Sequential Inversion by One-Step Mapping

• The paper presents one-step mapping as a direct inversion algorithm that bypasses sequential intermediate inverses, enhancing inversion efficiency.
• It introduces explicit construction methods for polynomial automorphisms using auxiliary sequences that guarantee global invertibility.
• The approach is applied in imaging, combinatorics, and hardware systems, demonstrating significant speedup and conceptual clarity over traditional methods.

Sequential inversion by one-step mapping refers to a broad class of algorithms that replace a conventional step-wise or pipeline inversion—where the inverse of a composite operation is assembled out of a cascade of sub-inverse mappings—by a direct mapping that inverts the entire sequence in a single mathematical or algorithmic step. This paradigm appears across algebraic, combinatorial, statistical, and computational domains, allowing for the efficient resolution of inverse problems that would otherwise typically require iterative, nested, or sequential application of intermediate inverses.

1. Theoretical Foundations and Core Paradigms

The essential idea underpinning sequential inversion by one-step mapping is the avoidance of intermediary decompositions. Instead of attacking each stage of a forward process separately (e.g., first inverting $A$, then $B$, to invert $B \circ A$), one constructs an auxiliary object—often through recursion, auxiliary sequences, or a direct nonlinear operator—that, upon vanishing or attaining a fixed structural form, guarantees invertibility and provides a formula for the global inverse.

In the context of affine polynomial automorphisms, for example, the one-step mapping algorithm constructs for each coordinate index $i$ a sequence of polynomials $P_i^{(k)}$ whose simultaneous truncation to zero characterizes invertibility. This yields an alternating-sum identity for reconstructing the global inverse without requiring resolution of intermediary sub-inverses or reduction to lower-degree cases (Adamus et al., 2015).

2. One-Step Mapping for Polynomial Automorphisms

Consider a polynomial map $F\colon \mathbb{A}^n \to \mathbb{A}^n$ over a field of characteristic zero, $F(X) = (F_1(X),\ldots,F_n(X))$. The inversion problem asks whether $F$ is a bijective morphism and, if so, seeks the explicit inverse $G(Y) = (G_1(Y),\ldots, G_n(Y))$.

The one-step algorithm proceeds as follows:

• Define for each $i$ a sequence of polynomials via

$$P_i^{(0)}(X) = X_i,\qquad P_i^{(k+1)}(X) = P_i^{(k)}\bigl(F(X)\bigr) - P_i^{(k)}(X)$$

• The invertibility of $F$ is characterized by the vanishing of each $P_i^{(m_i)}$ for some finite $m_i$: $F$ is an automorphism iff $\forall i\,\exists m_i: P_i^{(m_i)}\equiv 0$.
• The explicit inverse is then given by the alternating sum:

$$G_i(Y) = \sum_{\ell=0}^{m_i-1} (-1)^\ell P_i^{(\ell)}(Y)$$

This process yields a constructive, non-iterative procedure for both decision (testing invertibility) and explicit inversion. The associated alternating-sum identity is rigorously justified and degree-counting ensures truncation when $G_i$ is polynomial (Adamus et al., 2015).

3. Non-Iterative Inversion in Combinatorics and Discrete Structures

Analogous methods apply for structured combinatorial maps. For example, the general Order sweep map on Dyck paths transforms path diagrams by a deterministic sorting process. The one-step inverse is achieved by a two-phase algorithm: first, a vertical balancing procedure (VIB) produces a balanced, increasing diagram via a lift operation guided by row-counts; then, a horizontal labeling unravels the original path in a single pass via a rule-based relabeling determined by sweep order permutations. This fully generalizes earlier inversion algorithms for modular and rational $(m,n)$-Dyck path sweep maps, demonstrating that the "one-step" inversion is not tied to specific commutative structures (Wang et al., 2023).

