RandRAND: Randomized Range-Deflation Preconditioners

• RandRAND is a class of randomized deflation preconditioners that projects onto random subspaces to flatten large eigenvalues in shifted linear systems.
• It employs randomized range sketches and orthogonal projectors to replace explicit spectral computations, thereby reducing condition numbers for Krylov solvers.
• The method includes basis-less algorithms using Q-less QR and iterative projection, making it efficient for high-dimensional and memory-constrained applications.

Searching arXiv for papers on “RandRAND” and closely related work. RandRAND denotes a class of randomized deflation preconditioners for large linear systems. Introduced in “Preconditioning via Randomized Range Deflation (RandRAND),” it deflates the spectrum via efficient orthogonal projections onto random subspaces, without computing eigenpairs or low-rank approximations. The method targets systems of the form $(A+\mu I)x=b$, uses randomized range sketches of the shifted operator, and yields projector-based preconditioners whose condition number bounds depend only weakly on the problem size and reduce to a small constant when the dimension of the deflated subspace is comparable to the effective spectral dimension (Balabanov et al., 24 Sep 2025).

1. Problem setting and randomized range deflation

The basic object is the shifted matrix

$$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$

with the goal of accelerating Krylov solvers for

$(A+\mu I)x=b.$

The regime of interest is one in which $A$ or $A_\mu$ has a small number of large singular or eigenvalues compared to the rest. In that setting, classical deflation would build a basis of dominant eigenvectors and project against it. RandRAND replaces this explicit spectral computation by randomized range approximation (Balabanov et al., 24 Sep 2025).

The construction starts from a sketch matrix $\Omega\in\mathbb{R}^{n\times l}$, typically generated from an oblivious subspace embedding. For symmetric $A$,

$\Omega := A^q X^\top,$

and for non-symmetric $A$,

$\Omega := (A^\top A)^q X^\top,$

where $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$0 is a small power and $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$1 is an OSE. The randomized deflation space is then

$$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$2

If the dominant eigendirections or singular directions are well represented in $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$3, projecting onto that range suppresses the large-scale spectral components that drive poor conditioning.

This formulation is projection-based rather than low-rank-approximation-based. A plausible implication is that the method is naturally suited to settings where matvecs with $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$4 are available but explicit eigenspaces, factorizations, or matrix entries are not.

2. Orthogonal projectors and the three RandRAND preconditioners

Let $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$5 be the orthogonal projector onto $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$6, so that

$$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$7

With an explicit QR factorization

$$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$8

the projector is

$$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$9

A central identity is

$(A+\mu I)x=b.$0

and, when $(A+\mu I)x=b.$1,

$(A+\mu I)x=b.$2

These identities let RandRAND act on the deflated range without forming $(A+\mu I)x=b.$3 explicitly (Balabanov et al., 24 Sep 2025).

Three preconditioner families are introduced.

The R-RandRAND preconditioner for SPD $(A+\mu I)x=b.$4 is

$(A+\mu I)x=b.$5

with preconditioned operator

$(A+\mu I)x=b.$6

On $(A+\mu I)x=b.$7, one has $(A+\mu I)x=b.$8. Thus the large eigenvalues captured by $(A+\mu I)x=b.$9 are flattened to $A$0, while the orthogonal complement is governed by the projected operator $A$1.

The C-RandRAND preconditioner for SPD $A$2 is

$A$3

This is a correction-type construction: it leaves the complement essentially untouched and corrects the action of $A$4 on the projected range.

The G-RandRAND preconditioner for indefinite or non-symmetric systems is

$A$5

This form is SPD and is used as a left preconditioner for symmetric indefinite systems with MINRES and for general non-symmetric systems in left-preconditioned GMRES/BiCG-type methods.

The distinction among the three forms is structural. R-RandRAND modifies the operator directly on the projected range; C-RandRAND introduces an additive correction; G-RandRAND transfers the same idea to singular-value control and indefinite or non-normal regimes.

