Function Flattening: Concepts & Applications

• Function flattening is the process of converting complex, high-dimensional data objects into simpler, often matrix-like forms to reveal intrinsic structural properties.
• It facilitates advanced analyses in areas such as tensor decomposition, hierarchical clustering, geometric mapping, logical semantics, and phylogenetic inference.
• The methodology employs functorial and optimization techniques to systematically reduce dimensionality while preserving key features for further exploration.

Function flattening encompasses a range of methodologies in mathematics, computer science, combinatorics, logic, and geometry that transform complex, high-dimensional, or interactively-structured data objects, functions, or formulas into simpler—often matrix-like or pointwise—forms. In rigorous research contexts, this "flattening" is applied to tensors, hierarchical partitions, combinatorial objects, logical formulas, geometric mappings, and algebraic sheaves, typically aiming to isolate intrinsic properties, facilitate analysis using linear-algebraic or optimization techniques, or enforce locality and flatness in logical semantics.

1. Tensor Flattening and Flattening Rank in Combinatorics

A central paradigm of function flattening is found in tensor theory, particularly for a $d$-dimensional tensor $T: A_1 \times \dots \times A_d \to \mathbb{F}$, where $\mathbb{F}$ is a field. The $i$-flattening of $T$ treats $T$ as a matrix: one coordinate ($A_i$) indexes the rows, the remaining coordinates ($B_i = \prod_{j \neq i} A_j$) index the columns. The $i$-flattening rank, $\mathrm{frank}_i(T)$, is defined as the rank of this matrix.

The max-flattening rank, $$\mathrm{mfrank}(T) = \max_{i \in [d]} \mathrm{frank}_i(T)$$, measures the maximum rank over all possible flattenings. These ranks generalize classical notions of matrix rank to higher-order tensors, maintaining key properties such as subadditivity and monotonicity under subtensor restriction. Particularly, for "semi-diagonal" tensors—those with nonzero entries only on tuples with all entries equal and zero on all-distinct tuples—it is proved that $\mathrm{mfrank}(T) \geq |A|/(d-1)$, and this lower bound is tight (Correia et al., 2021).

This flattening methodology is used to generalize the Frankl–Wilson theorem (forbidden intersection sizes in set systems) and to establish improved bounds for rainbow matchings in $r$-uniform multi-hypergraphs, by encoding combinatorial structures as semi-diagonal tensors and leveraging the flattening rank as a dimension-counting argument.

2. Flattening in Hierarchical Clustering and Category Theory

In topological data analysis and applied category theory, function flattening refers to a functorial procedure that transforms multiparameter hierarchical clusterings into binary integer programs (Shiebler, 2021). Here, the flattening functor $F: \mathbb{MHP} \to \mathbb{BIP}$ encodes the combinatorial and parameter-wise relationships of clusterings—originally represented as hierarchical partitions or poset-structured families—into a set of binary variables and linear constraints that define feasible cluster selections. This enables leveraging established optimization techniques for solution and analysis.

Empirical results confirm that such flattenings preserve critical multi-scale and structural information, enabling effective clustering, and the category-theoretic (functorial) formulation ensures that changes in the clustering paradigm translate systematically to modifications in the optimization problem.

Furthermore, the integration of a Bayesian update algorithm for data-driven learning of clustering parameters, composed with the flattening functor, yields a system with provable consistency: the posteriors concentrate on parameter values producing "flattened" cluster structures matching the data's true organization.

3. Geometric and Algorithmic Aspects of Flattening

In computational geometry, function flattening arises in the mapping of polyhedral manifolds from $\mathbb{R}^3$ to the plane. It is proven that any finite 3D polyhedral manifold can be continuously flattened into 2D, preserving intrinsic metric and avoiding crossings, provided one allows countably infinite creases (Abel et al., 2021). The flattening operation here consists of slicing the manifold into prismoidal slabs along a dense set of planes, with local flattening gadgets (parameterized by geometric quantities such as dihedral and face angles) orchestrating a globally continuous, isometric, collision-free motion.

This result extends prior work, which was confined to convex or semi-orthogonal polyhedra, to arbitrary nonconvex or non-orientable manifolds. Applications include guaranteed texture mapping, origami design, deployable architectural surfaces, and metamaterial engineering.

A related class of flattening problems involves the planar embedding (flattening) of fixed-angle orthogonal chains, central to polymer science and theoretical molecular biology (Demaine et al., 2022). Flattening in this context means finding a noncrossing planar embedding for a linkage with fixed joint angles (either $90^\circ$ or $180^\circ$). While open chains can be always flattened, the flattening problem for closed chains or chains constrained to a bounding box is strongly NP-complete. An extension analyzes the hydrophobic–hydrophilic (HP) model of protein folding in the fixed-angle regime, proving the maximization of H–H contacts under planar flattening is also strongly NP-complete. This indicates inherent computational intractability in even simplified models of constrained folding and packing.

