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# Copyright (c) 2024, Albert Gu and Tri Dao.
"""Minimal implementation of SSD.
This is the same as Listing 1 from the paper.
"""
import torch
import torch.nn.functional as F
from einops import rearrange, repeat
from mamba_ssm.ops.triton.ssd_combined import mamba_chunk_scan_combined
def segsum_unstable(x):
"""Naive segment sum calculation."""
T = x.size(-1)
x_cumsum = torch.cumsum(x, dim=-1)
x_segsum = x_cumsum[..., :, None] - x_cumsum[..., None, :]
mask = torch.tril(torch.ones(T, T, device=x.device, dtype=bool), diagonal=0)
x_segsum = x_segsum.masked_fill(~mask, -torch.inf)
return x_segsum
def segsum(x):
"""More stable segment sum calculation."""
T = x.size(-1)
x = repeat(x, "... d -> ... d e", e=T)
mask = torch.tril(torch.ones(T, T, device=x.device, dtype=bool), diagonal=-1)
x = x.masked_fill(~mask, 0)
x_segsum = torch.cumsum(x, dim=-2)
mask = torch.tril(torch.ones(T, T, device=x.device, dtype=bool), diagonal=0)
x_segsum = x_segsum.masked_fill(~mask, -torch.inf)
return x_segsum
def ssd_minimal_discrete(X, A, B, C, block_len, initial_states=None):
"""
Arguments:
X: (batch, length, n_heads, d_head)
A: (batch, length, n_heads)
B: (batch, length, n_heads, d_state)
C: (batch, length, n_heads, d_state)
Return:
Y: (batch, length, n_heads, d_head)
"""
assert X.dtype == A.dtype == B.dtype == C.dtype
assert X.shape[1] % block_len == 0
# Rearrange into blocks/chunks
X, A, B, C = [rearrange(x, "b (c l) ... -> b c l ...", l=block_len) for x in (X, A, B, C)]
A = rearrange(A, "b c l h -> b h c l")
A_cumsum = torch.cumsum(A, dim=-1)
# 1. Compute the output for each intra-chunk (diagonal blocks)
L = torch.exp(segsum(A))
Y_diag = torch.einsum("bclhn,bcshn,bhcls,bcshp->bclhp", C, B, L, X)
# 2. Compute the state for each intra-chunk
# (right term of low-rank factorization of off-diagonal blocks; B terms)
decay_states = torch.exp((A_cumsum[:, :, :, -1:] - A_cumsum))
states = torch.einsum("bclhn,bhcl,bclhp->bchpn", B, decay_states, X)
# 3. Compute the inter-chunk SSM recurrence; produces correct SSM states at chunk boundaries
# (middle term of factorization of off-diag blocks; A terms)
if initial_states is None:
initial_states = torch.zeros_like(states[:, :1])
states = torch.cat([initial_states, states], dim=1)
decay_chunk = torch.exp(segsum(F.pad(A_cumsum[:, :, :, -1], (1, 0))))
new_states = torch.einsum("bhzc,bchpn->bzhpn", decay_chunk, states)
states, final_state = new_states[:, :-1], new_states[:, -1]
# 4. Compute state -> output conversion per chunk
# (left term of low-rank factorization of off-diagonal blocks; C terms)
state_decay_out = torch.exp(A_cumsum)
Y_off = torch.einsum('bclhn,bchpn,bhcl->bclhp', C, states, state_decay_out)
# Add output of intra-chunk and inter-chunk terms (diagonal and off-diagonal blocks)
Y = rearrange(Y_diag+Y_off, "b c l h p -> b (c l) h p")
return Y, final_state
# Simple test
def test_correctness():
torch.manual_seed(42)
## Dimensions
# Denoted (B, T, Q, D, P) in the paper
batch, seqlen, chunk_size, dim, headdim = 1, 2048, 64, 2048, 64
nheads = dim // headdim # (H) in the paper
ngroups = 1 # (G) in the paper
dstate = 64 # (N) in the paper
dtype = torch.float32
device = "cuda"
x = torch.randn(batch, seqlen, nheads, headdim, dtype=dtype, device=device)
dt = F.softplus(torch.randn(batch, seqlen, nheads, dtype=torch.float32, device=device) - 4).requires_grad_()
A = (-torch.exp(torch.rand(nheads, dtype=torch.float32, device=device))).requires_grad_()
B = torch.randn(batch, seqlen, ngroups, dstate, dtype=dtype, device=device)
C = torch.randn(batch, seqlen, ngroups, dstate, dtype=dtype, device=device)
D = torch.randn(nheads, dtype=dtype, device=device)
# Comparing fused version and minimal version
y = mamba_chunk_scan_combined(x, dt, A, B, C, chunk_size, D=None)
y_min, _ = ssd_minimal_discrete(x*dt.unsqueeze(-1), A*dt, B, C, chunk_size)
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