The Expression of a Tensor or a Polyadic as a Sum of Products

By F. L. Hitchcock. 1927

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Modeling Relational Data with Graph Convolutional Networks (2017, methodology): “We optimize for cross-entropy loss to push the model to score observable triples higher than the negative ones: L = − 1 (1 + ω)|Ê | ∑ (s,r,o,y)∈T y log l ( f(s, r, o) ) + (1− y) log ( 1− l ( f(s, r, o) )) , (7) where T is the total set of real and corrupted triples, l is the logistic sigmoid…”
Complex Embeddings for Simple Link Prediction (2016, methodology): “Canonical Polyadic (CP) decomposition (Hitchcock, 1927) on both relations as it has to push symmetric and antisymmetric patterns through the entity embeddings.”
PCA: The Basic Building Block of Chemometrics (2012, methodology): “For a complete and very pedagogical comparison of Tucker3 with PARAFAC, another multiway procedure, see Jiang [35].”
T-Basis: a Compact Representation for Neural Networks (2020, methodology): “For example, tensor decompositions such as the CP-decomposition (Hitchcock, 1927) and the Tucker decomposition (Tucker, 1963) (see review (Kolda & Bader, 2009)) have been used for compression purposes (Lebedev et al.”
Most Tensor Problems Are NP-Hard (2009, background): “More generally, the rank of a tensor A = JaijkK i,j,k=1 ∈ Fl×m×n, denoted rank(A), is defined as the minimum r for which A may be expressed as a sum of r rank-1 tensors [27, 28],”
Jacobi Algorithm for the Best Low Multilinear Rank Approximation of Symmetric Tensors (2013, background): “The concept of multilinear rank was first introduced in [25, 26] and is simply a generalization of the row and column rank of matrices to higher-order tensors.”
Canonical polyadic decomposition of third-order tensors: relaxed uniqueness conditions and algebraic algorithm (2015, methodology): “hen the CPD is not unique. Nevertheless, the third factor matrix can still be unique [7, Example 4.11]. The literature shows some variation in terminology. The CPD was introduced by F.L. Hitchcock in [16] and was later referred to as Canonical Decomposition (CANDECOMP) [2], Parallel Factor Model (PA”
Tensor Decomposition for Signal Processing and Machine Learning (2016, background): “Tensor decomposition has come a long way since Hitchcock ’27, [141], Cattell ’44 [142], and later Harshman ’70’72 [31], [32], Carroll and Chang [143], and Kruskal’s ’77 [35] seminal papers.”
Tensor Decompositions for temporal knowledge base completion (2020, background): “The Canonical Polyadic (CP) Decomposition (Hitchcock, 1927) is the tensor equivalent to the low-rank decomposition of a matrix.”
Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains (2014, background): “Recent papers have attempted to address the curse of dimensionality by using a low-parametric representation of tensors known as canonical polyadic decomposition or CANDECOMP/ PARAFAC, both notions being subsumed under the acronym CP [27,28].”
A Spectral Algorithm for Latent Dirichlet Allocation (2012, background): “This is known as a symmetric canonical polyadic decomposition [24,25,34] (or a Tucker decomposition with a diagonal core tensor [44]).”
Non-intrusive low-rank separated approximation of high-dimensional stochastic models (2012, background): “Separated representations, also known as canonical decompositions (CANDECOMP) or parallel factor analysis (PARAFAC), have been first introduced by Hitchcock [39] to represent a multi-way tensor as a finite sum of rank-one tensors.”
O(dlog N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling (2011, background): “The relation to a separable representation of polyadics was emphasized in [20, 6].”
How hard is the tensor rank (2016, background): “Deﬁnition 10.”
Deep Multitask Learning for Semantic Dependency Parsing (2017, methodology): “Using the fact that a tensor of rank at most r can be decomposed into a sum of r rank-1 tensors (Hitchcock, 1927), we reparameterize W to enforce the low-rank constraint by construction: where LA ∈ r × d are now our parameters. the [ ] j, : denotes j th row of a matrix.”