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An Improved Approximation Algorithm for the Column Subset Selection Problem
Christos Boutsidis, Michael W. Mahoney, Petros Drineas
TL;DR
CSSP asks for the best exactly-k-column subset of a matrix. The paper proposes a randomized-then-deterministic selection algorithm running in O(min{mn^2, m^2n}) time, with probability at least 0.8 achieving the stated Frobenius and spectral guarantees. The manuscript also records a corrected sampling-based version because a conference-paper bug affected its earlier spectral bound.
Problem
CSSP seeks exactly k columns whose projection captures an m × n matrix under spectral or Frobenius error, a hard combinatorial problem with practical feature-selection applications.
Method
The algorithm samples Θ(k log k) columns using top-k right-singular-subspace information, then deterministically selects exactly k columns from that sample.
Results
With probability at least 0.8, the algorithm runs in O(min{mn^2, m^2n}) time and returns exactly k columns with the paper's Frobenius and spectral approximation bounds.
Takeaways & Limitations
The approach improves algorithmic CSSP guarantees while remaining efficient in the matrix dimensions and independent of k in its stated running time.
Takeaways & Limitations
The manuscript notes that a conference-version bug affected the previously reported spectral bound, while the Frobenius bound remained unaffected; computing the top-k right singular subspace also remains necessary.
Abstract
from arXiv · showhide
We consider the problem of selecting the best subset of exactly $k$ columns from an $m \times n$ matrix $A$. We present and analyze a novel two-stage algorithm that runs in $O(\min\{mn^2,m^2n\})$ time and returns as output an $m \times k$ matrix $C$ consisting of exactly $k$ columns of $A$. In the first (randomized) stage, the algorithm randomly selects $Θ(k \log k)$ columns according to a judiciously-chosen probability distribution that depends on information in the top-$k$ right singular subspace of $A$. In the second (deterministic) stage, the algorithm applies a deterministic column-selection procedure to select and return exactly $k$ columns from the set of columns selected in the first stage. Let $C$ be the $m \times k$ matrix containing those $k$ columns, let $P_C$ denote the projection matrix onto the span of those columns, and let $A_k$ denote the best rank-$k$ approximation to the matrix $A$. Then, we prove that, with probability at least 0.8, $$ \FNorm{A - P_CA} \leq Θ(k \log^{1/2} k) \FNorm{A-A_k}. $$ This Frobenius norm bound is only a factor of $\sqrt{k \log k}$ worse than the best previously existing existential result and is roughly $O(\sqrt{k!})$ better than the best previous algorithmic result for the Frobenius norm version of this Column Subset Selection Problem (CSSP). We also prove that, with probability at least 0.8, $$ \TNorm{A - P_CA} \leq Θ(k \log^{1/2} k)\TNorm{A-A_k} + Θ(k^{3/4}\log^{1/4}k)\FNorm{A-A_k}. $$ This spectral norm bound is not directly comparable to the best previously existing bounds for the spectral norm version of this CSSP. Our bound depends on $\FNorm{A-A_k}$, whereas previous results depend on $\sqrt{n-k}\TNorm{A-A_k}$; if these two quantities are comparable, then our bound is asymptotically worse by a $(k \log k)^{1/4}$ factor.
1 Introduction
The paper frames CSSP as selecting exactly k columns that best capture a matrix, then introduces a two-stage algorithm with improved approximation guarantees and practical running time.
- Problem: CSSP selects exactly k columns of A whose projection best captures A under the spectral or Frobenius norm.The selected columns form C, and PC denotes projection onto their span.
- Motivation: The problem is motivated by unsupervised feature selection, including applications where actual features should reproduce structure identified by PCA.The paper also notes motivations such as visualization, reduced training time, avoiding overfitting, and data understanding.
- Method: The algorithm first samples Θ(k log k) columns using top-k right-singular-subspace information, then deterministically returns exactly k corresponding columns of A.The two stages combine randomized sampling with deterministic column selection.
- Main results: O(min{mn^2, m^2n}) time and probability at least 0.8 characterize the algorithm's stated efficiency and success guarantee.Its running time is linear in the larger dimension, quadratic in the smaller, and independent of k; success probability can be boosted by repetition.
- Main results: The Frobenius guarantee is roughly O(√k!) better than the best previous algorithmic result and only √(k log k) worse than the best existential result.The paper emphasizes improved algorithmic performance relative to prior CSSP results.
- Novelty: The paper's novelty is a nontrivial combination of theoretical-computer-science algorithmic developments with traditional numerical-linear-algebra techniques.This combination is presented as the basis for improved CSSP bounds.
2 Background and prior work
The background defines the norms, SVD notation, and projection benchmark used for CSSP, then situates the work among randomized sampling, volume sampling, and RRQR approaches.
- Notation: Θ notation denotes an asymptotically tight bound, while O notation denotes an asymptotic upper bound.These bounds are defined through constants and sufficiently large input sizes.
