The CANDECOMP PARAFAC (CP) decomposition is a leading method for the analysis of multiway data. The standard alternating least squares algorithm for the CP decomposition (CP-ALS) involves a series of highly overdetermined linear least squares problems. We extend randomized least squares methods to tensors and show the workload of CP-ALS can be drastically reduced without a sacrifice in quality. We introduce techniques for efficiently preprocessing, sampling, and computing randomized least squares on a dense tensor of arbitrary order, as well as an efficient sampling-based technique for checking the stopping condition. We also show more generally that the Khatri-Rao product (used within the CP-ALS iteration) produces conditions favorable for direct sampling. In numerical results, we see improvements in speed, reductions in memory requirements, and robustness with respect to initialization.
An alternative to sketching is to compress the tensor using lossy methods before computation. Zhou and Cichocki examine the effectiveness of performing a CP decomposition on a compressed representation of the data using the lossy Tucker decomposition to produce a smaller problem size @cite_9 . ParCube @cite_15 compresses the original tensor by directly sampling and performs a decomposition on the result.
@cite_9 @cite_15
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Tensor CANDECOMP PARAFAC (CP) decomposition has wide applications in statistical learning of latent variable models and in data mining. In this paper, we propose fast and randomized tensor CP decomposition algorithms based on sketching. We build on the idea of count sketches, but introduce many novel ideas which are unique to tensors. We develop novel methods for randomized computation of tensor contractions via FFTs, without explicitly forming the tensors. Such tensor contractions are encountered in decomposition methods such as tensor power iterations and alternating least squares. We also design novel colliding hashes for symmetric tensors to further save time in computing the sketches. We then combine these sketching ideas with existing whitening and tensor power iterative techniques to obtain the fastest algorithm on both sparse and dense tensors. The quality of approximation under our method does not depend on properties such as sparsity, uniformity of elements, etc. We apply the method for topic modeling and obtain competitive results. Tensor decompositions are promising tools for big data analytics as they bring multiple modes and aspects of data to a unified framework, which allows us to discover complex internal structures and correlations of data. Unfortunately most existing approaches are not designed to meet the major challenges posed by big data analytics. This paper attempts to improve the scalability of tensor decompositions and provides two contributions: A flexible and fast algorithm for the CP decomposition (FFCP) of tensors based on their Tucker compression; A distributed randomized Tucker decomposition approach for arbitrarily big tensors but with relatively low multilinear rank. These two algorithms can deal with huge tensors, even if they are dense. Extensive simulations provide empirical evidence of the validity and efficiency of the proposed algorithms. The Canonical Polyadic Decomposition (CPD) of tensors is a powerful tool for analyzing multi-way data and is used extensively to analyze very large and extremely sparse datasets. The bottleneck of computing the CPD is multiplying a sparse tensor by several dense matrices. Algorithms for tensor-matrix products fall into two classes. The first class saves floating point operations by storing a compressed tensor for each dimension of the data. These methods are fast but suffer high memory costs. The second class uses a single uncompressed tensor at the cost of additional floating point operations. In this work, we bridge the gap between the two approaches and introduce the compressed sparse fiber (CSF) a data structure for sparse tensors along with a novel parallel algorithm for tensor-matrix multiplication. CSF offers similar operation reductions as existing compressed methods while using only a single tensor structure. We validate our contributions with experiments comparing against state-of-the-art methods on a diverse set of datasets. Our work uses 58 less memory than the state-of-the-art while achieving 81 of the parallel performance on 16 threads. Tensor CANDECOMP PARAFAC (CP) decomposition is a powerful but computationally challenging tool in modern data analytics. In this paper, we show ways of sampling intermediate steps of alternating minimization algorithms for computing low rank tensor CP decompositions, leading to the sparse alternating least squares (SPALS) method. Specifically, we sample the the Khatri-Rao product, which arises as an intermediate object during the iterations of alternating least squares. This product captures the interactions between different tensor modes, and form the main computational bottleneck for solving many tensor related tasks. By exploiting the spectral structures of the matrix Khatri-Rao product, we provide efficient access to its statistical leverage scores. When applied to the tensor CP decomposition, our method leads to the first algorithm that runs in sublinear time per-iteration and approximates the output of deterministic alternating least squares algorithms. Empirical evaluations of this approach show significantly speedups over existing randomized and deterministic routines for performing CP decomposition. On a tensor of the size 2.4m by 6.6m by 92k with over 2 billion nonzeros formed by Amazon product reviews, our routine converges in two minutes to the same error as deterministic ALS. Modern applications such as Internet traffic, telecommunication records, and large-scale social networks generate massive amounts of data with multiple aspects and high dimensionalities. Tensors (i.e., multi-way arrays) provide a natural representation for such data. Consequently, tensor decompositions such as Tucker become important tools for summarization