Umap - Learning Module | OneNoughtOne

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Fuzzy Topological Representation

From Points to Topology

The genius of UMAP lies in its representation of data—not as a collection of points in space, but as a topological object that captures the intrinsic shape and connectivity of the underlying manifold. This representation, the fuzzy simplicial set, is UMAP's secret weapon for achieving embeddings that faithfully preserve both local neighborhoods and global structure.

While traditional dimensionality reduction methods obsess over distances and coordinates, UMAP asks a more fundamental question: What is the shape of this data? By "shape," we mean topology—the properties that remain invariant under continuous deformations like stretching and bending (but not tearing or gluing). Two clusters connected by a thin bridge have different topology than two fully separated clusters, even if the metric distances are similar.

This page explores the fuzzy topological representation in depth, building the mathematical machinery needed to understand UMAP's remarkable effectiveness.

What You Will Learn

By the end of this page, you will understand: (1) How fuzzy simplicial sets generalize classical topological constructions, (2) The weighted graph interpretation and its computational implications, (3) How membership strengths encode multi-scale topological information, and (4) The category-theoretic foundations that justify UMAP's approach.

Classical Simplicial Complexes

To appreciate fuzzy simplicial sets, we must first understand their classical counterparts. Simplicial complexes are the fundamental objects of algebraic topology, providing a discrete, combinatorial way to represent topological spaces.

Building Blocks: Simplices

An n-simplex is the n-dimensional generalization of a triangle:

0-simplex: A point (vertex)
1-simplex: A line segment (edge) with 2 vertices
2-simplex: A filled triangle with 3 vertices and 3 edges
3-simplex: A filled tetrahedron with 4 vertices, 6 edges, and 4 faces
n-simplex: The convex hull of (n+1) points in general position

Each n-simplex has faces of dimension 0 through (n-1). For example, a triangle (2-simplex) has three edges (1-simplices) and three vertices (0-simplices) as faces.

Simplex Face Counts
Simplex	Vertices	Edges	Triangles	Tetrahedra
0-simplex (vertex)	1	0	0	0
1-simplex (edge)	2	1	0	0
2-simplex (triangle)	3	3	1	0
3-simplex (tetrahedron)	4	6	4	1
n-simplex	n+1	C(n+1,2)	C(n+1,3)	C(n+1,4)

Simplicial Complexes:

A simplicial complex (K) is a collection of simplices satisfying two conditions:

Face closure: Every face of a simplex in (K) is also in (K)
Proper intersection: Two simplices in (K) intersect only along their common faces

These conditions ensure the complex "glues together" properly without overlapping interiors.

From Point Clouds to Complexes:

Given data points (X = {x_1, ..., x_n}), the Vietoris-Rips complex (VR_\epsilon(X)) at scale (\epsilon) contains:

All individual points as 0-simplices
An edge ([x_i, x_j]) whenever (d(x_i, x_j) \leq \epsilon)
A k-simplex ([x_{i_0}, ..., x_{i_k}]) whenever all pairwise distances are (\leq \epsilon)

This construction approximates the topology of the underlying manifold—but only at one specific scale (\epsilon).

The Scale Selection Problem

Choosing the right scale ε is critical and often impossible. Too small: the complex is disconnected (0-dimensional). Too large: everything connects to everything (trivial topology). Real data has structure at multiple scales simultaneously—a single threshold cannot capture this richness.

Extended Pseudo-Metric Spaces

UMAP's theoretical framework begins with a generalization of metric spaces that admits the value "infinity" for distances between disconnected points.

Definition: Extended Pseudo-Metric Space

An extended pseudo-metric space ((X, d)) is a set (X) with a function (d: X \times X \rightarrow [0, \infty]) satisfying:

Non-negativity: (d(x, y) \geq 0) for all (x, y)
Self-distance zero: (d(x, x) = 0) for all (x)
Symmetry: (d(x, y) = d(y, x)) for all (x, y)
Triangle inequality: (d(x, z) \leq d(x, y) + d(y, z)) for all (x, y, z)

Note: Unlike a true metric, (d(x, y) = 0) does not imply (x = y) (hence "pseudo"), and (d(x, y) = \infty) is allowed (hence "extended").

Why Extended Pseudo-Metrics?

Allowing infinite distances naturally encodes disconnected components: points with infinite distance between them cannot be in the same connected component. This maps directly to the topological notion of path-connectedness, bridging metric and topological concepts.

