Root-Laplacian Eigenmap: Graph Embedding

• Root-Laplacian eigenmap is a spectral embedding technique that uses the square-root of the graph Laplacian to enhance local cluster separation.
• It leverages heat kernel weighting and a generalized eigenproblem to construct low-distortion embeddings aligned with manifold geometry.
• The method improves robustness in spectral clustering and graph signal processing by compressing high frequencies and reducing spectral spillover.

The root-Laplacian eigenmap is a spectral embedding technique for graphs and high-dimensional data, in which embedding coordinates are constructed from the eigenvectors of the square root of the combinatorial graph Laplacian. This operator, known as the root-Laplacian, serves as a discrete analog to the square root of the Laplace–Beltrami operator on Riemannian manifolds in the Gromov sense. Root-Laplacian eigenmaps tightly connect with the geometry induced by the heat kernel, emphasizing locality, enhancing cluster boundary separation, and providing robustness in various applications such as spectral clustering and graph signal processing (Choudhury, 2023).

1. Definition of the Root-Laplacian Operator

Given an undirected graph $G = (V, E)$ on $n$ vertices, the relevant matrices are:

• Adjacency (weight) matrix $W \in \mathbb{R}^{n \times n}$, symmetric with $W_{ij} \ge 0$,
• Degree matrix $D = \mathrm{diag}(d_1, \ldots, d_n)$, where $d_i = \sum_j W_{ij}$,
• Combinatorial graph Laplacian

$L = D - W,$

which is symmetric and positive semidefinite.

Spectral decomposition gives

$$L = U \Lambda U^T, \quad \Lambda = \mathrm{diag}(\lambda_0, \ldots, \lambda_{n-1}),$$

with eigenvalues ordered $$0 = \lambda_0 \le \lambda_1 \le \cdots \le \lambda_{n-1}$$.

The root-Laplacian is the unique positive-semidefinite square root:

$$L^{1/2} = U \Lambda^{1/2} U^T, \quad \Lambda^{1/2} = \mathrm{diag}(\lambda_0^{1/2}, \ldots, \lambda_{n-1}^{1/2}).$$

2. Root-Laplacian Eigenmap Embedding Methodology

The root-Laplacian eigenmap embeds data via the following algorithm:

1. Graph construction: Build a neighborhood graph from a point cloud $\{p_i\}_{i=1}^n \subset \mathbb{R}^d$.
   • ε-ball graph: Connect $i \leftrightarrow j$ if $\|p_i - p_j\| < \varepsilon$.
   • k-nearest neighbors: Connect $i \leftrightarrow j$ if $i \in kNN(j)$ or $j \in kNN(i)$.
2. Edge weighting:
   • Heat kernel: $$W_{ij} = \exp\left(-\frac{\|p_i - p_j\|^2}{t}\right)$$ if $i \sim j$, 0 otherwise, for scale parameter $t > 0$.
   • Unit weights: $W_{ij} = 1$ for adjacent nodes in the limit $t \to \infty$.
3. Laplacian and root: Form $L = D - W$, compute $S = L^{1/2}$ by spectral or iterative methods.
4. Generalized eigenproblem:

$S f = \mu D f$

Equivalently, solve $L f = \lambda D f$ and set $\mu = \sqrt{\lambda}$.

1. Eigenpair selection: Order eigenpairs $(\mu_k, f_k)$ by non-decreasing $\mu_k$, discard the trivial $\mu_0 = 0$ eigenvector (constant vector), and retain the next $m$ eigenvectors.
2. Embedding:

$\Psi(i) = (f_1(i), f_2(i), \ldots, f_m(i))^T$

provides the $m$-dimensional spectral coordinates.

3. Theoretical Properties and Advantages

• Locality preservation: The root-Laplacian eigenmap minimizes the square-root Dirichlet energy,

$$\sqrt{\,\sum_{i,j}(f_i - f_j)^2 W_{ij}\,} = \|L^{1/2} f\|_2,$$

under normalization $f^T D f = 1$. This further suppresses long-range “short-circuits” compared to classic Laplacian eigenmaps, highlighting local clusters.

• Cluster sensitivity: Weighting spectrum by $\sqrt{\lambda}$ yields wider separation of small positive eigenvalues, enhancing cluster boundary sharpness in embedding space.
• Geometric correspondence: The mapping approximates the operator $\sqrt{\Delta_{LB}}$ (square-root Laplace–Beltrami), relating discrete and continuous embeddings through manifold heat-kernel geometry.
• Stability: Embeddings are invariant under isometric rescaling of input data and less sensitive to outliers, due to the local dominance in the square-root energy.
• Spectral distortion: The root spectrum compresses high frequencies, lowering dilation/distortion metrics as defined by Gromov.

4. Computational Complexity and Implementation

• Nearest-neighbor graph construction:
  • Exact kNN: $O(n^2)$ brute-force, $O(n \log n)$ using spatial indexing (e.g., kd-trees).
• Eigen-decomposition:
  • Full diagonalization: $O(n^3)$, impractical for large $n$.
  • Sparse iterative solvers (Lanczos, ARPACK): $O(m \cdot \#\text{nonzeros in } L)$ per iteration, where $m$ is target dimension.
• Memory requirements: Storing $W$ and $L$ as sparse matrices requires $O(n + |E|)$ space.
• Root computation: Typically, the explicit matrix root $L^{1/2}$ is not formed. Instead, solutions proceed via the standard generalized problem $L f = \lambda D f$ and eigenvalues are mapped as $\mu = \sqrt{\lambda}$, sidestepping explicit square-root realization.

5. Illustrative Examples and Empirical Observations

• Heat-maps: In random graphs, the Laplacian $L$ displays a broad spectrum of off-diagonal values. Conversely, its square root $L^{1/2}$ is more diagonally dominant, indicating enhanced local coupling.
• 2D embeddings: For two-dimensional point clouds, using heat-kernel weights ($\varepsilon = 0.32$, $t > 0$), root-Laplacian eigenmaps empirically display clearer cluster separation than classical Laplacian eigenmaps (see Figure 1 of (Choudhury, 2023)), with more distinct cluster boundaries and reduced spectral spillover.
• Graph signal processing: The graph Fourier transform (GFT) may be reinterpreted with eigenvectors of $L$, but ordered by frequencies $\nu_k = \sqrt{\lambda_k}$. A signal $x \in \mathbb{R}^n$ admits the expansion

$$\widehat{x}_k = \langle u_k, x \rangle;\qquad x = \sum_k \widehat{x}_k u_k,\quad \text{frequency} = \nu_k$$

Filtering in the root-Laplacian domain allows for smoother low-pass filtering and greater alignment with manifold-induced frequencies, applicable to domains such as IoT-based smart city sensing.

6. Connections, Limitations, and Broader Impact

Root-Laplacian eigenmaps extend the spectral embedding framework of Belkin and Niyogi by integrating the square-root Laplacian, which better preserves manifold geometry and local cluster integrity. The spectral distortion properties, as studied in the context of Gromov’s dilation metrics, position the root-Laplacian as a tool for lower-distortion embeddings in high-dimensional data analysis. The computational strategy relies on sparse matrix routines, and advances in efficient eigensolvers (e.g., ARPACK) enhance its scalability. A plausible implication is that root-Laplacian eigenmaps yield improved spectral clustering and signal processing performance in contexts where local structure dominates and spectral separation is paramount (Choudhury, 2023).