reasoning
Let me analyze this problem:
1. We need to fuse quaternion-to-rotation-matrix conversion with batched matrix multiplication and translation addition.
2. Input: points (B, N, 3), quat (B, 4), trans (B, 3)
3. Output: points @ R.T + t, where R is derived from quat
The mathematical operations per batch element and per point:
1. Normalize quaternion q = (w, x, y, z)
2. Compute rotation matrix elements:
m00 = 1 - 2*(y² + z²)
m01 = 2*(xy - wz)
m02 = 2*(xz + wy)
m10 = 2*(xy + wz)
m11 = 1 - 2*(x² + z²)
m12 = 2*(yz - wx)
m20 = 2*(xz - wy)
m21 = 2*(yz + wx)
m22 = 1 - 2*(x² + y²)
3. For each point (px, py, pz):
out_x = px*m00 + py*m10 + pz*m20 + tx
out_y = px*m01 + py*m11 + pz*m12 + ty
out_z = px*m02 + py*m12 + pz*m22 + tz
Wait, let me double check the matrix multiplication. The reference does:
```python
R = _quaternion_to_rotation_matrix(quat) # (B, 3, 3)
return torch.matmul(points, R.transpose(-1, -2)) + trans.unsqueeze(1)
```
So `points` is (B, N, 3) and R.transpose is (B, 3, 3). The matmul gives (B, N, 3).
For each point row vector p:
`p' = p @ R.T`
Where R.T is:
```
[m00, m10, m20]
[m01, m11, m21]
[m02, m12, m22]
```
So:
out_x = px*m00 + py*m01 + pz*m02 + tx
out_y = px*m10 + py*m11 + pz*m12 + ty
out_z = px*m20 + py*m21 + pz*m22 + tz
Wait, that's if R.T is the transpose of R. Let me look more carefully at R construction.
R is constructed as:
row0 = [m00, m01, m02]
row1 = [m10, m11, m12]
row2 = [m20, m21, m22]
So R = [[m00, m01, m02], [m10, m11, m12], [m20, m21, m22]]
Then R.T = [[m00, m10, m20], [m01, m11, m21], [m02, m12, m22]]
Then p @ R.T = [px, py, pz] @ [[m00, m10, m20], [m01, m11, m21], [m02, m12, m22]]
out_x = px*m00 + py*m01 + pz*m02 + tx
out_y = px*m10 + py*m11 + pz*m12 + ty
out_z = px*m20 + py*m21 + pz*m22 + tz
Wait no, I made a mistake. Let me trace matrix multiplication properly:
[p_x, p_y, p_z] * R.T means:
output_x = p_x * R.T[0,0] + p_y * R.T[1,0] + p_z * R.T[2,0]
= p_x * m00 + p_y * m01 + p_z * m02
output_y = p_x * R.T[0,1] + p_y * R.T[1,1] + p_z * R.T[2,1]
= p_x * m10 + p_y * m11 + p_z * m12
output_z = p_x * R.T[0,2] + p_y * R.T[1,2] + p_z * R.T[2,2]
= p_x * m20 + p_y * m21 + p_z * m22
Yes, that's right. Let me write the kernel.
For optimization on H100:
- Use vectorized loads/stores (float4 for float, or just float3)
- Each thread can process one point
- The quaternion is shared across all points in a batch, so we can load it once per batch
- Thread blocks organized as (N) threads per block, processing a batch element
Actually, since B=8 and N=512, we could either:
1. Have each block process one batch element with 512 threads
2. Have each thread process multiple points to increase arithmetic intensity
3. Use a 2D grid over (B, N)
For N=512, 512 threads = 16 warps which might be fine. But we might want each thread to process multiple points to better utilize H100's compute.
Actually with B=8 and N=512, there are only 4096 points total. That's not a lot for H100. We need to make sure work is well-distributed.
Let me think about memory bandwidth.
Input: points (B, N, 3) floats = 8 * 512 * 3 * 4 = 49KB
quat: 8 * 4 * 4 = 128B
trans: 8 * 3 * 4 = 96B
Output: same as points = 49KB
Total memory traffic: ~100KB. This is tiny! Even with 1 TB/s memory bandwidth, this would take ~0.1 microseconds. But kernel launch overhead is ~5-10 microseconds. So we probably can't beat PyTorch by much for this tiny input size, but we can try to be competitive.
Actually, wait - the problem says batch_size = 8 and num_points = 512 for testing, but the actual evaluation may use larger sizes. Let me check... It says "Your submission is evaluated on multiple input sizes, not just the one shown in the prompt."
So I should write a general kernel that handles arbitrary B, N.
