Sequential Principal Curves Analysis Toolbox (SPCA)
V. Laparra, G. Gutiérrez, S. Jimenez, G. Camps and J. Malo

Papers and Matlab Toolbox


SPCA is a manifold learning technique that identifies the curvilinear coordinates of a data set.
It defines an invertible transform that can be tuned to for NonLinear ICA (infomax) or optimal Vector Quantization (error minimization), and can be used in Dimensionality Reduction, Domain Adaptation and Classification problems.
The explicit form of the identified features (and associated nonlinear "filters") make it useful to model sensors in theoretical neuroscience.

Illustrative Results I:  learning nonlinear features 
Identification of curved features and effect of metric in SPCA in a curved 2D manifold. Note the different marginal PDFs in the direction perpendicular to the principal curve: Laplacian and Uniform PDFs of increasing variance.

SPCA (Euclidean Metric)
SPCA (Infomax Metric) SPCA (Error Minimization Metric)
RMSE = 0.55
Mutual Info = 0.27 bits
RMSE = 0.56
Mutual Info = 0.05 bits

RMSE = 0.53
Mutual Info = 0.06 bits

Infomax and error minimization through SPCA. 500 randomly selected samples of the sets in the first row were transformed using SPCA (second row) with different metrics. Additionally, Cartesian lattices in the response domain were inverted back to the input domain giving rise to the curved lattices in the top row. Results are analyzed in terms of independence (Mutual Information), and reconstruction error (RMSE). In each case, MI was computed in the corresponding transform domain, while the RMSE values refer to the quantization error in the original domain using the corresponding lattices as codebook. For the sake of reference, in the original domain results were MI = 0.75 bits, and and RMSE=0.63

Illustrative Results II: Image coding according to different optimization criteria

Related Papers

V1 non-linear properties emerge from local-to-global non-linear ICA

J. Malo and J. Gutiérrez
Network: Comp. Neural Systems. 17(1):  85-102 (2006)

The SPCA:  
Visual Aftereffects and Sensory Nonlinearities from a single Statistical Framework
V. Laparra, & J. Malo.
Frontiers in Human Neuroscience. Special Issue on Perceptual Illusions 2015  
A guide to the full supplementary material (description of the code, data, experiments and results).

Nonlinearities and Adaptation of Color Vision from Sequential Principal Curves Analysis.

V. Laparra, S. Jiménez, G. Camps & J. Malo.
Neural Computation, 24(10):2751-88, Oct 2012

Visual Discrimination and Adaptation using nonlinear unsupervised learning pdf
S. Jiménez, V. Laparra, & J. Malo.
Proc. SPIE. Human Vision and Elect. Imag. 2013

Full Technical Report on Sequential Principal Curves Analysis
V. Laparra & J. Malo. 
Technical Report IPL. Universitat de Valencia, 2015
A guide to the supplementary material (2012 version).

Principal Polynomial Analysis (PPA)
V. Laparra, S. Jimenez, D. Tuia, G. Camps-Valls and J. Malo
International Journal of Neural Systems, 24(7) Nov. 2014. pdmatPPA Toolbox

Dimensionality Reduction via Regression in Hyperspectral Imagery
V. Laparra, J. Malo and G. Camps-Valls
IEEE Journal on Selected Topics of Signal Processing. Vol. 9, Num. 9. September 2015. pdf  DRR Toolbox mat

Download Code

Once you download and decompress the toolbox, see the readme file demo_SPCA_toy_example_2D_manifold.m