3. Spectral effect, effective spectral dimension, and condition-number theory

The phrase “deflating the spectrum” is used in a specific sense. RandRAND does not zero out eigenvalues. Instead, it flattens the action of $A$6 on $A$7 to a controlled scale $A$8, while the complement is governed by the projected operator. For R-RandRAND,

$A$9

so the large eigenvalues represented in $A_\mu$0 are replaced by $A_\mu$1 (Balabanov et al., 24 Sep 2025).

Let

$A_\mu$2

For SPD $A_\mu$3, if

$A_\mu$4

for some $A_\mu$5, then

$A_\mu$6

This shows that the preconditioned condition number is controlled by the norm of the projected residual operator $A_\mu$7, rather than by the full spectral spread of $A_\mu$8.

The randomized theory then links $A_\mu$9 to the quality of the sketch. For Gaussian $\Omega\in\mathbb{R}^{n\times l}$0, $\Omega\in\mathbb{R}^{n\times l}$1, $\Omega\in\mathbb{R}^{n\times l}$2, $\Omega\in\mathbb{R}^{n\times l}$3, and $\Omega\in\mathbb{R}^{n\times l}$4, with probability at least $\Omega\in\mathbb{R}^{n\times l}$5,

$\Omega\in\mathbb{R}^{n\times l}$6

for $\Omega\in\mathbb{R}^{n\times l}$7, where $\Omega\in\mathbb{R}^{n\times l}$8, and $\Omega\in\mathbb{R}^{n\times l}$9 in expectation. For $A$0, the paper gives a corresponding bound involving $A$1 and $A$2. The dependence on $A$3 appears only through a mild factor of the form $A$4.

The analysis is organized around the effective spectral dimension. Informally, this is the number of eigenvalues above the regularization scale $A$5, or above a comparable threshold relative to the smallest shifted eigenvalue. If the sketch dimension is chosen comparable to that dimension, then the projected operator norm $A$6 becomes close to the trailing spectral scale, and the condition number can reduce to a small constant.

A further convergence statement is given for R-RandRAND. If

$A$7

then CG or MINRES on the preconditioned operator satisfies an iteration estimate of the form

$A$8

For sketch size $A$9, the paper states

$\Omega := A^q X^\top,$0

and, with probability at least $\Omega := A^q X^\top,$1,

$\Omega := A^q X^\top,$2

This suggests that the iteration count depends only weakly on $\Omega := A^q X^\top,$3 rather than on $\Omega := A^q X^\top,$4.

4. Basis-explicit and basis-less algorithms

An explicit implementation computes $\Omega := A^q X^\top,$5, stores $\Omega := A^q X^\top,$6, and applies

$\Omega := A^q X^\top,$7

For symmetric $\Omega := A^q X^\top,$8, the paper gives construction cost

$\Omega := A^q X^\top,$9

and application cost $A$0 per preconditioned operator call. This is viable when $A$1 is small and explicit orthogonalization is affordable (Balabanov et al., 24 Sep 2025).

A major contribution is the basis-less implementation. Instead of storing $A$2, RandRAND retains only Q-less information and uses

$A$3

This makes it possible to exploit fast randomized transforms and bypass explicit basis orthogonalization.

The first basis-less route is fast Q-less Cholesky QR. It forms

$A$4

then sets $A$5. With structured sketches such as SRHT or sparse embeddings, the Gram matrix can be built at cost

$A$6

However, the paper notes that for ill-conditioned $A$7, the orthogonality error of Cholesky QR scales like $A$8.

The stabilization strategy is preconditioned Cholesky QR. A second-level sketch $A$9 is used to form

$\Omega := (A^\top A)^q X^\top,$0

followed by

$\Omega := (A^\top A)^q X^\top,$1

If $\Omega := (A^\top A)^q X^\top,$2 is an $\Omega := (A^\top A)^q X^\top,$3-embedding with $\Omega := (A^\top A)^q X^\top,$4, then

$\Omega := (A^\top A)^q X^\top,$5

so the orthogonality error becomes $\Omega := (A^\top A)^q X^\top,$6.