For a general kernel approach:
- Grid: (B blocks in x, ceil(N/THREADS_PER_BLOCK) in y) or just linearize (B * ceil(N/THREADS_PER_BLOCK)) blocks
- Each block processes a subset of points for one batch element
- Threads in a block cooperate, but since each point is independent, it's embarrassingly parallel
Let me design the kernel:
```cuda
__global__ void fused_quat_transform_kernel(
const float* points,
const float* quat,
const float* trans,
float* out,
int B, int N) {
int b = blockIdx.x;
int tid = threadIdx.x + blockIdx.y * blockDim.x;
if (b >= B || tid >= N) return;
// Load quaternion for this batch
float qw = quat[b * 4 + 0];
float qx = quat[b * 4 + 1];
float qy = quat[b * 4 + 2];
float qz = quat[b * 4 + 3];
// Normalize
float norm = sqrtf(qw*qw + qx*qx + qy*qy + qz*qz);
float inv_norm = 1.0f / norm;
qw *= inv_norm;
qx *= inv_norm;
qy *= inv_norm;
qz *= inv_norm;
// Compute rotation matrix elements
float xx = qx * qx;
float yy = qy * qy;
float zz = qz * qz;
float xy = qx * qy;
float xz = qx * qz;
float yz = qy * qz;
float wx = qw * qx;
float wy = qw * qy;
float wz = qw * qz;
float m00 = 1.0f - 2.0f * (yy + zz);
float m01 = 2.0f * (xy - wz);
float m02 = 2.0f * (xz + wy);
float m10 = 2.0f * (xy + wz);
float m11 = 1.0f - 2.0f * (xx + zz);
float m12 = 2.0f * (yz - wx);
float m20 = 2.0f * (xz - wy);
float m21 = 2.0f * (yz + wx);
float m22 = 1.0f - 2.0f * (xx + yy);
// Load point
int idx = (b * N + tid) * 3;
float px = points[idx];
float py = points[idx + 1];
float pz = points[idx + 2];
// Transform
float tx = trans[b * 3 + 0];
float ty = trans[b * 3 + 1];
float tz = trans[b * 3 + 2];
float ox = px * m00 + py * m01 + pz * m02 + tx;
float oy = px * m10 + py * m11 + pz * m12 + ty;
float oz = px * m20 + py * m21 + pz * m22 + tz;
out[idx] = ox;
out[idx + 1] = oy;
out[idx + 2] = oz;
}
```
For grid organization, I could use 1D blocks:
- Each block processes one batch element
- Grid has B blocks
- Each block has enough threads to cover N points
Or 2D grid where blockIdx.x is batch and blockIdx.y is chunk of points.
For better occupancy, we should use a reasonable block size like 256 or 512 threads. But for small N like 512, using a whole block per batch is fine.
Wait, but what if N is not a multiple of block size? We should handle that.
Actually, for very small N, we might want multiple batch elements per block to improve occupancy. But let's start simple.
Another consideration: repeated computation of the rotation matrix. Since all threads in a block for the same batch element compute the same rotation matrix, we could compute it once and store in shared memory. But with only 9 floats and simple arithmetic, recomputing might be faster than shared memory access.
Actually, recomputing is cheaper than a shared memory read on modern GPUs for such simple arithmetic. The bottleneck would be memory bandwidth for points anyway.
Let me think about vectorized memory access. We can use float3 loads and stores to improve memory bandwidth. But float3 doesn't guarantee alignment. We can use float4 and waste some memory, or just use individual float loads.
Since points are (B, N, 3), the stride between consecutive points is 3 floats = 12 bytes, not aligned to 16 bytes. So we can't easily use float4 for point loads.
However, since the memory bandwidth is already very high and the compute is minimal, the exact memory access pattern might not matter much for larger N. For smaller N, we might need to worry about latency hiding.
Let me write a kernel that uses a 2D grid: blockIdx.x for batch, blockIdx.y for point chunks.
