1. Data structures#

spikes_times: data structure containing spikes times (\(\mathbf{t}_n^{(r)}\) in Eq. 6 of Duncker and Sahani [DS18]).

Double list of length n_trials by n_neurons such that spikes_times[r][n] is a list-like collection of spikes times for trials r and neuron n.
ind_points_locs: data structure containing inducing points locations (\(\mathbf{z}_k^{(r)}\) in Eq. 3 of Duncker and Sahani [DS18]).

List of length n_latents of PyTorch tensors of size (n_trials, n_ind_points, 1), such that ind_points_locs[k][r, :, 0] gives the inducing points locations for trial r and latent k.
leg_quad_points and leg_quad_weights: data structures containing the Legendre quadrature points and weights, respectively, used for the calculation of the integral in the first term of the expected posterior log-likelihood in Eq. 7 in Duncker and Sahani [DS18].

Both leg_quad_points and leg_quad_weights are tensors of size (n_trials, n_quad_elem, 1), such that leg_quad_points[r, :, 0] and leg_quad_weights[r, :, 0] give the quadrature points and weights, respectively, of trial r.
var_mean: mean of the variational distribution \(q(\mathbf{u}_k^{(r)})\) (\(\mathbf{m}_k^{(r)}\) in the paragraph above Eq. 4 of Duncker and Sahani [DS18]).

List of length n_latents of PyTorch tensors of size (n_trials, n_ind_points, 1), such that var_mean[k][r, :, 0] gives the variational mean for trial r and latent k.
var_chol: vectorized cholesky factor of the covariance of the variational distribution \(q(\mathbf{u}_k^{(r)})\) (\(S_k^{(r)}\) in the paragraph above Eq. 4 of Duncker and Sahani [DS18]).

List of length n_latents of PyTorch tensors of size (n_trials, n_ind_points * (n_ind_points + 1)/2, 1), such that var_chol[k][r, :, 0] gives the vectorized cholesky factor of the variational covariance for trial r and latent k.
emb_post_mean_quad: embedding posterior mean (\(\nu_n^{(r)}(t)\) in Eq. 5 of Duncker and Sahani [DS18]) evaluated at the Legendre quadrature points.

List of length n_latents of PyTorch tensor of size (n_trials, n_ind_points, 1), such that emb_post_mean[k][r, :, 0] gives the embedding posterior mean for trial r and latent k, evaluated at leg_quad_points[r, :, 0].
emb_post_var_quad: embedding posterior variance (\(\sigma_n^{(r)}(t, t)\) in Eq. 5 of Duncker and Sahani [DS18]) evaluated at the Legendre quadrature points.

List of length n_latents of PyTorch tensors of size (n_trials, n_ind_points, 1), such that emb_post_var[k][r, :, 0] gives the embedding posterior variance for trial r and latent k, evaluated at leg_quad_points[r, :, 0].
Kzz: kernel covariance function evaluated at inducing points locations (\(K_{zz}^{(kr)}\) in Eq. 5 of Duncker and Sahani [DS18]).

List of length n_latents of PyTorch tensors of PyTorch tensors of size (n_trials, n_ind_points, n_ind_points), such that Kzz[k][r, i, j] is the kernel covariance function for latent k evaluated at the ith and kth components of the inducing point locations for latent k and trial r (Kzz[k][r, i, j] = \(\kappa_k(\mathbf{z}_k^{(r)}[i], \mathbf{z}_k^{(r)}[j])\)).
Kzz_inv: lower-triangular cholesky factors (\(L^{(kr)}\)) of kernel covariance matrices evaluated at inducing points locations (\(K_{zz}^{(kr)}=L^{(kr)}\left(L^{(kr)}\right)^\intercal\)).

List of length n_latents of PyTorch tensors of PyTorch tensors of size (n_trials, n_ind_points, n_ind_points), such that Kzz_inv[k][r, :, :] is the lower-triangular cholesky factor \(L^{(kr)}\).
Ktz: kernel covariance function evaluated at the quadrature points and at the inducing points locations (\(\kappa_k(t, \mathbf{z}_k^{(r)})\) in Eq. 5 of Duncker and Sahani [DS18]).

List of length n_latents of PyTorch tensors of size (n_trials, n_quad_elem, n_ind_points), such that Kzz[k][r, i, j] is the kernel covariance function for latent k evaluated at the ith quadrature time for trial r (leg_quad_points[r, :, 0]) and at the jth inducing points location for trial r and latent k (Kzz[k][r, i, j] = \(\kappa_k(\text{leg_quad_points}[r, i, 0], \mathbf{z}_k^{(r)}[j]\)).

Ktt: kernel covariance function evaluated at quadrature points (\(\kappa_k(t, t)\) in Eq. 5 of Duncker and Sahani [DS18]).

Note: svGPFA does not need to evaluate \(\kappa_k(t, t')\) for \(t\neq t'\). It only needs to evaluate \(\kappa_k(t, t)\) to calculate the variance of the posterior embedding \(\sigma^2_n(t, t)\), which is used to compute \(\mathbb{E}_{q(h_n^{(r)})}\left[g(h_n^{(r)}(t))\right]\).

List of length n_latents of PyTorch tensors of size (n_trials, n_quad_elem, n_latents), such that Ktt[k][r, i, k] is the kernel variance function for latent k evaluated at the ith quadrature time for trial r (leg_quad_points[r, i, 0]). That is Ktt[k][r, i, k] = \(\kappa_k(\text{leg_quad_points[r, i, 0]}, \text{leg_quad_points[r, i, 0]})\).