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A Spectral-Domain Pseudo-Inverse Construction Method for Unitary Diagonalizable Linear Inverse Problems

Shengchang Chen

arXiv:2607.26951v1math.NAcs.NAphysics.geo-ph

TL;DR

Ill-conditioned or singular inverse problems need stabilization, and classical regularization and generalized-inverse approaches can be computationally difficult at large scale. The paper derives an analytical SVD and constructs a stable pseudo-inverse directly from unitary spectral structure. It proves bounded stability and convergence to the Moore–Penrose inverse, and reports an applied seismic migration method.

  • Problem

    Ill-conditioned or singular linear inverse problems require stabilization, while classical regularization and generalized-inverse methods face computational difficulties on large-scale problems.

    Source: §1 · pp. 1–2

  • Method

    The paper derives an analytical SVD for square unitary-diagonalizable matrices and constructs a spectral-domain pseudo-inverse from their transform structure.

    Source: §3 · p. 3, §7.1 · p. 5

  • Results

    The constructed operator is bounded and converges to the Moore–Penrose inverse; the method has also been used in an industrially applied seismic least-squares reverse-time migration method.

    Source: §1 · p. 2, §4.2 · p. 4

  • Takeaways & Limitations

    The construction applies to any unitary transform that diagonalizes the kernel matrix, offering a framework for such structured inverse problems.

    Source: §7.1 · p. 6

    The analytical SVD premise is restricted to square matrices diagonalizable by a unitary matrix.

    Source: §3 · p. 3

Abstract

from arXiv · show

Linear inverse problems are prevalent in geophysics, signal processing, image restoration, and medical imaging. Mathematically, they can be formulated as $Gm = d$. When $G$ is ill-conditioned or singular, the problem becomes ill-posed and requires regularization methods or generalized inverse methods for a stable solution. However, both types of methods encounter significant computational difficulties for large-scale linear inverse problems. In this paper, we establish a spectral-domain pseudo-inverse construction method for a class of linear inverse problems that can be diagonalized by unitary matrices. The core idea is to construct a stable pseudo-inverse operator directly in the transform domain, starting from the unitary diagonalization structure of the matrix. We first provide the analytic Singular Value Decomposition (SVD) of this class of matrices, clarifying the correspondence between spectral decomposition and SVD. On this basis, we define spectral-domain regularization filtering factors, construct a stable spectral-domain pseudo-inverse operator, and prove its bounded stability as well as its consistency in converging to the Moore--Penrose generalized inverse. This construction is numerically equivalent to zeroth-order Tikhonov regularization and converges to the Moore--Penrose generalized inverse as $\alpha \to 0^+$, but its methodological path differs from both. This paper reveals that the stable generalized inverse of a class of structured matrices can be directly constructed from their spectral decomposition, providing an efficient solution method for large-scale structured inverse problems. The Fourier transform case is a special instance of this method when the unitary matrix is taken as the discrete Fourier transform matrix.

1 Introduction

Ill-conditioned or singular linear inverse problems require stabilization, while regularization and generalized-inverse methods can be computationally difficult at large scale. This paper uses unitary diagonalization to construct a stable inverse directly in the transform domain.

1 Introduction

  • Both regularization and generalized-inverse methods face significant computational difficulties on large-scale problems, with typical complexity O(N^3).Tikhonov regularization and SVD-based Moore–Penrose inversion are the two classical approaches described.Source: §1 · pp. 1–2
  • For matrices diagonalizable by a unitary transform, the method constructs a stable generalized inverse from the spectral decomposition rather than proposing a new regularization method.The paper identifies this as a methodological complement to existing theories.Source: §1 · p. 2
  • The method provides an analytical SVD and constructs a spectral-domain pseudo-inverse whose bounded stability and convergence to the Moore–Penrose inverse are proved.It also clarifies the numerical equivalence of the construction to classical methods while distinguishing its methodological path.Source: §1 · p. 2
  • The construction has been used in an image-domain least-squares reverse-time migration method for seismic waves, applied in industrial oil and gas exploration processing in China.The application combines local space-invariant deconvolution with a wavenumber-domain pseudo-generalized inverse.Source: §1 · p. 2

2 Preliminaries

The preliminaries establish the SVD components and the unitary-transform structure used to describe matrices diagonalizable in the transform domain.

2.1 Singular Value Decomposition and Matrix Properties

  • An SVD expresses a matrix using unitary left and right factors and a real nonnegative diagonal matrix of singular values.The number of nonzero singular values equals the matrix rank.Source: §2.1 · p. 2

2.2 Unitary Matrices and Unitary Diagonalization

  • A matrix is unitary diagonalizable when it has the form G = T^HΛT, with unitary T and diagonal Λ whose entries are transform-domain spectral values.Unitary matrices satisfy T^HT = TT^H = I.Source: §2.2 · p. 3

3 Analytical Singular Value Decomposition of Unitary Diagonalizable Matrices

For a square matrix diagonalizable by a unitary transform, its SVD is determined analytically by the transform and spectral values. A phase factor is needed in the complex case.

