Table 1.
RPCA pseudo-code.
Figure 1.
Different basis types of RPCA preprocessing and Sparse Coding.
Sample receptive fields are scaled into range [0,1]. (A) no RPCA, columns of dictionary . (B) receptive fields learned after PCA pre-filtering: features show wavy, global structure. (C) Features (‘global filters’) of the low dimensional signal for the case
(dimension = 17). (D) reverse correlation of the full rank sparsified signal
yields stereotypical DoG-like filters with symmetric 2D structure. The figure shows the profile of the central section as a function of
. At higher values the negative basin around the peak gets deeper. (E) Randomly selected sparse coding filter sets (over-completeness is
,
and
) With increasing
the filters get smaller and more localized (i.e. cleaner). (F) For comparison, a set of sparse coding filters (
) and the corresponding linear approximations (normalized reverse correlation,
) are shown at
.
Figure 2.
Distribution of the shape parameters for the model and for the experimental data.
Receptive fields of simple cells in primary visual cortex, linearly approximated by spike triggered averaging. Data [8] are available at http://web.mac.com/darioringach/lab/Data.html. Our model filters show significant diversity in the fitted shapes similar to what has been found experimentally. While other models (e.g. [39], [40]) are also able to partially match the filters to the observed RFs, a significant difference is that our model uses highly overcomplete representations. For other differences, see the main text.
Figure 3.
The impact of on the signal decomposition and the overall quality of the sparse coding filters.
(A) The empirical distribution of the Gabor patch fitting error as a function of . Larger spread signifies deviation from ideal Gabor patch, often used as model shape for experimentally recorded receptive fields. The shift of the mean toward
as
increases is a consequence of the decrease of the average filter size. For each mean value a sample filter is shown demonstrating this shrinkage effect. (B) The dimension and the relative weight of L (the low dimensional signal) in the reconstruction as a function of
. Relevant range is where the dimensionality is low, yet L is able to capture most of the original signal. For image size 16×16 this range is about 0.3–0.8.
Figure 4.
Reconstruction quality as a function of the number of nonzero coding units and .
Reconstruction quality is measured by mean SNR: , where
runs over the inputs. Since RPCA is an additive decomposition, the reconstruction error is given as
. The total number of nonzero entries is given as the sum of the rank estimate of
and the preserved number of nonzero units (k) in the sparse overcomplete representation of the atypical part (
) of the RPCA output. Since sparseness level is automatically set by SCE, the following arbitrary values for k were chosen. For
and for
,
.
Figure 5.
RPCA on concatenated image sequences.
Left: The first 10 spatio-temporal filters of the low rank signal,L (rank ) are shown. Each filter is shown as a sequence of 16 frames of size 8×8 pixels. It can be seen that there are spatio-temporally separable as well as non-separable filters. All filters correspond to low frequency temporal or spatial changes Right: 10 selected spatio-temporal filters of the corresponding overcomplete sparse codes that display different spatio-temporal localization and dynamics. While many filters are similar to the presented ones, more training would be needed to achieve similar locality for the majority of filters at this input dimensionality (8×8×16) and level of overcompleteness (16×).
Figure 6.
A comparison of the amplitude spectra of the “atypical” output part of RPCA, the whitened input and the whitened ideal input.
This plot demonstrates that the particular whitening filter as used in [5], [40] can be seen as a linear approximation of the filtering properties of RPCA when only the atypical output is considered. The thick (red) line is the amplitude spectrum of the RPCA output. The dashed (blue) line with square markers is the amplitude spectrum of the training images filtered with the whitening filter. The thin (green) line serves as a reference: this is the amplitude spectrum of whitened ideal input which has an amplitude spectrum proportional to 1/frequency. Due to the limited input size, there is a natural cutoff at higher frequencies. (Since the size of the images is 16×16, the largest frequency is .) The whitening filter:
, where the cutoff frequency is
. The variances of the plots are due the artifacts caused by the rectangular sampling lattice. For comparison purposes the plots are rescaled onto
.
Table 2.
The pseudocode of the Subspace Pursuit method.
Table 3.
Pseudo-code of the subspace cross-entropy (SCE) method for Bernoulli distributions.