Families of Extended Pseudo-Metric Spaces:

A family of extended pseudo-metric spaces over a parameter (t \in [0, \infty)) assigns to each (t) an extended pseudo-metric (d_t) on the same set (X), subject to:

(d_0(x, y) = 0) for all (x, y) (at t=0, everything collapses)
(d_t(x, y) \leq d_s(x, y)) whenever (t \leq s) (distances are monotonic in t)

This captures the multi-scale nature of data structure: at small scales (t), points are very close or connected; at large scales, they separate.

Connection to UMAP:

For UMAP, each data point (x_i) defines its own local extended pseudo-metric:

$$d_i^t(x_j, x_k) = \max\left(0, \min(d_i(x_j), d_i(x_k)) - t\right)$$

where (d_i(x_j) = \frac{d(x_i, x_j) - \rho_i}{\sigma_i}) is the normalized local distance.

As (t) increases, the effective distances shrink—more points become "infinitely close" (distance 0). This is exactly the threshold behavior of building a Vietoris-Rips complex, but now parameterized continuously.

From Metrics to Topology via Functors

The mathematical elegance of UMAP emerges from category theory—a branch of mathematics that studies structure-preserving transformations between mathematical objects. While not strictly necessary to use UMAP, understanding this perspective illuminates why the algorithm works.

Categories and Functors:

A category consists of:

Objects: Mathematical structures (sets, groups, topological spaces, etc.)
Morphisms: Structure-preserving maps between objects
Composition: Morphisms can be composed associatively
Identity: Each object has an identity morphism

A functor is a structure-preserving map between categories. It transforms objects to objects and morphisms to morphisms, respecting composition and identities.

UMAP's Key Categories:

FinEPMet: Finite extended pseudo-metric spaces with non-expansive maps
FinFuzz: Finite fuzzy simplicial sets with fuzzy set morphisms
ΔTop: Standard topological simplices with continuous maps

The Big Picture

Category theory provides a rigorous way to say: "There's a canonical way to convert metric information into topological information, and this conversion respects the structure of both." The fuzzy simplicial set is the topological representation corresponding to a given metric representation.

The Fuzzy Singular Set Functor:

UMAP defines a functor (\mathcal{F}: \text{FinEPMet} \rightarrow \text{FinFuzz}) that transforms extended pseudo-metric spaces into fuzzy simplicial sets.

For an extended pseudo-metric space ((X, d)), the functor produces a fuzzy simplicial set where:

0-simplices (vertices): All points in (X), each with membership 1
1-simplices (edges): For each pair ((x_i, x_j)), membership strength:

$$\mu([x_i, x_j]) = e^{-d(x_i, x_j)}$$
Higher simplices: Can be defined but are typically ignored in practice

The exponential function converts distances to membership strengths:

Distance 0 → membership 1 (fully connected)
Distance ∞ → membership 0 (fully disconnected)
Intermediate distances → intermediate memberships

Adjunction with Realization:

The functor (\mathcal{F}) has a right adjoint—a "realization" functor that converts fuzzy simplicial sets back into extended pseudo-metric spaces. This adjunction shows that the two representations contain equivalent information, justifiable moving between metric and topological views.

category_perspective.py
Python
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"""
Categorical Perspective on UMAP's Fuzzy Simplicial Sets
 
This module illustrates the conceptual relationship between
extended pseudo-metric spaces and fuzzy simplicial sets.
"""
 
import numpy as np
from dataclasses import dataclass
from typing import Dict, Tuple, Set
 
@dataclass
class ExtendedPseudoMetricSpace:
    """
    A finite extended pseudo-metric space.
    
    Points are indexed 0 to n-1.
    Distances can be np.inf for disconnected points.
    """
    points: int  # Number of points
    distances: np.ndarray  # n x n distance matrix
    
    def __post_init__(self):
        assert self.distances.shape == (self.points, self.points)
        # Verify axioms
        assert np.allclose(np.diag(self.distances), 0)  # d(x,x) = 0
        assert np.allclose(self.distances, self.distances.T)  # Symmetry
        # Triangle inequality is assumed
 
 
@dataclass  
class FuzzySimplicialSet:
    """
    A fuzzy simplicial set (restricted to 0- and 1-simplices).
    
