Fast Inversion Algorithm

• Fast Inversion Algorithms are computational techniques that leverage mathematical structures, recursion, and low-rank approximations to compute matrix inverses more efficiently than traditional O(n³) methods.
• Block-recursive and structured-matrix approaches, such as Strassen’s algorithm and displacement methods, reduce computational complexity to as low as O(n²) or O(n log n) for specialized matrices.
• Emerging methods using randomized algorithms, iterative GPU acceleration, and quantum block-encoding are transforming applications in physics, geophysics, signal processing, and quantum computing.

A fast inversion algorithm is any computational scheme that computes the inverse of a matrix or an implicit transformation significantly more efficiently than classical $O(n^3)$ direct methods, typically by exploiting mathematical structure, hardware parallelism, or advanced recursion and approximation techniques. The domain now encompasses classical block-recursive and structured-matrix algorithms, randomized and low-rank approximations, displacement structure methods, fast Fourier-related inversion, quantum block-encoding inverses, and rapid iterative solvers tailored to modern hardware architectures and practical inversion settings.

1. Foundational Principles and Taxonomy

The central aim of fast inversion is the computation of $A^{-1}$ for a matrix $A \in \mathbb{C}^{n \times n}$, or more generally, the inverse of an operator or transform. The complexity barrier of naive Gaussian elimination ($O(n^3)$) motivates a variety of algorithmic strategies:

• Block-recursive and divide-and-conquer methods: Exploit recursive block structure, as in Strassen's inversion and its subcubic descendants. These achieve $O(n^\omega)$ for $\omega < 3$, where $\omega$ is the exponent of matrix multiplication (Tonks, 2019, Misra et al., 2018, Riahi, 2023).
• Structured-matrix methods: Harness algebraic structure such as displacement rank, toeplitzness, Hankel, Vandermonde, or quasiseparability to achieve $O(n^2)$ or better for special matrix families (Perera et al., 2014, Shishun et al., 2024).
• Low-rank and randomized algorithms: Formulate inversion as the solution of well-conditioned subproblems via randomized SVD, Krylov/Hankel decomposition, or interpolative decompositions, often paired with iterative regularization (Zhou et al., 2024, Vatankhah et al., 2017, Renaut et al., 2020).
• Iterative methods: Newton, Neumann, Chebyshev, and their generalizations (e.g., Nested Neumann) are used in settings where approximate inversion suffices, particularly on modern GPU hardware (Engsig et al., 2022).
• Transform-specific inversion: Develop fast (often $O(n\log n)$) schemes for the inversion of structured transforms, including Abel, Fourier-related, pseudo-polar, and nonlinear Fourier (Micheli, 2022, &&&10&&&, Vaibhav, 2017).
• Quantum fast inversion: Implement block-encoded quantum circuits that realize the matrix inverse as an efficiently-repeatable primitive, enabling preconditioned quantum linear system solvers (Tong et al., 2020).

2. Block-Recursive and Fast Multiplicative Approaches

Block-recursive inversion, introduced by Strassen and formalized by Bunch and Hopcroft, is foundational. Matrices are partitioned recursively; Schur complements and block operations reduce the inversion to a logarithmic-depth tree of smaller problems, each solved by the same method or ultimately by a direct approach for small sizes.

• Strassen's algorithm achieves $O(n^{\log_2 7})\approx O(n^{2.807})$ inversion complexity, dominating classical $O(n^3)$ Gaussian elimination or LU-based schemes for sufficiently large $n$ (Tonks, 2019, Misra et al., 2018).
• Combinatorial and Recurrent Formalisms: Modern research shows that for (block-)triangular matrices, explicit combinatorial formulas for entries enable highly parallel, block-recursive inversion (as in the COMBRIT method), with leading constants dramatically lower than classic Strassen inversion. For general matrices, block-splitting plus fast multiplication asymptotically dominates for large $n$ (Riahi, 2023).

These approaches are extensively leveraged in distributed systems; e.g., the SPIN method demonstrates halving leading constants of block-multiplications per recursion and clear empirical wall-time advantages in Spark environments (Misra et al., 2018).

