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REVIEW 2 major objections 5 minor 17 references

Direct methods for Moore-Penrose pseudoinverses overcome conditioning, rank, and densification barriers and work for large sparse multilinear regression.

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2026-07-14 12:46 UTC pith:IRDOYT7Z

load-bearing objection Solid packaging of bidiagonal direct routes for the MP (and group) pseudoinverse; theorems hold, code exists, sparse-scale claims still under-demonstrated. the 2 major comments →

arxiv 2607.10302 v1 pith:IRDOYT7Z submitted 2026-07-11 math.NA cs.NA

Direct Methods for Calculating PseudoInverses

Jeff Knisley This is my paper
classification math.NA cs.NA MSC 15A0965F2065F50
keywords pseudoinversesregressionsparsitybidiagonalizationMoore-PenroseLanczosgroup inverseleast squares
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper shows that direct methods for the Moore-Penrose pseudoinverse, long dismissed as rank-revealing, ill-conditioned, or density-creating, can be made practical for large-scale and sparse least-squares problems. The strategy reduces any matrix to a bidiagonal factor via Householder or Lanczos processes, then inverts that bidiagonal by one of three explicit techniques: CK deflation into invertible and nilpotent blocks, dual-normal tridiagonal solves, or in-place formulas on the natural zero-split blocks. A fourth construction reinterprets Lanczos itself as successive rank-one updates that converge to a weak pseudoinverse. Because the expensive work stays on the sparse bidiagonal and Woodbury tearing keeps storage linear, the resulting pseudoinverse can be applied repeatedly without ever forming a dense matrix. Readers who care about multilinear regression on sparse data therefore gain a family of direct solvers that sit between full SVD and pure iterative methods.

Core claim

Direct methods applied after bidiagonalization overcome the three classic limitations of conditioning (squared condition numbers of AᵀA), exact rank revelation, and densification of sparse matrices, and are therefore appropriate for large-scale or sparse multilinear regression.

What carries the argument

Bidiagonal deflation (Theorem 1) that splits an m imes(m+1) bidiagonal into an invertible bidiagonal C and a superdiagonal K, together with the three concrete solvers (CK, dual-normal Thomas, inplace blocks) and the Lanczos rank-one update construction of Theorem 3.

Load-bearing premise

The methods assume that bidiagonalizing the original large sparse matrix remains practical and does not destroy the sparsity that makes the later O(n) or O(n^{2}) work advantageous.

What would settle it

Time and residual-accuracy comparisons of the three bidiagonal methods against SVD-based pinv on a suite of sparse regression matrices of growing size and density; if the direct methods lose both speed and accuracy once the matrices become realistically sparse, the practicality claim fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A sparse matrix can keep a matrix-free or low-memory pseudoinverse that never densifies.
  • Multilinear problems with many right-hand sides can form the pseudoinverse once and reuse it.
  • Woodbury tearing of the bidiagonal reduces inverse storage from quadratic to linear.
  • The same bidiagonal toolkit yields both Moore-Penrose and group inverses via the Jordan-Wielandt embedding.
  • Complex regularization Re((A+iεI)^{-1}) supplies a rapidly converging family of approximations without squared condition numbers.

Where Pith is reading between the lines

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

  • The block formulas and Woodbury tearing are natural candidates for hardware-level sparse tensor cores already used in large language models.
  • The same CK and inplace machinery can be specialized to graph-Laplacian or attention-mask sparsity patterns that appear in modern transformers.
  • Because the methods only require existing LAPACK bidiagonalization kernels, they can be dropped into production libraries with modest engineering cost.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper argues that direct (non-SVD) methods for the Moore-Penrose and group pseudoinverses can overcome the classic practical limitations of conditioning (squared condition numbers of AᵀA/AAT), rank-revealing requirements, and densification of sparse matrices. It reduces the problem to bidiagonal matrices via Householder or Lanczos bidiagonalization, then supplies three constructive routes for the bidiagonal pseudoinverse (CK deflation by Givens chasing into invertible C and superdiagonal K, dual-normal Thomas solves on the block-tridiagonal BBᵀ, and explicit in-place formulas for the three possible square/rectangular blocks after splitting on zero etaⱼ and etaⱼ=eta_{j}=0). A Lanczos reinterpretation as successive rank-one updates that produce a weak (1,2,3) pseudoinverse is also given, together with a complex-regularization limit that recovers the group inverse of the Jordan-Wielandt augmentation. Theorems 1–3, the simplified Woodbury tear (11), and competitive timings versus scipy.linalg.pinv (Table 2) are offered as evidence that the methods are suitable for large-scale or sparse multilinear regression.

Significance. If the practical claims hold, the work supplies a useful algorithmic toolkit for applications that repeatedly apply a pseudoinverse (or a weak pseudoinverse) to many right-hand sides and that cannot afford a full SVD or a dense matrix. The explicit block formulas, the matrix-free back-substitution realization of C^{-1}, the Woodbury reduction of storage to O(r), and the publicly available Cython/LAPACK implementations are concrete engineering contributions. The reduction of the Moore-Penrose case to a group inverse of a sparse symmetric augmentation is also a clean conceptual observation that may seed further regularization algorithms. The paper therefore has clear value for numerical linear algebra and sparse least-squares practice even if the large-scale sparse validation remains incomplete.

