Struct CpDecomposition 

pub struct CpDecomposition<R: Runtime> {
    pub factors: Vec<Tensor<R>>,
    pub weights: Tensor<R>,
}

CP/PARAFAC decomposition result: T ≈ Σᵣ λᵣ × a₁ʳ ⊗ a₂ʳ ⊗ … ⊗ aₙʳ

The Canonical Polyadic (CP) decomposition, also known as PARAFAC (Parallel Factor Analysis), factors an N-dimensional tensor as a sum of R rank-one tensors.

Mathematical Definition

For a tensor T of shape [I₁, I₂, ..., Iₙ], the CP decomposition is:

T ≈ Σᵣ₌₁ᴿ λᵣ × a₁ʳ ⊗ a₂ʳ ⊗ ... ⊗ aₙʳ

where:

• R is the CP rank (number of components)
• λᵣ are scalar weights (can be absorbed into factors)
• aₖʳ are column vectors of factor matrix Aₖ [Iₖ, R]
• ⊗ denotes outer product

Matrix Notation

Using factor matrices A₁, A₂, ..., Aₙ where Aₖ has shape [Iₖ, R]:

T_{(1)} ≈ A₁ @ diag(λ) @ (Aₙ ⊙ ... ⊙ A₂)ᵀ

where ⊙ denotes Khatri-Rao (column-wise Kronecker) product.

Element-wise Formula

T[i₁, i₂, ..., iₙ] ≈ Σᵣ₌₁ᴿ λᵣ × A₁[i₁, r] × A₂[i₂, r] × ... × Aₙ[iₙ, r]

Properties

• Uniqueness: Unique under mild conditions (unlike Tucker)
• Interpretability: Each component is a rank-1 tensor (separable)
• Compression: Parameters = R × (1 + Σ Iₖ) vs ∏ Iₖ for full tensor
• Diagonal core: Equivalent to Tucker with superdiagonal core

Use Cases

• Chemometrics (fluorescence spectroscopy)
• Psychometrics (multi-trait, multi-method)
• Signal processing (source separation)
• Recommender systems (user-item-context)
• Topic modeling

Fields

factors: Vec<Tensor<R>>

Factor matrices [A₁, A₂, ..., Aₙ], one per mode

Each Aₖ has shape [Iₖ, R] where Iₖ is the dimension along mode k and R is the CP rank.

Column r of each factor matrix corresponds to component r.

weights: Tensor<R>

Weights (lambda) for each rank-1 component [R]

Scalar weights that can optionally be absorbed into factor matrices. For normalized factors (columns have unit norm), weights capture the “strength” of each component.