3. Structure- and Displacement-Based Inversion

For families of structured matrices, displacement structure or fast elimination via recurrence relations yield inversion schemes of $O(n^2)$ or better:

• Quasiseparable Vandermonde-like Matrices: Matrices obeying low-loop recurrence relations admit fast Gaussian elimination and Traub-style inversion via displacement operators, culminating in $O(n^2)$ operations for broad classes previously considered intractable except by dense algorithms. Stability is managed by structure-preserving partial pivoting (Perera et al., 2014).
• Hermitian Small Matrices: For $3 \times 3$ Hermitian matrices (ubiquitous in PolSAR), analytical cofactor-based inversion avoids square roots and delivers a $>2\times$ measured speedup over Cholesky, with $\sim28\%$ fewer operations. GPU data layouts and parallel strategies straightforwardly exploit this structure (Coelho et al., 2018).
• Block Hankel and Block Krylov: For sparse matrices over finite fields, block-Krylov decompositions reduce inversion to a block-Hankel structure, efficiently invertible thanks to its low Sylvester displacement rank and modern fast rectangular multiplication, achieving $O(n^{2.2131})$ expected time (Casacuberta et al., 2021).
• Toeplitz/FFT Embedding: When the forward matrix is block Toeplitz with Toeplitz blocks (BTTB), e.g., gravity and magnetic inversion, embedding into block-circulant matrices enables $O(n\log n)$ matvecs via FFT. Projected-subspace iterative solvers (e.g., RSVD, GKB) efficiently recover solutions for models as large as $n \sim 10^6$ (Renaut et al., 2020, Vatankhah et al., 2017).

4. Fast Inversion in Transform and Signal Spaces

Many inverse problems involve structured transforms instead of explicit matrices; efficient inversion methods have been developed for these scenarios:

• Abel Transform: The inversion leverages a Legendre-Fourier correspondence: via a periodic extension, the Legendre coefficients of the reconstructed function equal FFT coefficients of a related function. A single FFT and stable regularization produce an overall $O(N\log N)$ inverse, with rigorous stability estimates, robust even for discontinuous or noisy data (Micheli, 2022).
• 3D Pseudo-Polar Fourier Transform (PPFT): Direct inversion is achieved in $O(n^3\log n)$ by an onion-peeling grid resampling coupled to a three-pass separable Toeplitz inversion strategy, outperforming all prior iterative PPFT inversion approaches in both speed and accuracy on large volumes (Averbuch et al., 2015).
• Nonlinear Fourier Transform (NFT): Fast layer-peeling (LP) combined with a fast Darboux transform (FDT) achieves inverse NFT in $O(KN+N\log^2N)$ for $N$ samples and $K$ solitons, yielding second-order convergence (with respect to $N$) and scalability to hundreds of discrete eigenvalues with controlled numerical errors (Vaibhav, 2017).

5. Low-Rank and Randomized Approaches

Low-rank factorizations and randomized sampling enable inversion of matrices that admit strong spectral decay or localized structure:

• ISDF-SMW for Dielectric Matrices: In the GW approximation, the dielectric matrix is approximated by interpolative separable density fitting (ISDF), yielding a low-rank factorization $\chi_0 \approx \Theta M \Theta^\top$. Application of Sherman-Morrison-Woodbury (SMW) provides an $O(N_r N_e^2)$ inversion cost (with $N_r\gg N_e$), yielding $30$--$80\times$ speedup versus optimized dense codes with $<0.05$ eV energy error (Zhou et al., 2024).
• Randomized SVD and Subspace/Surrogate Approaches: In large data-driven forward models (e.g., gravity/magnetic inversion), RSVD can extract dominant spectral directions, reducing full-system inversion to small surrogates. Model update and regularization use UPRE on the surrogate; $q \sim m/10$--$m/6$ captures most invertibility at $O(qmn)$ per iteration, drastically reducing overall wall-clock even for ill-posed systems (Vatankhah et al., 2017, Renaut et al., 2020).

6. Iterative and GPU-Accelerated Fast Inversion

Iterative inversion methods are practical for large or sparse systems and are amenable to GPU acceleration:

• Nested Neumann (NN) Recursion: A generalized scheme that connects Neumann series, Newton iteration, and Chebyshev iteration, parameterized by an inception depth $L$ and nests $i$, with the iteration

$$\phi^{(i+1)}_L = \sum_{n=0}^L (I-\phi^{(i)}_L \tilde W)^n \phi^{(i)}_L$$

converging for $\|\tilde W-I\|_2<1$, and yielding high-order convergence for $L\geq 2$. For $L=2$, $i=37$ suffices for $10^{-6}$ relative error at $\mathrm{cond}(A)\sim 10^6$. Dense and sparse implementations realize the method's potential on GPU architectures (Engsig et al., 2022).