```cuda
__global__ void fused_quat_transform_kernel(
const float* __restrict__ points,
const float* __restrict__ quat,
const float* __restrict__ trans,
float* __restrict__ out,
int B, int N) {
int b = blockIdx.x;
int tid = blockIdx.y * blockDim.x + threadIdx.x;
if (b >= B || tid >= N) return;
// Compute rotation matrix (recomputed per thread, but same for all threads in block)
// Could put in registers - cheap
float qw = quat[b * 4 + 0];
float qx = quat[b * 4 + 1];
float qy = quat[b * 4 + 2];
float qz = quat[b * 4 + 3];
float norm = sqrtf(qw*qw + qx*qx + qy*qy + qz*qz);
float inv_norm = 1.0f / norm;
qw *= inv_norm; qx *= inv_norm; qy *= inv_norm; qz *= inv_norm;
float xx = qx * qx, yy = qy * qy, zz = qz * qz;
float xy = qx * qy, xz = qx * qz, yz = qy * qz;
float wx = qw * qx, wy = qw * qy, wz = qw * qz;
float m00 = 1.0f - 2.0f * (yy + zz);
float m01 = 2.0f * (xy - wz);
float m02 = 2.0f * (xz + wy);
float m10 = 2.0f * (xy + wz);
float m11 = 1.0f - 2.0f * (xx + zz);
float m12 = 2.0f * (yz - wx);
float m20 = 2.0f * (xz - wy);
float m21 = 2.0f * (yz + wx);
float m22 = 1.0f - 2.0f * (xx + yy);
int base = (b * N + tid) * 3;
float px = points[base];
float py = points[base + 1];
float pz = points[base + 2];
float tx = trans[b * 3 + 0];
float ty = trans[b * 3 + 1];
float tz = trans[b * 3 + 2];
out[base] = px * m00 + py * m01 + pz * m02 + tx;
out[base + 1] = px * m10 + py * m11 + pz * m12 + ty;
out[base + 2] = px * m20 + py * m21 + pz * m22 + tz;
}
```
Now let me think about parallelization strategy for different sizes:
- For small N and B: we want small blocks, perhaps processing multiple points per thread
- For large N and B: one point per thread works well
A common strategy is to use a grid-stride loop where each thread processes multiple points:
```cuda
int b = blockIdx.x;
int tid = threadIdx.x;
int blockSize = blockDim.x;
for (int i = tid; i < N; i += blockSize) {
// process point i for batch b
}
```
This has the advantage that a single block processes one batch element, and the number of threads can be fixed (e.g., 256) regardless of N. This provides good occupancy.
But wait, if N = 512 and we use 256 threads, each thread processes 2 points. If N = 100000 and we use 256 threads, each thread processes ~390 points. That's fine because each point is independent.
The downside is that consecutive threads no longer access consecutive memory locations if they're processing points at stride blockSize. But for float3 loads, consecutive threads within a warp (first iteration) will read consecutive memory locations. However, with a grid-stride loop, threads 0, 256, 512... don't read consecutive addresses in the first iteration.
Actually, thread 0 reads index 0, thread 1 reads index 1, etc. within the first iteration. So warp 0 reads addresses 0, 1, 2, ... which are contiguous. That's good! The next iteration reads addresses 256, 257, ... which are also contiguous.
Wait, but the memory for points is (B, N, 3). For a fixed batch b, the addresses for point i are at base + i*3. So threads 0-31 read:
- point 0: base, base+1, base+2
- point 1: base+3, base+4, base+5
- ...
- point 31: base+93, base+94, base+95
These are contiguous floats. But each thread reads 3 floats. The access pattern is not perfectly coalesced because thread 0 reads addresses 0,1,2 while thread 1 reads 3,4,5. With float loads, each thread issues 3 separate 4-byte loads. If the cache line is 128 bytes, then threads 0-31 cover 96 consecutive bytes. Some loads will be coalesced.
Actually on H100, the L1 cache is 128 bytes per sector. For 32 threads in a warp reading consecutive 4-byte values:
- Thread 0: bytes 0, 4, 8
- Thread 1: bytes 12, 16, 20
- etc.
Covering 0 to 95 in total. But thread 0 reads 0,4,8 which might be in one sector, and thread 1 reads 12,16,20 which is in another sector, etc. This might not be perfectly coalesced.
An alternative: we could use structured loads or reinterpret the (N, 3) as 3 separate arrays (x, y, z) or transpose to (3, N) in memory. But we can't control the input layout.
Actually, if we use `float4` loads with careful indexing... no, (N, 3) doesn't align to 16 bytes.
For better memory access, we could transpose the point data on the fly or use vector types. But this might not be worth it forB=8, N=512.
Let me also think about occupancy. On H100, we want many warps per SM. With B=8 and maybe 1 block per batch element, that's only 8 blocks total. With 132 SMs on H100, we'd only use 8 SMs. To improve this, we could launch more blocks by splitting the N dimension.
Better grid configuration:
- Grid: (B, ceil(N / block_size))
- Block: (block_size,)
- Each block processes a chunk of points for one batch element
For B=8, N=512, block_size=256: grid = (8, 2) = 16 blocks. Still not great.
For larger N, say N=100000, grid = (8, 391) = 3128 blocks. Good.
