Krylov Subspace-Based Filters

• Krylov subspace-based filters are filtering constructions that use low-dimensional Krylov representations to amplify, attenuate, isolate, or validate operator-dependent components.
• They encompass various implementations—polynomial, rational, and adaptive filtering—each balancing spectral sharpness with numerical stability for tasks like eigenvalue computation and graph processing.
• This approach not only enhances regularization in ill-posed inverse problems but also supports diagnostic and reliability checks in non-Hermitian and quantum settings.

Krylov subspace-based filters are filtering constructions whose action is represented, approximated, or diagnosed through Krylov subspaces of the form $$\mathcal{K}_m(A,b)=\mathrm{span}\{b,Ab,\ldots,A^{m-1}b\}$$. In the literature, the term covers several technically distinct but algebraically related objects: polynomial filters implemented by matrix powers, rational filters implemented by shifted resolvents, projected regularization filters built on Lanczos or Golub–Kahan bases, and, in some settings, post-hoc reliability filters applied to projected generalized eigenvalue problems (Huang et al., 2024, Wang et al., 27 Mar 2025, Drum et al., 29 Nov 2025, Oliveira et al., 13 Apr 2026). This suggests a unifying view in which filtering is not identified solely with an explicit spectral transfer function, but with any mechanism that uses a low-dimensional Krylov representation to amplify, attenuate, isolate, or validate operator-dependent components.

1. Algebraic foundations

A central fact is that degree-$K$ polynomial filters over a propagation matrix $P$ lie in the order-$(K+1)$ Krylov subspace $\mathcal{K}_{K+1}(P,x)$, so diverse polynomial graph filters have equivalent expressive power once re-expressed in the monomial basis (Huang et al., 2024). In the network-analysis setting, this is written explicitly as

$y=p(A)v_0=\sum_{t=0}^{m-1} c_t A^t v_0,$

with the normalized Krylov matrix providing a basis for implementation and interpretation (Frost, 2024). In large-scale generalized eigenvalue problems, the corresponding rational form replaces powers by shifted inverses,

$$r(K,M)V=\sum_{j=1}^{p}\omega_j (K-\zeta_j M)^{-1}MV,$$

so that rational filters approximate spectral projectors by linear combinations of resolvents (Wang et al., 27 Mar 2025).

The same algebraic viewpoint appears in dynamical sampling, where the altered Krylov subspace

$$\mathcal{A}\mathcal{K}_m(A;B,x)=\mathrm{span}\{Ax,ABx,\ldots,AB^{m-1}x\}$$

supports annihilating polynomials whose roots recover eigenvalues visible through the sampling operator (Aldroubi et al., 2014). Across these formulations, the filter is determined by coefficients on a basis generated from repeated operator action or shifted solves. This suggests that “Krylov subspace-based filter” is best treated as a structural category rather than a single algorithmic template.

2. Rational interval filters for eigenvalue computation

For the symmetric definite generalized eigenvalue problem

$Kx=\lambda Mx,$

with $K$ and $K$0 large, sparse, real symmetric, and positive definite, rational filtering is used to compute all eigenpairs with eigenvalues in a prescribed interval $K$1, typically $K$2 for vibration spectra (Wang et al., 27 Mar 2025). The filter transfer function is

$K$3

and is designed so that $K$4 for $K$5 and $K$6 for $K$7. Filter sharpness is quantified by the separation factor $K$8.

The paper "A shifted Laplace rational filter for large-scale eigenvalue problems" proposes the shifted Laplace rational filter (SLRF), placing poles on two rays

$K$9

with abscissae and weights obtained by minimizing a weighted $P$0 distance to a step function on $P$1 (Wang et al., 27 Mar 2025). The design criterion is explicitly contradictory in structural vibration: poles close to the interval sharpen the filter and accelerate outer subspace convergence, whereas poles farther from the real axis improve conditioning of the shifted systems and reduce Krylov cost. The paper formalizes this through the condition-number inequality

$P$2

showing monotonic conditioning improvement as $P$3 increases.

