Generalized Interlacing Families: New Error Bounds for CUR Matrix Decompositions (2512.07903v1)

Abstract: This paper introduces the concept of generalized interlacing families of polynomials, which extends the classical interlacing polynomial method to handle polynomials of varying degrees. We establish a fundamental property for these families, proving the existence of a polynomial with a desired degree whose smallest root is greater than or equal to the smallest root of the expected polynomial. Applying this framework to the generalized CUR matrix approximation problem, we derive a theoretical upper bound on the spectral norm of a residual matrix, expressed in terms of the largest root of the expected polynomial. We then explore two important special cases: the classical CUR matrix decompositions and the row subset selection problem. For classical CUR matrix decompositions, we derive an explicit upper bound for the largest root of the expected polynomial. This yields a tighter spectral norm error bound for the residual matrix compared to many existing results. Furthermore, we present a deterministic polynomial-time algorithm for solving the classical CUR problem under certain matrix conditions. For the row subset selection problem, we establish the first known spectral norm error bound. This paper extends the applicability of interlacing families and deepens the theoretical foundations of CUR matrix decompositions and related approximation problems.

• The paper presents a novel approach by generalizing interlacing families of polynomials to obtain spectral norm error bounds for CUR decompositions.
• It introduces deterministic algorithms with polynomial-time complexity that achieve guaranteed performance in both matrix and row subset selection.
• The new error bounds improve upon classical techniques by leveraging singular value analysis and recursive rank-one update methods for tighter estimates.

Generalized Interlacing Families and Error Bounds for CUR Matrix Decompositions

Introduction

The paper "Generalized Interlacing Families: New Error Bounds for CUR Matrix Decompositions" (2512.07903) introduces the concept of generalized interlacing families of polynomials to extend and refine the theory underpinning matrix approximation via CUR decompositions. By advancing the classical interlacing polynomial methodology, which has proved essential in spectral graph theory, combinatorial optimization, and matrix selection problems, the authors provide a flexible approach that accommodates polynomial families with varying degrees, thereby enabling new theoretical analyses for matrix selection.

The principal contribution is a spectral norm error bound for the CUR decomposition based on polynomial root analysis, accompanied by deterministic algorithms for subset selection with guaranteed performance. The work fills critical theoretical gaps, including delivering the first spectral norm guarantee for the row subset selection problem.

Generalized Interlacing Families: Formalism and Properties

Traditional interlacing family constructions require each member polynomial to share the same degree. The generalized interlacing family concept removes this restriction, allowing for degrees to vary—an essential generalization for the analysis of CUR-type approximations. The central technical achievement is to prove that, within such a family, there always exists a polynomial of the root degree whose smallest root is no less than that of the expected polynomial.

Let $\mathcal{P}_k$ denote a family $\{p_{S,W}\}$ of polynomials indexed by $k$-subsets $S, W$ of rows and columns, associated with data and source matrices. The generalized interlacing framework allows for polynomial degrees corresponding to ranks of selected submatrices, and shows that convex combinations and tree structures built from these polynomials will maintain essential interlacing and root properties for spectral norm bounding.

CUR Matrix Approximation and Error Bounds

CUR matrix decompositions approximate a target matrix $\boldsymbol{A}\in\mathbb{R}^{n\times d}$ as $$\boldsymbol{A} \approx \boldsymbol{C} \cdot \boldsymbol{U} \cdot \boldsymbol{R}$$ where $\boldsymbol{C}$ and $\boldsymbol{R}$ are selected columns and rows, and $\boldsymbol{U}$ mediates the reconstruction. The classical case restricts $\boldsymbol{C}, \boldsymbol{R}, \boldsymbol{U}$ to be submatrices of $\boldsymbol{A}$, while the generalized CUR allows arbitrary sources.