But for the reference case B=8, N=512, 16 blocks is still not enough to fill all SMs. Maybe we should process multiple batch elements per block, or use more threads per block?
Actually, for very small sizes, the kernel might be memory-bound with low occupancy, but since the total work is tiny, it doesn't matter much. PyTorch also won't do great on 8*512 points.
Wait, I should also consider if there's a way to process the problem with fewer memory operations. The reference does:
1. Load quat (B, 4)
2. Compute R (B, 3, 3)
3. Store R (B, 3, 3)
4. Load points (B, N, 3) and R (B, 3, 3)
5. Compute matmul
6. Load trans (B, 3)
7. Add and store output
Our fused kernel does:
1. Load quat (one float4 per batch element)
2. Compute R in registers
3. For each point: load 3 floats, compute, store 3 floats
This avoids storing/loading the intermediate R matrix, which saves memory bandwidth.
Let me think about block size. I'll use 256 threads per block, which is a good balance. For 2D grid:
- blockIdx.x = batch index
- blockIdx.y = chunk index within batch
Actually, let me also consider using a 1D grid where blocks process (batch, chunk) combinations linearized. This can be more flexible.
Actually 2D is fine. Let me write it.
Also, for correctness, I need to handle the normalization with eps=1e-8. The reference uses `F.normalize(q, dim=-1, eps=1e-8)`.
F.normalize computes: `q / max(norm, eps)`
So if norm < eps, it divides by eps instead. In our kernel:
```cuda
float norm = sqrtf(qw*qw + qx*qx + qy*qy + qz*qz);
float inv_norm = 1.0f / fmaxf(norm, 1e-8f);
```
Actually, looking at PyTorch's implementation more carefully:
```python
q = F.normalize(q, dim=-1, eps=1e-8)
```
This normalizes each row. It computes `q / sqrt(sum(q^2) + eps)`... wait no, that's not right either.
PyTorch's `F.normalize` with `eps=1e-8` computes: `input / max(norm, eps)` where `norm = sqrt(sum(input^2))`.
Actually let me double check. In PyTorch, `normalize` computes:
```python
norm = input.norm(p, dim, keepdim=True)
return input / norm.clamp(min=eps)
```
Wait, that's `clamp(min=eps)`, meaning if norm < eps, use eps. But actually more precisely, the clamp is applied before division, so it becomes `input / max(norm, eps)`.
But wait, clamp(min=eps) clamps the norm to a minimum of eps, so yes, `max(norm, eps)`.
Hmm, but in very old PyTorch versions, `normalize` used `sqrt(sum + eps)`:
```python
norm = input.norm(p, dim, keepdim=True)
return input / norm
```
where `eps` was added inside the norm computation? Let me check...
Actually for `F.normalize`, looking at the documentation: if the norm is less than `eps`, it returns the input divided by `eps`.
Wait, no. Let me look more carefully. The actual implementation of `normalize` is:
```python
def normalize(input, p=2, dim=1, eps=1e-12, out=None):
norm = input.norm(p, dim, keepdim=True)
return input / norm.clamp(min=eps)
```
So yes, it's `input / max(norm, eps)`.
But wait, there's a subtlety. For p=2, `norm = sqrt(sum(x_i^2))`. If norm is 0, then `max(0, eps) = eps`, and the result is `0 / eps = 0`. So a zero quaternion would remain zero, which is fine since it doesn't have a defined rotation anyway.
Wait, actually `0 / eps = 0`. And then `w, x, y, z` would all be 0, and the rotation matrix would be computed from all zeros. With our formulas:
- xx=yy=zz=0, etc.
- m00=1, m01=0, m02=0, ...
- m11=1, m22=1
So a zero quaternion gives the identity matrix, interestingly.
Let me implement the kernel with `fmaxf(norm, 1e-8f)`.
One more thing: `torch.utils.cpp_extension.load_inline`. I need to make sure the setup is correct.