    Represents topology as a weighted graph where edge weights
    are membership strengths in [0, 1].
    """
    vertices: int  # Number of 0-simplices
    edge_weights: Dict[Tuple[int, int], float]  # (i,j) -> membership
    
    def get_weight(self, i: int, j: int) -> float:
        """Get symmetric edge weight."""
        if i == j:
            return 1.0  # Self-loops have full membership
        key = (min(i, j), max(i, j))
        return self.edge_weights.get(key, 0.0)
 
 
def fuzzy_singular_set(metric_space: ExtendedPseudoMetricSpace) -> FuzzySimplicialSet:
    """
    The Fuzzy Singular Set Functor: FinEPMet -> FinFuzz
    
    Converts an extended pseudo-metric space to a fuzzy simplicial set.
    
    This is the core transformation at the heart of UMAP's theoretical
    framework. It canonically converts metric information (distances)
    into topological information (connectivity/membership).
    
    Mathematical basis:
    - Distances map to membership via exponential decay: exp(-d)
    - This preserves the monoid structure of [0, ∞]
    - Distance 0 → membership 1 (connected)
    - Distance ∞ → membership 0 (disconnected)
    """
    n = metric_space.points
    edge_weights = {}
    
    for i in range(n):
        for j in range(i + 1, n):
            d = metric_space.distances[i, j]
            
            if np.isinf(d):
                # Infinite distance -> zero membership
                weight = 0.0
            else:
                # Finite distance -> exponential decay
                weight = np.exp(-d)
            
            if weight > 1e-10:  # Threshold very small weights
                edge_weights[(i, j)] = weight
    
    return FuzzySimplicialSet(vertices=n, edge_weights=edge_weights)
 
 
def realization(fuzzy_set: FuzzySimplicialSet) -> ExtendedPseudoMetricSpace:
    """
    The Realization Functor: FinFuzz -> FinEPMet
    
    Converts a fuzzy simplicial set back to an extended pseudo-metric space.
    This is the right adjoint to the fuzzy singular set functor.
    
    Mathematical basis:
    - Membership maps to distance via negative log: -log(μ)
    - Membership 1 → distance 0
    - Membership 0 → distance ∞
    """
    n = fuzzy_set.vertices
    distances = np.full((n, n), np.inf)
    np.fill_diagonal(distances, 0.0)
    
    for (i, j), weight in fuzzy_set.edge_weights.items():
        if weight > 0:
            d = -np.log(weight)
            distances[i, j] = d
            distances[j, i] = d
    
    return ExtendedPseudoMetricSpace(points=n, distances=distances)
 
 
# Demonstrate the functor relationship
if __name__ == "__main__":
    # Create a simple metric space (4 points in 2D)
    points = np.array([[0, 0], [1, 0], [0, 1], [5, 5]])
    from scipy.spatial.distance import cdist
    distances = cdist(points, points)
    
    epmet = ExtendedPseudoMetricSpace(points=4, distances=distances)
    
    print("Original distances:")
    print(distances.round(2))
    
    # Apply functor: Metric -> Topology
    fuzzy = fuzzy_singular_set(epmet)
    
    print("\nFuzzy edge weights (topology):")
    for (i, j), w in sorted(fuzzy.edge_weights.items()):
        print(f"  Edge ({i}, {j}): {w:.4f}")
    
    # Apply adjoint: Topology -> Metric
    recovered = realization(fuzzy)
    
    print("\nRecovered distances:")
    print(recovered.distances.round(2))
    
    # Note: Recovered distances equal original where edges exist

The Weighted Graph Interpretation

While the category-theoretic foundations provide mathematical rigor, the practical computation of UMAP works with a more concrete object: weighted graphs.

Fuzzy Simplicial Set as Weighted Graph:

Restricting to 0- and 1-simplices (vertices and edges), a fuzzy simplicial set becomes a weighted undirected graph:

Nodes: Data points ({x_1, ..., x_n})
Edges: Pairs of points with non-zero membership
Weights: Membership strengths (\mu([x_i, x_j]) \in [0, 1])

This representation is computationally efficient (sparse adjacency matrix) and directly usable for optimization.