- Notation: The SVD supplies top and bottom singular vectors and values, with Vk containing the top k right singular vectors used in the paper's sampling framework.The notation also distinguishes the best rank-k approximation Ak and the residual components.
- CSSP benchmark: Ak is the best rank-k approximation to A under both spectral and Frobenius norms, providing the benchmark against which projection residuals are compared.PC projects onto the span of selected columns, while A+ denotes the Moore–Penrose pseudoinverse.
- RRQR methods: RRQR-based methods provide deterministic CSSP guarantees through selecting columns associated with a rank-revealing QR factorization.The factorization uses orthonormal Q, upper-triangular R, and a permutation matrix Π, with polynomially bounded quality functions.
- Randomized methods: Prior randomized methods often trade the number of selected columns against approximation accuracy and focus mainly on Frobenius-norm guarantees.Some methods select more than k columns to obtain stronger error bounds.
- Sampling methods: Volume sampling gives an existential exactly-k-column result, but the known way to find those columns is exhaustive search; adaptive sampling provides an efficient approximation.The cited adaptive method runs in O(mnk + kn).
3 A two-stage algorithm for the CSSP
Algorithm 1 combines randomized subspace-informed sampling with deterministic selection to return exactly k columns of A. Its improved guarantees rely on applying the deterministic procedure to a smaller, well-spread matrix, while requiring computation of the top-k right singular subspace.
- Randomized stage: The randomized stage samples Θ(k log k) columns from V_k^T using a nonuniform distribution informed by A’s top-k right singular subspace.Sampling is performed in independent trials with replacement, so columns may be selected more than once and are rescaled by 1/√(c p_i).
- Deterministic stage: The deterministic stage selects exactly k columns from the sampled set and returns the corresponding m × k submatrix C = AS_1S_2.It applies Algorithm 4 of to the rescaled matrix V_k^T S_1D_1, producing the selection matrix S_2.
- Running time: The algorithm’s theoretical running time is O(min{mn^2, m^2n}), with the deterministic stage operating on a sampled matrix rather than A’s n columns.The analysis uses V_k, and iterative algorithms could accelerate this computation in practice.
- Sampling distribution: The sampling probabilities can omit their second term while preserving the Frobenius norm bound of Theorem 1.Thus, the full probability distribution is not necessary for that guarantee.
- Scope and constraints: The approach still requires knowledge of V_k, and identifying suitable distributions that avoid computing the top-k right singular subspace remains open.Algorithm 4 also requires padding because its input has fewer rows than columns before deterministic selection.
- Rationale: Applying the deterministic procedure to the lower-dimensional matrix V_k^T S_1D_1 improves the bounds because its columns are sampled to be spread out according to subspace-based leverage information.The relevant probabilities correspond, up to scaling, to diagonal entries of the projection onto the top-k right singular subspace.
4 Proof of Theorem 1
The proof establishes that randomized sampling followed by deterministic selection yields full rank and the stated Frobenius- and spectral-norm CSSP guarantees with probability at least 0.8.
- Rank preservation: Θ(k log k) randomized samples make V_k^T S_1D_1 full rank with probability at least 0.9.The proof uses a matrix perturbation result and a Rudelson–Vershynin-type theorem to establish this rank condition.
- Deterministic selection: The deterministic second stage selects k linearly independent columns from the randomized sample, forming C = AS_1S_2.The construction of S_2 preserves linear independence in the sampled top-k right singular subspace.
- Residual analysis: The approximation analysis replaces the optimal pseudoinverse expression with one involving A_k, then bounds the residual using unitarily invariant norm inequalities.The argument applies to both spectral and Frobenius norms and uses γ_2 = ∥A − A_k∥_ξ.
- Frobenius norm: The Frobenius-norm theorem combines Lemmas 2, 3, and 5, and c = Θ(k log k) gives the stated bound with probability at least 0.8.The probability follows from a union bound over two events, each failing with probability at most 0.1.
Appendix
The appendix presents the Exactly(c) algorithm, which samples rescaled columns according to probabilities satisfying a leverage-style condition and provides an approximation guarantee.
- Exactly(c) algorithm: The Exactly(c) algorithm constructs C from c rescaled columns sampled independently according to probabilities p_i.The sampling matrix is initialized column by column, with each selected column rescaled by the inverse square root of its sampling probability.
- Guarantee: Theorem 2 assumes ∥A∥_2 ≤ 1 and requires the sampling probabilities to satisfy a lower-bound condition involving a constant β ∈ (0, 1].The theorem also introduces an accuracy parameter ε ∈ (0, 1) and a sample-size condition.
- Guarantee: The theorem’s constant c_0 is identified as the unknown constant from Theorem 3.1 of Rudelson and Vershynin’s result.This constant enters the sufficient sampling-size requirement.