ParaLSTM¶

Fidelity: paper-faithful
Sources: Danieli et al., arXiv:2510.21450, eq. 3.1b, 3.3; CIFG peephole (Greff et al. 2017).
Code: pararnn.cells.para_lstm.ParaLSTM

State¶

Two slots, layout (..., 2, d_h): index 0 = cell \(c\), index 1 = hidden \(h\).
hidden_slot = 1 for readout.

Recurrence¶

Diagonal recurrent \(a_f, a_z, a_o\) and peepholes \(c_f, c_o\). Packed input Linear \(d_{\mathrm{in}}\to 3 d_h\) → \((f_x, z_x, o_x)\).

\[ \begin{aligned} f_t &= \sigma(a_f \odot h_{t-1} + c_f \odot c_{t-1} + f_x), \\ z_t &= \tanh(a_z \odot h_{t-1} + z_x), \\ c_t &= f_t \odot c_{t-1} + (1-f_t)\odot z_t, \\ o_t &= \sigma(a_o \odot h_{t-1} + c_o \odot c_t + o_x), \\ h_t &= o_t \odot \tanh(c_t). \end{aligned} \]

Coupled input-forget: input gate is \(1-f_t\).

Jacobian class¶

jac_structure='block2' — per-channel \(2\times 2\) blocks over \((c,h)\) (eq. 3.2b with diagonal \(A,C\)).

Parallel path¶

Newton + scan on the stacked state. App. A guess from \(f(0,x_t)\). Typical \(K{=}3\). Default max_recurrent_norm=0.5 (App. C.1).

Deviations¶

None relative to Danieli CIFG peephole diag-\(A,C\) presentation used in this library.

Reproduce (sequential)¶

import torch
from pararnn import ParaLSTM
from pararnn.layout import LSTM_CELL, LSTM_HIDDEN

cell = ParaLSTM(32, 32)
state = torch.zeros(2, 2, 32)  # (batch, slots, d_h)
x = torch.randn(2, 32)
state = cell.step(state, x)
h = state[..., LSTM_HIDDEN, :]

Agreement¶

Wrap with ParaRNN and call verify_agreement (numerics contract).