Diagonal-Plus-Low-Rank Transition Matrices

• DPLR matrices are defined as the sum of a diagonal matrix and a low-rank term, balancing numerical efficiency with the ability to model global interactions.
• They support fast eigenvalue computation, matrix inversion, and determinant evaluations through structured QR iterations and Givens rotation algorithms.
• Their versatile framework advances applications in scientific computing, machine learning, and control theory by enabling scalable operator sketching and efficient system updates.

A Diagonal-Plus-Low-Rank (DPLR) transition matrix is a matrix expressible as the sum of a diagonal matrix and a low-rank term, a structure that arises in numerous computational mathematics, scientific computing, machine learning, and control-theoretic contexts. This hybrid structure provides a tractable balance between expressiveness (modeling nontrivial global interactions via the low-rank factor) and numerical efficiency (thanks to the diagonal component’s simplicity). The DPLR framework generalizes diagonal-plus-rank-one (DPR1) and more general diagonal-plus-low-rank constructs, and it plays a pivotal role in fast eigenvalue computations, matrix equations, approximate diagonalization, and large-scale operator sketching.

1. Mathematical Structure and Definitions

DPLR matrices are defined as

$A = D + UV^*$

where $D$ is a diagonal matrix (entries in $\mathbb{R}$ or $\mathbb{C}$), and $U, V \in \mathbb{C}^{n \times k}$ (with $k \ll n$), or variations thereof (including symmetric, Hermitian, or real restrictions). The rank $k$ controls the complexity of the non-diagonal perturbation. Specialized instances include:

• Diagonal-plus-rank-one (DPR1): $k=1$
• Arrowhead matrices: a closely-related structure appearing in certain eigenproblems
• Block-DPLR: block diagonal plus block low-rank, relevant for structured systems and block Markov models

A DPLR matrix is also a prototype of a $(k_1,k_2)$-quasiseparable matrix, with strictly controlled ranks in the off-diagonal blocks—fundamental in fast linear algebra (Bini et al., 2015).

2. Fast Algorithms and Structural Exploitation

The DPLR structure enables algorithmic breakthroughs in computational efficiency. Key results include:

• Hessenberg reduction for DPLR matrices (Bini et al., 2015): A novel algorithm reduces $A = D + UV^*$ to upper Hessenberg form $H=QAQ^*$ via a sequence of Givens rotations, maintaining $(1,2k-1)$-quasiseparability and achieving

$O(n^2 k)\text{ arithmetic operations}$

for the reduction. Each shifted QR step on this structured Hessenberg matrix is $O(nk^2)$.

• Fast eigenvalue computation: Via structured QR iterations, eigenvalues of DPLR matrices can be computed at much lower cost than with general unstructured matrices. Applications to polynomial rootfinding and companion matrices are especially prominent (Bevilacqua et al., 2018).
• Efficient inversion, multiplication, and determinants: For DPR1 (rank-1) and arrowhead forms, explicit formulas enable computation of inverses, matrix-vector products, and determinants in $O(n)$ operations for real, complex, quaternionic, or block matrices (Stor et al., 2022):

| Operation | Arrowhead / DPR1 Structure | Complexity | | -------------- | ----------------------------------------------------------------- | ------------ | | $Az$ | $w_i = \delta_i z_i + x_i \rho (y^\star x)$ | $O(n)$ | | $\det(A)$ | $\prod \delta_i (1 + y^\star \Delta^{-1} x \rho)$ | $O(n)$ | | $A^{-1}$ | Explicit DPR1/arrowhead inverse formulae | $O(n)$ |

This unification carries over seamlessly to block and non-commutative fields (e.g., quaternions).

3. Structurally-Aware Numerical Methods

DPLR transition matrices are a fundamental case for the application of data-sparse factorizations, dynamical low-rank algorithms, and structured sketching techniques:

• Quasiseparable and Givens-vector representations: These are used to preserve and exploit the structure at each step of the reduction/Hessenbergization process (Bini et al., 2015).
• Data-sparse factorizations and embedding: The LFR (Lower-FRontlier) representation leverages a minimal parameterization for DPLR matrices in the context of QR iterations (Bevilacqua et al., 2018), reducing both arithmetic cost and memory overhead.
• Dynamical low-rank and projection techniques: For time-dependent DPLR matrices (e.g., in Riccati-like differential equations, covariance evolution), closed-form orthogonal projections onto the DPLR manifold enable stable evolution with linear-in-$d$ cost, giving invertible approximants and enabling tractable filtering and inference in high-dimensional settings (Bonnabel et al., 1 Jul 2024).

