I Introduction
Sparse representations of data have been widely used in a variety of information processing tasks such as data compression, feature extraction, data classification, signal denoising and inpainting, and audio processing
[1, 2, 3]. One of the powerful techniques to obtain sparse representations is dictionary learning (DL) which can be formulated as(1) 
We wish to find an overcomplete basis with unitnorm columns and dictionary coefficient matrix such that each observation is represented by a linear combination of no more than columns of . Since this problem is not convex, it is typically solved by alternating minimization; is updated using a fixed and then, is updated using a fixed [3].
In traditional DL literature, when dealing with multidimensional data, high order data (tensors of order
or higher) are vectorized and stacked in columns of an observation matrix
: the structure of the data is not considered in the dictionary underlying the data. In this case, any standard DL method can be used to find sparse representations of data. This simplistic method disregards the multidimensional structure in the data and does not capture the correlation and structure along different dimensions in each original “data point.”On the other hand, in structured DL methods for tensor data, the multidimentional structure in data is taken into account. There exists a class of DL algorithms for tensor data that are based on the Tucker decomposition [4] of tensors. The resulting “Kronecker structured” DL methods (KSDL) assume the dictionary consists of the Kronecker product [5] of smaller subdictionaries. Such algorithms represent tensor data using many fewer parameters compared to vectorized DL techniques [6, 7, 8, 9]
. This is due to the fact that the number of degrees of freedom in the KSDL problem is significantly less than the traditional DL problem; this suggests that dictionary recovery is possible with smaller sample complexity using KSDL methods
[10, 11]. These works provide KSDL algorithms to represent second order [6, 7, 8] and 3rdorder tensors [9]. In [6], for example, the KSDL objective function is minimized using a Riemannian conjugate gradient method along with a nonmonotonic line search. In other methods, the subdictionaries composing the KS dictionary are updated alternately; in [7], an approach similar to KSVD [3] is employed that uses higherorder SVD (HOSVD) [12] to alternately update coordinate dictionaries for second order tensor data and in [9], gradient descent is used to alternately update coordinate dictionaries for third order tensor data. The dictionary update stage in these algorithms involves solving a nonconvex minimization problem. The central challenge in the theoretical analysis of such KSDL solvers is that the dictionary update stage is nonconvex. Furthermore, these explicit models are specialized to KS problems: they do not extend to more general structures in the underlying dictionary. In contrast, Dantas et al. [8] recently proposed an algorithm to learn the sum of KS dictionaries that represent second order tensor data by adding a regularizer in the objective function.In this paper, we propose a novel algorithm called “STARK” to learn KS dictionaries for thorder tensor data (). Our method involves adding a regularization term to the objective function of the DL problem defined in (1). The motivation for this term comes from the following realization: elements of any KS matrix can be rearranged to form a rank
tensor. Thus, enforcing a rank1 constraint on such rearrangement of the dictionary results in a KS dictionary. To this end, we take advantage of lowrank tensor estimation literature
[13, 14, 15, 16, 17] to add a convex regularizer that imposes lowrankness on the rearrangement tensor. This formulation has the advantage that it can be used to learn KS dictionaries as well as the case where the underlying dictionary is better approximated by sum of KS dictionaries. Our method can learn dictionaries of arbitrary tensor order; in the case of second order tensor data our general formulation coincides with that of Danita’s et al. [8].We conduct numerical experiments to validate the performance of our algorithm. We use STARK for representation of third order synthetic and real tensor data and demonstrate that STARK outperforms vectorized DL technique KSVD [3] and KSDL technique KHOSVD [7] for small sample sizes.
Notation Convention: Underlined bold uppercase, bold uppercase and lowercase letters are used to denote tensors, matrices and vectors, respectively. Lowercase letters denote scalars. We denote the Kronecker product and outer product by and , respectively, while denotes the mode product between a tensor and a matrix [18]. Norms are given by subscripts, so and are the and norms of , while , , and are the spectral, Frobenius, and nuclear norms of , respectively. A slice of a tensor is a dimensional section defined by fixing all but two of its indices. Particularly, a frontal slice of a dimensional tensor is defined by fixing the third index.
Ii TuckerBased KSDL
According to the Tucker decomposition of tensors, an th order data tensor can be decomposed in the following form:
(2) 
where denotes the core tensor and ’s denote transformation matrices along each mode of . The vectorized version of can be written as
(3) 
where and [18]. Here, the structure in tensor data is being exploited using the Kronecker product of transformation matrices. Stacking vectorized data points in columns of a matrix , we get
(4) 
which is similar to the conventional DL problem, except that the dictionary is Kronecker structured.
