Multi-View Co-Regularized Spectral Clustering

class polyview.cluster.mv_coreg_sc.MultiViewCoRegSpectralClustering(*args: Any, **kwargs: Any)

Bases: BaseMultiViewClusterer

Multi-view co-regularized spectral clustering algorithm.

Parameters:

  • n_clusters (int, default=2) – The number of clusters to form.

  • n_init (int, default=10) – Number of time the k-means algorithm will be run with different centroid seeds.

  • max_iter (int, default=50) – Maximum number of iterations of the alternating optimization.

  • v_lambda (float, default=0.01) – Weight of the agreement term between the embeddings of different views. Its suitable scale depends on laplacian: values around 0.01 suit the normalized Laplacian, whose eigenvalues lie in [0, 2], whereas the unnormalized Laplacian scales with the node degrees.

  • affinity (str, default='rbf') – Kernel to use for computing the affinity matrix. Should be a valid metric for sklearn.metrics.pairwise.pairwise_kernels.

  • eigen_solver ({'auto', 'dense', 'arpack'}, default='auto') – Solver for the n_clusters smallest eigenvectors computed at each iteration. The co-regularized Laplacians are applied matrix-free, without forming the (n, n) co-regularization terms. ‘dense’ materializes them and uses LAPACK restricted to the requested eigenpairs; ‘arpack’ uses an iterative Lanczos solver warm-started from the previous embedding. ‘auto’ picks ‘arpack’ when n_samples > 200, ‘dense’ otherwise.

  • random_state (int or None, default=None) – Determines random number generation for centroid initialization. Use an int to make the randomness deterministic.

  • tol (float, default=1e-6) – Stop when the change of the objective between two iterations is below tol times its initial value.

  • laplacian ({"normalized", "unnormalized"}, default="normalized") – Graph Laplacian of each view: the symmetric normalized Laplacian I - D^{-1/2} A D^{-1/2} used by Kumar et al. (2011), or the unnormalized Laplacian D - A. The eigenvectors of the unnormalized Laplacian can concentrate on a few low-degree samples, which degrades the clustering on some datasets.

  • gamma (float, 'median' or None, default=None) – Kernel coefficient passed to the affinity. None uses the sklearn default (1 / n_features for ‘rbf’). 'median' uses the median heuristic: gamma = 1 / (2 * median(||x_i - x_j||^2)) for ‘rbf’ (1 / median(||x_i - x_j||_1) for ‘laplacian’).

embedding_

The concatenated spectral embeddings from all views after fitting.

Type:

np.ndarray of shape (n_samples, n_clusters * n_views)

view_embeddings_

The orthonormal spectral embedding U_v learned for each view.

Type:

list of np.ndarray of shape (n_samples, n_clusters)

objective_

The objective function values at each iteration of the optimization process.

Type:

list of float

n_iter_

Number of iterations run.

Type:

int

labels_

Cluster labels for each sample after fitting.

Type:

np.ndarray of shape (n_samples,)

Notes

Each view v has a spectral embedding U_v with orthonormal columns. Following the pairwise co-regularization of Kumar et al. (2011), the method minimizes

J = sum_v tr(U_v^T L_v U_v) - v_lambda * sum_{v<w} tr(U_v U_v^T U_w U_w^T),

i.e. the spectral clustering objective of each view minus the agreement between the embeddings of each pair of views, which is maximal when they span the same subspace. The views are updated one at a time: given the other embeddings, the optimal U_v is formed by the eigenvectors of L_v - v_lambda * sum_{w != v} U_w U_w^T with the smallest eigenvalues, so that J never increases. The final clusters are obtained by k-means on the row-normalized concatenation of the embeddings. By default, L_v = I - D_v^{-1/2} A_v D_v^{-1/2} is the symmetric normalized Laplacian of the affinity matrix A_v of view v, as in Kumar et al.; the unnormalized Laplacian D_v - A_v is available with laplacian="unnormalized".

References

  • Kumar, A. et al. (2011). Co-regularized Multi-view Spectral Clustering. Advances in Neural Information Processing Systems 24.

fit(views: List[numpy.ndarray], y=None) → None

Fits the multi-view co-regularized spectral clustering model to the provided views.

Parameters:

views (list of np.ndarray) – List of data matrices for each view, where each matrix has shape (n_samples, n_features_v).

fit_predict(views: List[numpy.ndarray], y=None) → numpy.ndarray

Fit and return cluster labels.

Parameters:

  • views (list of array-like)

  • y (ignored)

Returns:

labels

Return type:

ndarray of shape (n_samples,)