Invertible Reverse-Ordering Transforms

• Invertible reverse-ordering transforms are bijective mappings that reverse the intrinsic order of mathematical objects like convex functions and matrices.
• They manifest in key examples such as the Fenchel conjugation and the reverse order law for the Moore–Penrose pseudoinverse, with explicit algebraic formulations.
• Algorithmic implementations of these transforms enable efficient inversion in diverse fields including data compression, optimization, and combinatorial analysis.

An invertible reverse-ordering transform is an invertible (bijective) transformation that reverses or inverts a fundamental ordering structure, typically defined on a space of mathematical objects such as strings, functions, operators, convex functions, or other algebraic structures. Such transforms are critical both for theoretical investigations—where reversibility and order-duality are structurally central—and for practical algorithms in areas such as combinatorics, information geometry, optimization, and data compression.

1. Formal Definition and General Properties

An invertible reverse-ordering transform T on a set $\mathcal{F}$ equipped with an order $\leq$ is a bijective mapping $T:\mathcal{F}\to\mathcal{F}'$ such that:

• $F_1 \leq F_2 \implies T F_2 \leq T F_1$ (T is order-reversing),
• $T$ is invertible and $T^{-1}$ is also order-reversing (the fully invertible case).

In the context of convex functions, this is exemplified by transforms $T$ satisfying:

$$F_1 \leq F_2 \implies T F_2 \leq T F_1, \quad T F_1 \leq T F_2 \implies F_2 \leq F_1,$$

where $F_1, F_2$ are convex lower-semicontinuous functions (see (Nielsen, 28 Jul 2025, 1212.1120)).

Invertible reverse-ordering transforms arise in a variety of contexts:

• As dualities (e.g., Fenchel–Legendre conjugation in convex analysis),
• As invertible variants of sorting (e.g., in bijective string transforms for text compression (1201.3077)),
• In linear algebra and multilinear algebra via reverse-order laws for generalized inverses (Kędzierski, 3 Apr 2024, Panigrahy et al., 2017, Sahoo et al., 2019),
• In order theory and combinatorics (e.g., reversible posets (Kurilić et al., 2017), invertible sweep maps (Wang et al., 2023)).

2. Canonical Examples and Structural Theorems

Convex Analysis: Fenchel-Legendre Conjugation and Its Generalizations

The only fully invertible reverse-ordering transform (modulo affine changes and positive scalings) on the space of lower-semicontinuous proper convex functions is the Fenchel conjugation $$\mathcal{L}(F)(\eta) = \sup_{\theta} (\langle \theta,\eta\rangle - F(\theta))$$ and its affine-deformed generalizations (1212.1120, Nielsen, 28 Jul 2025):

$$(TF)(\eta) = \lambda (LF)(E\eta + f) + \langle \eta, g \rangle + h$$

with parameters $\lambda > 0$, $E$ invertible, $f,g$ vectors, $h$ scalar. Every such generalized Legendre transform is itself an ordinary Legendre transform of an affine-deformed function:

$$F_P(\theta) = \lambda F(A\theta+b) + \langle \theta,c \rangle + d$$

and the inverse transform corresponds to a "dual" affine deformation. The full classification from Artstein-Avidan and Milman asserts that, essentially, all such invertible reverse-ordering transforms are generalized (affine-deformed) convex conjugates (Nielsen, 28 Jul 2025, 1212.1120).

Algebraic and Matrix Theory: The Reverse Order Law

In (tensor) matrix analysis, an invertible reverse-ordering transform may take the form:

$(AB)^+ = B^+ A^+,$

i.e., the Moore–Penrose pseudoinverse of a product equals the product of pseudoinverses in reverse order. Detailed necessary and sufficient conditions for when matrices or tensors satisfy this law (including characterizations in terms of the singular spaces, principal angles, or commutativity) have been given (Kędzierski, 3 Apr 2024, Panigrahy et al., 2017, Sahoo et al., 2019). For example:

$$(AB)^+ = B^+A^+ \iff C(A^*AB) = C(BB^*A^*) = C(A^*) \cap C(B),$$

and, in tensors, the analogous property holds under the Einstein product when specific commutativity and range-inclusion conditions are met.

Combinatorics: Sweeps, Bijective String Sorting, and Poset Reversibility

Invertible reverse-ordering also arises in combinatorics:

• The bijective string sorting transform ($\text{\textdollar}$-transform) splits the string into Lyndon factors, sorts all cyclic rotations, and outputs a string whose inversion is always possible; this contrasts with the original Burrows-Wheeler transform, which requires auxiliary information for inversion (1201.3077).
• The "Order sweep map" on Dyck paths generalizes the classical sweep maps by precisely controlling the reverse-ordering at critical levels with parameterized permutations, yielding a family of bijections whose inverse is given by a geometric and combinatorial algorithm (Wang et al., 2023).
• In order theory, a poset (e.g., a disjoint union of well orders or their reverses) is reversible if and only if its structure is rigid enough that every bijective homomorphism is an automorphism, with the mathematical characterization given in terms of order type sequences and natural number invariants (Kurilić et al., 2017).

3. Mathematical Frameworks and Explicit Formulas

Invertible reverse-ordering transforms frequently have explicit algebraic or algorithmic forms.

