Bilinear Factorization: Theory and Applications

• Bilinear factorization is a method that represents matrices or tensors as products of lower-rank factors, typically in the form X = ABᵀ.
• It underpins various applications in signal processing, matrix completion, and deep learning by capturing pairwise feature interactions while reducing computation and storage.
• The approach facilitates norm-optimal representations and scalable optimization, connecting operator theory, statistical inference, and combinatorial algorithms.

A bilinear factorization expresses a linear or multilinear object—such as a matrix, tensor, operator, or map—explicitly as a product or sum of products of lower-rank matrices or factors. Formally, bilinear factorization seeks representations of the form $X = AB^T$ or, in more structured contexts, decompositions that minimize some norm or regularizer over the factors, often subject to additional constraints. This paradigm arises in diverse settings, including statistical modeling, linear algebra, optimization, signal processing, operator theory, algebraic structures, polynomial systems, and deep learning. Core motivations are to capture second-order or pairwise feature interactions efficiently, enforce low-rank structure, and reduce computation and storage complexity. Bilinear factorizations also provide norm-optimal representations and are central to modern algorithmic approaches for high-dimensional inference, model compression, and scalable learning.

1. Mathematical Foundations and Canonical Forms

Bilinear factorization is rooted in the representation of a matrix or bilinear form as a product involving lower-dimensional factors. The canonical case is the approximation (or exact representation) $X = AB^T$ for $X \in \mathbb{R}^{m \times n}$, $A \in \mathbb{R}^{m \times r}$, and $B \in \mathbb{R}^{n \times r}$. This decomposition is exact for any $X$ of rank at most $r$. The singular value decomposition (SVD) is a classical instance, yielding the orthogonally optimal bilinear factorization associated with the spectral and Frobenius norms.

The general theory encompasses several induced norms, non-Euclidean Banach settings, and completely bounded norms in operator spaces. Fundamental results such as Grothendieck's inequalities (Christensen, 2023, Christensen, 2023) connect the minimal norm of a bilinear factorization to classical constants and provide unique or near-unique optimal factorizations in several senses. Operator-theoretic settings distinguish between ordinary and completely bounded norms, leading to distinct but closely related matrix factorization identities (see Tables 1 and 2 in (Christensen, 2023)).

In statistical and convex optimization, bilinear parameterization is used to represent matrix norms, e.g., nuclear or Schatten-$q$ norms, as minima over factorizations with specific structure and penalties (Qin et al., 2024, Örnhag et al., 2018). The nuclear norm, for instance, admits the equivalent formulation

$$\|X\|_* = \min_{X=AB^T} \frac{1}{2}\|A\|_F^2 + \frac{1}{2}\|B\|_F^2.$$

2. Bilinear Factorization in Statistical Learning and Machine Learning

Bilinear models are omnipresent in matrix completion, collaborative filtering, robust PCA, and representation learning. Random effects models for binary response data (BIRE) use bilinear factorization to model user-item interactions via latent factors:

$$y_{ij} \sim \mathrm{Bernoulli}(p_{ij}),\quad s_{ij} = f(x_{ij}) + \alpha_i + \beta_j + u_i^T v_j$$

where $u_i, v_j$ are per-user and per-item factors, and the log-odds $s_{ij}$ combines fixed and random (bilinear) effects (Khanna et al., 2012). Inference in these models with massive or imbalanced data leverages scalable parallel algorithms (MapReduce, ensemble), and highly accurate sampling-based methods (ARS) are required to avoid the overshrinkage that plagues variational approximations.

For robust PCA and matrix completion, nonconvex quasi-norms and weighted Schatten-$q$ quasi-norms are regularized via bilinear parameterization, enabling efficient ADMM algorithms and improved computational scalability (Qin et al., 2024). The bilinear form smooths otherwise nonsmooth regularizers, allowing the use of second-order methods (LM/VarPro) for efficient optimization (Örnhag et al., 2018).

In deep learning, bilinear layers model second-order feature interactions:

$$y = b + w^T x + x^T F^T F x \qquad (F \in \mathbb{R}^{k \times n})$$

with low-rank $F$ to control parameter growth and computational budget (Li et al., 2016). DropFactor regularization randomly drops bilinear rank-1 factors to improve generalization. These layers improve empirical performance in vision tasks with modest increase in parameters and flops.

3. Bilinear Factorization in Optimization and Message Passing

Bilinear factorization underpins scalable optimization for semidefinite programming and related large-scale convex problems. Replacing an $n \times n$ PSD matrix $Z$ by $Z = X Y^T$ recasts the problem into a biconvex surrogate:

$F(X,Y;\gamma) = f(XY^T) + (\gamma/2)\|X-Y\|_F^2$

which admits efficient block-wise alternating minimization and strong theoretical connections to the Burer-Monteiro quadratic factorization for sufficiently large $\gamma$ (Hu, 2018).

These ideas extend to inference under probabilistic and statistical generative models. Message passing (e.g., AMP, BiG-AMP) on factor graphs with bilinear cores enables scalable MAP or MMSE inference in matrix/tensor models, e.g., hyperspectral unmixing (Vila et al., 2015). Recent developments use hybrid vector message passing (HVMP) combining expectation propagation and variational message passing, operating on matrix-valued variables, resulting in loop-free, Gaussian-message factor graphs with superior convergence and NMSE performance (Jiang et al., 2024).

