3D Cubical Complex Filtration

• 3D Cubical Complex Filtration is a hierarchical construction of nested cubical complexes used to approximate Rips filtrations for persistent homology in 3D point sets.
• It employs shifted, scaled integer lattices and cubical maps to define active vertices and faces, ensuring computational efficiency with at most 216n cells per scale.
• Acyclic carrier theory and scale balancing yield strong interleavings between cubical and Rips filtrations in both L∞ and L2 metrics, achieving an overall approximation factor of about 2.63.

A 3D cubical complex filtration is a discrete, tower-like sequence of axis-aligned cubical complexes built to efficiently approximate the persistent homology of metric data sets in $\mathbb{R}^3$. Specifically, such filtrations can provide strong, quantifiable approximations to Rips filtrations in both $L_\infty$ and $L_2$ metrics, with rigorous algorithmic complexity bounds and approximation guarantees derived from acyclic carrier theory and scale balancing. The construction leverages integer lattice grids and cubical maps, yielding both an efficient representation and effective computational tractability for large data sets (Choudhary et al., 2021).

1. Shifted, Scaled Integer Lattices and Grid Construction

Given a finite point set $P \subset \mathbb{R}^3$, the construction is initialized by defining key geometric parameters: $$\mathrm{CP} := \min_{p \neq q \in P} \|p-q\|_\infty$$ and $\mathrm{diam}(P) := \max_{p,q} \|p - q\|_\infty$. The fundamental scale parameter is set as $\lambda := \mathrm{CP}/9$, and the discrete set of filtration scales is $$I = \{\alpha_s = \lambda \cdot 2^s \mid s \in \mathbb{Z}\}$$.

For each scale index $s$, a regular grid $G_{\alpha_s}$ in $\mathbb{R}^3$ is constructed:

• Base Case ($s=0$): $G_{\alpha_0} = \lambda \cdot \mathbb{Z}^3$.
• Up-step ($s \to s+1$): Select an "origin" $O_{\alpha_s} \in G_{\alpha_s}$. Define $G_{\alpha_{s+1}}$ as $$2 \cdot (G_{\alpha_s} - O_{\alpha_s}) + O_{\alpha_{s+1}} + (\alpha_s/2) \cdot (\varepsilon_1, \varepsilon_2, \varepsilon_3)$$, where each $\varepsilon_i = \pm 1$ is chosen randomly and independently.
• Down-step ($s \to s-1$): Analogously, divide by 2 and shift by $\pm (\alpha_{s-1}/2)$ in each coordinate.

Each grid $G_{\alpha_s}$ forms a nested structure such that every point in $G_{\alpha_s}$ is associated with a unique Voronoi cell of $G_{\alpha_{s+1}}$. The axis-aligned cubical complex $T_{\alpha_s}$ built over $G_{\alpha_s}$ is the full cubical complex whose cubes (including faces of all dimensions 0–3) are Cartesian products of intervals of length $0$ or $\alpha_s$ in each coordinate.

2. Scale Parameters and Approximation to Rips Filtration

The goal is to approximate the $L_\infty$-Rips filtration $R_\alpha(P)$, whose simplices correspond to subsets of $P$ with $L_\infty$-diameter at most $\alpha$. The 3D cubical complex filtration operates only at discrete scales $\alpha_s$, but is extended piecewise-constantly to all $\alpha \in [\alpha_s, \alpha_{s+1})$.

A core result is the existence of a strong 2-interleaving between the barcode of $(H(U_{2\alpha}))_{\alpha \geq 0}$ and the barcode of the Rips filtration $(H(R_\alpha))_{\alpha \geq 0}$. In particular, $H(U_{2\alpha}) \simeq H(R_\alpha)$ up to a factor 2 in the bottleneck metric.

When translating from $L_\infty$ to Euclidean ($L_2$) metrics, since $$\|p - q\|_\infty \leq \|p - q\|_2 \leq \sqrt{3}\|p - q\|_\infty$$, the filtrations are $\sqrt{3}$-interleaved. Applying a "scale balancing" technique improves this approximation ratio to $3^{1/4}$, so the overall approximation factor to the Euclidean Rips barcode is $2 \cdot 3^{1/4} \approx 2.63$.

3. Construction of the Cubical Complexes $U_{\alpha_s}$

A. Active Vertices

Each input point $p \in P$ is mapped to $a_s(p)$, the unique grid vertex in $G_{\alpha_s}$ whose Voronoi cell contains $p$. The set $$V_{\alpha_s} = \text{image}(a_s) \subseteq G_{\alpha_s}$$ is termed the set of active vertices.

B. Active and Secondary Faces

A cube-face $f$ of $T_{\alpha_s}$ is active if $f \cap V_{\alpha_s} \neq \emptyset$ and the active vertices of $f$ are not all contained in a single proper facet of $f$. Any face of an active face $f$ that does not satisfy this second property is called secondary.

