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Six-axis & CP Decompositions

Updated 30 June 2025

Six-axis and CP decompositions are techniques that factorize high-dimensional tensors into minimal rank-one terms, providing a clear framework for structural analysis and identifiability.
Advances like ALS variants, QR/SVD-based methods, and Gauss-Newton optimizations improve stability, convergence, and scalability in complex tensor computations.
These methods are pivotal in practical applications such as signal processing, quantum state classification, and optimization, transforming intricate tensor operations into manageable tasks.

Six-axis and CP Decompositions refer to interrelated lines of research at the intersection of tensor algebra, optimization, and structured matrix analysis. While “six-axis” can, in some contexts, denote order-6 tensor decompositions or the characterization of quantum/mixed states by six distinct structural representations, the core of the literature focuses on Canonical Polyadic (CP) decompositions (also known as CANDECOMP/PARAFAC) and their role in matrix and tensor factorization, algorithmic stability, and the structural understanding of positive semidefinite, completely positive, or partially symmetric objects.

1. Canonical Polyadic (CP) Decomposition: Foundations and Structural Properties

The CP decomposition represents a tensor as a minimal sum of rank-one terms. Formally, for a tensor $\mathcal{T} \in \mathbb{K}^{I_1 \times I_2 \times \cdots \times I_N}$ ,

$\mathcal{T} = \sum_{r=1}^R \mathbf{a}^{(1)}_r \circ \mathbf{a}^{(2)}_r \circ \cdots \circ \mathbf{a}^{(N)}_r,$

where $\mathbf{a}^{(n)}_r \in \mathbb{K}^{I_n}$ are the factor vectors and $R$ is the minimal such number, defining the tensor rank. The CP model generalizes the notion of matrix rank (SVD) to higher-order arrays and is central in signal processing, data mining, chemometrics, and quantum information.

A critical structural result is the essential uniqueness of the CP decomposition under mild conditions, such as Kruskal’s condition: $k_{A^{(1)}} + k_{A^{(2)}} + \cdots + k_{A^{(N)}} \geq 2R + (N - 1),$ where $k_{A^{(n)}}$ is the $k$ -rank of the $n$ -th factor matrix. This property underpins the interpretability and identifiability of tensor components in applications.

2. Algorithmic Advances: Stability, Scalability, and Advanced Optimization

Algorithm development for CP decomposition has addressed numerical instability, computational bottlenecks, and convergence behavior—particularly salient for higher-order tensors (six-axis and beyond).

Alternating Least Squares (ALS) is the most widespread method, alternating updates of factor matrices via least-squares fits. Performance is sensitive to factor collinearity, ill-conditioning, and initialization.
QR- and SVD-based ALS mitigate instability by replacing normal equation solves with QR decomposition or SVD. These yield more stable updates, especially in high-rank or ill-conditioned contexts, without substantial cost increases when the target rank is moderate (2112.10855, 2503.18759).
Dimension tree and branch reutilization accelerate multi-TTM contractions in ALS-QR, reducing leading-order computational cost by up to 33% for third and fourth order tensors (2503.18759).
Gauss-Newton and Newton-like methods recast CP as nonlinear least-squares and leverage structured tensor contractions for implicit matrix computations, achieving superior convergence on challenging problems and strong parallel scalability (1910.12331).
Randomized sketching and Tucker+CP pipelines combine low-rank projection with compressed CP decomposition, achieving both speed and improved accuracy for large and sparse tensors (2104.01101).
Alternating Mahalanobis Distance Minimization (AMDM) introduces a new, adaptive metric for residue minimization, improving conditioning and yielding superlinear local convergence for exact decompositions (2204.07208).