A third route is factorization-free iterative projection. Instead of factoring $\Omega := (A^\top A)^q X^\top,$7, the projection step is recast as a least-squares problem: $\Omega := (A^\top A)^q X^\top,$8 With $\Omega := (A^\top A)^q X^\top,$9, this becomes

$$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$00

and can be solved by iterative sketch-based least-squares methods, such as Newton-Sketch-type refinement. This route is particularly attractive in GPU or streaming settings.

5. Numerical stability, related methods, and empirical behavior

The numerical-stability analysis centers on projector accuracy, loss of orthogonality, and preservation of symmetry or SPD structure. The paper proposes several remedies: preconditioned Cholesky QR, repeated application of $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$01 in the projected operator, and refined projector approximations such as

$$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$02

which reduce projector error from first order to second order in $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$03 (Balabanov et al., 24 Sep 2025).

RandRAND is closely related to two existing lines of work. Relative to classical deflation, it serves the same spectral role but does not compute eigenvectors. Relative to Nyström-based randomized preconditioning, it uses the same randomized information in a different way. The paper rewrites the Nyström pseudo-inverse as

$$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$04

and observes that

$$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$05

may be viewed as “Nyström pseudo-inverse plus a projector.” The theoretical distinction is that Nyström analysis is driven by $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$06, whereas RandRAND is driven by projection errors such as $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$07. The paper states that, for heavy-tailed spectra, Nyström’s bound may be worse by a factor

$$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$08

The empirical study covers synthetic SPD and indefinite matrices, ridge regression with random Fourier features, kernel ridge regression, portfolio optimization, PDE-constrained inverse problems, and limited-memory kernel ridge regression. The reported pattern is consistent across these settings. For synthetic SPD systems, RandRAND reduces condition numbers by 2–4 orders of magnitude for moderate $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$09. For portfolio optimization, RandRAND cuts iterations by approximately $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$10–$$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$11, and R-RandRAND gives approximately $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$12–$$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$13 fewer iterations than Nyström and C-RandRAND. In basis-less PDE inverse and limited-memory kernel settings, basis-less RandRAND remains robust, while basis-less Nyström is reported as numerically unstable or prone to breakdown for large $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$14. This suggests that the projection-based design is especially advantageous when memory is limited or when explicit orthogonalization is infeasible.

6. Terminology and other uses of the label

In the supplied arXiv literature, “RandRAND” is not uniformly used for a single concept. The formal named method is the randomized range-deflation preconditioner described above (Balabanov et al., 24 Sep 2025). The label is also used informally or hypothetically in several distinct contexts.

In “Batched Ranged Random Integer Generation,” the paper itself states that it does not name the algorithm “RandRAND,” but the label is used there to denote efficient unbiased conversion of 64-bit random words into multiple independent bounded integers by full-width multiplication and a single rejection criterion (Brackett-Rozinsky et al., 2024). In “The RAM equivalent of P vs. RP,” the core primitive is $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$15, which returns a uniformly distributed random integer in the range $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$16; the supplied discussion explicitly treats “RandRAND” as shorthand for the equivalence between $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$17 and $$A_\mu := A+\mu I,\qquad A\in\mathbb{R}^{n\times n},\ \mu\in\mathbb{R},$$18 in suitably equipped RAM models (Brand, 2013). Two random-number-generation papers use the term only as a hypothetical design label: one for a chaos-based PRNG built from the Rössler attractor and a Maiorana-type bent Boolean function (Stoyanov et al., 2017), and one for an enhanced-NRBG architecture that combines a quantum entropy source with SHA-256 Hash-DRBG and Toeplitz hashing (Aggarwal et al., 2021).

This suggests that, in current arXiv usage, “RandRAND” is a polysemous label. In the strict bibliographic sense, it refers to randomized range deflation for preconditioning large linear systems. In adjacent discussions, it also serves as an informal name for ranged random integer generation, RAND-based RAM stochasticity, or hypothetical RNG architectures.