Let me draft the full code:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline
cuda_source = '''
#include <torch/extension.h>
#include <cuda_runtime.h>
__global__ void fused_quat_transform_kernel(
const float* __restrict__ points,
const float* __restrict__ quat,
const float* __restrict__ trans,
float* __restrict__ out,
int B, int N) {
int b = blockIdx.x;
int tid = blockIdx.y * blockDim.x + threadIdx.x;
if (b >= B || tid >= N) return;
float qw = quat[b * 4 + 0];
float qx = quat[b * 4 + 1];
float qy = quat[b * 4 + 2];
float qz = quat[b * 4 + 3];
float norm = sqrtf(qw*qw + qx*qx + qy*qy + qz*qz);
float inv_norm = 1.0f / fmaxf(norm, 1e-8f);
qw *= inv_norm;
qx *= inv_norm;
qy *= inv_norm;
qz *= inv_norm;
float xx = qx * qx;
float yy = qy * qy;
float zz = qz * qz;
float xy = qx * qy;
float xz = qx * qz;
float yz = qy * qz;
float wx = qw * qx;
float wy = qw * qy;
float wz = qw * qz;
float m00 = 1.0f - 2.0f * (yy + zz);
float m01 = 2.0f * (xy - wz);
float m02 = 2.0f * (xz + wy);
float m10 = 2.0f * (xy + wz);
float m11 = 1.0f - 2.0f * (xx + zz);
float m12 = 2.0f * (yz - wx);
float m20 = 2.0f * (xz - wy);
float m21 = 2.0f * (yz + wx);
float m22 = 1.0f - 2.0f * (xx + yy);
int base = (b * N + tid) * 3;
float px = points[base];
float py = points[base + 1];
float pz = points[base + 2];
float tx = trans[b * 3 + 0];
float ty = trans[b * 3 + 1];
float tz = trans[b * 3 + 2];
out[base] = px * m00 + py * m01 + pz * m02 + tx;
out[base + 1] = px * m10 + py * m11 + pz * m12 + ty;
out[base + 2] = px * m20 + py * m21 + pz * m22 + tz;
}
torch::Tensor fused_quat_transform(
torch::Tensor points,
torch::Tensor quat,
torch::Tensor trans) {
auto B = points.size(0);
auto N = points.size(1);
auto out = torch::empty_like(points);
const int threads_per_block = 256;
dim3 block(threads_per_block);
dim3 grid(B, (N + threads_per_block - 1) / threads_per_block);
fused_quat_transform_kernel<<<grid, block>>>(
points.data_ptr<float>(),
quat.data_ptr<float>(),
trans.data_ptr<float>(),
out.data_ptr<float>(),
B, N);
return out;
}
'''
class ModelNew(nn.Module):
def __init__(self):
super().__init__()
def forward(self, points, quat, trans):
return fused_quat_transform(points, quat, trans)
```
Wait, I need to be careful. The `functions=[...]` argument to `load_inline` should list the function names. Also, I can't write my own `PYBIND11_MODULE`. Let me use `load_inline` correctly.
Actually, with `load_inline`, I need:
- `cuda_sources`: the CUDA source code
- `functions`: list of function names to expose
- `extra_cuda_cflags`: optional compilation flags
The CUDA source should only contain the kernel and the wrapper function. `load_inline` auto-generates the pybind11 bindings.
Let me also consider some optimizations:
1. Using `__ldg` for read-only data
2. Processing the quaternion normalization once per batch and storing R in shared memory
3. Vectorized memory access
4. Grid-stride loops for better occupancy
For small B and N, the overhead of computing R per thread is negligible compared to the memory operations. But let me try adding shared memory for R:
```cuda
__shared__ float s_R[9];
__shared__ float s_t[3];
if (threadIdx.x == 0) {
// compute R and t, store in shared memory
}
__syncthreads();
```
This might help a tiny bit for large numbers of threads, but for 256 threads, redoing 20 FMAs per thread is basically free. Let me skip shared memory and keep it simple.
Actually, wait. Let me reconsider the memory access pattern. If each thread does scalar loads of points[base], points[base+1], points[base+2], then consecutive threads in a warp load consecutive floats. But each thread makes 3 separate loads. On H100, these might be coalesced into a few transactions.
If I reinterpret the points tensor as having shape (B*N*3,), consecutive threads load consecutive floats, which is great for L2 cache bandwidth.
Let me think about another optimization: can I use vector types for better bandwidth? float3 doesn't exist as a native CUDA type for guaranteed alignment. But I can define a struct:
```cuda
struct float3 {
float x, y, z;
};
```
And load it as `reinterpret_cast<const float3*>(points)[idx]`. But (B, N, 3) at offset `b*N + tid` gives a float3 starting at byte offset `(b*N + tid) * 12`. This is aligned to 4 bytes but not 16 bytes. float3 requires 12 bytes. For a warp of 32 threads, each loads a 12-byte struct.
On modern GPUs, unaligned vector loads of 12 bytes can still be fairly efficient because the hardware handles misalignment.
Actually, I could try using `float4` and reading 4 floats, then ignoring the 4th. But the alignment requirements for float4 are 16 bytes. Since each point is 12 bytes, only point indices that are multiples of 4/3? No, 12n mod 16 = 0 only when n is multiple of 4. So every 4th point is aligned. We could handle this with a branch, but it's probably not worth it.