3. Completely Positive (CP) Matrices and Decomposition Rank

The cp-rank of a symmetric nonnegative matrix $Q$ is the minimal $r$ such that $Q = UU^T$ for some nonnegative $U \in \mathbb{R}_+^{n \times r}$ . In binary quadratic programming (BQP) with quadratic constraints of low cp-rank, this structural property enables polynomial-time approximation schemes. For instance, every constraint $x^T Q x \leq C^2$ can be rewritten as $\| U^T x \|_2^2 \leq C^2$ , reducing feasibility to membership in a normed ball in low dimension (1411.5050).

This property is crucial in transforming and approximating hard optimization problems:

PTAS for maximization: For packing-type BQP with linear or certain nonlinear objectives and cp-rank constraints, a PTAS is available if the number of constraints and cp-rank are fixed.
Approximation for submodular objectives: Via geometric and knapsack reductions, submodular maximization subject to cp-rank quadratic constraints admits $1/4 - \epsilon$ or $1 - 1/e - \epsilon$ approximations.
QPTAS for minimization: For covering-type BQP and one cp-constraint, quasi-polynomial-time $(1+\epsilon)$ approximations are possible.

Problem Type	Constraint	Objective	Approximation
Packing, linear objective	cp-rank $r$ , $m$	linear	PTAS
Packing, submodular	cp-rank $r$ , $m$	submodular	$1/4-\epsilon$
Covering, linear (minimize)	cp-rank $r$ , $m=1$	linear	QPTAS
Quadratic/nonlinear objective	cp-rank $r$ , $m$	quadratic/nonlinear	PTAS

The tractability of these approximations is fundamentally controlled by the cp-rank; if it is not constant, no such schemes are possible in general (1411.5050).

4. Six Decompositions of One-dimensional Mixed States and Correspondence with Matrix Factorizations

In quantum information, particularly in the paper of matrix product density operators (MPDOs) and mixed states, there exists a six-axis framework of decompositions, each corresponding to a classical matrix factorization or rank concept (1907.03664). The six decompositions are:

MPDO: minimal factorization ( $\mathrm{rank}(M)$ ),
Local purification: psd factorization ( $\mathrm{psd\text{-}rank}(M)$ ),
Separable decomposition: nonnegative factorization ( $\mathrm{rank}_+(M)$ ),
Translationally invariant (t.i.) MPDO: symmetric factorization,
t.i. separable: completely positive (cp) factorization ( $\mathrm{cp\text{-}rank}(M)$ ),
t.i. local purification: completely positive semidefinite transposed (cpsdt) factorization.

For a bipartite, computational basis-diagonal state $p = \sum_{ij} m_{ij} |i,j\rangle\langle i,j|$ , these structural forms correspond directly to well-founded linear algebraic and convex rank concepts. There exist proven exponential gaps between different ranks (e.g., cp vs. psd rank), implying fundamental limits on the compressibility and simulation of quantum states.

5. Applications in Tensor Computation, Signal Processing, and Quantum Information

CP and related decompositions serve as foundational tools in a broad array of scientific and engineering domains:

Matrix multiplication algorithms: Numerical CP decomposition methods have revealed new low-rank factorizations for matrix product tensors; for example, the $3 \times 3 \times 2$ multiplication tensor is decomposed with only 15 scalar multiplications (1603.01372).
Sparse optimization for CP decomposition: LASSO and group-lasso based techniques efficiently recover CP decompositions using the fewest rank-1 terms, facilitating explicit formulas for classical algebraic structures such as the $4 \times 4$ determinant tensor, which is expressed as a sum of 12 rank-1 tensors (2305.13964).
Quantum state invariants and equivalence: For multipartite quantum systems, the CP decomposition enables a classification of states under local unitary (LU) equivalence by analyzing orbits of factor matrices under local orthogonal transformations and constructing invariants such as tensor and $k$ -ranks, Gram matrix traces, and norms of matricizations (2205.06422).
Low-rank completion and relaxations: For constrained CP and partial symmetry settings, such as conjugate partial-symmetric tensors, matrix unfoldings and CP rank bounds enable tractable nuclear norm surrogates for otherwise intractable tensor rank minimizations (2111.03238).