major comments (2)
  1. §5 and the Abstract claim that the three bidiagonal routes (plus Woodbury tearing (11)) are appropriate for large-scale or sparse multilinear regression and that they avoid densification. The only numerical evidence is Table 2 (dense m,n ≤ 1000 versus scipy.linalg.pinv). No sparse matrix suite, fill-in statistics for gebrd/orgbr or Lanczos, memory profiles, or comparison against LSQR/sparse direct solvers is supplied. Without such evidence the central practical claim remains unsubstantiated; at minimum a sparse benchmark section (or a clear restriction of the claim to the dense/bidiagonal regime) is required.
  2. §4, Theorem 3 constructs a weak (1,2,3) pseudoinverse by successive rank-one updates, then notes that condition 4 can be restored by a post-processing formula involving an arbitrary Z. For the least-squares applications emphasized in §1–2 (Eqs. (1)–(2)), the weak inverse already suffices, but the manuscript never states this limitation cleanly nor quantifies the extra cost or numerical effect of restoring the fourth Penrose condition when it is needed. A short clarifying paragraph and, if possible, a numerical illustration would remove the ambiguity.
minor comments (5)
  1. Figure 1 is typeset with many unreadable control characters and overlapping fractions; it should be regenerated as a clean LaTeX array or TikZ diagram.
  2. §3, after Theorem 1: the claim that “the study of pseudoinverses can be reduced to the study of pseudoinverses of bidiagonal matrices” is slightly overstated for the Drazin case (index >1); a one-sentence qualification would be accurate.
  3. Table 2 caption and surrounding text: report absolute wall-clock times (or FLOPs) in addition to the relative factors so that readers can judge absolute performance.
  4. Several bibliographic entries (e.g., the ICML 2025 Schwarz-Schur paper) appear to be future or arXiv-only; verify publication status and supply stable identifiers.
  5. Notation: both A^i and A_i are used for pseudoinverses of different types; a short notational table would help.

Circularity Check

0 steps flagged

No significant circularity: all derivations are constructive from the Penrose/group axioms and standard bidiagonal identities.

full rationale

The paper opens from the four Moore-Penrose axioms (Table 1) and the three group-inverse axioms, then builds explicit algorithms: Givens deflation of a bidiagonal into invertible C plus superdiagonal K (Theorem 1), closed-form block inverses (Proposition 1), dual-normal Thomas solves on the natural block-tridiagonal of BBᵀ, and the successive rank-one update X_k = X_{k-1} + w_k (A w_k)ᵀ that is verified by direct expansion to satisfy conditions 1–3 (Theorem 3). The error bound of Theorem 2 follows from the orthogonal decomposition ∥A∥_F^{2} = ∥B_k∥_F^{2} + ∥R_k∥_F^{2} together with R_k V_k = 0, without external premises. Woodbury tearing (Eq. 11) is an algebraic identity for the chosen tear geometry. No parameters are fitted to data, no uniqueness result is imported from the author’s prior papers, and the GitHub code is merely an implementation artifact. Consequently every load-bearing step is independent of its own conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The work rests almost entirely on standard linear-algebra facts (Penrose axioms, existence of SVD/Schur/bidiagonal forms, uniqueness of Drazin/group inverses, Woodbury identity, Lanczos recurrence). No free parameters are fitted; the only modeling choices are algorithmic (when to tear, when to threshold small diagonals). Invented entities are absent; all objects are classical matrices and their pseudoinverses.

axioms (5)
  • standard math The four Penrose conditions uniquely characterize the Moore-Penrose pseudoinverse; conditions 1 and 3 alone already solve the normal equations.
    Invoked throughout §2 and used to justify that weak (1,2,3) pseudoinverses suffice for least-squares (Eqs. 1–2).
  • standard math Any matrix admits a bidiagonalization A = U B Vᵀ (Householder or Lanczos) with U, V orthogonal (or with orthonormal columns).
    Foundation of all three direct methods (§3–§4); existence is classical.
  • standard math For a superdiagonal nilpotent N one has N† = N^{-T} and N^D = 0; the Drazin inverse annihilates the nilpotent part of the Jordan form.
    Used to reduce pseudoinverses of bidiagonals to the invertible block C (§3, Theorem 1).
  • standard math Woodbury matrix identity holds for low-rank updates; when the update severs only superdiagonal entries of a bidiagonal the correction simplifies to B^{-1} = B̂^{-1} - B̂^{-1} U Vᵀ B̂^{-1}.
    Eqs. 10–11; enables linear storage for the pseudoinverse of large bidiagonals.
  • domain assumption Large sparse multilinear regression problems remain of practical interest and benefit from matrix-free or low-memory pseudoinverses.
    Stated in §1 and §7 as the motivating application regime; not proved but widely accepted in the ML/numerical literature cited.

pith-pipeline@v1.1.0-grok45 · 21921 in / 2998 out tokens · 40697 ms · 2026-07-14T12:46:36.291289+00:00 · methodology

0 comments
read the original abstract

The Moore-Penrose pseudoinverse of a matrix can be defined and calculated using its singular value decomposition. There are also direct methods for computing matrix pseudo-inverses (those that avoid eigenvalue computations), but these are often rank-revealing, poorly conditioned, or otherwise limited in practice. In this paper, we demonstrate that direct methods can overcome these limitations. In particular, we reinterpret several existing direct methods and introduce new variations that are appropriate for large scale or sparse multilinear regression applications.

Figures

Figures reproduced from arXiv: 2607.10302 by Jeff Knisley.

Figure 1
Figure 1. Figure 1: The in-place MP pseudoinverse of the bidiagonal matrix [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

discussion (0)

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Reference graph

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