tensorly.contrib.sparse.decomposition.robust_pca

robust_pca(X, mask=None, tol=1e-06, reg_E=1.0, reg_J=1.0, mu_init=0.0001, mu_max=10000000000.0, learning_rate=1.1, n_iter_max=100, return_errors=False, verbose=1)

Robust Tensor PCA via ALM with support for missing values

Decomposes a tensor X into the sum of a low-rank component D and a sparse component E.

Parameters:

Xndarray

tensor data of shape (n_samples, N1, …, NS)

maskndarray

array of booleans with the same shape as X should be zero where the values are missing and 1 everywhere else

tolfloat

convergence value

reg_Efloat, optional, default is 1

regularisation on the sparse part E

reg_Jfloat, optional, default is 1

regularisation on the low rank part D

mu_initfloat, optional, default is 10e-5

initial value for mu

mu_maxfloat, optional, default is 10e9

maximal value for mu

learning_ratefloat, optional, default is 1.1

percentage increase of mu at each iteration

n_iter_maxint, optional, default is 100

maximum number of iteration

return_errorsbool, default is False

if True, additionally returns the reconstruction errors

verboseint, default is 1

level of verbosity

Returns:

(D, E) or (D, E, rec_errors)

Robust decomposition of X

DX-like array

low-rank part

EX-like array

sparse error part

rec_errorslist of errors

only returned if return_errors is True

Notes

The problem we solve is, for an input tensor $\tilde{X}$:

$$\begin{array}{rlrl}& \underset{\{J_i\},\tilde{D},\tilde{E}}{min}& & \sum _{i=1}^N{\text{reg}}_J\Vert J_i{\Vert }_{\ast }+{\text{reg}}_E\Vert E{\Vert }_1\\ & \text{subject to}& & \tilde{X}=\tilde{A}+\tilde{E}\\ & & & A_{[i]}=J_i,\text{ for each }i\in \{1,2,\cdots ,N\}\end{array}$$