Crossed-Hyperplane Filtering: Foundations & Applications

• Crossed-hyperplane filtering is a method that employs intersecting hyperplanes to partition Euclidean space into distinct atoms for classification and pattern recognition.
• The polar complex and bitflip-invariant structure provide topological and combinatorial criteria that ensure stability and realizability in neural and geometric systems.
• The duality between hyperplane and hyperspherical boundaries enables flexible transformations between Euclidean and spherical domains, enhancing kernel methods and machine learning algorithms.

Crossed-hyperplane filtering refers to the use of multiple intersecting hyperplanes to partition, classify, or filter a domain—typically a Euclidean space—by organizing data according to which regions, or "atoms," lie between or beyond these hyperplanes. This concept subsumes the geometric, topological, and combinatorial properties of hyperplane codes and extends to transformations involving hyperspheres, enabling dual representations in spherical and Euclidean spaces with significant implications for neural codes, kernel methods, discriminative boundary construction, and pattern recognition systems.

1. Foundational Framework: Hyperplane Codes and Filtering

A hyperplane code is defined via the intersection patterns of half-spaces in $\mathbb{R}^d$. Each hyperplane is given by %%%%1%%%%, with open half-spaces $H^+ = \{x: w \cdot x - h > 0\}$ and $H^- = \{x: w \cdot x - h < 0\}$. For an open convex domain $X \subset \mathbb{R}^d$ and a collection of half-spaces $\mathcal{U} = \{U_1, \dots, U_n\}$, $U_i = H_i^+ \cap X$, the atom for $\sigma \subset [n]$ is $$A_\sigma = (\bigcap_{i \in \sigma} U_i) \setminus (\bigcup_{j \notin \sigma} U_j)$$. The hyperplane code is then $$\mathcal{C} = \{\sigma \subset [n]: A_\sigma \ne \emptyset \} \subset 2^{[n]}$$.

Stable hyperplane codes arise from arrangements where the intersection of any subset of hyperplanes is generic: if $X \cap (\bigcap_{i \in \sigma} H_i) \ne \emptyset$, then $\dim(\bigcap_{i \in \sigma} H_i) = d - |\sigma|$ and each nonempty atom has nonempty interior. Stability confers robustness to small perturbations of hyperplane parameters and invariance under "bitflip"—reversal of hyperplane orientation.

In a neural modeling context, the pattern of neuron activation following a layer of linear threshold neurons partitions the input space by crossed hyperplanes, and the resulting convex code is determined precisely by these intersection patterns.

2. The Polar Complex and Bitflip-Invariant Structure

The polar complex, associated to any combinatorial code $\mathcal{C} \subset 2^{[n]}$, is a pure $(n-1)$-dimensional simplicial complex constructed from two disjoint copies of the vertex set: "unbarred" $[n] = \{1,\dots,n\}$ and "barred" $\{\bar{i}: i \in [n]\}$. For each codeword $\sigma$, define its facet as $$\Sigma(\sigma) = \{i: i \in \sigma \} \cup \{\bar{i}: i \notin \sigma\}$$. The polar complex is $$\Gamma(\mathcal{C}) = \Delta(\{ \Sigma(\sigma): \sigma \in \mathcal{C} \})$$.

The polar complex captures both "on" and "off" states of the crossing hyperplanes, while bitflip actions $g \in (\mathbb{Z}_2)^n$ transform $\mathcal{C}$ and induce isomorphic polar complexes. In the special case $\mathcal{C} = 2^{[n]}$, $\Gamma(2^{[n]})$ realizes the boundary of the cross-polytope. For stable arrangements, the polar complex is also isomorphic to the nerve of the covering by positive and negative half-spaces, connecting the geometric realization with combinatorial topology and supporting further analysis via Stanley–Reisner theory.

3. Shellability: Topological Filtration and Obstruction Removal

Shellability is a property of pure simplicial complexes, exemplified by a facet ordering such that the intersection of a new facet $F_i$ with the union of previous facets is pure of dimension $d-1$. For polar complexes of stable hyperplane codes, shellability is established by sweeping a hyperplane through the domain and ordering atoms accordingly.