The filter is embedded in block subspace iteration by solving, for each pole,

$P$4

forming

$P$5

$P$6-orthonormalizing $P$7, and then performing Rayleigh–Ritz projection (Wang et al., 27 Mar 2025). Because the shifts are complex and the matrices are non-Hermitian, the paper uses BiCGStab and notes GMRES or QMR as alternatives; with real shifts that remain indefinite, MINRES or SYMMLQ may be appropriate. Numerical experiments on a 2D beam, a 3D hollow platform, and a 3D fish-like body show that SLRF often achieved the lowest average sparse matrix–vector products per outer iteration and competitive total cost; Gauss–Legendre was slightly better on the hollow platform model, while BiCGStab stagnation occurred more frequently for Gauss–Chebyshev (Wang et al., 27 Mar 2025).

A recurrent misconception is that the sharpest rational filter at fixed pole count is necessarily the most effective eigensolver. The SLRF results show that, when inner solves are iterative and inexact, end-to-end efficiency is governed by a balance between spectral separation and shifted-system solvability rather than filter sharpness alone.

3. Polynomial graph and graph-spectral filters

In graph learning and graph signal processing, Krylov subspace-based filters are predominantly polynomial. For an undirected graph with normalized Laplacian $P$8, spectral filters of ChebNet, GPR-GNN, BernNet, and JacobiConv can all be written as linear combinations of $P$9 for an appropriate propagation matrix $(K+1)$0, and thus belong to the same Krylov subspace (Huang et al., 2024). The paper’s Proposition 1 states that all $(K+1)$1-order polynomial filters with propagation matrix $(K+1)$2 belong to $(K+1)$3, while Proposition 2 states that for a signal of grade $(K+1)$4, the spectral optimal filter equals a degree-$(K+1)$5 polynomial in $(K+1)$6.

The paper "Optimizing Polynomial Graph Filters: A Novel Adaptive Krylov Subspace Approach" introduces adaptive propagation

$(K+1)$7

derived from a heat-diffusion discretization, and proves that the eigenvalues of $(K+1)$8 are monotonically increasing in $(K+1)$9 (Huang et al., 2024). This gives explicit spectrum controllability. At the same time, Theorem 1 shows that $\mathcal{K}_{K+1}(P,x)$0, a rank-one stable propagation matrix, so large $\mathcal{K}_{K+1}(P,x)$1 induces rank-one behavior. This is the paper’s formal statement of the oversmoothing limitation of fixed polynomial bases.

A complementary interpretation is given by Krylov subspace trajectories, defined as rows of the Krylov matrix

$\mathcal{K}_{K+1}(P,x)$2

Any polynomial graph filter is a linear combination of the Krylov columns, and the row trajectories $\mathcal{K}_{K+1}(P,x)$3 encode path-length–resolved node responses (Frost, 2024). With a non-random initial vector, the coefficients $\mathcal{K}_{K+1}(P,x)$4 determine whether a filter emphasizes local propagation, long-range propagation, or a custom spectral response. The oscillation statistic

$\mathcal{K}_{K+1}(P,x)$5

quantifies non-monotone trajectory behavior and, in the paper’s interpretation, spectral interference among components (Frost, 2024).

In graph denoising, the same principle appears as polynomial approximation of $\mathcal{K}_{K+1}(P,x)$6 without eigendecomposition:

$\mathcal{K}_{K+1}(P,x)$7

Chebyshev recurrences, Lanczos polynomial approximations, flexible CG, and LOBPCG all produce Krylov-generated low-pass filters on graph Laplacians (Knyazev et al., 2015). Bilateral and guided filters become graph operators $\mathcal{K}_{K+1}(P,x)$8 and $\mathcal{K}_{K+1}(P,x)$9, respectively, so repeated filtering is already a Krylov polynomial in the underlying graph operator.

These results establish a strong equivalence between polynomial graph filtering and Krylov basis design. This suggests that many purported differences among graph filters arise less from expressive power than from basis conditioning, optimization dynamics, and the spectral geometry induced by the chosen propagation matrix.

4. Regularization filters for ill-posed inverse problems

For linear discrete ill-posed problems, Krylov subspace-based filters appear as regularizing iterates. LSQR and CGLS admit the filtered SVD expansion

$y=p(A)v_0=\sum_{t=0}^{m-1} c_t A^t v_0,$0

with filter factors

$y=p(A)v_0=\sum_{t=0}^{m-1} c_t A^t v_0,$1

where $y=p(A)v_0=\sum_{t=0}^{m-1} c_t A^t v_0,$2 are Ritz values from Lanczos bidiagonalization (Jia, 2018). The iteration count $y=p(A)v_0=\sum_{t=0}^{m-1} c_t A^t v_0,$3 is therefore the regularization parameter. The same paper proves a $y=p(A)v_0=\sum_{t=0}^{m-1} c_t A^t v_0,$4 theorem relating the Krylov subspace $y=p(A)v_0=\sum_{t=0}^{m-1} c_t A^t v_0,$5 to the dominant right singular subspace, and ties the onset of semi-convergence to the behavior of the smallest Ritz values (Jia, 2018).

