K-Nearest Hyperplanes Model

• K-Nearest Hyperplanes (KNH) is a graph learning framework that represents nodes as m-dimensional affine subspaces, enabling effective multi-view data fusion.
• It uses precise subspace distance metrics, including projection and Grassmannian distances, to measure dissimilarity between hyperplanes.
• Empirical results in misinformation detection demonstrate that KNH outperforms traditional K-NN methods in precision, recall, and overall accuracy.

The K-Nearest Hyperplanes (KNH) model is a generalization of classic graph-based learning frameworks, in which each node is represented not by a datapoint but by an $m$-dimensional Euclidean subspace—an affine “m-flat” or hyperplane. KNH enables multi-view modeling of entities, constructing graphs whose nodes encode higher-order structures across heterogeneous feature sets. Key applications include misinformation detection, where empirical results demonstrate quantitative gains over traditional K-Nearest Neighbors (K-NN) approaches (Abdali et al., 2021).

1. Nodes as m-Flats in Euclidean Space

Whereas the classic K-NN graph defines each node as a point $x_i \in \mathbb{R}^d$, KNH represents each entity $i$ by an $m$-dimensional affine subspace (“flat” or hyperplane) $F_i \subset \mathbb{R}^d$:

$$F_i = \{\, x \in \mathbb{R}^d : x = U_i \alpha + c_i,\ \alpha \in \mathbb{R}^m\,\}$$

Here, $U_i \in \mathbb{R}^{d \times m}$ contains orthonormal columns spanning the subspace, and $c_i \in \mathbb{R}^d$ is an anchor point (e.g., the centroid of projected views). When $m = 0$, the construction reduces to classic node-as-point K-NN. For $m = d-1$, each node corresponds to a hyperplane in $x_i \in \mathbb{R}^d$0. This representation is particularly well-suited to settings with multiple “views” or feature groups per entity.

2. Subspace Distance Metrics

KNH replaces standard Euclidean pointwise distances with measures of dissimilarity between affine subspaces. Two principal metrics are employed:

• Projection Distance: For $x_i \in \mathbb{R}^d$1, $x_i \in \mathbb{R}^d$2 with anchors $x_i \in \mathbb{R}^d$3, $x_i \in \mathbb{R}^d$4 and projectors $x_i \in \mathbb{R}^d$5,

$x_i \in \mathbb{R}^d$6

This metric captures both orientation and relative translation. It is the preferred measure in KNH for heterogeneous, multi-view data.

• Grassmannian (Subspace-Angle) Distance: Let $x_i \in \mathbb{R}^d$7 be principal angles between $x_i \in \mathbb{R}^d$8 and $x_i \in \mathbb{R}^d$9 via $i$0,

$i$1

This metric focuses solely on relative orientation in Grassmannian manifolds.

3. Construction of the KNH Graph

Given multi-view-fused hyperplanes $i$2 for $i$3, a weighted undirected graph is constructed based on $i$4-nearest-neighbor relations using the chosen subspace distance. The adjacency matrix $i$5 is built as follows:

1. Compute the pairwise distance matrix $i$6.
2. For each node $i$7, select $i$8 as the indices of the $i$9 smallest nonzero $m$0.
3. Set edge weight $m$1 for $m$2 and $m$3 otherwise; kernel choice may vary.
4. Symmetrize the adjacency: $m$4.
5. Optionally row-normalize $m$5.

This construction collapses to the standard K-NN graph when $m$6, making KNH a strict generalization.

4. Multi-View Data Fusion and Subspace Estimation

KNH is inherently designed to harness multi-view data, where each entity is described by $m$7 distinct feature sets or modalities. Let $m$8 denote the $m$9-th view for all $F_i \subset \mathbb{R}^d$0 entities. The goal is a common $F_i \subset \mathbb{R}^d$1-dimensional latent structure across views, achievable via Tensor Canonical Correlation Analysis (TCCA):

• Compute an order-$F_i \subset \mathbb{R}^d$2 covariance tensor from the $F_i \subset \mathbb{R}^d$3 view matrices.
• Solve the TCCA minimization via rank-$F_i \subset \mathbb{R}^d$4 CP decomposition:

$F_i \subset \mathbb{R}^d$5

• For each entity $F_i \subset \mathbb{R}^d$6, concatenate projected vectors $F_i \subset \mathbb{R}^d$7 from each view:

$F_i \subset \mathbb{R}^d$8

• Orthonormalize $F_i \subset \mathbb{R}^d$9 (e.g., via SVD or QR).