C. Definition of $U_{\alpha_s}$

The cubical complex $U_{\alpha_s}$ is the subcomplex of $T_{\alpha_s}$ whose cubes consist of all active faces (of any dimension) and their secondary subfaces.

D. Cubical Maps Between Scales

A cubical map $g_s : U_{\alpha_s} \to U_{\alpha_{s+1}}$ is defined: each vertex $x \in G_{\alpha_s}$ is mapped to the unique $y \in G_{\alpha_{s+1}}$ whose Voronoi cell contains $x$. For each cube $$\gamma = [x_1, x_1 + m_1] \times [x_2, x_2 + m_2] \times [x_3, x_3 + m_3]$$, $$g_s(\gamma) = [g_s(x_1), g_s(x_1) + m_1'] \times [g_s(x_2), g_s(x_2) + m_2'] \times [g_s(x_3), g_s(x_3) + m_3']$$ where each $m_i'$ is $0$ or $\alpha_{s+1}$. The map preserves activeness.

4. Acyclic Carriers and Approximation Guarantees

Two acyclic carriers are developed to relate the homological features of the cubical filtration and the Rips filtration:

• Carrier $C_1^\alpha$ (From $R_\alpha$ to $U_{2\alpha}$): For a simplex $\sigma = \{p_0, ..., p_k\}$ in $R_\alpha$, the points $a_{2\alpha}(p_i)$ all lie in some face $f \subset T_{2\alpha}$. The carrier assigns to $\sigma$ the subcomplex of $U_{2\alpha}$ formed by all active and secondary faces in $f$. This carrier is nonempty and acyclic.
• Carrier $C_2^\alpha$ (From $U_\alpha$ to $R_\alpha$): For each cube $\gamma \in U_\alpha$, $C_2^\alpha(\gamma)$ is the simplex on $\{p \in P \mid a_\alpha(p)$ is a vertex of $\gamma\}$. Since the $L_\infty$-diameter is bounded by $\alpha$, this is a valid simplex in $R_\alpha$.

By the Acyclic Carrier Theorem, these carriers induce augmentation-preserving chain maps $c_1^\alpha: C_*(R_\alpha) \to C_*(U_{2\alpha})$ and $c_2^\alpha: C_*(U_\alpha) \to C_*(R_\alpha)$. They satisfy homological commutative diagrams demonstrating a strong $2$-interleaving up to scale, after applying scale balancing as necessary (Choudhary et al., 2021).

5. Complexity Bounds and Scalability

A precise size bound for the cubical approximation is established. For $d=3$, the total number of cubical cells added across all scales is at most $n \cdot 6^3 = 216n$. Consequently, each complex $U_{\alpha_s}$ at any level contains at most $216n$ cubes. The algorithmic complexity per scale is as follows:

Operation	Complexity	Explanation
Compute $a_s(p)$ for $p \in P$	$O(n)$	Hashing into grid
Enumerate active vertices $V_\alpha$	$O(n)$	$|V_\alpha| \leq n$
For $v \in V_\alpha$, check 27 neighbors	$O(n)$	At most $27 |V_\alpha|$ checks
Map all cubes by $g_s$	$O(n)$	Proportional to cube count
Output new cubes	$O(n)$	At most $216n$ new cubes per level

There are $O(\log(\mathrm{diam}/\mathrm{CP}))$ scales, so the overall expected running time is $O(n \log \Delta)$ and space complexity is linear in $n$.

6. Scale Balancing and Extension to Euclidean Metric

The filtration yields a strong 2-interleaving with the $L_\infty$-Rips tower. Since $$\|p - q\|_\infty \leq \|p - q\|_2 \leq \sqrt{3}\|p - q\|_\infty$$, the $L_2$- and $L_\infty$-Rips filtrations are strongly $\sqrt{3}$-interleaved. Applying a "scale balancing" technique described in (Choudhary et al., 2021) refines this to a strong $3^{1/4} \approx 1.316$-interleaving. Therefore, the overall approximation factor from the cubical filtration to the Euclidean Rips barcode is $2 \cdot 3^{1/4} \approx 2.63$.

7. Summary and Significance

The 3D cubical complex filtration framework produces a combinatorially efficient and algebraically sound approximation of Rips filtrations for topological analysis of point cloud data. The approach systematically constructs a tower of cubical complexes, employs acyclic carriers for rigorous homological approximation, and applies scale balancing for optimal approximation in the Euclidean setting. The method guarantees at most $216n$ cells per scale, $O(n \log \Delta)$ time, and extends directly to higher dimensions with $n \cdot 6^d$ cubical cells, preserving the essential topological features with quantifiable approximation factors (Choudhary et al., 2021).