065|
In addition, we apply SLTR in high-dimensional settings. We let the shape of data to be and varies from 50 to 400, with 50 increments. The sparsity level is and the noise factor is set to . The MSE values shown in Table (1) indicate that SLTR obtains the best solution under almost all the conditions. Note that for , even the MSE of SLTR is , which is a little worse than the MSE of Remurs (0.9190), this value is much better than others.
Finally, to test the stability of SLTR and its sensitiveness to the sparsity level, we generate simulated datasets varying the sparsity level from . The shape of each sample is
and a totally of 160 samples are generated. We generate different datasets on a different trail. The averaged time cost and the variance of 20 trials are reported in Figure (2). Apparently, under every setting, SLTR outperforms baselines. Moreover, the time cost variance of SLTR is at most , which is much less than it of SURF and Remurs (at least ). Although LR methods also have small time variance, they might obtain a worse solution than SLTR (see Table (1)). Therefore, SLTR is faster than other methods on datasets with different sparsity levels and it is more stable.
7.3 Experiments on real-world data
In this section, we perform fMRI classification tasks on CMU2008 datasets  with 9 projects in total. Each sample of this dataset is a 3-mode tensor of size (71553 voxels). This classification task aims to predict human activities associated with recognizing the meanings of nouns. Following [10, 20], we focus on classifications of binary classes: “tools” and “animals”. Here, the class “tool” combines observations from “tool” and “furniture”, class “animal” combines observations from “animal” and “insect” in the CMU2008 dataset. Like simulated experiments, values of tuning parameters of each method are selected through 5-fold cross-validation. For each subject, we split the entire dataset into the training dataset and testing dataset with the proportion of and respectively and use AUC to evaluate classification results. Results are shown in Table 3, which indicates that SLTR obtains the best time cost among all the cases. Notice that although SURF has the lowest time cost, the AUC of its solution is always around 0.5 and drops below 0.5 sometimes. Hence, we think the classification result of SURF is unacceptable. One interesting result occurs on project #1, where linear regression methods obtain a much better solution. We think the reason might be that in this subject, the voxels are independent, hence, the data has no latent structure. Anyway, on a real-world fMRI dataset, SLTR can obtain an acceptable solution with the least time cost.
|seq. time||par. time||AUC||time||AUC||time||AUC||time||AUC||time||AUC|
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