3 Analytical Singular Value Decomposition of Unitary Diagonalizable Matrices

  • The SVD of a unitary-diagonalizable matrix has an explicit form determined by its unitary transform and spectral values, without numerical computation.The factors are U = T^HP, Σ = |Λ|, and V^H = T.Source: §3 · p. 3
  • For complex spectral values, the phase diagonal matrix P is necessary to maintain a rigorous decomposition; replacing Λ directly with |Λ| is not sufficient.The proof substitutes Λ = PΣ into the unitary diagonalization.Source: §3 · p. 3
  • The analytical SVD requires the matrix to be square, restricting the stated premise to finite-dimensional unitary-diagonalizable systems.The paper connects this premise to large-scale structured-problem applications and notes that it avoids logical gaps under non-square special cases.Source: §3 · p. 3

4 Construction and Properties of the Spectral-Domain Pseudo-Inverse

The paper represents the exact generalized inverse spectrally and stabilizes small spectral components with a filtered pseudo-inverse. The operator is bounded for positive regularization and converges to the Moore–Penrose inverse.

4.1 Spectral Representation of the Exact Generalized Inverse

  • When spectral values are very small, the exact inverse can amplify noise, motivating stabilization of those components.The Moore–Penrose generalized inverse is represented in the spectral domain for the unitary-diagonalizable matrix.Source: §4.1 · p. 3

4.2 Spectral-Domain Pseudo-Inverse Operator

  • For α > 0, the method defines spectral-domain filter factors and constructs a pseudo-inverse operator directly in the transform domain.The paper calls it a “pseudo-generalized inverse” to emphasize its constructive path, not approximation; it does not rely on numerical SVD.Source: §4.2 · p. 4
  • The filter is bounded: x/(x^2 + α) reaches its maximum 1/(2√α) at x = √α.This bound establishes stability for positive α.Source: §4.2 · p. 4
  • As α approaches zero from above, the constructed operator converges to the Moore–Penrose inverse, with zero components where spectral values vanish.This establishes consistency of the spectral-domain construction.Source: §4.2 · p. 4

5 Special Case: The Fourier Transform Case

With the discrete Fourier transform as the unitary matrix, the method becomes frequency-domain deconvolution. For circulant convolution, it gives a stable spectral-division form, and cited Fourier-based methods are special cases.

5 Special Case: The Fourier Transform Case

  • Setting T = F reduces the method to frequency-domain deconvolution under the Fourier transform.Source: §5 · p. 4
  • For a circulant convolution matrix, the spectral-domain pseudo-inverse becomes a stable frequency-domain spectral-division form.The paper identifies earlier Fourier-transform methods as special cases of the general method with T = F.Source: §5 · p. 4
  • Chen, Chen and Wang, Chen and Xiao, and Mallat are cited as prior Fourier-transform special cases of the general method.Source: §5 · p. 4

6 Relationship with Classical Methods

The spectral-domain pseudo-inverse is numerically equivalent to zeroth-order Tikhonov regularization, but its construction proceeds directly from spectral decomposition. Compared with TSVD and Wiener filtering, it uses continuous attenuation and a deterministic regularization parameter.

6.1 Relationship with Tikhonov Regularization

  • The spectral-domain pseudo-inverse is numerically equivalent to zeroth-order Tikhonov regularization but is methodologically distinct.Its value is the direct construction of a stable generalized inverse from spectral decomposition for structured matrices.Source: §6.1 · p. 5
  • The paper frames its construction as a complement to classical methods: those address general problems, while this method addresses structured problems.Source: §6.1 · p. 5

6.2 Relationship with TSVD and Wiener Filtering

  • Unlike TSVD’s hard truncation of small singular values, the method uses continuous attenuation for smoother filtering.Source: §6.2 · p. 5
  • Although its filtering form resembles a Wiener filter, α is a deterministic regularization parameter rather than a statistical model parameter.Source: §6.2 · p. 5

7 Conclusions and Future Work

The conclusions establish an analytical SVD and a bounded-stable spectral-domain pseudo-inverse for matrices diagonalizable by unitary transforms. The paper also identifies extensions to spatially varying systems, nonlinear inverse problems, and adaptive spectral modification as future work.

7.1 Conclusions

  • For G = T^HΛT, the analytical SVD is U = T^HP, Σ = |Λ|, V^H = T, with phase diagonal P needed for rigor in complex spectral cases.Source: §7.1 · p. 5
  • The spectral-domain pseudo-inverse operator is constructed with bounded stability.Source: §7.1 · p. 5
  • The method is numerically equivalent to Tikhonov regularization but methodologically independent, extending the Fourier-transform case to general unitary transforms.Source: §7.1 · p. 5
  • The framework applies to any unitary transform that diagonalizes the kernel matrix.Source: §7.1 · p. 6

7.2 Future Work

  • Spatially varying systems are proposed for future treatment by combining local space-invariant approximations with this method.Source: §7.2 · p. 6
  • Future work proposes extending unitary diagonalization to weakly nonlinear inverse problems or using it as a preconditioner for nonlinear iterative algorithms.Source: §7.2 · p. 6
  • The paper proposes investigating optimal spectral modification functions based on signal and noise characteristics.Source: §7.2 · p. 6
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