and analysis. One major challenge is how to deal with high-dimensional, sparse data. In other words, how do we compute decompositions of tensors where most of the entries of the tensor are zero. Specialized techniques are needed for computing the Tucker decompositions for sparse tensors because standard algorithms do not account for the sparsity of the data. As a result, a surprising phenomenon is observed by practitioners: Despite the fact that there is enough memory to store both the input tensors and the factorized output tensors, memory overflows occur during the tensor factorization process. To address this intermediate blowup problem, we propose Memory-Efficient Tucker (MET). Based on the available memory, MET adaptively selects the right execution strategy during the decomposition. We provide quantitative and qualitative evaluation of MET on real tensors. It achieves over 1000X space reduction without sacrificing speed; it also allows us to work with much larger tensors that were too big to handle before. Finally, we demonstrate a data mining case-study using MET. We propose a method for unsupervised linear feature extraction through tensor decomposition. The linear feature extraction can be formulated as a canonical polyadic decomposition (CPD) of a third-order tensor when transformation matrix is constrained to be equal to the Khatri-Rao product of two matrices. Therefore, standard algorithms for computing CPD decomposition can be used for feature extraction. The proposed method is validated on publicly available low-resolutionmass spectra of cancerous and non-cancerous samples. Obtained results indicate that this approach could be of practical importance in analysis of protein expression profiles. Given an input tensor, its CANDECOMP PARAFAC decomposition (or CPD) is a low-rank representation. CPDs are of particular interest in data analysis and mining, especially when the data tensor is sparse and of higher order (dimension). This paper focuses on the central bottleneck of a CPD algorithm, which is evaluating a sequence of matricized tensor times Khatri-Rao products (MTTKRPs). To speed up the MTTKRP sequence, we propose a novel, adaptive tensor memoization algorithm, AdaTM. Besides removing redundant computations within the MTTKRP sequence, which potentially reduces its overall asymptotic complexity, our technique also allows a user to make a space-time tradeoff by automatically tuning algorithmic and machine parameters using a model-driven framework. Our method improves as the tensor order grows, making its performance more scalable for higher-order data problems. We show speedups of up to 8× and 820× on real sparse data tensors with orders as high as 85 over the SPLATT package and Tensor Toolbox library respectively; and on a full CPD algorithm (CP-ALS), AdaTM can be up to 8× faster than state-of-the-art method implemented in SPLATT. CANDECOMP PARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank . In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. To address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian treatment by incorporating a sparsity-inducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters, resulting in automatic rank determination. To learn the model, we develop an efficient deterministic Bayesian inference algorithm, which scales linearly with data size. Our method is characterized as a tuning parameter-free approach, which can effectively infer underlying multilinear factors with a low-rank constraint, while also providing predictive distributions over missing entries. Extensive simulations on synthetic data illustrate the intrinsic capability of our method to recover the ground-truth of CP rank and prevent the overfitting problem, even when a large amount of entries are missing. Moreover, the results from real-world applications, including image inpainting and facial image synthesis, demonstrate that our method outperforms state-of-the-art approaches for both tensor factorization and tensor completion in terms of predictive performance. Large CNNs have delivered impressive performance in various computer vision applications. But the storage and computation requirements make it problematic for deploying these models on mobile devices. Recently, tensor decompositions have been used for speeding up CNNs. In this paper, we further develop the tensor decomposition technique. We propose a new algorithm for computing the low-rank tensor decomposition for removing the redundancy in the convolution kernels. The algorithm finds the exact global optimizer of the decomposition and is more effective than iterative methods. Based on the decomposition, we further propose a new method for training low-rank constrained CNNs from scratch. Interestingly, while achieving a significant speedup, sometimes the low-rank constrained CNNs delivers significantly better performance than their non-constrained counterparts. On the CIFAR-10 dataset, the proposed low-rank NIN model achieves @math accuracy (without data augmentation), which also improves upon state-of-the-art result. We evaluated the proposed method on CIFAR-10 and ILSVRC12 datasets for a variety of modern CNNs, including AlexNet, NIN, VGG and GoogleNet with success. For example, the forward time of VGG-16 is reduced by half while the performance is still comparable. Empirical success suggests that low-rank tensor decompositions can be a very useful tool for speeding up large CNNs. For @math -dimensional tensors with possibly large @math , an hierarchical data structure, called the Tree-Tucker format, is presented as an alternative to the canonical decomposition. It has asymptotically the same (and often even smaller) number of representation parameters and viable stability properties. The approach involves a recursive construction described by a tree with the leafs corresponding to the Tucker decompositions of three-dimensional tensors, and is based on a sequence of SVDs for the recursively obtained unfolding matrices and on the auxiliary dimensions added to the initial “spatial” dimensions. It is shown how this format can be