Empirical error-confidence curves for neural network and Gaussian classifiers.

"Error-Confidence" measures the probability that the proportion of errors made by a classifier will be within epsilon of EB, the optimal (Bayes) error. Probably Almost Bayes (PAB) theory attempts to quantify how this confidence increases with the number of training samples. We investigate the relationship empirically by comparing average error versus number of training patterns (m) for linear and neural network classifiers. On Gaussian problems, the resulting EC curves demonstrate that the PAB bounds are extremely conservative. Asymptotic statistics predicts a linear relationship between the logarithms of the average error and the number of training patterns. For low Bayes error rates we found excellent agreement between the prediction and the linear discriminant performance. At higher Bayes error rates we still found a linear relationship, but with a shallower slope than the predicted-1. When the underlying true model is a three-layer network, the EC curves show a greater dependence on classifier capacity, and the linear predictions no longer seem to hold.

Do Gabor functions provide appropriate descriptions of visual cortical receptive fields?

Several recent theoretical models for human spatial vision posit that cortical receptive fields act to minimize simultaneously the product of the standard deviation of the sensitivities to position (delta chi) and to spatial frequency (delta omega) in accord with the uncertainty principle from Fourier analysis. The receptive-field functions resulting from this approach--one-dimensional or two-dimensional Gabor elementary functions--have been shown by others to fit measured receptive fields adequately in some species. However, only complex-valued Gabor functions minimize this product, and these cannot be fitted to single-cell receptive fields. We point out that the derivations of others have an implied metric or measure of positional and spatial-frequency uncertainties and that there is an infinitely large class of such metrics, many of which yield other receptive-field functions that are quite plausible biologically. We review neurophysiological measurements of others and analyze psychophysical masking data and find that in many cases receptive-field functions other than Gabor functions fit better. We conclude that there are insufficient theoretical demonstrations and experimental data to favor Gabor functions over any of a number of other plausible receptive-field functions.

Temporal impulse responses from flicker sensitivities.

We derive visual temporal impulse response functions from the large-field sinusoidal flicker sensitivity measurements made at several luminances by Kelly [J. Opt. Soc. Am. 51, 422 (1961)]. To this end we recognize response function causality and use Kramers-Krönig relations to reconstruct the (unmeasured) temporal phase spectrum and hence also the impulse response functions. At high luminances our derived functions are triphasic, while at low luminances they are biphasic. These functions are model independent, rely on no free parameters, and are consistent with certain theoretical model-dependent functions as well as psychophysical measurements by others.

Receptive-field symmetry probed using converging gratings.

Contrast-sensitivity functions (CSF's) for converging and diverging gratings were obtained under voluntary fixation at several field widths and under retinal stabilization at one field width. In general, these types of gratings had similar CSF's at all temporal frequencies tested, a result that can be explained by plausible spatiotemporal receptive-field models possessing spatial antisymmetry but not by models possessing pure (even) spatial symmetry. Phase condition at the central line in converging and diverging gratings affected grating detectability at high spatial frequencies, as expected.