Projected iterated Tikhonov (PIT) refines this picture by projecting with Golub–Kahan bidiagonalization and then applying preconditioned Landweber/Tikhonov iteration inside the reduced space. After $y=p(A)v_0=\sum_{t=0}^{m-1} c_t A^t v_0,$6 GKB steps, the stationary filter factors are

$y=p(A)v_0=\sum_{t=0}^{m-1} c_t A^t v_0,$7

while the nonstationary version obeys a recursive update in $y=p(A)v_0=\sum_{t=0}^{m-1} c_t A^t v_0,$8 selected by the discrepancy principle (Drum et al., 29 Nov 2025). The paper shows that these factors “open faster” than unpreconditioned Landweber and that, with reorthogonalization, fp64, fp32, and fp16 implementations can yield nearly indistinguishable relative reconstruction errors on the reported 2D image problems (Drum et al., 29 Nov 2025).

A closely related interpretation is developed in "Hybrid CG-Tikhonov as a filtration of the CG Lanczos vectors," where the hybrid iterate is written as

$y=p(A)v_0=\sum_{t=0}^{m-1} c_t A^t v_0,$9

The coefficients $$r(K,M)V=\sum_{j=1}^{p}\omega_j (K-\zeta_j M)^{-1}MV,$$0 are called Lanczos filters and satisfy

$$r(K,M)V=\sum_{j=1}^{p}\omega_j (K-\zeta_j M)^{-1}MV,$$1

so later Lanczos directions are damped more strongly as the Tikhonov parameter grows (Gerth et al., 29 May 2025). At Ritz values, the reduced filter behaves like $$r(K,M)V=\sum_{j=1}^{p}\omega_j (K-\zeta_j M)^{-1}MV,$$2, mirroring the classical Tikhonov factor $$r(K,M)V=\sum_{j=1}^{p}\omega_j (K-\zeta_j M)^{-1}MV,$$3.

One practical consequence is that Krylov filtering in inverse problems is not merely a matter of stopping early. It can also be understood as explicit damping of basis vectors or projected singular components. This suggests a continuum between “iterative regularization” and “hybrid regularization,” rather than a strict methodological divide.

5. Non-Hermitian and adaptive filtered Krylov variants

For non-symmetric eigenvalue problems, the filtered Krylov paradigm must contend with non-normality and genuinely complex spectral regions. The paper "On Relaxed Filtered Krylov Subspace Method for Non-Symmetric Eigenvalue Problems" defines the relaxed filtered Krylov subspace

$$r(K,M)V=\sum_{j=1}^{p}\omega_j (K-\zeta_j M)^{-1}MV,$$4

where both the polynomial filter $$r(K,M)V=\sum_{j=1}^{p}\omega_j (K-\zeta_j M)^{-1}MV,$$5 and the filtered direction $$r(K,M)V=\sum_{j=1}^{p}\omega_j (K-\zeta_j M)^{-1}MV,$$6 are updated dynamically (Miao et al., 2020). The complex Chebyshev filter is constructed on an ellipse $$r(K,M)V=\sum_{j=1}^{p}\omega_j (K-\zeta_j M)^{-1}MV,$$7 and normalized by

$$r(K,M)V=\sum_{j=1}^{p}\omega_j (K-\zeta_j M)^{-1}MV,$$8

with a three-term recurrence for efficient application. The paper proves bounds on the damping coefficient $$r(K,M)V=\sum_{j=1}^{p}\omega_j (K-\zeta_j M)^{-1}MV,$$9 and recommends “fat” ellipses when targeting eigenvalues with the largest real parts (Miao et al., 2020). Numerically, RFKS consistently outperformed CD and FKS, and CD outperformed FKS.