applied to the problem of multidimensional convolution. Convincing numerical examples are given. Abstract Tensors are valuable tools to represent Electroencephalogram (EEG) data. Tucker decomposition is the most used tensor decomposition in multidimensional discriminant analysis and tensor extension of Linear Discriminant Analysis (LDA), called Higher Order Discriminant Analysis (HODA), is a popular tensor discriminant method used for analyzing Event Related Potentials (ERP). In this paper, we introduce a new tensor-based feature reduction technique, named Higher Order Spectral Regression Discriminant Analysis (HOSRDA), for use in a classification framework for ERP detection. The proposed method (HOSRDA) is a tensor extension of Spectral Regression Discriminant Analysis (SRDA) and casts the eigenproblem of HODA to a regression problem. The formulation of HOSRDA can open a new framework for adding different regularization constraints in higher order feature reduction problem. Additionally, when the dimension and number of samples is very large, the regression problem can be solved via efficient iterative algorithms. We applied HOSRDA on data of a P300 speller from BCI competition III and reached average character detection accuracy of 96.5 for the two subjects. HOSRDA outperforms almost all of other reported methods on this dataset. Additionally, the results of our method are fairly comparable with those of other methods when 5 and 10 repetitions are used in the P300 speller paradigm. Abstract Tensor decomposition, the higher-order analogue to singular value decomposition, has emerged as a useful tool for finding relationships in large, sparse, multidimensional data. As this technique matures and is applied to increasingly larger data sets, the need for high performance implementations becomes critical. A better understanding of the performance characteristics of tensor decomposition on large and sparse tensors can help drive the development of such implementations. In this work, we perform an objective empirical evaluation of three state of the art parallel tools that implement the Canonical Decomposition Parallel Factorization tensor decomposition algorithm using alternating least squares fitting (CP-ALS): SPLATT, DFacTo, and ENSIGN. We conduct performance studies across a variety of data sets and evaluate the tools with respect to total memory required, processor stall cycles, execution time, data distribution, and communication patterns. Furthermore, we investigate the performance of the implementations on tensors with up to 6 dimensions and when executing high rank decompositions. We find that tensor data structure layout and distribution choices can result in differences as large as 14.6x with respect to memory usage and 39.17x with respect to execution time. We provide an outline of a distributed heterogeneous CP-ALS implementation that addresses the performance issues we observe. Abstract: Large CNNs have delivered impressive performance in various computer vision applications. But the storage and computation requirements make it problematic for deploying these models on mobile devices. Recently, tensor decompositions have been used for speeding up CNNs. In this paper, we further develop the tensor decomposition technique. We propose a new algorithm for computing the low-rank tensor decomposition for removing the redundancy in the convolution kernels. The algorithm finds the exact global optimizer of the decomposition and is more effective than iterative methods. Based on the decomposition, we further propose a new method for training low-rank constrained CNNs from scratch. Interestingly, while achieving a significant speedup, sometimes the low-rank constrained CNNs delivers significantly better performance than their non-constrained counterparts. On the CIFAR-10 dataset, the proposed low-rank NIN model achieves @math accuracy (without data augmentation), which also improves upon state-of-the-art result. We evaluated the proposed method on CIFAR-10 and ILSVRC12 datasets for a variety of modern CNNs, including AlexNet, NIN, VGG and GoogleNet with success. For example, the forward time of VGG-16 is reduced by half while the performance is still comparable. Empirical success suggests that low-rank tensor decompositions can be a very useful tool for speeding up large CNNs. In this paper we propose novel methods for compression and recovery of multilinear data under limited sampling. We exploit the recently proposed tensor- Singular Value Decomposition (t-SVD)[1], which is a group theoretic framework for tensor decomposition. In contrast to popular existing tensor decomposition techniques such as higher-order SVD (HOSVD), t-SVD has optimality properties similar to the truncated SVD for matrices. Based on t-SVD, we first construct novel tensor-rank like measures to characterize informational and structural complexity of multilinear data. Following that we outline a complexity penalized algorithm for tensor completion from missing entries. As an application, 3-D and 4-D (color) video data compression and recovery are considered. We show that videos with linear camera motion can be represented more efficiently using t-SVD compared to traditional approaches based on vectorizing or flattening of the tensors. Application of the proposed tensor completion algorithm for video recovery from missing entries is shown to yield a superior performance over existing methods. In conclusion we point out several research directions and implications to online prediction of multilinear data. A recent work (Wang et. al., NIPS 2015) gives the fastest known algorithms for orthogonal tensor decomposition with provable guarantees. Their algorithm is based on computing sketches of the input tensor, which requires reading the entire input. We show in a number of cases one can achieve the same theoretical guarantees in sublinear time, i.e., even without reading most of the input tensor. Instead of using sketches to estimate inner products in tensor decomposition algorithms, we use importance sampling. To achieve sublinear time, we need to