In the next section, we present our proposed KSDL algorithm called “STructured dictionAry learning through RanK1 Tensor recovery” (STARK), which implicitly enforces Kronecker structure on the dictionary being learned by means of a regularizer in the DL objective function.
We note that STARK can be used to enforce a more general structure called lowseparationrank (LSR) structure. An LSR matrix can be written as sum of a few KS matrices:
(5) 
where the factor matrices have the same size for a fixed , and is the separation rank of [19].
Iii Enforcing Structure via Regularization
To motivate the idea behind STARK, let us consider . It turns out that the elements of can be rearranged to form , where for [5]. Figure 1 illustrates this rearrangement for . Similarly, for , we can write , where each frontal slice of the tensor is a scaled copy of . Following a similar procedure, we can show that if , then a certain “rearrangement” of is the rank tensor where for . This suggests that in the structured DL problem, we can impose the LSR structure (KS when ) on the dictionary being learned by minimizing the rank of .
Since tensor rank is a nonconvex function, in order to make this DL problem convex with respect to , we use a commonly used convex proxy for the tensor rank function, the sumtracenorm [15], which is defined as the average of the trace (nuclear) norms of the unfoldings of the tensor: Using this convex relaxation for the rank function, the KSDL problem has the following form:
(6) 
where the columns of have unit norm. We use alternating minimization to solve this nonconvex problem. To minimize the objective function in (6) with respect to , we can use any of the standard sparse coding methods. In simulations, we use orthogonal matching pursuit (OMP) [20, 21]. To update the KS dictionary , we use the alternating direction method of multipliers (ADMM) [22]. We describe this dictionary update step in the next section.
Iv Structured Dictionary Update Using ADMM
In this section, we discuss the dictionary update step of solving problem (6), which can be stated as
(7) 
The main issue in solving the convex dictionary update problem (7) is dealing with the interdependent nuclear norms. This makes optimization methods that use gradient information challenging. Inspired by many works in the literature on lowrank tensor estimation [14, 13, 15, 16], we instead suggest the following reformulation of (7):
(8) 
In this formulation, although the nuclear norms are associated with one another through the introduced constraint, we can decouple the minimization problem into separate subproblems. In particular, we can solve the objective function (IV) using ADMM, which involves decoupling the problem into independent subproblems by forming the following augmented Lagrangian function:
(9) 
where and . Here, the inner product of two tensors is defined as the inner product of their vectorizations.
Finally, to find the gradient of (IV) with respect to , we rewrite the Lagrangian function in the following form
(10) 
Here, we defined and , where and is a permutation matrix such that .
In the rest of this section, we briefly discuss derivation of the permutation matrix as well as the update steps of ADMM. Due to lack of space, we leave the details to an extended version of this paper.
Iva The Permutation Matrix
The permutation matrix
represents a linear transformation that maps the elements of
to . Given index of and the corresponding mapped index of , our strategy for finding the permutation matrix is to define as a function of . To this end, we first find the the corresponding row and column indices of matrix from the th element of . Then, we find the index of the element of interest on the th order rearranged tensor , and finally, we find its location on . Note that the permutation matrix is only a function of the dimensions of the factor matrices. We leave the formal explanation of this procedure to an extended version of this paper.IvB ADMM Update Rules
Recall that each iteration of ADMM consists of the following steps [22]:
(11)  
(12)  
(13) 
where is the vertical concatenation of instances of , the identity operator on .
The solution to problem (11) is found by taking the gradient of with respect to and setting it to zero. Suppressing the iteration index for ease of notation, we have
where denotes the adjoint of the linear operator [13]. Setting the gradient to zero results in
(14) 
Equivalently, we have,
(15) 
Therefore, the update rule for is
(16) 
To update , we can break the second step (12) into solving independent subproblems (suppressing the index ):
The objective function of this problem can be reformulated as
(17) 
The minimizer of an objective function of the form (IVB) is
(18) 
where is the shrinkage operator that applies softthresholding at level
on the singular values of
(see Theorem 2.1 in [23] for details). Therefore,(19) 
where is the inverse of the unfolding operator. The summary of our DL method is provided in Algorithm 1.
V Numerical Experiments
We compare the performance of STARK with two methods: KSVD [3], as a nonstructured DL method, and KHOSVD [7], which is a structured DL method that explicitly enforces Kronecker structure on the dictionary. We compare the performance of these methods are compared for synthetic dimensional and dimensional data as well as dimensional realworld data.
