Largest Angle Path Distance (LAPD)

• LAPD is a geometric metric on simplex graphs that measures the maximum dihedral angle along a path to capture intrinsic manifold structures.
• It minimizes worst-case angular deviations between adjacent simplices, enabling robust separation of intersecting or curved low-dimensional manifolds even in noisy data.
• Empirical studies on synthetic and real datasets demonstrate LAPD’s capacity to enhance clustering accuracy compared to standard Euclidean and intrinsic distance metrics.

Largest Angle Path Distance (LAPD) is a geometric metric defined on graphs of simplices, designed primarily for multi-manifold clustering (MMC) and related tasks requiring the discrimination of data sampled from intersecting low-dimensional manifolds. Distinct from conventional path metrics that sum edge weights, LAPD is an infinity-path or bottleneck metric: it measures, along a path in the graph, the maximum encountered dihedral angle between adjacent simplices and chooses the path minimizing this worst-case angular deviation. This approach captures intrinsic geometric transitions, enabling robust cluster separation even in data with noise, curvature, and intersecting structures (2507.10710).

1. Foundational Concept

LAPD originates in the context of clustering data sampled near unions of manifolds—sets possibly intersecting at small angles—where standard Euclidean distances or intrinsic path metrics fail to distinguish distinct components. By focusing on the turning angle required to transit between local geometric elements (“simplices”) within the data, LAPD leverages the observation that within a single manifold, these angles remain small, while paths connecting different manifolds must negotiate at least one large dihedral angle.

Formally, LAPD is constructed on a weighted simplex graph:

• Nodes: d-simplices extracted from the data (segments, triangles, etc.).
• Edges: Connect simplices sharing a (d–1)-face.
• Edge weights: Dihedral angles (or a function thereof) between adjacent simplices reflect the local geometric “bending.”

The LAPD between two simplices is defined as the minimal, over all connecting paths, of the maximum edge weight on the path:

$$\mathrm{LAPD}(\Delta_i, \Delta_j) = \min_{\Gamma} \max_{t} W_S(\Gamma_t, \Gamma_{t+1}),$$

where $W_S$ denotes the weight (dihedral angle-derived) function (2507.10710).

2. Mathematical Formulation

The simplex graph $\mathcal{G}_S = (S, E_S, W_S)$ is constructed as follows:

• Simplex Adjacency: Two $d$-simplices $\Delta_i, \Delta_j \in S$ are adjacent if they share exactly $d$ vertices (a $(d-1)$-face).
• Edge Weight (One-sided):

$W_S(\Delta_i, \Delta_j) = \pi - \theta_{ij},$

where $\theta_{ij}$ is the dihedral angle between simplices. For nearly coplanar (flat) pairs, $W_S$ is near zero.

• Alternative (Two-sided) Weights:

$$W_S^{(2)}(\Delta_i, \Delta_j) = \min\{\pi - \theta_{ij},\, \theta_{ij}\},$$

which makes edge weights small for both almost $0$ and almost $\pi$ angles, allowing for “backflips.”

Crucially, only “good” simplices (according to distortion metrics on lengths and volumes) are retained; these are filtered according to bounds on aspect ratio and normalized volume:

$$e \leq \|\boldsymbol{x}_i - \boldsymbol{x}_j\| \leq \frac{e}{q}, \quad r_0 V_0 e^d \leq \mathrm{vol}_d(\Delta) \leq V_0 \left(\frac{e}{q}\right)^d$$

for all $i, j$ in $\Delta$, where $V_0$ is the volume of a regular $d$-simplex of unit edge length (2507.10710).

3. Algorithmic Pipeline

The LAPD methodology comprises the following stages:

1. Locality Graph Construction: An annular $k$-nearest neighbor graph $\mathcal{G}_X$ is built to identify candidate neighbors at an appropriate scale ($e>0$), excluding distances too close (to avoid noise).
2. Simplex Construction and Filtering: Candidate $d$-simplices are formed from $k$-NN neighborhoods, retaining those with suitable aspect ratio and normalized volume.
3. Weighted Simplex Graph: Valid simplices serve as nodes; edges connect simplices sharing a $(d-1)$-face, and dihedral angle-based weights are assigned.
4. LAPD Computation: For pairs of simplices, the infinity-shortest path (i.e., path minimizing the largest edge) is computed, yielding the LAPD metric.
5. Denoising: Simplices potentially mixing manifold labels (near intersections) are removed based on their neighborhood LAPD values exceeding a threshold, enhancing separation.
6. Clustering: The denoised simplex graph undergoes clustering (often hierarchical or single-linkage). Data points inherit a label by majority vote among the simplices containing them (2507.10710).