Adaptation also appears in reduced-rank signal processing. In the APSM-Krylov framework, the filter is parameterized as $$\mathcal{A}\mathcal{K}_m(A;B,x)=\mathrm{span}\{Ax,ABx,\ldots,AB^{m-1}x\}$$0 with $$\mathcal{A}\mathcal{K}_m(A;B,x)=\mathrm{span}\{Ax,ABx,\ldots,AB^{m-1}x\}$$1 an orthonormal basis of $$\mathcal{A}\mathcal{K}_m(A;B,x)=\mathrm{span}\{Ax,ABx,\ldots,AB^{m-1}x\}$$2, but the reduced coefficients are updated by parallel subgradient projections onto data-consistency sets rather than by minimizing an estimated quadratic form (Lamare et al., 2013). The paper emphasizes that this tracks the optimal point in the sense of minimizing the true mean square error in the Krylov subspace even when the estimated statistics become erroneous. In the simplest case with $$\mathcal{A}\mathcal{K}_m(A;B,x)=\mathrm{span}\{Ax,ABx,\ldots,AB^{m-1}x\}$$3 and $$\mathcal{A}\mathcal{K}_m(A;B,x)=\mathrm{span}\{Ax,ABx,\ldots,AB^{m-1}x\}$$4, the reduced update becomes an NLMS iteration inside the Krylov subspace (Lamare et al., 2013).

The common theme is that the “filter” may itself be adaptive: the polynomial, the starting vector, the target subspace, or the projection rule changes with the iteration. This weakens any attempt to define Krylov filters solely through fixed transfer functions.

6. Diagnostics, spectral identification, and broader meanings of filtering

Not all Krylov subspace-based filters are forward transforms. In dynamical sampling, altered Krylov sequences are used to identify annihilating polynomials and recover spectral information from spatiotemporal samples. For a general sampler $$\mathcal{A}\mathcal{K}_m(A;B,x)=\mathrm{span}\{Ax,ABx,\ldots,AB^{m-1}x\}$$5, $$\mathcal{A}\mathcal{K}_m(A;B,x)=\mathrm{span}\{Ax,ABx,\ldots,AB^{m-1}x\}$$6 dynamical samples suffice to recover the $$\mathcal{A}\mathcal{K}_m(A;B,x)=\mathrm{span}\{Ax,ABx,\ldots,AB^{m-1}x\}$$7-annihilator, and for an unknown convolution operator under uniform subsampling $$\mathcal{A}\mathcal{K}_m(A;B,x)=\mathrm{span}\{Ax,ABx,\ldots,AB^{m-1}x\}$$8, the spectrum $$\mathcal{A}\mathcal{K}_m(A;B,x)=\mathrm{span}\{Ax,ABx,\ldots,AB^{m-1}x\}$$9 can be recovered from $Kx=\lambda Mx,$0 temporal samples for almost all $Kx=\lambda Mx,$1 (Aldroubi et al., 2014). In this setting, the filter is the annihilating polynomial whose roots encode the observable spectrum.

A different reinterpretation arises in quantum Krylov methods. The paper "Tackling instabilities of quantum Krylov subspace methods: an analysis of the numerical and statistical errors" introduces the imaginary filter

$Kx=\lambda Mx,$2

for QBKS-H, with acceptance threshold $Kx=\lambda Mx,$3, and the unitary filter

$Kx=\lambda Mx,$4

for QBKS-U, with threshold

$Kx=\lambda Mx,$5

These filters do not transform the operator; they test whether the projected eigenvalues preserve the physical structure expected from Hermiticity or unitarity (Oliveira et al., 13 Apr 2026). The paper further states that $Kx=\lambda Mx,$6 alone is not a reliable indicator of solution accuracy or reliability, and that under realistic shot noise statistical fluctuations dominate unless regularization or filtering is used (Oliveira et al., 13 Apr 2026).

This broader usage is significant because it separates two meanings of “filter.” In classical numerical linear algebra and graph processing, a filter usually shapes spectral content. In quantum Krylov analysis, the proposed filters are reliability scores and acceptance tests. The shared element is still Krylov projection, but the operational role has shifted from spectral transformation to solution validation.

Krylov subspace-based filters therefore form a heterogeneous but coherent class. They include polynomial and rational approximants, projected regularization mechanisms, adaptive reduced-rank updates, annihilating polynomials for spectral recovery, and diagnostic filters for projected generalized eigenproblems. Across these domains, the decisive design variables are the same: basis generation, spectral or geometric weighting, conditioning of the projected problem, and the trade-off between approximation fidelity and numerical stability.