Algorithm A.1.

Feature weighted receptive field method. In the following, tensors are boldfaced and the dimensions of a tensor are always displayed in square bracket after its name. A change in the order of the dimension implies a transpose of the tensor and an arrow in the dimensions indicates broadcasting to a certain size.

1: procedure Generate modelspace tensor(ℱ, 𝒢): # Evaluate the spatial integrals of Eq 2 for all feature maps (ℱ) and pooling field (𝒢) candidates and resolutions.

2:    for (Fl, gl) ∈ (ℱ, 𝒢) do

3:      Ml ← TensorDot(Fl[n, Kl, xl, yl], gl[G, xl, yl], axis = [[2, 3], [1, 2]])

4:    M ← Concatenate((Ml[n, Kl, G], ∀l ∈ L), axis = 1)

5:    M ← Z-Score(M, axis = 0)

6:    return M[n, K, G]

7:

8: procedure Generate voxel predictions(M, w):

9:    R̂ ← Batched-TensorDot(M[G, n, K], w[G, V, K], axis = [[2], [2]])

10:    return R̂[n, V, G]

11:

12: procedure Loss(M, R, w):

13:    R̂ ← Generate voxel predictions(M, w)

14:    return L2-norm(R̂[n, V, G] − R[n, V, 1 → G])[V, G]

15:

16: procedure Optimize FWRF model parameters(M, R, winit): # M is the precalculated model-space tensor and R is the target voxel activity.

17:    mbest ← zeros[V] # best models initialization

18:    wbest ← zeros[V, K] # best weights initialization

19:    sbest ← inf[V] # best scores initialization

20:    for Vb ∈ Batch(V) do # take a batch of voxels

21:      for Gb ∈ Batch(G) do # take a batch of candidates f.p.f.

22:        wb ← winit[1 → Gb, Vb, K]

23:        for e ∈ 1‥epochs do

24:          sb ← zeros[Vb, Gb] # scores for this batch

25:          for nb ∈ Batch(ntrain) do # take a batch from the training samples

26:             ${\mathbf{w}}_b\leftarrow {\mathbf{w}}_b-\lambda \frac{d\text{Loss}}{d\mathbf{w}}(\mathbf{M}[n_b,K,G_b],\mathbf{R}[n_b,V_b],{\mathbf{w}}_b[G_b,V_b,K])$

27:          for nb ∈ Batch(nholdout) do # take a batch from the holdout samples

28:            sb ← sb + Loss(M[nb, K, Gb], R[nb, Vb], wb[Gb, Vb, K])

29:          for υ ∈ Element(Vb) do # for all voxels

30:            g ← argmin(sb[υ, Gb]) # get the best model for this batch

31:            if sb[υ, g] < sbest[υ] then

32:              mbest[υ] ← g

33:              wbest[υ, K] ← wb[g, υ, K]

34:              sbest[υ] ← sb[υ, g]

35:    return mbest, wbest, sbest