Shellability of $\Gamma(\mathcal{C})$ has powerful implications: it guarantees the structure admits no weak or strong bitflip local obstructions, no sphere link obstructions, and no chamber obstructions—removing all previously known topological obstacles to realization as a stable hyperplane code. This unification makes shellability a necessary (and efficiently testable) condition for a combinatorial code to be realizable via crossed-hyperplane filtering.

4. Inversive Geometry: Hyperplanes and Hyperballs Duality

Crossed-hyperplane filtering can be generalized to include hyperspherical boundaries using inversive geometry, stereographic projections, and explicit transformations. Through spherical embedding $S(X, v, s)$ and its inverse $S^{-1}(Y, v, s)$, ordinary Euclidean data can be mapped onto a sphere such that hyperplane separation corresponds to hyperspherical cap separation, and vice versa. This cap–ball duality allows interchange between discriminative boundaries: a cap on the sphere arises from a hyperplane, while its inverse image in $\mathbb{R}^d$ is a hyperball with an explicitly given center and radius.

Transformations are provided in closed-form: for $v = (0,\dots,0,1)$ and scale $s$,

• $S(x, v, s) = 2s \cdot ( x / (\|x\|^2 + s^2) ) - v$
• $$S^{-1}(y, v, s) = s \cdot (y_{1,\dots,d}/ (1 + y_{d+1}))$$

Under these embeddings, the mapping between hyperplanes (spherical caps) and hyperballs is equally explicit, allowing rapid switching between geometric domains in machine learning architectures and similarity search algorithms.

5. Filtering, Kernel Functions, and Computational Tests

Explicit formulae relate pre- and post-embedding inner products and Euclidean distances:

• For $x, y \in \mathbb{R}^d$, their embeddings yield

$$\langle \hat{x}, \hat{y} \rangle = 1 - 2s^2 \frac{ \|x\|^2 - 2\langle x, y \rangle + \|y\|^2 }{ (\|x\|^2 + s^2)(\|y\|^2 + s^2) }$$

$$\|\hat{x}-\hat{y}\|^2 = 4s^2 \frac{ \|x-y\|^2 }{ (\|x\|^2+s^2)(\|y\|^2+s^2) }$$

These relationships enable implicit kernel calculations without explicit embedding/unembedding steps.

Algebraic invariants from Stanley–Reisner ideals and multigraded Betti numbers derived from the polar complex serve as computational tests for realizability. Deviations from expected Betti numbers or failure of shellability reveal that a code cannot arise from crossed-hyperplane filtering, thus filtering out unfit combinatorial signatures for neural or geometric systems.

6. Applications, Dimensionality Considerations, and Extensions

Crossed-hyperplane filtering admits diverse applications:

• Neural classifier structures, where each neuron corresponds to filtering via a specific hyperplane and the output code encodes data partitioning.
• Kernel methods, where data is mapped to spherical domains for optimized inner-product computations and more effective similarity searching (e.g., HIOB, ScaNN).
• Support Vector Machines, which, when operated on spherical embedded data, exhibit effective equivalence to Support Vector Data Description using hyperballs.

Projection strategies guarantee the embedding of data onto hemispheres, with scale $s$ chosen such that $s \ge \max_{x \in X} \|x\|$, ensuring coverage. To optimize data distribution on the sphere, the intrinsic dimensionality estimator ($ABID$) is recommended, with $s$ swept over a range proportional to mean vector norm.

Neural network architectures can employ spherical activation by replacing the first layer with spherical embedding functions, resulting in faster convergence and "softer" decision boundaries compared to conventional ReLU-based partitions.

7. Contextual Significance and Theoretical Synthesis

Crossed-hyperplane filtering not only generalizes the construction of convex codes via hyperplane intersections but—through the polar complex and inversive geometry—systematizes all known combinatorial, algebraic, and topological necessary conditions for such filtering to be geometrically sound and robust. Shellability of the associated polar complex unifies prior obstruction criteria and offers algorithmic verifiability. The geometric duality between hyperplanes and hyperballs enables flexible yet rigorous construction of discriminative boundaries in high-dimensional algorithmic contexts.

A plausible implication is that the frameworks developed for crossed-hyperplane filtering provide standardized criteria for the design and validation of neural architectures, kernel methods, and pattern recognition systems that rely on intersection-based filtering, and set benchmarks for combinatorial codes to be realized in applied geometric and topological data analysis (Itskov et al., 2018, Thordsen et al., 28 May 2024).