4. Role in Polynomial Eigenvalue Problems and Statistical Modeling

DPLR matrices arise as a result of linearization in polynomial eigenvalue problems, especially in companion forms: $P(x) = \sum_{i=0}^d P_i x^i,\qquad A = D + UV^*$ where $A$ is obtained via linearization, and $d$ is the degree. Fast DPLR-aware reduction algorithms deliver an overall complexity of $O(nk^2) = O(dm^2)$ for matrix polynomials of size $dm \times dm$, with accuracy verified to near-machine precision (Bini et al., 2015).

In statistical modeling and signal processing, DPLR decompositions underlie factor analysis and minimum trace factor analysis (MTFA) (Saunderson et al., 2012). Given $X = D + L$ (with $D$ unknown diagonal and $L$ low-rank PSD), convex optimization-based MTFA admits recovery guarantees controlled by the coherence $\mu(\mathcal{U})$ of the column space, with sharp threshold $\mu(\mathcal{U}) < \frac{1}{2}$ for unique recovery.

5. DPLR Approximation and Matrix Sketching

For large-scale operators accessible only by matrix-vector products, DPLR (sometimes labeled LoRD—Low-Rank plus Diagonal) approximations have received dedicated sketching methods:

• SKETCHLORD (Fernandez et al., 28 Sep 2025) executes joint low-rank and diagonal recovery from a small number of MVPs via nuclear norm minimization constrained by sketching equations:

$$\min_L \frac{1}{2} \|\widetilde{Y} - (LS)(I - \frac{1}{p} 11^T)\|_F^2 + \lambda \|L\|_*$$

Diagonal extraction follows by deflation. This joint approach is provably and empirically superior to any sequential diagonal/low-rank strategies for matrices genuinely of DPLR form, and is well-suited to Hessians and operators in deep learning.

• Compression and fast application: Joint estimation ensures fidelity across both low-rank and diagonal features, critical for best-approximate preconditioners and surrogates in large-scale linear algebra.

6. Extensions, Limitations, and Numerical Stability

• Field Generality: Explicit inversion and determinant formulas for DPLR/DPR1 extend directly to real, complex, block, and even quaternionic fields, with adaptations provided for non-commutative settings (Stor et al., 2022).
• Iterative Optimization and Approximate Diagonalization: In simultaneous diagonalization and approximate joint diagonalization problems, DPLR structure is closely related to the output of structured low-rank approximation (ATDS algorithm) (Akema et al., 2020). Alternating projection solvers leveraging Kronecker sum and low-rankness provide guarantees of convergence to DPLR (diagonalizable) forms, especially surpassing Jacobi-like iterations in robustness and accuracy.
• Numerical Stability: High accuracy is demonstrable in practical computations, with error close to machine epsilon and robustness even at large $n$ and moderate-to-high $k$, subject to occasional reorthogonalization for best stability (Bini et al., 2015).
• Scalability: The transition from $O(n^3)$ to $O(n^2k)$ (or lower) computational cost is critical for modern applications, bridging the gap between theory and practice in large matrix computations, and accommodating large-scale scientific and data-driven problems.

7. Applications Across Scientific Computing and Machine Learning

• Modeling field interactions in recommendation systems: DPLR decompositions are exploited in Field-weighted Factorization Machines, replacing explicit full interaction matrices with a diagonal-plus-symmetric-low-rank form. This reduces inference costs from $O(m^2k)$ to $O(\rho |I| k)$, enabling low-latency deployment without significant loss in accuracy (Shtoff et al., 22 Jul 2024).
• Efficient Riccati and Kalman filtering: Time-evolving DPLR approximations provide stable, scalable updates for covariance matrices in control-theoretic filtering and statistical Bayesian computation, maintaining invertibility and tractability (Bonnabel et al., 1 Jul 2024).
• Approximate simultaneous diagonalization: In analysis of matrix tuples and multivariate decoupling, structured low-rank approximation algorithms enforce a DPLR output in the diagonalizing basis, achieving theoretical recovery guarantees absent in traditional iterative schemes (Akema et al., 2020).
• Large-scale operator approximation: In high-dimensional Hessian approximation, sketching-based DPLR recovery enables efficient, principled operator surrogates for downstream tasks (Fernandez et al., 28 Sep 2025).

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DPLR transition matrices thus simultaneously provide a framework for capturing global and local linear structure, enabling development of scalable, structure-exploiting algorithms for fundamental problems throughout computational mathematics, machine learning, engineering, and signal processing. Their algorithmic and structural analysis continues to catalyze advances for both theoretical insight and practical large-scale computation.