6. Convergence Theory and Acceleration of CP-ALS for High-Order Tensors

The convergence properties of standard CP-ALS have been quantified for both orthogonally decomposable and incoherently decomposable tensors (2505.14037):

For $N$ -th order orthogonally decomposable tensors, ALs achieves superlinear (polynomial order $N-1$ ) local convergence, measured in the angle between factor vectors,

$\epsilon_k \leq [c(\kappa, R) \cdot \epsilon_{k-1}]^{N-1}.$

For incoherent factors, convergence is linear, with explicit dependence on the (small) mutual coherence $\mu$ .
SVD-based coherence reduction schemes, in which factors are periodically orthogonalized, can empirically accelerate and stabilize ALS, particularly effective where near-orthogonality is nearly satisfied but not exact.

These theoretical advances directly inform practice for six-axis (six-way) CP decompositions, ensuring predictable behavior and guiding initialization and block update choices.

7. Implementation Considerations: Performance, Scalability, and Algorithm Selection

Algorithm choice is context-dependent:

ALS is simple and fast but fragile to ill-conditioning and can stagnate for difficult problems; QR-ALS and SVD-ALS remove these weaknesses at no substantial cost when $R$ is small (2112.10855, 2503.18759).
Dimension tree and branch reutilization are essential for efficient TTM operations for high-order/large-tensor scenarios (2503.18759).
Gauss-Newton/CG methods parallelize well on distributed-memory architectures and overcome ALS “swamps,” at the cost of per-iteration complexity (1910.12331).
Hybrid Mahalanobis/ALS minimizations and coherence reduction allow explicit trade-offs between solution stability and objective fitness (2204.07208, 2505.14037).

Method	Numerical Stability	Scalability	Best Use Cases
ALS	Sensitive	Excellent (MTTKRP, TTM)	Well-conditioned, moderate $R$
ALS-QR/SVD	Robust	Fast if $R$ is small	Ill-conditioned, high-order
Gauss-Newton/CG	Highly robust	Distributed, large tensor	High rank, high accuracy, large $N$
Sketched/Randomized	Robust	Large/sparse data	Data analytics, sparse tensors
SVD-based acceleration	Very robust	Moderate	Nearly orthogonal tensors

Effective deployment of six-axis and CP decompositions requires careful alignment of algorithmic structure to tensor properties and computational resources.

Six-axis and CP decomposition research unifies algebraic, combinatorial, and algorithmic techniques to efficiently decompose, analyze, and optimize high-dimensional arrays—enabling advances in optimization theory, scientific computing, quantum information, and algorithmic linear algebra. Structural properties such as cp-rank, positive semidefiniteness, and symmetry, along with advanced algorithmic strategies (QR, SVD, randomized sketching, coherence reduction, branch reutilization) continue to shape both the theoretical landscape and practical performance envelope of tensor decomposition methods.

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References (12)

CP Decomposition for Tensors via Alternating Least Squares with QR Decomposition (2021)

Efficient QR-Based CP Decomposition Acceleration via Dimension Tree and Extrapolation (2025)

Comparison of Accuracy and Scalability of Gauss-Newton and Alternating Least Squares for CP Decomposition (2019)

Fast and Accurate Randomized Algorithms for Low-rank Tensor Decompositions (2021)

Alternating Mahalanobis Distance Minimization for Stable and Accurate CP Decomposition (2022)

Approximation Schemes for Binary Quadratic Programming Problems with Low cp-Rank Decompositions (2014)

Mixed states in one spatial dimension: decompositions and correspondence with nonnegative matrices (2019)

Numerical CP Decomposition of Some Difficult Tensors (2016)

Finding tensor decompositions with sparse optimization (2023)

10.

Local unitary equivalence of generic multi-qubits based on the CP decomposition (2022)

11.

The low-rank approximation of fourth-order partial-symmetric and conjugate partial-symmetric tensor (2021)

12.

Convergence of the alternating least squares algorithm for CP tensor decompositions (2025)