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Tensor
Multidimensional data structure generalizing vectors and matrices, represented by an N-order array where N is the number of dimensions or modes.
CANDECOMP/PARAFAC (CP) Decomposition
Tensor factorization decomposing a tensor into a minimal sum of rank-one tensors, representing latent interactions between different modes.
Unique Mode Decomposition (Unique PARAFAC)
Variant of CP decomposition imposing a uniqueness constraint on the factors of each mode to improve interpretability and solution stability.
Tucker Decomposition
Tensor factorization decomposing a tensor into a core tensor and factor matrices for each mode, allowing flexible data compression.
Core Tensor
In Tucker decomposition, a smaller multidimensional tensor that captures interactions between the principal components of each mode.
Tensor Rank
Minimum number of rank-one tensors required to express a tensor as a sum, generalizing the concept of matrix rank to multidimensional structures.
n-mode Matrix Product
Fundamental operation multiplying a tensor by a matrix along a specific mode, essential for tensor transformations and decompositions.
Higher-Order Singular Value Decomposition (HOSVD)
Extension of SVD to tensors, computing orthogonal factor matrices for each mode and the corresponding core tensor in Tucker decomposition.
Coupled Matrix/Tensor Factorization
Approach simultaneously factorizing multiple tensors and/or matrices by sharing common factors to integrate heterogeneous data sources.
Tensor Train (TT) Decomposition
Decomposition representing a high-order tensor as a chain of third-order tensors (cores), drastically reducing the number of parameters for high-dimensional tensors.
Hierarchical Tucker Decomposition
Recursive tensor decomposition structure organizing core tensors in a tree for adaptive and efficient compression of multidimensional data.
Tensor Completion
Problem of estimating missing entries of a tensor by exploiting its underlying low-rank structure, generalizing matrix completion.
Non-negative CP Decomposition
Variant of CP decomposition constraining all factors to be non-negative, ensuring additive interpretability of components for applications like signal analysis.
Tensor Regularization
Techniques adding constraints on tensor factors (such as sparsity or smoothing) to prevent overfitting and improve model generalization.
Unfolding (Matricization)
Operation reorganizing tensor elements into a matrix by concatenating its fibers along a specific mode, enabling the application of matrix algorithms.
Tensor Fiber
Vector obtained by fixing all indices of a tensor except one, representing a column, row, or tube depending on the selected mode for analysis.
Tensor Slice
Lower-order sub-tensor obtained by fixing one or more indices, producing a matrix or vector for the analysis of local structures.
Block Diagonal Tensor Decomposition
Constrained factorization method where the core tensor is structured in diagonal blocks, revealing clusters or groupings in multidimensional data.
Tensor Orthogonality
Property of factors in certain decompositions where the factor matrices for different modes are orthogonal, simplifying interpretation and computation.
Tensor Ring (TR) Decomposition
Generalization of Tensor Train decomposition where the core tensors are connected in a ring, offering a more flexible representation for certain types of data.