Let me also think about `__restrict__` and `__ldg`. Using `const float* __restrict__` helps the compiler optimize. `__ldg` is for read-only data through the texture cache, but on modern GPUs, L1 cache is usually fine.
One more thing: I should check whether `torch::empty_like(points)` preserves the correct strides. For contiguous tensors, which points should be, this is fine.
Let me write the first version and test it.
Actually, I want to add a few compiler flags for better performance:
- `-O3`
- `--use_fast_math` (this might change precision though... the tolerance is 1e-4 for fp32, so fast math should be fine)
Wait, does `--use_fast_math` affect `sqrtf`? Yes, it uses a faster but less precise approximation. For normalize with tolerance 1e-4, it should still be fine. But let me be careful.
Actually, the reference uses `F.normalize` which uses standard division, and standard `sqrt`. If I use `--use_fast_math`, I might get slightly different results. Let me test without it first.
Another idea: for very large N, I could process multiple points per thread. This reduces the relative overhead of quaternion computation per point. Let me use a grid-stride loop:
```cuda
int b = blockIdx.x;
for (int tid = blockIdx.y * blockDim.x + threadIdx.x; tid < N; tid += blockDim.x * gridDim.y) {
// process
}
```
Hmm, but with 2D grid, the stride in the y-dimension is `gridDim.y * blockDim.x`. The issue is that for small N with small gridDim.y, this doesn't do much. For large gridDim.y, it helps with occupancy.
Actually, I think the simplest approach where each block covers a range of points and each thread processes one point is fine for a first attempt.
Wait, let me think about this differently. With 2D grid (B, num_blocks_per_batch), if B is small, we have few SMs occupied. If we instead linearize the problem as a 1D grid over (B * N) elements, and have each block process a contiguous chunk, we get better SM utilization when B is small and N is large.
For B=8, N=512:
- 1D grid with block_size=256: total elements = 4096, num_blocks = 16
- 2D grid with block_size=256: num_blocks = 8 * 2 = 16
Same total blocks.
For B=1, N=100000:
- 1D grid: num_blocks = 391
- 2D grid: num_blocks = 391
Same.
For B=8, N=100000:
- 1D grid: num_blocks = 3125
- 2D grid: num_blocks = 8 * 391 = 3128
Essentially the same. The main difference is scheduling: 1D might distribute blocks more evenly across SMs.
But 2D is conceptually simpler. Let me use 1D with linear indexing instead:
```cuda
int total = B * N;
int tid = blockIdx.x * blockDim.x + threadIdx.x;
int stride = gridDim.x * blockDim.x;
for (int idx = tid; idx < total; idx += stride) {
int b = idx / N;
int n = idx % N;
// process point n in batch b
}
```
This is a grid-stride loop over all (B, N) points. For small total, we might have more blocks than needed, but that's fine. Actually, we should compute gridDim based on total elements:
```cuda
int total_elements = B * N;
int num_blocks = (total_elements + threads_per_block - 1) / threads_per_block;
```
But for total_elements = 4096 and threads_per_block = 256, num_blocks = 16. Still 16 blocks.
The grid-stride loop doesn't help much unless we intentionally launch more blocks than needed. We could launch a fixed number of blocks based on the GPU's SM count. But that's getting complicated.
Let me stick with the straightforward approach. For the H100 with 132 SMs, even 16 blocks means we're using < 15% of SMs. But each block has 256 threads = 8 warps. With 32 resident blocks per SM, we could theoretically have 32 * 132 = 4224 blocks. We only have 16. But each block only processes 256 elements (or 512 if N=512 and we use 2 blocks per batch).
Hmm. For B=8, N=512, total work is 4096 points. Each point requires ~30 FLOPs. Total FLOPs: ~123K. At 67 TFLOPS, that's 1.8 nanoseconds of compute. The kernel will be entirely memory or launch-bound. There's not much we can do. PyTorch will also be memory bound for this size.
But the evaluation might use larger sizes! For B=64, N=65536:
- Total elements = 4,194,304
- Points memory = 64 * 65536 * 3 * 4 = 50.3 MB
- Our kernel reads 50MB and writes 50MB, plus tiny quat/trans
- At 3.35 TB/s, memory time = ~30 microseconds
- PyTorch has to write/read R (64 * 9 * 4 = 2.3KB, negligible)
- But PyTorch does 3 separate ops: normalize, matrix construction, matmul+add
- The matmul would dominate and is compute-heavy for large N
For large N, a fused kernel saves memory bandwidth of intermediate R storage/loading. The matmul in PyTorch is probably optimized with tensor cores for (N, 3) @ (3, 3), which is tiny. It might still be memory bound.