Synthetic Data
For synthetic data, we randomly generate the dictionary and the sparse coefficient matrix to construct the observation matrix . We generate a KS dictionary as (and for dimensional data) with unitnorm columns according to the following procedure. The elements of the subdictionaries through
are chosen i.i.d from a Gaussian distribution
, and then the columns of the subdictionaries are normalized. For generating , we select the locations of the nonzero elements of each column uniformly at random. The values of those elements are sampled i.i.d. from . In the learning process, the dictionary is initialized using random columns of the observation matrix . The experiments were run for 20 Monte Carlo iterations and for various training sample sizes and the resulting dictionaries were tested on a set of test samples. For 2ndorder tensor data we selected , , , , , and for 3rdorder tensor data we selected , , , , , , and .The results of our experiments on synthetic data are shown in Figure 2. We compared our method to KSVD and KHOSVD. For KHOSVD, we used algorithm provided by the authors, which actually enforces a KhatriRao structure on the dictionary rather than KS. STARK outperforms both KSVD and KHOSVD for all training sample sizes, especially when the number of training samples is small. The improvement over KSVD can be attributed to the lower sample complexity of structured DL models. We conjecture that one reason for the improvement over KHOSVD is that the dictionary update in STARK is a convex problem and thus the algorithm is less prone to getting stuck in a poor local minimum.
Real Data
For these experiments, we compare the denoising performance of the three methods on two RGB images, Peppers and Lena, which are and tensors, respectively. We corrupt the images using additive white Gaussian noise with . To construct the training data set, we extract overlapping patches from each image and treat each patch as a data point. Then we compare the denoising performances of the methods based on the resulting peak signal to noise ratio (PSNR) of the reconstructed images [24].
Figure 3 shows the results averaged over 10 Monte Carlo iterations. We can see the denoising performance of STARK is superior to both KSVD and KHOSVD for all training sample sizes. This is in part due to the fact that for realworld data, the underlying dictionaries may not be KS. STARK allows to have rank higher than , meaning the algorithm can use a larger number of parameters ( times as many when ) to approximate the true dictionary.
Vi Conclusion
In this paper we showed that the Kronecker product of matrices can be rearranged to form an th order rank1 tensor. Based on this, we proposed a novel structured dictionary learning method for multidimensional data that enforces LSR structure in the dictionary through imposing a lowrank structure on the rearranged tensor. In particular, Kronecker structure can be enforced by imposing a rank1 constraint on the rearranged tensor. Since the lowrank tensor recovery problem is a nonconvex problem, we resort to solving its convex relaxation, namely, minimizing the sumtracenorm of the rearranged tensor, which is a convex proxy for tensor rank. We used ADMM for solving the dictionary update stage of this structured DL problem. Our experiments on both synthetic and real data showed that when the sample size is small, our method considerably outperforms both KSVD, which returns unstructured dictionaries, and KHOSVD, a KSDL method that directly finds the subdictionaries.
References
 [1] K. KreutzDelgado, J. F. Murray, B. D. Rao, K. Engan, T.W. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representation,” Neural computation, vol. 15, no. 2, pp. 349–396, February 2003. [Online]. Available: https://doi.org/10.1162/089976603762552951

[2]
M. Elad, J.L. Starck, P. Querre, and D. L. Donoho, “Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA),”
Appl. and Computational Harmonic Anal., vol. 19, no. 3, pp. 340–358, November 2005. [Online]. Available: https://doi.org/10.1016/j.acha.2005.03.005  [3] M. Aharon, M. Elad, and A. Bruckstein, “SVD: An algorithm for designing overcomplete dictionaries for sparse representation,” IEEE Trans. Signal Process., vol. 54, no. 11, pp. 4311–4322, November 2006. [Online]. Available: https://doi.org/10.1109/TSP.2006.881199
 [4] L. R. Tucker, “Implications of factor analysis of threeway matrices for measurement of change,” Problems in Measuring Change, pp. 122–137, 1963.
