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arxiv: 2604.23193 · v1 · submitted 2026-04-25 · 💻 cs.DS · cs.LG· cs.NA· math.NA· math.PR· stat.ML

Well-Conditioned Oblivious Perturbations in Linear Space

Shabarish Chenakkod , Micha{\l} Derezi\'nski , Xiaoyu Dong , Mark Rudelson This is my paper

Pith reviewed 2026-05-08 07:27 UTC · model grok-4.3

classification 💻 cs.DS cs.LGcs.NAmath.NAmath.PRstat.ML
keywords smoothed analysiscondition numbermatrix perturbationlinear systemsconjugate gradientrandom matricessingular valuesoblivious perturbations
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The pith

Perturbing any matrix with O(n) random numbers in O(log n) bits reduces its condition number to O(n), matching full Gaussian noise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that a new oblivious perturbation construction achieves the same conditioning guarantee as adding independent Gaussian noise to every entry, but stores and generates only O(n) random values each using O(log n) bits. A sympathetic reader cares because this removes the quadratic space barrier that has limited practical use of smoothed-analysis techniques in matrix algorithms. The result directly yields an algorithm for solving n by n linear systems to constant backward error in linear space using only O(n) matrix-vector products via perturbed conjugate gradient. The construction works by first applying a fixed dense pattern matrix that turns any sparse vector dense, then adding a sparse perturbation whose nonzero locations and values are chosen dependently and nonuniformly. New analytic tools are introduced to prove that the smallest singular value stays at least inverse-linear in n with high probability.

Core claim

Any n by n deterministic matrix A satisfies that with high probability the perturbed matrix A plus a specially constructed random perturbation has smallest singular value at least 1 over O(n). The perturbation is formed by adding a deterministic pattern matrix to a sparse random matrix whose O(n) nonzero entries are located in a nonuniform pattern and drawn from dependent distributions; the pattern matrix ensures that every sparse vector is mapped to a dense one, which in turn forces the random perturbation to interact with the entire row space.

What carries the argument

The pattern matrix (a fixed dense deterministic matrix mapping every sparse vector to a dense vector) together with a sparse dependent perturbation whose nonzero entries occupy a nonuniform support pattern.

If this is right

  • An n by n linear system can be solved to arbitrarily small constant backward error using O(n) matrix-vector products while storing only linear space.
  • The perturbed conjugate-gradient method achieves the improved iteration count without requiring quadratic storage for the random perturbation.
  • Any matrix algorithm whose complexity depends on the condition number can be made efficient in linear space by applying the same perturbation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same pattern-plus-sparse idea may extend to other iterative solvers such as GMRES or to eigenvalue problems where only a few matrix-vector products are affordable.
  • Because the perturbation is oblivious and uses few random bits, it could be useful in streaming or external-memory settings where full Gaussian matrices cannot be materialized.
  • The new singular-value bounds for dependent random matrices might be applied to analyze other structured random constructions appearing in randomized numerical linear algebra.

Load-bearing premise

The new lower bounds on the smallest singular value continue to hold when the random matrix has the specific dependent and nonuniformly supported entries arising from the pattern-plus-sparse construction.

What would settle it

Exhibiting one explicit n by n deterministic matrix for which the smallest singular value of the perturbed matrix falls below 1 over 100n for large n would disprove the conditioning claim.

Figures

Figures reproduced from arXiv: 2604.23193 by Mark Rudelson, Micha{\l} Derezi\'nski, Shabarish Chenakkod, Xiaoyu Dong.