4. Applications in Numerical and Computational Systems

Digital Memcomputing Machines (DMMs), realized as networks of self-organizing logic gates (SOLGs), achieve one-step numerical inversion for fixed-point scalars and extend to matrix inversion. Here, instead of applying Newton–Raphson or iterative techniques, the entire inversion is encoded in the topology and logic constraints of the hardware circuit: the satisfying assignment of voltages across interconnected Boolean constraints simultaneously encodes the solution in a single convergent relaxation, with $O(n^2)$ gate complexity for $n$-bit precision. Extension to matrix inversion or linear systems is achieved by parallel replication and appropriate constraint wiring, ensuring direct one-step convergence for all components (Manukian et al., 2016).

5. Direct One-Step Inversion in Imaging and Data Science

In modern computational imaging and machine learning, one-step inversion replaces classic sequential pipelines—e.g., denoising followed by deblurring, or background removal plus deconvolution—by an integrated mapping learned end-to-end or solved as a single optimization.

Prominent examples include:

• Weakly-supervised single-step Quantitative Susceptibility Mapping (wTFI) in MRI, which fuses background field removal and field-to-source inversion into a single neural network mapping $$\chi, \hat{\Phi}_{\text{loc}} = F_\theta(\Phi_{\text{tot}})$$, leveraging multi-task loss to enforce physical and structural consistency in a single pass. This avoids intermediate erosion artifacts and produces high-quality susceptibility maps (Liu et al., 2020).
• One-step inversion methods in dual-energy spectral CT, where nonlinear forward operators relating tissue/material images to measured sinograms are inverted directly by primal-dual hybrid gradient methods (NL-PDHGM), bypassing intermediary material decomposition, and regularizing all variables jointly in the high-dimensional space (Jolivet et al., 2021).

6. Sequential Inversion and Learning-based Single-Step Techniques

Recent advances in diffusion models and deep learning have established one-step inversion as an outstandingly efficient alternative to multi-step reverse diffusion or iterative denoising protocols.

Examples include:

• OSI (One-step Inversion) for diffusion watermark extraction, which reformulates extraction as a single learnable sign classification mapping from images to watermark code, replacing multi-step (e.g., 50-step) diffusion inversion with a single forward pass, and achieving 20× speedup, improved accuracy, and greater payload capacity (Chen et al., 10 Feb 2026).
• SwiftEdit for text-guided image editing, implementing a one-step inversion network $f_\theta$ to map VAE-latent $x$ to plausible noise $\hat{\varepsilon}$ in one pass, such that a one-step generator $G_{\mathrm{IP}}$ reconstructs or edits the image using just two passes. This contrasts sharply with traditional T-step DDIM/NPI inversion pipelines, yielding 50×–500× acceleration without substantial fidelity loss (Nguyen et al., 2024).

7. Structural Implications and Classification

The one-step mapping paradigm not only optimizes inversion—computationally and conceptually—but also exposes hierarchical and structural properties of automorphism or symmetry groups. For polynomial automorphisms, sequences' truncation depths stratify Aut$(\mathbb{A}^n)$ by complexity, giving a filtration $\mathcal{P}_d$ based on the iteration depth at which each $P_i^{(k)}$ vanishes (e.g., $\mathcal{P}_2$ for quasi-translations). This classification effect extends to the study of invertible combinatorial bijections, signal-processing transformations, and hardware-embedded inversion logic (Adamus et al., 2015, Wang et al., 2023, Manukian et al., 2016).

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References:

• (Adamus et al., 2015) "An inversion algorithm for polynomial maps"
• (Wang et al., 2023) "Inverting the General Order Sweep Map"
• (Manukian et al., 2016) "Memcomputing Numerical Inversion with Self-Organizing Logic Gates"
• (Liu et al., 2020) "Weakly-supervised Learning for Single-step Quantitative Susceptibility Mapping"
• (Jolivet et al., 2021) "One-step inversion algorithms for spectral CT, with application to dynamic Cone Beam CT"
• (Chen et al., 10 Feb 2026) "OSI: One-step Inversion Excels in Extracting Diffusion Watermarks"
• (Nguyen et al., 2024) "SwiftEdit: Lightning Fast Text-Guided Image Editing via One-Step Diffusion"