Fig 1.
Sparsity-promoting cost functions c considered and their corresponding thresholding operators.
A: Plot of 1D cost functions considered. B: Corresponding thresholding operators.
Fig 2.
CEL0, and to a lesser extent, ℓ1/2, and hard thresholding produce sparser codes than ISTA.
A: MSE between the reconstructed and the actual image for the last 500 batches as a function of the thresholding method. B: Distribution of activity of the units for the image stimuli presented. The middle of the vertical lines represent the mean number of active units per image patch, their length the standard deviation. C: Distribution of the amplitudes of the active units D:MSE as a function of the number of active units. To get these data points we varied the λ parameter for each algorithm.
Fig 3.
CEL0 and, to a lesser extent, ℓ1/2, and hard thresholding have better reconstruction error than ISTA for all dictionary sizes tested (from approximately 2 to 20 degrees of overcompleteness).
MSE of the different thresholding methods for several dictionary sizes. The parameter λ has been adjusted for each dictionary size and algorithm so that the sparsity level is approximately stable (see S5 Fig). Each time we run 1600 batches of 250 image patches (in total 400000 patches), and took the mean and standard deviation of the reconstruction error of the last 100 batches.
Fig 4.
In contrast to macaque V1 neurons that have a uniform circular variance distributions, the units of all thresholding algorithms show a distinct peak in their circular variance distribution that shifts to the right (more broadly tuned neurons) as the units in the dictionary increase.
Distribution of circular variance values for the ϕ learned by the different thresholding algorithms (the area in all cases was normalized to sum to 1) with V1 experimental data from [24] included for A: 500 units and B: 2000 units.
Fig 5.
The largest number of ϕ vectors responding maximally to a particular orientation are the ones with a preference towards the vertical orientation, with this subset also showing the sharpest orientation tuning (as indicated by their circular variance).
A: Polar plots of the proportion of ϕ vectors responding maximally to an orientation for ISTA, CEL0, ℓ1/2, and hard thresholding. B: Polar plots of the mean circular variance of ϕ vectors binned according to their preferred orientation.
Fig 6.
Sampling of RFs of different aspect ratios from our thresholding algorithms and recordings on macaques’ V1 and V2.
A: Sampling of RFs generated by CEL0. As we go down the rows their aspect ratio (defined as the width (SD) of the Gaussian envelope parallel to the axis of the Gabor over the width orthogonal to it) increases. The same RF organization applies to the rest of the Figures. B: RFs generated by ISTA. C: RF subunits from electrophysiological recordings in V1. D: Same as (C) for V2 (data courtesy of Liang She).
Fig 7.
Spatial properties of RFs generated by thresholding algorithms and in macaques’ V1 and V2.
A: Width (SD) of the Gaussian envelope along the parallel axis of the Gabor as a function of the width orthogonal to it (both normalized by the Gabor’s period) for the fits of the RFs generated by ISTA and CEL0. B: Same as (A) for CEL0 and an instance of ISTA with the same sparsity level as CEL0 (sparse ISTA). C: Same as (A) for ISTA and an instance of CEL0 with the same sparsity level as ISTA (dense CEL0). D: Same as (A) for the RF subunits from recordings in macaque V1 and V2.
Fig 8.
Control for probing the effect of sparsity on the RFs.
MSE values computed between the reconstructed and the actual image for the last 500 batches as a function of the mean number of active units for two instances of ISTA and CEL0. Horizontal lines indicate the standard deviation of the mean number of units, vertical the standard deviation of the MSE.