Receptive Field Inference with Localized Priors
Figure 3
Estimated filters and prior covariances for ALD methods.
(Same example filter as shown in Fig. 2). Left column shows the true filter (dotted black) and ALD estimates (red) replotted from the right-most column of Fig. 2. Top: Space-localized estimate. The estimated prior variance (black trace, middle) is a Gaussian form that controls the falloff in amplitude of filter coefficients (red) as a function of position. The prior covariance (right) is a diagonal matrix with this Gaussian along the diagonal. The prior is thus independent with location-dependent variance. Middle: Frequency-localized estimate. A Gaussian form (reflected around the origin due to symmetries of the Fourier transform) specifies the prior variance as a function of frequency (black trace, middle). The Fourier power of the filter estimate (red) drops quickly to zero outside the estimated region. The prior covariance matrix (right) is diagonal in the Fourier domain, meaning the Fourier coefficients are independent with frequency-dependent variance. Bottom: Space and frequency localized estimate. The estimated prior covariance matrix is not diagonal in spacetime or frequency, but takes the form of a “sandwich matrix” that combines the prior covariances from ALDs and ALDf (see text). The resulting prior covariance matrix can be visualized in either the spacetime domain (left) or the Fourier domain (right). It is localized (has a local region of large prior variance) in both coordinate frames, but has strong dependencies (off-diagonal elements), particularly across space.