A table summarizes canonical instances:

Context	Core Transform	Invertibility Condition / Formula
Convex Functions (Fenchel)	$F \mapsto F^*$	Affine pre/post-processing; $\mathcal{L}_P(F) = \mathcal{L}(F_{P^\diamond})$ (Nielsen, 28 Jul 2025)
Matrices (Moore-Penrose)	$(AB)^+ = B^+A^+$	$C(A^*AB)=C(BB^*A^*)$; specific SVD-based construction (Kędzierski, 3 Apr 2024)
Tensors (Einstein product)	$(A *_n B)^{\dagger}$	Sufficient commutativity/range-inclusion (Panigrahy et al., 2017, Sahoo et al., 2019)
Strings (BWT / Lyndon)	$$-transform | Lyndon factorization, bijective cyclic rotation sorting ([1201.3077]) | | Dyck Paths | Order sweep map | Sequences of permutations parameterize generalized invertible maps 2307.15357 | Formulas in convex analysis (see [2507.20577]) relate the generalized Legendre transform to a "diamond" involution on the deformation parameters. In the context of operator transforms on functions, mutually inverse transforms can also act by dual envelope constructions, e.g.,$$

Ff = \sup_{t>0} \frac{f(xt)}{t+1},\qquad Gf = \inf_{t>0} f(xt)\cdot (1+t), \quad \text{with } GF[f] = \text{concave majorant}(f)

$$

(Protasov et al., 2021).

4. Algorithmic Constructs and Structural Invertibility

Several works provide algorithmic criteria or constructions for ensuring invertible reverse-ordering, often exploiting cycle decompositions, permutations, or specific data encodings:

• In bijective string sorting and related BWT variants, threads/cycles are exploited to map and invert unique rotational decompositions via permutations (1201.3077, Köppl et al., 2020).
• In generalized sweep maps on Dyck paths, specialized algorithms (e.g., Algorithm VIB) construct balanced path diagrams and induce reversible transformations (Wang et al., 2023).
• In functional programming, invertibility of tail-recursive function transforms is enforced through explicit tracking of call configurations; global injectivity (instead of local reversibility) is sufficient if restricted to feasible operational domains (Kristensen et al., 2023).

The common theme is that invertibility in reverse-ordering transforms is achieved either by decomposing the input structure into uniquely labeled, invertible components (e.g., Lyndon words, path diagrams, singular vector spaces) or by restricting data flows according to structural constraints (e.g., permissible configuration spaces).

5. Applications Across Disciplines

Invertible reverse-ordering transforms are central to a range of theoretical developments and practical applications:

• Data Compression: The bijective string sorting transform enables better compression and removes the need for auxiliary indices, with empirical gains documented on standard compression corpora (1201.3077, Köppl et al., 2020).
• Convex Optimization and Duality: The uniqueness and structure of the Fenchel conjugation underpins dual formulations and optimality conditions (1212.1120, Nielsen, 28 Jul 2025).
• Multilinear and Linear Algebra: Characterization results for the Moore–Penrose pseudoinverse and core inverses of matrices/tensors facilitate efficient solutions for linear and multilinear systems (Kędzierski, 3 Apr 2024, Panigrahy et al., 2017, Sahoo et al., 2019), and have direct computational consequences for solving equations and structure-exploiting algorithms.
• Combinatorics: Generalized order sweep maps produce bijections vital for enumerative formulae and deeper combinatorial analysis, including applications to $q,t$-Catalan combinatorics (Wang et al., 2023).
• Information Geometry: The paper of generalized Legendre transforms clarifies the invariance and coordinate-freedom in dually flat statistical manifolds and relates directly to the structure of exponential families (Nielsen, 28 Jul 2025).

6. Geometric and Structural Interpretation

From an abstract standpoint, invertible reverse-ordering transforms reflect dualities or symmetry properties intrinsic to the mathematical structures being studied:

• In convex function spaces, all such transforms can be viewed as equivalence classes of convex functions modulo affine deformation—generalized conjugation acts canonically on these classes (Nielsen, 28 Jul 2025).
• Information geometry manifests these dualities at the level of dually flat manifolds, where affine coordinates and potential functions admit affine ambiguity, and the Legendre transform realizes the canonical dual coordinate change (parameter transformations correspond to scaling, shifting, and affine composition).
• In multilinear algebra, the geometric content of the reverse order law for generalized inverses is encoded in the principal angles between subspaces: the law holds when these angles are all zero or $\pi/2$, i.e., when the relevant subspaces are either coincident or orthogonal (Kędzierski, 3 Apr 2024).
• In order theory, reversibility of unions of well orders reflects the absence of "nonisomorphic" condensations, enforcing a rigid structure resistant to nontrivial automorphisms that reverse or permute order components (Kurilić et al., 2017).

7. Open Questions and Future Perspectives

Although exhaustive characterizations exist in some domains, several directions remain active:

• Extending full characterizations to nonreflexive Banach spaces and more general function spaces (1212.1120).
• Generalizing reverse-order laws for weighted or non-Einstein tensor products and to broader classes of generalized inverses (Panigrahy et al., 2017, Sahoo et al., 2019).
• Developing further algorithmic frameworks for efficient inversion and application of order-reversing transforms in nontraditional domains (e.g., sequence models or nonabelian group actions).
• Clarifying the set-theoretic and geometric universality of these transforms when subject to more relaxed or partial invertibility assumptions.

A plausible implication is that invertibility in reverse-ordering transforms continues to serve as a guiding principle in unifying dualities, symmetry operations, and efficient data representations across mathematics, theoretical computer science, and information geometry.