4. Operator Theory, Norm-Optimality, and Duality

Operator-space theory provides a deep setting for bilinear factorization. Christensen's theorems (Christensen, 2023, Christensen, 2023) establish explicit norm-optimal factorizations in four senses: cb-operator, cb-bilinear-form, Schur, and bilinear-Schur. The classical Grothendieck inequalities furnish upper bounds for completely bounded norms in terms of standard operator or form norms. For $X \in M(m,n)$, the optimal factorization identities are as follows:

Structure	Factorization	Minimal Product (Norm)
cb-operator	$X = A \,\mathrm{diag}(\varphi)$	$\|A\|\|\varphi\|_2$
cb-bilinear form	$X = \mathrm{diag}(\eta) B \mathrm{diag}(\varphi)$	$\|\eta\|_2\|B\|\|\varphi\|_2$
Schur multiplier	$X = L^* R$	$\|L\|_c \|R\|_c$
bilinear-Schur	$X = \mathrm{diag}(\eta) L^* R$	$\|\eta\|_2\|L\|_c\|R\|_c$

Uniqueness results hold for all but the Schur case (where additional conditions provide uniqueness), and polars under trace-duality intertwine the bilinear and Schur norm balls. The framework unifies the analysis of operator norm inequalities, Schur multipliers, and matrix factorizations.

5. Algebraic and Combinatorial Aspects

Bilinear factorization extends beyond numerical or analytical linear algebra to encompass algebraic and combinatorial structures. In the theory of associative algebras over a commutative ring, a "bilinear factorization" means realizing a given algebra $R$ as a weak wreath product $B \, \underline{\otimes} \, A$ via algebra maps $\alpha: A \to R$, $\beta: B \to R$, with a splitting of the canonical bimodule projection $B \otimes A \to R$ (Böhm et al., 2011). Bicategorical frameworks establish a biequivalence between the categories of weak distributive laws and bilinear factorizations.

In combinatorics and harmonic analysis, weak factorization of Hardy spaces (and dual BMO spaces) is realized by decomposing functions as sums of bilinear forms involving singular integrals (e.g., Riesz transforms), with precise norm control and atomic representations (Duong et al., 2015).

Factorization theory also appears in the context of biorthogonal polynomials, where the Gauss-Borel (or $LU$) factorization of the Gram matrix defines the associated families and kernels. Structural consequences include explicit construction of Christoffel-Darboux kernels, recurrence relations, and perturbation formulas (Mañas, 2019).

6. Applications and Practical Schemes

Applications of bilinear factorization span scientific computation, statistical inference, data mining, signal and image processing, and optimization. Key examples include:

• Deep neural networks: Factorized Bilinear (FB) layers improve vision benchmarks by modeling pairwise feature interactions with modest resource impact (Li et al., 2016).
• Multi-view and multi-modal learning: Bilinear factorizations with shared encoding matrices, coupled with per-view orthonormality constraints and trace-norm penalties, enforce consensus subspace structure in clustering (Zheng et al., 2019).
• Message-passing and graphical models: Bilinear factorization enables efficient inference in large-scale bilinear systems, e.g., hyperspectral unmixing (Vila et al., 2015), generalized bilinear factorization with HVMP (Jiang et al., 2024).
• Low-rank modeling: Bilinear parameterizations are foundational in matrix completion, robust PCA, and rank-constrained regularization, enabling efficient ADMM- and second-order algorithms (Qin et al., 2024, Örnhag et al., 2018).
• Deterministic parallel algorithms: Many randomized estimation and derandomization tasks for combinatorial optimization (MIS, discrepancy, automata-fooling) admit efficient PRAM algorithms once recast via bilinear expectation factorization (Harris, 2017).

7. Theoretical Generalizations and Future Directions

Recent developments indicate broad directions for theoretical generalization and rigorous analysis:

• Theory of uniqueness and optimality: Recent operator-space results establish necessary and sufficient conditions for the uniqueness of norm-minimal factorizations, linking to dual extremality and polar sets (Christensen, 2023).
• Robustness and heavy-tailed data: Robust bilinear factor analysis builds on matrix-variate $t$ distributions, providing high-breakdown estimators and closed-form Fisher information for contaminated or heavy-tailed matrix data (Ma et al., 2024).
• Generalization to non-Euclidean settings: Bilinear factorization in Banach spaces using induced norms extends the classical SVD, yielding a taxonomy of centroid, taxicab, extreme, and spectral decompositions, with applications to new variants of multidimensional scaling (Choulakian, 2015).
• Superintegrability and character factorization: In integrable matrix models, bilinear factorization of character correlators via commuting families of differential operators diagonalizes infinite families of invariant correlators, revealing deep algebraic and combinatorial structures (Mironov et al., 2022).

The field continues to expand into tensor factorization, quantum information, automated reasoning for operator and algebraic structures, and high-dimensional inference, with bilinear factorization providing a unifying and computationally tractable perspective.