Graph Properties and Topological Interpretation
Graph Property	Topological Meaning	UMAP Significance
High edge weight	Strong topological connection	Points should be close in embedding
Low edge weight	Weak topological connection	Points can separate in embedding
Zero edge weight	No topological connection	Points are topologically independent
Connected component	Path-connected region	Cluster or continuous region
Bridge edge (high betweenness)	Topological neck	Transition between clusters
Dense subgraph	Locally dense region	Core of a cluster

Multi-Scale Information in Edge Weights:

The beauty of the fuzzy representation is that edge weights encode information about all scales simultaneously:

Weight near 1: Points are neighbors at every scale—always considered connected
Weight near 0.5: Points are neighbors at some scales but not others—intermediate relationship
Weight near 0: Points are rarely neighbors—only connected at very large scales

This smooth encoding avoids the discrete threshold problem of classical simplicial complexes.

graph_analysis.py
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import numpy as np
from scipy.sparse import csr_matrix
from scipy.sparse.csgraph import connected_components, minimum_spanning_tree
import networkx as nx
 
def analyze_fuzzy_graph(weights: csr_matrix, labels=None):
    """
    Analyze the topological properties of a UMAP fuzzy graph.
    
    Parameters:
    -----------
    weights : csr_matrix
        Sparse adjacency matrix with fuzzy edge weights
    labels : array-like, optional
        Ground truth labels for cluster analysis
    
    Returns:
    --------
    analysis : dict
        Various topological metrics and interpretations
    """
    n = weights.shape[0]
    
    # Basic statistics
    nnz = weights.nnz
    density = nnz / (n * n)
    mean_weight = weights.data.mean() if nnz > 0 else 0
    
    # Connected components (at various thresholds)
    thresholds = [0.01, 0.1, 0.3, 0.5]
    components_by_threshold = {}
    for thresh in thresholds:
        binary = (weights > thresh).astype(int)
        n_components, labels_cc = connected_components(binary)
        components_by_threshold[thresh] = n_components
    
    # Weight distribution analysis
    weight_quantiles = np.percentile(weights.data, [25, 50, 75, 90, 99])
    
    # Convert to NetworkX for advanced analysis
    G = nx.from_scipy_sparse_array(weights)
    
    # Clustering coefficient (local structure)
    avg_clustering = nx.average_clustering(G, weight='weight')
    
    # Analysis results
    analysis = {
        'n_nodes': n,
        'n_edges': nnz // 2,  # Symmetric matrix
        'edge_density': density,
        'mean_edge_weight': mean_weight,
        'weight_quantiles': dict(zip(['25%', '50%', '75%', '90%', '99%'], weight_quantiles)),
        'components_by_threshold': components_by_threshold,
        'average_clustering': avg_clustering,
    }
    
    # If labels provided, analyze cluster structure
    if labels is not None:
        unique_labels = np.unique(labels)
        n_clusters = len(unique_labels)
        
        # Intra-cluster vs inter-cluster weights
        intra_weights = []
        inter_weights = []
        
        rows, cols = weights.nonzero()
        for idx in range(len(rows)):
            i, j = rows[idx], cols[idx]
            if i < j:  # Avoid counting twice
                w = weights[i, j]
                if labels[i] == labels[j]:
                    intra_weights.append(w)
                else:
                    inter_weights.append(w)
        
        analysis['n_true_clusters'] = n_clusters
        analysis['mean_intra_cluster_weight'] = np.mean(intra_weights) if intra_weights else 0
        analysis['mean_inter_cluster_weight'] = np.mean(inter_weights) if inter_weights else 0
        analysis['weight_ratio'] = (
            analysis['mean_intra_cluster_weight'] / 
            max(analysis['mean_inter_cluster_weight'], 1e-10)
        )
    
    return analysis
 
 
def visualize_weight_distribution(weights: csr_matrix, ax=None):
    """
    Visualize the fuzzy edge weight distribution.
    
    Multi-scale structure manifests as specific patterns:
    - Uniform data: weights concentrated around one value
    - Multi-scale clusters: multi-modal weight distribution
    - Hierarchical structure: gradual weight falloff
    """
    import matplotlib.pyplot as plt
    
    if ax is None:
        fig, ax = plt.subplots(figsize=(10, 4))
    
    nonzero_weights = weights.data
    
    # Histogram of edge weights
    ax.hist(nonzero_weights, bins=50, edgecolor='white', alpha=0.7)
    ax.set_xlabel('Edge Weight (Membership Strength)')
    ax.set_ylabel('Frequency')
    ax.set_title('Fuzzy Graph Weight Distribution')
    
    # Add annotation for interpretation
    median = np.median(nonzero_weights)
    ax.axvline(median, color='red', linestyle='--', label=f'Median: {median:.3f}')
    ax.legend()
    
    return ax

Constructing Local Fuzzy Sets

UMAP constructs the global fuzzy simplicial set by first building local fuzzy sets around each point, then combining them. This two-phase process respects the manifold's locally varying geometry.