The result is one $$F_i = \{\, x \in \mathbb{R}^d : x = U_i \alpha + c_i,\ \alpha \in \mathbb{R}^m\,\}$$0-flat per node capturing correlations and complementarities across views.

5. Application to Misinformation Detection

The KNH adjacency matrix $$F_i = \{\, x \in \mathbb{R}^d : x = U_i \alpha + c_i,\ \alpha \in \mathbb{R}^m\,\}$$1 is used as input to graph-based semi-supervised classifiers. Two principal approaches are highlighted:

• Graph Convolutional Networks (GCN): Nodes are featurized by $$F_i = \{\, x \in \mathbb{R}^d : x = U_i \alpha + c_i,\ \alpha \in \mathbb{R}^m\,\}$$2 or other metadata. Layer-wise propagation uses the normalized adjacency:

$$F_i = \{\, x \in \mathbb{R}^d : x = U_i \alpha + c_i,\ \alpha \in \mathbb{R}^m\,\}$$3

with softmax and cross-entropy loss on labeled nodes.

• Belief-Propagation-Style Diffusion: Initialize label vector $$F_i = \{\, x \in \mathbb{R}^d : x = U_i \alpha + c_i,\ \alpha \in \mathbb{R}^m\,\}$$4 for real/fake/unknown, iterate

$$F_i = \{\, x \in \mathbb{R}^d : x = U_i \alpha + c_i,\ \alpha \in \mathbb{R}^m\,\}$$5

until convergence, then threshold $$F_i = \{\, x \in \mathbb{R}^d : x = U_i \alpha + c_i,\ \alpha \in \mathbb{R}^m\,\}$$6 to assign labels. In experiments, 40% labels are fixed, the remainder predicted.

On real-world datasets, including Twitter (with features such as word-cooccurrence tensors and HTML-tag matrices) and FakeNewsNet/PolitiFact (user-news and publisher-news feature matrices), this procedure yielded consistently enhanced classification metrics for KNH over standard K-NN (Abdali et al., 2021).

6. Theoretical Properties and Computational Complexity

• Reduction to K-NN: KNH generalizes K-NN, reducing exactly to classic K-NN when each flat is $$F_i = \{\, x \in \mathbb{R}^d : x = U_i \alpha + c_i,\ \alpha \in \mathbb{R}^m\,\}$$7 and Euclidean distances are used.
• Complexity:
  • TCCA/CP decomposition: $$F_i = \{\, x \in \mathbb{R}^d : x = U_i \alpha + c_i,\ \alpha \in \mathbb{R}^m\,\}$$8 for $$F_i = \{\, x \in \mathbb{R}^d : x = U_i \alpha + c_i,\ \alpha \in \mathbb{R}^m\,\}$$9 ALS iterations and rank $U_i \in \mathbb{R}^{d \times m}$0; practically independent of $U_i \in \mathbb{R}^{d \times m}$1 (excluding tensor formation).
  • Orthonormalization: $U_i \in \mathbb{R}^{d \times m}$2 overall.
  • Pairwise distances: $U_i \in \mathbb{R}^{d \times m}$3.
  • $U_i \in \mathbb{R}^{d \times m}$4-nearest selection: $U_i \in \mathbb{R}^{d \times m}$5 (or $U_i \in \mathbb{R}^{d \times m}$6 with selection algorithms).
• Scalability: Overall complexity is $U_i \in \mathbb{R}^{d \times m}$7, on par with exact K-NN graph construction, aside from multi-view fusion costs.

7. Empirical Evaluation and Comparative Performance

Empirical evaluation on semi-supervised fake-news classification demonstrated the following comparative metrics (10 random splits, 40% label rate):

Dataset	Method	Precision	Recall	F1	Accuracy
Twitter (R=20,K=15)	KNN (plain)	0.687	0.683	0.684	0.694
	KNN+CCA	0.691	0.682	0.686	0.697
	KNH	0.709	0.720	0.713	0.719
FakeNewsNet (R=30,K=20)	KNN	0.836	0.742	0.789	0.780
	KNN+CCA	0.833	0.747	0.787	0.777
	KNH	0.875	0.808	0.839	0.830

Increasing the “flat” dimension $U_i \in \mathbb{R}^{d \times m}$8 and neighborhood size $U_i \in \mathbb{R}^{d \times m}$9 further amplifies KNH’s advantage, suggesting subspace-based nodes more effectively exploit complex, multi-view structures compared to pointwise approaches. This provides empirical justification for preferring KNH in high-dimensional, multi-view graph learning contexts (Abdali et al., 2021).