Figure 1
Figure 1. Figure 1: Illustration of the pattern matrix generation model for n = 5. (2) After conditioning on V1 and V3, for any fixed unit vector x, V x has independent coordi￾nates. Each coordinate (V x)i is a linear combination of the coordinates of x with 4-wise independent Rademacher coefficients. This is enough to show that |(V x)i | is greater than some constant with constant probability. The independence of rows allows… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of R2 = H ⊙R¯ 2 for one realization. Since Row1(H) = 0, the first row of R¯ 2 is zeroed in R2. columns. This way, we separate out the random signs Xl,n in the last column which are independent from the part that we condition on. More precisely, recalling that Yn denotes the last column of M and Z is a unit vector orthogonal to the subspace spanned by the first n − 1 columns, we can write ⟨Yn, Z⟩ = view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of H¯ with n = 8, K = 2, L = 3. Each column i selects an independent uniform subset Ji ⊂ [n] of size K and sets h¯ j,i = 1Ji (j). 1 L X1,1 1 L X1,2 1 L X1,3 1 L X1,4 0 0 0 0 0 0 0 0 1 L X1,5 0 0 1 L X1,8 0 0 0 0 0 1 L X1,6 0 0 0 0 0 0 0 0 1 L X1,7 0 1 L X2,1 0 0 0 1 L X2,5 0 0 0 0 1 L X2,2 0 0 0 1 L X2,6 0 0 0 0 1 L X2,3 0 0 0 1 L X2,7 0 0 0 0 1 L X2,4 0 0 0 1 L X2,8 i = 1 i = 2 i = 3 i = 4 i … view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of R¯ 2 with n = 8, K = 2, L = 3. Here, we enumerate Ji = {j(i, 1) < · · · < j(i, K)} and set ¯rj(i,ℓ),i = 1 LXℓ,i for ℓ ∈ [K], else 0. To reduce the norm of the random perturbation, we remove the heavy rows of H¯ and R¯ 2 to construct the matrices H and R2. Formally, we set Rowj (H) = Rowj (H¯ ) if the number of non-zero entries in Rowj (H¯ ) does not exceed L and Rowj (H) = 0 otherwise. Note… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of constructing H from H¯ with n = 8, K = 2, L = 3. Here, we keep row j if ∥Rowj (H¯ )∥0 ≤ L, else set it to 0. Finally, set R2 = H ⊙ R¯ 2 where ⊙ stands for the entry-wise (Hadamard) product, which was illustrated earlier in view at source ↗
read the original abstract

Perturbing a deterministic $n$-dimensional matrix with small Gaussian noise is a cornerstone of smoothed analysis of algorithms [Spielman and Teng, JACM 2004], as it reduces the condition number of the input to $O(n)$, and with it the complexity of many matrix algorithms. However, when deployed algorithmically, these perturbations are expensive due to the cost of generating and storing $n^2$ Gaussian random variables. We propose a perturbation that requires generating and storing $O(n)$ random numbers in $O(\log n)$ bits of precision, and reduces the condition number of any deterministic matrix to $O(n)$, matching Gaussian perturbations. Our result in particular implies a better complexity for the perturbed conjugate gradient algorithm, showing that we can solve an $n\times n$ linear system in linear space to within an arbitrarily small constant backward error using $O(n)$ matrix-vector products. In our construction, we introduce the concept of a pattern matrix, which is a dense deterministic matrix that maps all sparse vectors into dense vectors, and we combine it with a sparse perturbation whose entries are dependent and located in a non-uniform fashion. In order to analyze this construction, we develop new techniques for lower bounding the smallest singular value of a random matrix with dependent entries.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces an oblivious perturbation for n×n matrices that uses a fixed dense 'pattern matrix' P (which maps sparse vectors to dense ones) combined with a sparse perturbation S whose O(n) nonzero entries are dependent and placed non-uniformly. It claims that this construction requires only O(n) random numbers of O(log n) bits of precision, reduces the condition number of any deterministic matrix to O(n) with high probability (matching Gaussian perturbations), and yields an improved perturbed conjugate-gradient algorithm that solves linear systems in linear space using O(n) matrix-vector products to arbitrarily small constant backward error. The analysis relies on new lower bounds for the smallest singular value of random matrices with dependent, non-uniformly located entries.

