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Local Convergence of the Alternating Least Squares Algorithm For Canonical Tensor Approximation
, 2011
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Temporal Collaborative Filtering with Bayesian Probabilistic Tensor Factorization
"... Realworld relational data are seldom stationary, yet traditional collaborative filtering algorithms generally rely on this assumption. Motivated by our sales prediction problem, we propose a factorbased algorithm that is able to take time into account. By introducing additional factors for time, w ..."
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Cited by 61 (2 self)
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Realworld relational data are seldom stationary, yet traditional collaborative filtering algorithms generally rely on this assumption. Motivated by our sales prediction problem, we propose a factorbased algorithm that is able to take time into account. By introducing additional factors for time, we formalize this problem as a tensor factorization with a special constraint on the time dimension. Further, we provide a fully Bayesian treatment to avoid tuning parameters and achieve automatic model complexity control. To learn the model we develop an efficient sampling procedure that is capable of analyzing largescale data sets. This new algorithm, called Bayesian Probabilistic Tensor Factorization (BPTF), is evaluated on several realworld problems including sales prediction and movie recommendation. Empirical results demonstrate the superiority of our temporal model. 1
On local convergence of alternating schemes for optimization of convex problems in the tensor train format
 SIAM J. Numer. Anal
"... Abstract. Alternating linear schemes (ALS), with the Alternating Least Squares algorithm a notable special case, provide one of the simplest and most popular choices for the treatment of optimization tasks by tensor methods. An according adaptation of ALS for the recent TT ( = tensor train) format ( ..."
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Cited by 56 (4 self)
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Abstract. Alternating linear schemes (ALS), with the Alternating Least Squares algorithm a notable special case, provide one of the simplest and most popular choices for the treatment of optimization tasks by tensor methods. An according adaptation of ALS for the recent TT ( = tensor train) format (Oseledets, 2011), known in quantum computations as matrix product states, has recently been investigated in (Holtz, Rohwedder, Schneider, 2012). With the present work, the positive practical experience with TTALS is backed up with an according local linear convergence theory for the optimization of convex functionals J. The main assumption entering the proof is that the redundancy introduced by the TT parametrization τ matches the null space of the Hessian of the induced functional j = J ◦ τ, and we give conditions under which this assumption can be expected to hold. In particular, this is the case if the TT rank has been correctly estimated. The case of nonconvex functionals J is also shortly discussed. Key words. ALS, highdimensional optimization, local convergence, matrix product states, nonlinear GaussSeidel, tensor product approximation, TT decomposition AMS subject classifications. 15A69, 65K10, 90C06
the Spectral Algorithm.
"... Here we give a brief introduction to tensor algebra (for more details, see [2]). A tensor is a multidimensional array, and its order is the number of dimensions, also known as modes. In this paper, vectors (tensors of order one) are denoted by boldface lowercase letters, e.g., a. Matrices (tensors o ..."
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Cited by 53 (9 self)
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Here we give a brief introduction to tensor algebra (for more details, see [2]). A tensor is a multidimensional array, and its order is the number of dimensions, also known as modes. In this paper, vectors (tensors of order one) are denoted by boldface lowercase letters, e.g., a. Matrices (tensors of order two) are denoted by boldface capital letters, e.g., A. Higherorder tensors (order three or higher) are denoted by boldface caligraphic letters, e.g., T. Scalars are denoted by lowercase letters, e.g., a. Subarrays of a tensor are formed when a subset of the indices is fixed. Particularly, a fiber is defined by fixing every index but one. Fibers are the higherorder analogue of matrix rows and columns. A colon is used to indicate all elements of a mode. Thus, the jth column of a matrix A is A(:, j), and the ith row of A is A(i,:). Analogously, the moden fiber of a Nth order tensor T is then denoted as T (i1, i2,..., in−1,:, in+1,..., iN). Tensors can be multiplied together. For matrices and vectors, we will use standard notation for their multiplications, e.g., Ba and AB. For tensors of higher order, we are particularly interested in multiplying a tensor by matrices and vectors. The nmode matrix product is the multiplication of a tensor with a matrix in mode n of the tensor. Let T ∈ R I1×I2×...×IN be an Nth order tensor and A ∈ R J×In be a matrix. Then T ′ = T ×n A ∈ R I1×...In−1×J×In+1×...×IN, (1) where the entries T ′ (i1,..., in−1, j, in+1,..., iN) are defined as ∑ In
Triplerank: Ranking semantic web data by tensor decomposition
 In ISWC
, 2009
"... Abstract. The Semantic Web fosters novel applications targeting a more efficient and satisfying exploitation of the data available on the web, e.g. faceted browsing of linked open data. Large amounts and high diversity of knowledge in the Semantic Web pose the challenging question of appropriate rel ..."