 [5] C. F. Van Loan, “The ubiquitous Kronecker product,” J. Computational and Appl. Math., vol. 123, no. 1, pp. 85–100, November 2000. [Online]. Available: https://doi.org/10.1016/S03770427(00)003939

[6]
S. Hawe, M. Seibert, and M. Kleinsteuber, “Separable dictionary learning,” in
Proc. IEEE Conf. Comput. Vision and Pattern Recognition (CVPR)
, June 2013, pp. 438–445. [Online]. Available: https://doi.org/10.1109/CVPR.2013.63  [7] F. Roemer, G. Del Galdo, and M. Haardt, “Tensorbased algorithms for learning multidimensional separable dictionaries,” in Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process. (ICASSP), May 2014, pp. 3963–3967. [Online]. Available: https://doi.org/10.1109/ICASSP.2014.6854345
 [8] C. F. Dantas, M. N. da Costa, and R. da Rocha Lopes, “Learning dictionaries as a sum of Kronecker products,” IEEE Signal Process. Lett., vol. 24, no. 5, pp. 559–563, March 2017. [Online]. Available: https://doi.org/10.1109/LSP.2017.2681159
 [9] S. Zubair and W. Wang, “Tensor dictionary learning with sparse Tucker decomposition,” in Proc. IEEE 18th Int. Conf. Digital Signal Process. (DSP), July 2013, pp. 1–6. [Online]. Available: https://doi.org/10.1109/ICDSP.2013.6622725
 [10] Z. Shakeri, W. U. Bajwa, and A. D. Sarwate, “Minimax lower bounds for Kroneckerstructured dictionary learning,” in Proc. 2016 IEEE Int. Symp. Inf. Theory, July 2016, pp. 1148–1152. [Online]. Available: https://doi.org/10.1109/ISIT.2016.7541479
 [11] ——, “Minimax lower bounds on dictionary learning for tensor data,” arXiv preprint arXiv:1608.02792, August 2016. [Online]. Available: https://arxiv.org/abs/1608.02792

[12]
L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear singular value decomposition,”
SIAM J. Matrix Analy. and Applicat., vol. 21, no. 4, pp. 1253–1278, 2000. [Online]. Available: https://doi.org/10.1137/S0895479896305696  [13] S. Gandy, B. Recht, and I. Yamada, “Tensor completion and lownrank tensor recovery via convex optimization,” Inverse Problems, vol. 27, no. 2, p. 025010, January 2011. [Online]. Available: https://doi.org/10.1088/02665611/27/2/025010
 [14] B. RomeraParedes, H. Aung, N. BianchiBerthouze, and M. Pontil, “Multilinear multitask learning,” in Proc. 30th Int. Conf. Mach. Learn. (ICML), vol. 28, no. 3, Atlanta, Georgia, USA, June 2013, pp. 1444–1452. [Online]. Available: http://proceedings.mlr.press/v28/romeraparedes13.html

[15]
K. Wimalawarne, M. Sugiyama, and R. Tomioka, “Multitask learning meets tensor factorization: Task imputation via convex optimization,” in
Proc. Advances in Neural Inform. Process. Syst. (NIPS), 2014, pp. 2825–2833. [Online]. Available: http://dl.acm.org/citation.cfm?id=2969033.2969142  [16] B. Huang, C. Mu, D. Goldfarb, and J. Wright, “Provable lowrank tensor recovery,” OptimizationOnline, vol. 4252, p. 2, February 2014. [Online]. Available: http://www.optimizationonline.org/DB_FILE/2014/02/4252.pdf
 [17] J. Liu, P. Musialski, P. Wonka, and J. Ye, “Tensor completion for estimating missing values in visual data,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 35, no. 1, pp. 208–220, January 2013. [Online]. Available: http://doi.org/10.1109/TPAMI.2012.39
 [18] T. G. Kolda and B. W. Bader, “Tensor decompositions and applications,” SIAM Review, vol. 51, no. 3, pp. 455–500, August 2009. [Online]. Available: https://doi.org/10.1137/07070111X
 [19] T. Tsiligkaridis and A. O. Hero, “Covariance estimation in high dimensions via Kronecker product expansions,” IEEE Trans. Signal Process., vol. 61, no. 21, pp. 5347–5360, November 2013. [Online]. Available: https://doi.org/10.1109/TSP.2013.2279355
 [20] Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in Proc. 27th Asilomar Conf. Signals, Syst. and Comput., November 1993, pp. 40–44 vol.1. [Online]. Available: https://doi.org/10.1109/ACSSC.1993.342465
 [21] J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory, vol. 53, no. 12, pp. 4655–4666, December 2007. [Online]. Available: https://doi.org/10.1109/TIT.2007.909108
 [22] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn., vol. 3, no. 1, pp. 1–122, January 2011. [Online]. Available: https://doi.org/10.1561/2200000016
 [23] J.F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM J. Optimization, vol. 20, no. 4, pp. 1956–1982, March 2010. [Online]. Available: https://doi.org/10.1137/080738970
 [24] A. Hore and D. Ziou, “Image quality metrics: PSNR vs. SSIM,” in Proc. IEEE int. conf. Pattern recognition (ICPR), August 2010, pp. 2366–2369. [Online]. Available: https://doi.org/10.1109/ICPR.2010.579
Comments
There are no comments yet.