Mdl = rica (X, Q) learns Q features from the $N\; \times \; P$ data matrix X (rows are observations, columns are predictors) and returns a ReconstructionICA object Mdl. Apply the learned transformation to data with transform (Mdl, X), which returns X * Mdl.TransformWeights.

The $P\; \times$Q weight matrix (with unit-length columns) minimizes the objective

 Lambda * ||X * W * W' - X||_F^2
       + sum (sum (g (X * W)))

over the transformation weights W, combining a reconstruction cost with a sparsity contrast g applied elementwise and selected by 'ContrastFcn'.

Name/Value pairs:

'IterationLimit'

Maximum number of iterations (default 1000).

'Lambda'

Weight of the reconstruction term (default 1).

'Standardize'

Logical; center and scale each predictor before fitting (default false).

'ContrastFcn'

The sparsity contrast g applied to each element $z$ of $$X × W: 'logcosh' (default), which is $0.5\; \times \; log\; (cosh\; (2\; \times \; z))$; 'exp', which is $-exp\; (-z^2\; /\; 2)$; or 'sqrt', which is $sqrt\; (z^2\; +\; 1e-8)$, a smooth stand-in for $abs\; (z)$.

'InitialTransformWeights'

A $P\; \times$Q initial value for the weights. The default is random.

'GradientTolerance', 'StepTolerance'

Stop once the gradient’s or the step’s infinity norm falls to or below the given value (default 1e-6 each). They govern the fit only under 'Solver', 'lbfgs'; the 'quasinewton' solver runs to its own tighter internal tolerances and records these without acting on them.

'Solver'

'quasinewton' (default) minimizes through Octave’s fminunc, which carries a full inverse Hessian. 'lbfgs' selects the limited-memory BFGS solver MATLAB uses, holding as many curvature pairs as the transform has parameters, and is several times faster here. It stops where 'GradientTolerance' and 'StepTolerance' say to, so a value tighter than the default carries it further.

Note on reproducibility

The RICA objective is not convex and is minimized by a quasi-Newton solver, so the learned weights depend on the starting point and the solver, and are only defined up to a permutation and sign of the feature columns. Different runs (or different software, including MATLAB) may return different weights that nonetheless describe an equally valid feature transformation. Fix 'InitialTransformWeights' for a reproducible result.

See also: ReconstructionICA, sparsefilt, pca

Source Code: rica