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Cited by 53 (0 self)
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Abstract. The Semantic Web fosters novel applications targeting a more efficient and satisfying exploitation of the data available on the web, e.g. faceted browsing of linked open data. Large amounts and high diversity of knowledge in the Semantic Web pose the challenging question of appropriate relevance ranking for producing finegrained and rich descriptions of the available data, e.g. to guide the user along most promising knowledge aspects. Existing methods for graphbased authority ranking lack support for finegrained latent coherence between resources and predicates (i.e. support for link semantics in the linked data model). In this paper, we present TripleRank, a novel approach for faceted authority ranking in the context of RDF knowledge bases. TripleRank captures the additional latent semantics of Semantic Web data by means of statistical methods in order to produce richer descriptions of the available data. We model the Semantic Web by a 3dimensional tensor that enables the seamless representation of arbitrary semantic links. For the analysis of that model, we apply the PARAFAC decomposition, which can be seen as a multimodal counterpart to Web authority ranking with HITS. The result are groupings of resources and predicates that characterize their authority and navigational (hub) properties with respect to identified topics. We have applied TripleRank to multiple data sets from the linked open data community and gathered encouraging feedback in a user evaluation where TripleRank results have been exploited in a faceted browsing scenario. 1
Factorizing YAGO: scalable machine learning for linked data
 In WWW
, 2012
"... Vast amounts of structured information have been published in the Semantic Web’s Linked Open Data (LOD) cloud and their size is still growing rapidly. Yet, access to this information via reasoning and querying is sometimes difficult, due to LOD’s size, partial data inconsistencies and inherent noisi ..."
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Cited by 49 (15 self)
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Vast amounts of structured information have been published in the Semantic Web’s Linked Open Data (LOD) cloud and their size is still growing rapidly. Yet, access to this information via reasoning and querying is sometimes difficult, due to LOD’s size, partial data inconsistencies and inherent noisiness. Machine Learning offers an alternative approach to exploiting LOD’s data with the advantages that Machine Learning algorithms are typically robust to both noise and data inconsistencies and are able to efficiently utilize nondeterministic dependencies in the data. From a Machine Learning point of view, LOD is challenging due to its relational nature and its scale. Here, we present an efficient approach to relational learning on LOD data, based on the factorization of a sparse tensor that scales to data consisting
Fast Local Algorithms for Large Scale Nonnegative Matrix and Tensor Factorizations
, 2008
"... Nonnegative matrix factorization (NMF) and its extensions such as Nonnegative Tensor Factorization (NTF) have become prominent techniques for blind sources separation (BSS), analysis of image databases, data mining and other information retrieval and clustering applications. In this paper we propose ..."
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Cited by 49 (13 self)
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Nonnegative matrix factorization (NMF) and its extensions such as Nonnegative Tensor Factorization (NTF) have become prominent techniques for blind sources separation (BSS), analysis of image databases, data mining and other information retrieval and clustering applications. In this paper we propose a family of efficient algorithms for NMF/NTF, as well as sparse nonnegative coding and representation, that has many potential applications in computational neuroscience, multisensory processing, compressed sensing and multidimensional data analysis. We have developed a class of optimized local algorithms which are referred to as Hierarchical Alternating Least Squares (HALS) algorithms. For these purposes, we have performed sequential constrained minimization on a set of squared Euclidean distances. We then extend this approach to robust cost functions using the Alpha and Beta divergences and derive flexible update rules. Our algorithms are locally stable and work well for NMFbased blind source separation (BSS) not only for the overdetermined case but also for an underdetermined (overcomplete) case (i.e., for a system which has less sensors than sources) if data are sufficiently sparse. The NMF learning rules are extended and generalized for Nth order nonnegative tensor factorization (NTF). Moreover, these algorithms can be tuned to different noise statistics by adjusting a single parameter. Extensive experimental results confirm the accuracy and computational performance of the developed algorithms, especially, with usage of multilayer hierarchical NMF approach [3].
Most tensor problems are NP hard
 CORR
, 2009
"... The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has ..."
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Cited by 45 (6 self)
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The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has attracted a lot of attention recently. We examine here the computational tractability of some core problems in numerical multilinear algebra. We show that tensor analogues of several standard problems that are readily computable in the matrix (i.e. 2tensor) case are NP hard. Our list here includes: determining the feasibility of a system of bilinear equations, determining an eigenvalue, a singular value, or the spectral norm of a 3tensor, determining a best rank1 approximation to a 3tensor, determining the rank of a 3tensor over R or C. Hence making tensor computations feasible is likely to be a challenge.
Statistical Performance of Convex Tensor Decomposition
"... We analyze the statistical performance of a recently proposed convex tensor decomposition algorithm. Conventionally tensor decomposition has been formulated as nonconvex optimization problems, which hindered the analysis of their performance. We show under some conditions that the mean squared erro ..."
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Cited by 36 (5 self)
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We analyze the statistical performance of a recently proposed convex tensor decomposition algorithm. Conventionally tensor decomposition has been formulated as nonconvex optimization problems, which hindered the analysis of their performance. We show under some conditions that the mean squared error of the convex method scales linearly with the quantity we call the normalized rank of the true tensor. The current analysis naturally extends the analysis of convex lowrank matrix estimation to tensors. Furthermore, we show through numerical experiments that our theory can precisely predict the scaling behaviour in practice. 1