Significance. If the central singular-value claim holds, the result would be significant for smoothed analysis of algorithms: it removes the quadratic space and randomness cost of Gaussian perturbations while preserving the O(n) condition-number guarantee, directly improving the complexity of matrix algorithms such as conjugate gradient. The introduction of new techniques for bounding singular values of dependent-entry matrices is a technical contribution that could apply more broadly. The linear-space algorithmic implication is concrete and falsifiable.

major comments (2)
  1. The O(n) condition-number guarantee (and therefore the claimed matching with Gaussian smoothed analysis) rests entirely on a new lower bound showing that σ_min(P + S) ≥ 1/poly(n) with high probability. The paper develops techniques for dependent non-uniform entries, but it is not clear from the argument whether those techniques survive the specific dependence structure and location bias induced by the pattern-matrix-plus-sparse construction; a gap here would invalidate the central claim. This is the load-bearing step identified in the abstract and the singular-value analysis section.
  2. The statement that the construction 'matches Gaussian performance' is not accompanied by explicit constants or a direct comparison theorem; without these, it is difficult to verify that the O(n) bound is of the same order as the Spielman-Teng Gaussian result rather than merely the same asymptotic class.
minor comments (2)
  1. The definition of the pattern matrix and the precise support pattern of the sparse perturbation should be stated with an explicit example for small n to clarify the non-uniform placement.
  2. The bit-precision claim (O(log n) bits) is stated in the abstract but would benefit from a short paragraph explaining how the random numbers are generated and stored without hidden logarithmic factors.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the load-bearing aspects of the proof and the comparison to prior work. We address both major comments below and will incorporate clarifications in the revision.

read point-by-point responses

  1. Referee: The O(n) condition-number guarantee (and therefore the claimed matching with Gaussian smoothed analysis) rests entirely on a new lower bound showing that σ_min(P + S) ≥ 1/poly(n) with high probability. The paper develops techniques for dependent non-uniform entries, but it is not clear from the argument whether those techniques survive the specific dependence structure and location bias induced by the pattern-matrix-plus-sparse construction; a gap here would invalidate the central claim. This is the load-bearing step identified in the abstract and the singular-value analysis section.

    Authors: We appreciate the referee's focus on this central step. The proof in Section 4 first uses the pattern matrix P to map any sparse vector to a dense one, ensuring that the support bias in S cannot align with a sparse left singular vector. The dependence among the O(n) entries of S is then handled by a union bound over a net of candidate singular vectors combined with a matrix Chernoff bound that accounts for the limited dependence (each row of S depends on at most O(1) other rows). The location bias is absorbed into the failure probability by weighting the net according to the non-uniform distribution. We agree that the exposition can be tightened; the revision will add a short subsection (new Section 4.3) that explicitly traces how the dependence and bias are controlled in the argument for Theorem 4.2. revision: partial

  2. Referee: The statement that the construction 'matches Gaussian performance' is not accompanied by explicit constants or a direct comparison theorem; without these, it is difficult to verify that the O(n) bound is of the same order as the Spielman-Teng Gaussian result rather than merely the same asymptotic class.

    Authors: We agree that an explicit side-by-side statement is helpful. Both results establish that the condition number is O(n) with probability 1-1/n^Ω(1). Our Theorem 1.1 gives the same O(n) bound (with the hidden constant independent of the input matrix) and the same polynomial failure probability as the Spielman-Teng analysis of Gaussian perturbations. The revision will insert a short comparison paragraph after Theorem 1.1 that quotes the relevant Spielman-Teng bound and notes that the asymptotic order and success probability match exactly, while the leading constants differ by small factors that are not optimized in either work. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new techniques for dependent singular-value bounds are self-contained

full rationale

The derivation introduces a pattern matrix plus sparse dependent perturbation and develops independent lower-bound techniques for the smallest singular value under that dependence structure. No equations reduce by construction to fitted parameters, no load-bearing steps rely on self-citations, and the O(n) condition-number claim is not renamed from prior results. The analysis is presented as extending smoothed analysis with new probabilistic tools rather than assuming the target bound.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper introduces one new entity (pattern matrix) and relies on standard random-matrix theory plus newly developed singular-value bounds for dependent entries.

axioms (1)
  • standard math Standard properties of singular values and matrix norms continue to apply when random entries are dependent and non-uniformly located.
    Invoked when lower-bounding the smallest singular value of the perturbed matrix.
invented entities (1)
  • pattern matrix no independent evidence
    purpose: Dense deterministic matrix that maps every sparse vector to a dense vector
    Core new object used to ensure the perturbation remains effective with only O(n) random numbers.

pith-pipeline@v0.9.0 · 5556 in / 1215 out tokens · 37654 ms · 2026-05-08T07:27:09.034523+00:00 · methodology

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