Phase 1: Local Extended Pseudo-Metrics

For each point (x_i), UMAP defines a local distance function:

$$d_i(x_j) = \begin{cases} 0 & \text{if } j = i \ \max\left(0, \frac{d(x_i, x_j) - \rho_i}{\sigma_i}\right) & \text{otherwise} \end{cases}$$

where:

(d(x_i, x_j)) is the ambient distance (typically Euclidean)
(\rho_i) is the distance to (x_i)'s nearest neighbor
(\sigma_i) is a normalization factor

Role of Each Parameter

•ρ_i (rho): Ensures local connectivity. By subtracting the nearest-neighbor distance, at least one neighbor has normalized distance 0, guaranteeing each point connects to its local neighborhood.
•σ_i (sigma): Adapts to local density. Found by binary search to achieve a target sum of membership strengths (controlled by n_neighbors). Dense regions get small σ (compressing distances); sparse regions get large σ (expanding distances).
•n_neighbors: Controls the locality scale. More neighbors → larger effective σ → smoother topology. Fewer neighbors → more local → potentially disconnected components.

Phase 2: Local Fuzzy Set Construction

The local distance function (d_i) becomes a local fuzzy set via exponential transformation:

$$\mu_i(x_j) = e^{-d_i(x_j)} = e^{-\frac{\max(0, d(x_i, x_j) - \rho_i)}{\sigma_i}}$$

Properties of this construction:

(\mu_i(x_i) = 1) (self-membership)
(\mu_i(x_{\text{nearest}}) = 1) (nearest neighbor has full membership)
Membership decays exponentially with normalized distance
The sum (\sum_{j \neq i} \mu_i(x_j) \approx \log_2(k)) where (k) is n_neighbors

Asymmetry is Intentional:

Note that (\mu_i(x_j) \neq \mu_j(x_i)) in general. This reflects the different local geometries at each point. If (x_i) is in a dense region and (x_j) is in a sparse region, the relationship looks different from each perspective.

The Smooth Maximum Function

The max(0, d - ρ) operation ensures non-negative normalized distances. This is a "soft" way to encode that the nearest neighbor should have maximum membership (1.0), regardless of the actual ambient distance. It's equivalent to shifting the origin of each local coordinate system to the nearest neighbor.

The Fuzzy Union and Symmetrization

With local fuzzy sets constructed for each point, UMAP must combine them into a single global representation. This is achieved through the fuzzy set union operation.

The Problem: Asymmetric Local Views

Each local fuzzy set (\mu_i) gives (x_i)'s view of its relationships. But we need a symmetric global graph for optimization. How do we reconcile (\mu_i(x_j)) and (\mu_j(x_i))?

The Solution: Probabilistic T-Conorm

UMAP uses the fuzzy union (probabilistic OR):

$$\mu(x_i, x_j) = \mu_i(x_j) + \mu_j(x_i) - \mu_i(x_j) \cdot \mu_j(x_i)$$

More intuitively, with (a = \mu_i(x_j)) and (b = \mu_j(x_i)):

$$\mu = a + b - ab = 1 - (1-a)(1-b)$$

This is the probability that at least one of two independent events occurs—exactly the probabilistic union!

Properties of Fuzzy Union

•Symmetry: μ(a,b) = μ(b,a)
•Idempotence: μ(a,a) = a (sort of)
•Monotonicity: Increasing either input increases output
•Bounded: Output always in [0, 1]
•Maximum preservation: If either input is 1, output is 1

Why Not Other Operations?