Wait, for batched matmul (B, N, 3) @ (B, 3, 3), each output point reads 3 floats from points and 3 floats from R (which is shared across the batch dimension). So each output point reads 3 + 3/32 (approximately) floats, since R is reused across warp threads.
In our fused kernel, each thread reads 3 floats from points and computes R from quat (4 floats). The quat fits in L1 cache. So fused kernel reads: 3 points + ~0 quat overhead per output point.
PyTorch's approach:
1. Normalize quat: reads 4 floats, writes 4 floats per batch
2. Construct R: reads 4 floats, writes 9 floats per batch
3. Matmul + add: reads 3 points + 9 R floats + 3 trans floats, writes 3 output floats per point
Total memory per point: points read once (3 floats), quat read twice (8 floats amortized), R written and read (18 floats amortized), trans read once (3 floats amortized), output written (3 floats).
Amortized over large N: ~3 + 3 + ~0 + ~0 = 6 floats per point for step 3, plus 18/B per point for R, plus 8/B for quat. For large B and N, this is ~6 floats per point.
Our fused kernel: ~3 reads + 3 writes = 6 floats per point. Similar!
But wait, PyTorch's matmul might not be highly optimized for tiny 3x3 matrices. The overhead of cuBLAS/cuDNN for such small matrices might be significant. For our fused kernel, it's just a simple linear operation, which gives us an advantage on the overhead side.
Let me also think whether we can use warp-level optimizations or tensor cores. Tensor cores for a 1x3 @ 3x3 multiplication per point would be massive overkill and slower due to data movement. Better to just do it in registers.
OK, I think the simple kernel will be competitive. Let me write it and test.
Actually, let me think about one more optimization: unrolling the point processing loop when each thread handles multiple points. But for now, one point per thread with 1D or 2D grid is fine.
Let me write the code.
Wait, I just realized something. The grid-stride loop over a 1D grid is actually very standard and gives good flexibility. Let me do:
```cuda
__global__ void kernel(...) {
int idx = blockIdx.x * blockDim.x + threadIdx.x;
int stride = gridDim.x * blockDim.x;
for (int i = idx; i < B * N; i += stride) {
int b = i / N;
int n = i % N;
// process
}
}
```
The number of blocks can be computed as:
```cuda
int num_blocks = min((B * N + threads_per_block - 1) / threads_per_block, MAX_BLOCKS);
```
But I don't know MAX_BLOCKS. I can just use the standard formula. For large problems, this fills the GPU well. For small problems, occupancy is low but the total work is small.
Actually, for good H100 utilization, we should aim for at least a few thousand active warps. With 256 threads = 8 warps per block, to get 3000 warps we'd need ~375 blocks. But that's only achievable with large B*N. For small problems, it doesn't matter because overhead dominates.
Let me just write the kernel and test it.
One more thought: should I process multiple points per thread to improve the computation-to-memory ratio? For example, each thread could process 4 consecutive points:
```cuda
for (int n = start; n < end; n += 4) {
// load 4 points
// process all with same R
}
```
For 4 points, we load 12 floats and do ~120 FLOPs (30 per point), then store 12 floats. Memory bandwidth = 24*4 = 96 bytes. Arithmetic intensity = 120 / 96 = 1.25 FLOPs/byte. H100 has 67 TFLOPS / 3.35 TB/s = 20 FLOPs/byte for compute-bound. So we're still memory bound. Processing more points per thread doesn't change this fundamentally, but it reduces the relative overhead of quat loads.
Let me just use a simple kernel and see how it performs.
Actually, another approach: since each point transform is a small matrix-vector multiply, we can potentially vectorize the computation across 4 points using float4 or compute multiple output channels in parallel. But this is probably not worth it.
Let me write the code now.
For `torch.utils.cpp_extension.load_inline`, I need to be careful about the API. The function signature should match what PyTorch expects.