The following pseudocode outlines the generic LAPD computation phase for a fixed pair:

def lapd_minimax_path(G, start, end): # G: weighted simplex graph, using dihedral angle-derived weights # start, end: simplex indices # Returns minimum bottleneck path between start and end from queue import PriorityQueue visited = set() pq = PriorityQueue() pq.put((0, start)) # (current_max_weight, node) while not pq.empty(): curr_max, curr = pq.get() if curr == end: return curr_max if curr in visited: continue visited.add(curr) for neighbor, weight in G.neighbors(curr): pq.put((max(curr_max, weight), neighbor)) return float('inf')

In practice, scalable multi-source computations are implemented via approximations akin to single-linkage dendrograms.

4. Theoretical Properties and Guarantees

LAPD is analyzed probabilistically under the assumption that data are sampled randomly from tubular neighborhoods around $K$ smooth $d$-dimensional manifolds, with maximal noise scale $\tau$ and minimal intersection angle $\Theta$.

Key properties:

• wLAPD (within-manifold): $$\displaystyle \max_{(\Delta, \Delta')} \mathrm{LAPD}(\Delta, \Delta') \lesssim \tau/e$$, where $e$ is the simplex side length.
• bLAPD (between-manifold): $$\displaystyle \min_{(\Delta, \Delta' \text{ from different manifolds})} \mathrm{LAPD}(\Delta, \Delta') \gtrsim \Theta$$.

For sufficiently separated manifolds ($\Theta$ not too small and $e \gg \tau$), and after denoising mixed simplices, one obtains a distribution of path distances with a clear gap: $\mathrm{wLAPD} \ll \mathrm{bLAPD}$. This is foundational for robust cluster recovery (2507.10710).

Formal theorems in the paper prove, with high probability, sufficient separation for correct cluster assignments.

5. Empirical Performance and Applications

Extensive experiments on synthetic and real datasets validate the approach:

• Synthetic Data: Intersecting subspaces, bent curves, and manifold unions with added noise—performance matches theory regarding separation and robustness to τ and Θ.
• Real-world Data: Image databases including COIL20, USPS, and MNIST serve to evaluate high-dimensional clustering performance. LAPD achieves superior or competitive accuracy, demonstrates robustness to curvature and noise, and is capable of inferring cluster numbers automatically using persistence-based techniques.
• Comparison: The method consistently outperforms or matches subspace clustering (e.g., LocPCA), path-based clustering, and deep MMC methods, particularly for intersecting or near-parallel manifold structures (2507.10710).

6. Computational Complexity and Practical Implementation

Implementation strategies yield scalability for large $n$ and $D$:

• Nearest-neighbor and Graph Building: $O(D n \log n)$ using spatial indexing.
• Simplex Enumeration: $O(n B^d)$; tractable for small $d$ ($B$=number of neighbors).
• Infinity-path Distance Computation: Approximate single-linkage graphs eliminate the need for complete $O(|S|^2)$ pairwise computations; quasi-linear in $|S|$.
• Denoising and Clustering: Hierarchical clustering, $k$-NN searches, and subsequent steps efficiently map labels back to original data (2507.10710).

The dependence on simplex dimension is exponential, but in practical MMC, $d$ is typically moderate (e.g., $d \leq 5$). The choice of parameters—scale $e$, aspect ratio $q$, volume threshold $r_0$, and denoising thresholds—controls the granularity and sensitivity of the analysis.

7. Relations to Path Planning and Angle Constraints

While LAPD is primarily introduced in the context of MMC in (2507.10710), the concept of optimizing or constraining largest turning angles appears in path planning on grids. For example, the LIAN algorithm (1506.01864) and modifications of Theta* (1401.3843) focus on bounding or minimizing the maximum turning angle along a path for smooth trajectories in robotics and navigation. There, the largest angle encountered—analogous to the LAPD along a path—serves as both a feasibility criterion and an optimization target. These connections highlight the broader applicability of bottleneck angular metrics beyond MMC, including robotics, computer vision, and geometric data analysis.

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In summary, the Largest Angle Path Distance offers a principled, geometric, and scalable approach for distinguishing inter-manifold from intra-manifold relations in sampled data, via a minimax angular path metric. Its rigorous theoretical analysis and robust empirical performance establish LAPD as a central tool for MMC and related geometric data tasks (2507.10710).