•Average: Would underweight strong asymmetric relationships
•Maximum: Would lose information about bidirectional strength
•Minimum: Would be too restrictive (intersection, not union)
•Product: Would severely downweight moderate relationships

Numerical Examples:

(\mu_i(x_j))	(\mu_j(x_i))	Fuzzy Union	Interpretation
0.9	0.1	0.91	Strong from one side → strong overall
0.5	0.5	0.75	Moderate both ways → fairly strong
0.3	0.3	0.51	Weak both ways → still moderate
0.1	0.1	0.19	Weak both ways → weak overall
1.0	0.0	1.00	Perfect from one side → perfect overall

The fuzzy union is "generous"—it gives the benefit of the doubt to connections. If either perspective suggests a relationship, it's preserved.

Connection to t-SNE Symmetrization

t-SNE uses a different symmetrization: p_ij = (p_i|j + p_j|i) / 2n. This averages conditional probabilities and normalizes globally. UMAP's fuzzy union is more aggressive at preserving connections and operates locally without global normalization—one reason UMAP better preserves global structure.

Encoding Topological Features

The fuzzy simplicial set encodes rich topological information about the data. Understanding what features are captured helps interpret UMAP embeddings correctly.

Connectivity and Clusters:

Connected components in the fuzzy graph (at any threshold) correspond to clusters in the data. The edge weights encode how "connected" clusters are:

Disjoint clusters: Connected components with zero inter-component weights
Overlapping clusters: Single component with varying internal weights
Hierarchical clusters: Nested structure visible at different weight thresholds

Density Information:

Local density is encoded in the σ values and, indirectly, in local edge weight patterns:

Dense regions: Many high-weight edges, small σ values
Sparse regions: Fewer edges, weights decay quickly with distance
Density boundaries: Asymmetric weight patterns at density transitions

Topological Features Preserved by UMAP
Feature	Fuzzy Representation	Embedding Consequence
Number of clusters	Number of connected components	Separated regions in 2D
Cluster size	Size of connected component	Relative area in embedding
Cluster density	Edge weight distribution	Point density variation
Cluster shape (roughly)	Local graph structure	Approximate shape preserved
Inter-cluster distances (relative)	Inter-component edge weights	Relative separation distances
Intrinsic dimensionality	Local neighborhood structure	Local neighborhood structure

Holes and Loops:

In persistent homology terms, 1-dimensional holes (loops) are detected by cycles in the graph that are not filled by higher simplices. UMAP's 1-skeleton (edge-only) representation can capture:

Loops in data: Cycles of high-weight edges
Holes in coverage: Missing edges in otherwise dense regions

However, because UMAP typically uses only 0- and 1-simplices, higher-dimensional topological features (voids, etc.) are not explicitly captured—though they may be implicitly reflected in global edge weight patterns.

What Topology Doesn't Capture

Topological equivalence is weaker than geometric similarity. Two sets can be topologically equivalent but geometrically very different (e.g., a circle and an ellipse). UMAP preserves topology, not exact geometry. Distances within clusters are meaningful only relatively, not absolutely. Never compare absolute distances across different UMAP runs or between clusters.

Summary: The Fuzzy Topological Representation

We have now thoroughly explored the fuzzy topological representation that underlies UMAP's effectiveness. Let's consolidate the key concepts:

Key Takeaways

•Fuzzy vs. Crisp: Unlike classical simplicial complexes with binary inclusion, fuzzy simplicial sets assign continuous membership strengths, elegantly handling multi-scale structure.
•Category Theory Foundation: Functors between extended pseudo-metric spaces and fuzzy simplicial sets provide a principled bridge between metric and topological representations.
•Weighted Graph Form: Computationally, the fuzzy simplicial set is a weighted undirected graph—sparse, efficient, and directly optimizable.
•Local Construction: Each point contributes a local fuzzy set based on its adapted distance function, respecting local geometry.
•Fuzzy Union: The probabilistic t-conorm combines asymmetric local views into a symmetric global representation, generously preserving connections.
•Topological Encoding: The graph structure captures clusters, connectivity, density patterns, and relative relationships—the essential 'shape' of the data.

What's Next:

With both the high-dimensional fuzzy simplicial set fully constructed, the next page dives into how UMAP optimizes the low-dimensional embedding. We'll explore the cross-entropy loss function that measures the divergence between high- and low-dimensional topological representations, and the stochastic gradient descent procedure that finds embeddings minimizing this divergence.

Page Complete

You now understand UMAP's fuzzy topological representation in depth—from classical simplicial complexes through category-theoretic functors to the practical weighted graph implementation. This representation is what allows UMAP to capture the true 'shape' of high-dimensional data and reproduce it faithfully in low dimensions.