Struct KernelPcaProjection 

pub struct KernelPcaProjection { /* private fields */ }

Corpus-fitted projection via kernel PCA with a Gaussian (RBF) kernel.

Mathematical background

Standard PCA finds the 3 directions of maximum linear variance. Kernel PCA first maps data into an infinite-dimensional feature space F via the kernel trick, then performs PCA there. With the Gaussian kernel k(x, y) = exp(−‖x−y‖²/(2σ²)), every data point Φ(x) lies on a hypersphere S in F (since k(x,x) = 1 for all x). This is a natural fit for SphereQL’s spherical geometry.

The key advantage over linear PCA: kernel PCA captures non-linear manifold structure (curved clusters, rings, spirals) that linear PCA crushes flat. For embedding spaces with complex semantic geometry, this preserves more meaningful neighborhood relationships.

Limit behaviour

• σ → ∞: kernel PCA converges to standard PCA (Hoffmann, Appendix A).
• σ → 0: all points become orthogonal in F; PCA is meaningless.

Complexity

• Fitting: O(n²·d) to build the kernel matrix + O(n²·q·iters) for power iteration on the n×n centered kernel matrix.
• Projection: O(n·d) per embedding (n kernel evaluations).
• Memory: O(n·d) for training data + O(n) per eigenvector.

References

• Schölkopf, Smola, Müller. “Nonlinear component analysis as a kernel eigenvalue problem.” Neural Computation 10 (1998) 1299–1319.
• Hoffmann. “Kernel PCA for novelty detection.” Pattern Recognition 40 (2007) 863–874.

Implementations

impl KernelPcaProjection

pub fn fit(embeddings: &[Embedding], radial: RadialStrategy) -> Self

Fit kernel PCA with automatic σ selection.

σ is set to the median pairwise Euclidean distance on the normalised embeddings divided by √2, so that the kernel value at the median distance is exp(−1) ≈ 0.37. This is a standard heuristic in the kernel methods literature.

pub fn fit_with_sigma( embeddings: &[Embedding], sigma: f64, radial: RadialStrategy, ) -> Self

Fit kernel PCA with an explicit kernel width σ.

Use this when you have domain knowledge about the appropriate scale, or when benchmarking different σ values.

pub fn fit_default(embeddings: &[Embedding]) -> Self

Convenience: fit with default radial strategy and auto σ.

pub fn with_volumetric(self, enabled: bool) -> Self

Enable volumetric mode: r comes from the kernel PCA projection magnitude instead of the embedding magnitude.

pub fn sigma(&self) -> f64

The Gaussian kernel width used for this projection.

pub fn num_training_points(&self) -> usize

Number of training points stored (needed for kernel evaluations).

pub fn explained_variance_ratio(&self) -> f64

The fraction of total feature-space variance captured by the top-3 kernel principal components.

Analogous to PcaProjection::explained_variance_ratio() but in the (infinite-dimensional) Gaussian feature space.

pub fn eigenvalues(&self) -> [f64; 3]

The top-3 eigenvalues of the centred kernel matrix.

Trait Implementations

impl Clone for KernelPcaProjection

fn clone(&self) -> KernelPcaProjection

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Projection for KernelPcaProjection

fn project(&self, embedding: &Embedding) -> SphericalPoint

fn project_rich(&self, embedding: &Embedding) -> ProjectedPoint

Project with rich metadata: certainty, intensity, projection magnitude.

fn dimensionality(&self) -> usize