Factorized receptive field in time and space

[BalzaniEdoardo]

In order to fit a spatio-temporal receptive field of a neuron (for example in response to a visual input) we may need to assume that the temporal and the spatial contribution to avoid over-parametrization.

The problem of non-factorized receptive fileds is the input would be a tensor of shape $(T, M, N)$, with T the time points, M,N samples or number of basis. this can be very high dimensional and prone to over-fitting.

If we assume spatial and temporal factorization, we reduce to $T + MN$, which is more manageble, however the log-rate will look something like:

$$\log(\mu) = \mathbf{\alpha}^\top M \beta + \text{other predictors}$$

This is a quadratic form in $\alpha \in \mathbb{R}^T$ and $\beta \in \mathbb{R}^{MN}$, i.e. is non-linear model and the parameters interacts. One way to fit this is by alternating gradient descent (alternating between $\alpha$ and $\beta$); each sub-problem is linear and convex.