The generalized interlacing theory enables the derivation of explicit spectral norm error bounds for the CUR residual: $$\|\boldsymbol{A} - \boldsymbol{C}_{:,W} (\boldsymbol{U}_{S,W})^{-1} \boldsymbol{R}_{S,:}\|_2^2 \leq \max\mathrm{root}\, [P_k(-x;\boldsymbol{A},\boldsymbol{C},\boldsymbol{U},\boldsymbol{R})]$$ where $P_k$ denotes the expected polynomial derived from averaging the $p_{S,W}$ over subset choices.

Classical CUR Case

For $$\boldsymbol{C} = \boldsymbol{U} = \boldsymbol{R} = \boldsymbol{A}$$, the spectral norm error is bounded by the largest root of a transformed polynomial linked to the singular values of $\boldsymbol{A}$, utilizing flip and Laguerre derivative operators: $$P_k(-x;\boldsymbol{A},\boldsymbol{A},\boldsymbol{A},\boldsymbol{A}) = \frac{(-1)^{d-k}(k!)^2}{\mathrm{Flip}\circ [\partial_x x \partial_x]^k \circ \mathrm{Flip}} \det[x\boldsymbol{I}_d - \boldsymbol{A}^T \boldsymbol{A}]$$

The resulting spectral norm error bound is

$$\|\boldsymbol{A} - \boldsymbol{A}_{:,W}(\boldsymbol{A}_{S,W})^{-1}\boldsymbol{A}_{S,:}\|_2^2 \leq \frac{(t-k+1)t}{(1-\sqrt{\alpha_k})^2} \|\boldsymbol{A} - \boldsymbol{A}_k\|_2^2$$

where $\alpha_k$ is a function of leading singular values, improving upon previous $(k+1)^2(t-k)$-factor bounds.

Row Subset Selection

The paper establishes, for the first time, a spectral norm guarantee for the row subset selection problem: $$\|\boldsymbol{A} - \boldsymbol{C}(\boldsymbol{C}_{S,:})^{-1}\boldsymbol{A}_{S,:}\|_2^2 \leq (1+kr)\|\boldsymbol{A} - \boldsymbol{C} \boldsymbol{C}^\dagger \boldsymbol{A}\|_2^2$$ where $$r = \mathrm{rank}(\boldsymbol{A} - \boldsymbol{C}\boldsymbol{C}^{\dagger}\boldsymbol{A})$$.

Algorithmic Implications

A deterministic polynomial-time selection algorithm is presented for classical CUR under the condition that all square submatrices of $\boldsymbol{A}$ of size at most $k$ are invertible. The algorithm leverages recursive rank-one update formulas for the residual and efficiently computes approximations for the largest roots of characteristic polynomials via Sturm sequences and binary search. The output attains the theoretical error bound up to a controlled numerical error.

The time complexity for computing a $k$-submatrix is $O(kn^2d^2 + knd^{w+1}\log(d\vee1/\varepsilon))$ per error parameter $\varepsilon$, where $w$ is the matrix multiplication exponent.

Theoretical and Practical Implications

The extension to generalized interlacing families allows the interlacing polynomial method to subsume matrix selection and CUR decomposition problems with singular submatrices and varying rank conditions, providing robust existence, uniqueness, and spectral norm bounds previously inaccessible to classical techniques. The results have implications for deterministic, interpretable matrix approximation in machine learning, signal processing, and data mining.

The theoretical developments support future research in several directions:

• Tightening root bounds of expected polynomials for more refined spectral estimates
• Extending polynomial-time algorithms to fully general settings (possibly singular submatrices)
• Applying generalized interlacing families to other matrix approximation problems such as sparse coding and distributed selection

Conclusion

This work advances the formalism of interlacing polynomial families by generalizing degree constraints, significantly strengthening the performance guarantees available for CUR and row/column selection tasks. The explicit spectral norm bounds and deterministic polynomial-time algorithms for subset selection situate the interlacing polynomial method as a foundational tool for matrix approximation and interpretable model selection. The potential for further theoretical and algorithmic advances is substantial, both for matrix analysis and related areas of theoretical computer science and high-dimensional statistics.