Face-Splitting Product

Overview

The face-splitting product (denoted by ⊖ in this package) is a row-wise Kronecker product operation. Given two matrices $\mathbf{A} \in \mathbb{R}^{m \times n}$ and $\mathbf{B} \in \mathbb{R}^{m \times p}$ with the same number of rows, their face-splitting product $\mathbf{A} \ominus \mathbf{B}$ is an $m \times np$ matrix where each row is the Kronecker product of the corresponding rows of $\mathbf{A}$ and $\mathbf{B}$:

\[\mathbf{A} \ominus \mathbf{B} = \begin{bmatrix} (\mathbf{a}_1^\top \otimes \mathbf{b}_1^\top)^\top \\[0.2em] (\mathbf{a}_2^\top \otimes \mathbf{b}_2^\top)^\top \\[0.2em] \vdots \\[0.2em] (\mathbf{a}_m^\top \otimes \mathbf{b}_m^\top)^\top \end{bmatrix}\]

where $\mathbf{a}_i^\top$ and $\mathbf{b}_i^\top$ are the $i$-th rows of $\mathbf{A}$ and $\mathbf{B}$, respectively.

Standard Face-Splitting Product

Usage Examples

using UniqueKronecker

A = [1 2; 3 4]
B = [5 6; 7 8]

# Standard face-splitting product
fs = face_split(A, B)
# Equivalent to:
# Row 1: kron([1,2], [5,6]) = [5, 6, 10, 12]
# Row 2: kron([3,4], [7,8]) = [21, 24, 28, 32]

Unicode Operator

You can use the Unicode operator ⊖:

fs = A ⊖ B  # Same as face_split(A, B)

Self-Product

When called with a single matrix argument, it computes the face-splitting product of the matrix with itself:

fs_self = face_split(A)  # Same as face_split(A, A)
fs_self = A ⊖ A          # or using the operator

Multiple Matrices

The face-splitting product can be extended to multiple matrices:

A = [1 2; 3 4]
B = [5 6; 7 8]
C = [9 10; 11 12]

fs_multi = face_split(A, B, C)
fs_multi = A ⊖ B ⊖ C  # Using operators

Power Form

You can compute the $d$-th face-splitting power of a matrix, where each row is raised to the Kronecker power $d$:

A = [1 2; 3 4]

# Second face-splitting power
fs_power2 = face_split(A, 2)  # Each row: kron(A[i,:], A[i,:])
fs_power2 = A ⊖ 2

# Third face-splitting power
fs_power3 = face_split(A, 3)  # Each row: kron(kron(A[i,:], A[i,:]), A[i,:])
fs_power3 = A ⊖ 3

Unique Face-Splitting Product

The unique face-splitting product (denoted by ⧁ in this package) eliminates redundant terms in each row by applying the unique Kronecker product to each row pair:

\[\mathbf{A}\,⧁\,\mathbf{B} = \begin{bmatrix} (\mathbf{a}_1^\top \oslash \mathbf{b}_1^\top)^\top \\[0.2em] (\mathbf{a}_2^\top \oslash \mathbf{b}_2^\top)^\top \\[0.2em] \vdots \\[0.2em] (\mathbf{a}_m^\top \oslash \mathbf{b}_m^\top)^\top \end{bmatrix}\]

Usage Examples

using UniqueKronecker

A = [1 2; 3 4]
B = [5 6; 7 8]

# Unique face-splitting product
ufs = unique_face_split(A, B)
ufs = A ⧁ B  # Using the operator

# For self-product with redundancies eliminated
ufs_self = unique_face_split(A)
ufs_self = A ⧁ A

Power Form with Unique Products

A = [1 2; 3 4]

# Second unique face-splitting power
ufs_power2 = unique_face_split(A, 2)
ufs_power2 = A ⧁ 2

# Third unique face-splitting power
ufs_power3 = unique_face_split(A, 3)
ufs_power3 = A ⧁ 3

Applications

Matrix Polynomial Operations

The face-splitting product is useful for representing polynomial operations on matrix rows:

# Data matrix
X = rand(100, 5)

# Create quadratic features row-wise (each sample independently)
X_quad = unique_face_split(X, 2)

# Each row of X_quad contains unique quadratic terms for that sample

Dynamical Systems

In polynomial dynamical systems, face-splitting products can represent state transformations:

# State trajectory matrix (each row is a time point)
X = rand(50, 3)

# Create polynomial features for each time point
X_poly = unique_face_split(X, 2)

# Use in dynamical system identification
# X_dot = X_poly * coefficients

Structured Matrix Computations

For operations requiring row-wise polynomial expansions:

# Input data with multiple observations
A = rand(10, 4)

# Create second-order row-wise features
A2 = face_split(A, 2)

# Each row of A2 contains all pairwise products for that observation

Properties

Size

For $\mathbf{A} \in \mathbb{R}^{m \times n}$ and $\mathbf{B} \in \mathbb{R}^{m \times p}$:

• Standard face-splitting: $\mathbf{A} \ominus \mathbf{B} \in \mathbb{R}^{m \times np}$
• Unique face-splitting (when $n = p$): Result has $\frac{n(n+1)}{2}$ columns for the self-product

Relationship to Kronecker Product

The face-splitting product can be viewed as a row-wise application of the Kronecker product:

A = [1 2; 3 4]
B = [5 6; 7 8]

# Face-splitting
fs = A ⊖ B

# Manual construction
fs_manual = vcat([transpose(kron(A[i,:], B[i,:])) for i in 1:size(A,1)]...)

@assert fs == fs_manual

Comparison: Khatri-Rao vs Face-Splitting

API Reference

face_split(A::AbstractMatrix, B::AbstractMatrix)

face_split(mats::AbstractMatrix...)

face_split(A::AbstractMatrix, d::Integer)

unique_face_split(A::AbstractMatrix, B::AbstractMatrix)

unique_face_split(mats::AbstractMatrix...)

unique_face_split(A::AbstractMatrix, d::Integer)

A ⊖ B

A ⧁ B