know the norms of tensor slices, and we show how to do this in a number of important cases. For symmetric tensors T = ∑ki=1 λiui⊗p with λi > 0 for all i, we estimate such norms in sublinear time whenever p is even. For the important case of p = 3 and small values of k, we can also estimate such norms. For asymmetric tensors sublinear time is not possible in general, but we show if the tensor slice norms are just slightly below || T ||F then sublinear time is again possible. One of the main strengths of our work is empirical - in a number of cases our algorithm is orders of magnitude faster than existing methods with the same accuracy. Many idealized problems in signal processing, machine learning, and statistics can be reduced to the problem of finding the symmetric canonical decomposition of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing inspiration from the matrix case, the successive rank-one approximation (SROA) scheme has been proposed and shown to yield this tensor decomposition exactly, and a plethora of numerical methods have thus been developed for the tensor rank-one approximation problem. In practice, however, the inevitable errors (say) from estimation, computation, and modeling necessitate that the input tensor can only be assumed to be a nearly SOD tensor---i.e., a symmetric tensor slightly perturbed from the underlying SOD tensor. This paper shows that even in the presence of perturbation, SROA can still robustly recover the symmetric canonical decomposition of the underlying tensor. It is shown that when the perturbation error is small enough, the approximation errors do not accumulate with ... Abstract: We propose a simple two-step approach for speeding up convolution layers within large convolutional neural networks based on tensor decomposition and discriminative fine-tuning. Given a layer, we use non-linear least squares to compute a low-rank CP-decomposition of the 4D convolution kernel tensor into a sum of a small number of rank-one tensors. At the second step, this decomposition is used to replace the original convolutional layer with a sequence of four convolutional layers with small kernels. After such replacement, the entire network is fine-tuned on the training data using standard backpropagation process. We evaluate this approach on two CNNs and show that it is competitive with previous approaches, leading to higher obtained CPU speedups at the cost of lower accuracy drops for the smaller of the two networks. Thus, for the 36-class character classification CNN, our approach obtains a 8.5x CPU speedup of the whole network with only minor accuracy drop (1 from 91 to 90 ). For the standard ImageNet architecture (AlexNet), the approach speeds up the second convolution layer by a factor of 4x at the cost of @math increase of the overall top-5 classification error. We propose a simple two-step approach for speeding up convolution layers within large convolutional neural networks based on tensor decomposition and discriminative fine-tuning. Given a layer, we use non-linear least squares to compute a low-rank CP-decomposition of the 4D convolution kernel tensor into a sum of a small number of rank-one tensors. At the second step, this decomposition is used to replace the original convolutional layer with a sequence of four convolutional layers with small kernels. After such replacement, the entire network is fine-tuned on the training data using standard backpropagation process. We evaluate this approach on two CNNs and show that it is competitive with previous approaches, leading to higher obtained CPU speedups at the cost of lower accuracy drops for the smaller of the two networks. Thus, for the 36-class character classification CNN, our approach obtains a 8.5x CPU speedup of the whole network with only minor accuracy drop (1 from 91 to 90 ). For the standard ImageNet architecture (AlexNet), the approach speeds up the second convolution layer by a factor of 4x at the cost of @math increase of the overall top-5 classification error. We investigate an efficient parallelization of the most common iterative sparse tensor decomposition algorithms on distributed memory systems. A key operation in each iteration of these algorithms is the matricized tensor times Khatri-Rao product (MTTKRP). This operation amounts to element-wise vector multiplication and reduction depending on the sparsity of the tensor. We investigate a fine and a coarse-grain task definition for this operation, and propose hypergraph partitioning-based methods for these task definitions to achieve the load balance as well as reduce the communication requirements. We also design a distributed memory sparse tensor library, HyperTensor, which implements a well-known algorithm for the CANDECOMP- PARAFAC (CP) tensor decomposition using the task definitions and the associated partitioning methods. We use this library to test the proposed implementation of MTTKRP in CP decomposition context, and report scalability results up to 1024 MPI ranks. We observed up to 194 fold speedups using 512 MPI processes on a well-known real world data, and significantly better performance results with respect to a state of the art implementation. Confronted with the high-dimensional tensor-like visual data, we derive a method for the decomposition of an observed tensor into a low-dimensional structure plus unbounded but sparse irregular patterns. The optimal rank-(R1,R2, ...Rn) tensor decomposition model that we propose in this paper, could automatically explore the low-dimensional structure of the tensor data, seeking optimal dimension and basis for each mode and separating the irregular patterns. Consequently, our method accounts for the implicit multi-factor structure of tensor-like visual data in an explicit and concise manner. In addition, the optimal tensor decomposition is formulated as a convex optimization through relaxation technique. We then develop a block coordinate descent (BCD) based algorithm to efficiently solve the problem. In experiments, we show several applications of our method in computer vision and the results are promising.