• Slice-within-Gibbs for High-Dimensional Inverse Problems: For Bayesian inversion in ill-posed or high-dimensional contexts, fast Gibbs samplers with generalized slice sampling efficiently handle non-Gaussian priors (e.g., TV, $\ell_p^q$), achieving MCMC mixing rates competitive with specialized direct samplers even in $n\sim 10^5$--$10^6$ dimensions (Lucka, 2016).

7. Quantum and Domain-Specific Fast Inversion Primitives

Quantum computing provides new primitives for inversion:

• Fast Inversion as a Block-Encoding: For normal or Hermitian $A$, a quantum circuit directly block-encodes $A^{-1}$ by controlled rotations on the eigenvalue register (see the “INV” unit), post-selects on ancilla, and uses amplitude amplification. This primitive serves as a preconditioner for quantum linear system algorithms, removing explicit $\kappa(A)$-dependence in the case where $A^{-1}$ is efficiently block-encodable and $A$ is well-conditioned. Applications include Green's function computations for quantum many-body systems (Tong et al., 2020).

8. Specialized and Hybrid Algorithms

A range of problem-specific fast inversion strategies address further domains:

• Complex-to-Real Frobenius Inversion: The inversion of $A+iB$ over $\mathbb{C}$ can be reduced to real-field operations by the formula $$(A+iB)^{-1} = (A+BA^{-1}B)^{-1} - i A^{-1}B(A+BA^{-1}B)^{-1}$$, requiring two real inversions plus three products. For Hermitian PD matrices, two Cholesky inversions and symmetric multiplications yield a measured $10$–$20\%$ speedup for $n=3000$–$6000$ with negligible accuracy loss. This approach is optimal among algorithms over $\mathbb{R}$ in the sense of straight-line program rank (Dai et al., 2022).
• Diffusion and Generative Model Inversion: Invertible generative models (e.g., DDIM inversion) see continual accelerations. Negative-prompt inversion and EasyInv, e.g., avoid expensive optimization by cleverly choosing prompt embeddings or reinjecting latent information at scheduled timesteps, achieving $>30\times$ speedup over null-text inversion and nearly matching its statistical fidelity for realistic image resolutions (Miyake et al., 2023, Zhang et al., 2024).

9. Applications and Impact across Domains

Fast inversion algorithms are central in:

• Applied physics and engineering: Rapid in-place inversion of small Hermitian matrices in PolSAR, real-time spectral editing, and electromagnetic field modeling (Coelho et al., 2018, Zhou et al., 2024).
• Large-scale geophysics and imaging: Focusing inversion for gravity, magnetic, and tomographic data exploits block Toeplitz structure, RSVD surrogates, and iterative projection methods to reconstruct $10^6$-parameter models on standard workstations (Renaut et al., 2020, Vatankhah et al., 2017).
• Signal processing, integral transforms, and computational mathematics: Direct FFT/NUFFT-based inversion is used for Abel and pseudo-polar/Fourier transforms at $O(n\log n)$–$O(n^3\log n)$ cost and high precision (Micheli, 2022, Averbuch et al., 2015).
• Quantum simulation and quantum computing: Block-encoding-based fast inversion as a preconditioner for QLSA in quantum chemistry and physics simulation workflows (Tong et al., 2020).

10. Challenges and Prospects

While the complexity of matrix inversion has been dramatically reduced in structured and randomized frameworks, several challenges remain:

• Crossover scale and tuning: For classical fast multiplication/inversion, constants and implementation burden favor standard methods for small $n$; tuning recursion thresholds and subspace ranks remain implementation-dependent (Misra et al., 2018, Riahi, 2023).
• Numerical stability: Block-recursive, displacement, and structured algorithms often require explicit stabilization (e.g., partial pivoting, regularization, spectrum truncation) to avoid accuracy loss near singularity (Perera et al., 2014, Coelho et al., 2018, Casacuberta et al., 2021).
• Generality: Highly specialized methods (e.g., those based on displacement structure or FFT) may not transfer out-of-structure; care is required when extending to irregular or massively sparse matrices (Renaut et al., 2020, Casacuberta et al., 2021).
• Quantum resource constraints: Fast inversion block-encodings on quantum hardware require that classical oracles for diagonalization, eigenvalue registers, and controlled rotations be efficiently realizable (Tong et al., 2020).

A plausible implication is that hybrid algorithms—combining structure-exploiting, randomized, and parallel iterative elements—will become increasingly dominant for very large, application-driven inversion tasks. Continued progress in fast matrix multiplication, quantum circuit synthesis, and fast transform inversion will further push the efficiency limits for future inversion algorithms.