4. Function Flattening in Logic: Team-Based Semantics

In team-based logics, the flattening operator $F$ enforces pointwise (flat) satisfaction of formulas with respect to team semantics (Durand et al., 27 May 2025). For a team $X$ and formula $\varphi$, $F\varphi$ is satisfied on $X$ exactly when $\varphi$ is satisfied on all singleton subsets: $$M \models_X F\varphi \iff \forall s \in X,\, M \models_{\{s\}} \varphi.$$ This operator "erases" any non-local, team-dependent dependencies, inclusions, or exclusions, reducing the satisfaction condition to individual assignments. It is idempotent and acts as an identity on formulas already flat; for complex team-based atoms such as dependence or inclusion, $F$ converts them to trivialities (always true or equivalent to classical logic).

An important discovery is that $F$ increases expressivity in certain restricted fragments (for example, allowing the expression of graph non-connectedness in the unary inclusion or anonymity versions), but is "safe"—does not increase expressivity—when added to dependence logic, exclusion, or independence logic. This establishes a nuanced landscape of the impact of flattening on expressive power.

5. Flattening Matrices in Phylogenetic Inference

In phylogenetics, function flattening is implemented by mapping joint site pattern probability tensors to "flattening matrices" (Snyman et al., 2021). For a set of taxa split into subsets $A$ and $B$, the flattening matrix $flat_{(A|B)}$ arranges the joint probabilities such that rows correspond to patterns on $A$ and columns to $B$.

A fundamental result establishes that the rank of this flattening matrix is $r^{\ell_T(A|B)}$, where $r$ is the number of character states and $\ell_T(A|B)$ the parsimony length (minimum number of substitutions required to explain the $A|B$ split). This sharpening of earlier lower and upper bounds links an algebraic invariant (the matrix rank) to a key combinatorial measure. It provides a precise algebraic criterion for reconstructing tree splits and enhances techniques based on singular value decomposition and phylogenetic invariants.

6. Combinatorial and Algebraic Flattening: Bijections and Algorithmics

The idea of function flattening appears in combinatorics via "flattened partitions" and "flattened parking functions" (Elder et al., 2022). A parking function, or permutation, is "flattened" if, upon decomposing into runs (maximal chains of weak ascents), the first entries of the runs form a weakly increasing sequence. The paper introduces insertion-based flattening (e.g., $S$-insertion), studies enumerative invariants, and constructs bijections with set partitions, revealing that some flattened parking function classes correspond to classical objects such as Catalan numbers and Bell numbers. Despite progress, the comprehensive enumeration problem remains open.

In algebraic geometry, flattening is realized via a functorial procedure that "flattens" a coherent sheaf (or its family) over a base by iteratively blowing up along centers defined by the "fitted flatifier" ideal (McQuillan, 1 Dec 2024). The proper transform of the sheaf becomes flat after finitely many such operations, and the construction is functorial and commutes with smooth base change—essential for its application to moduli problems and for proving algebraisation theorems for formal deformations. This flattening procedure can be used to control the coherence of nil radicals and resolve questions about the equivalence of algebraizability for formal schemes and their reductions.

7. Comparative Analysis: Methodological Diversity and Unified Themes

Across these domains, flattening serves the common role of projecting or transforming complex or high-order data—whether tensors, combinatorial objects, hierarchical structures, geometric mappings, logical formulas, or algebraic sheaves—into forms where structure can be revealed, analyzed, or optimized with existing mathematical tools. Key recurring technical features include:

• Reduction of dimensionality (tensor to matrix, team to singleton, manifold to plane)
• Enforcement or measurement of locality/flatness (pointwise satisfaction, parsimony length, algebraic flatness)
• Encoding of hierarchical or parameterized structure into linear or combinatorial optimization frameworks
• Functoriality or canonical procedures ensuring base-change invariance

Distinct methodological advances—tensor rank bounds in combinatorics (Correia et al., 2021), functorial flattening and Bayesian learning in clustering (Shiebler, 2021), geometric flattenings with infinite crease configurations (Abel et al., 2021), computational complexity of packing and folding (Demaine et al., 2022), and logical expressivity via semantic flattening (Durand et al., 27 May 2025)—demonstrate the applicability and necessity of flattening beyond its original algebraic-geometric and matrix-theoretic antecedents.

Function flattening thus provides a foundational toolkit across mathematics, logic, combinatorics, geometry, and algorithmics, with core unifying principles adapted to the structural specifics of each domain.