Here's the code I plan to submit for testing:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline
CUDA_SOURCE = r"""
#include <torch/extension.h>
#include <cuda_runtime.h>
__global__ void fused_quat_transform_kernel(
const float* __restrict__ points,
const float* __restrict__ quat,
const float* __restrict__ trans,
float* __restrict__ out,
int B, int N) {
int idx = blockIdx.x * blockDim.x + threadIdx.x;
int stride = gridDim.x * blockDim.x;
for (int linear = idx; linear < B * N; linear += stride) {
int b = linear / N;
int n = linear % N;
// Load quaternion
float qw = quat[b * 4 + 0];
float qx = quat[b * 4 + 1];
float qy = quat[b * 4 + 2];
float qz = quat[b * 4 + 3];
// Normalize
float norm = sqrtf(qw*qw + qx*qx + qy*qy + qz*qz);
float inv_norm = 1.0f / fmaxf(norm, 1e-8f);
qw *= inv_norm;
qx *= inv_norm;
qy *= inv_norm;
qz *= inv_norm;
// Compute rotation matrix
float xx = qx * qx;
float yy = qy * qy;
float zz = qz * qz;
float xy = qx * qy;
float xz = qx * qz;
float yz = qy * qz;
float wx = qw * qx;
float wy = qw * qy;
float wz = qw * qz;
float m00 = 1.0f - 2.0f * (yy + zz);
float m01 = 2.0f * (xy - wz);
float m02 = 2.0f * (xz + wy);
float m10 = 2.0f * (xy + wz);
float m11 = 1.0f - 2.0f * (xx + zz);
float m12 = 2.0f * (yz - wx);
float m20 = 2.0f * (xz - wy);
float m21 = 2.0f * (yz + wx);
float m22 = 1.0f - 2.0f * (xx + yy);
int base = (b * N + n) * 3;
float px = points[base];
float py = points[base + 1];
float pz = points[base + 2];
float tx = trans[b * 3 + 0];
float ty = trans[b * 3 + 1];
float tz = trans[b * 3 + 2];
out[base] = px * m00 + py * m01 + pz * m02 + tx;
out[base + 1] = px * m10 + py * m11 + pz * m12 + ty;
out[base + 2] = px * m20 + py * m21 + pz * m22 + tz;
}
}
torch::Tensor fused_quat_transform(
torch::Tensor points,
torch::Tensor quat,
torch::Tensor trans) {
int B = points.size(0);
int N = points.size(1);
auto out = torch::empty_like(points);
const int threads_per_block = 256;
int total_elements = B * N;
int num_blocks = (total_elements + threads_per_block - 1) / threads_per_block;
// Cap num_blocks to a reasonable value for occupancy
// But don't reduce below what's needed for correctness
num_blocks = max(num_blocks, 1);
fused_quat_transform_kernel<<<num_blocks, threads_per_block>>>(
points.data_ptr<float>(),
quat.data_ptr<float>(),
trans.data_ptr<float>(),
out.data_ptr<float>(),
B, N);
return out;
}
"""
fused_ext = load_inline(
name="fused_quat_transform",
cpp_sources="",
cuda_sources=CUDA_SOURCE,
functions=["fused_quat_transform"],
extra_cuda_cflags=["-O3", "--use_fast_math"],
verbose=False,
)
class ModelNew(nn.Module):
def __init__(self):
super().__init__()
def forward(
self,
points: torch.Tensor,
quat: torch.Tensor,
trans: torch.Tensor,
) -> torch.Tensor:
return fused_ext.fused_quat_transform(points, quat, trans)
```
Wait, `--use_fast_math` might cause precision issues. Let me not use it for the first test.
Also, I want to double check the division logic. `fmaxf(norm, 1e-8f)` gives the max of norm and 1e-8. If norm < 1e-8, we divide by 1e-8, same as PyTorch's `clamp(min=eps)`.
Wait, I should also think about whether PyTorch's normalize might have different behavior. Let me check:
```python
q = F.normalize(q, dim=-1, eps=1e-8)
```
Looking at PyTorch source more carefully:
For `F.normalize(input, p=2, dim=1, eps=1e-12, out=None)`:
```python
if out is None:
denom = input.norm(p, dim, keepdim=True).clamp_min(eps)
return input / denom
```
So `denom = max(norm, eps)` and result is `input / denom`. This is exactly what I implemented.
One potential issue: for very small norms, dividing by `eps` gives small values. For example, if q = (1e-9, 0, 0, 0), norm = 1e-9, denom = 1e-8, result = (0.1, 0, 0, 0). In my kernel, same thing. Good.
Actually, there might be a more subtle issue: `sqrtf` on very small values. But `qw*qw + ...` for subnormal numbers might underflow to 0. If all components are very small, norm = 0, and we divide by eps. Result would be the input scaled by 1/eps. If input is all subnormals and we multiply by 1e8, we might get normal or denormal numbers. This could differ from PyTorch if PyTorch does the computation in double precision or handles subnormals differently.
For practical purposes with random inputs, this shouldn't be an issue. But let me be aware of it.
Also, the compiler might reorder floating point operations slightly differently from PyTorch. The tolerance is 1e-4, so small differences should be OK.
Hmm, actually I should also think about warp divergence. Each thread in a warp for the same batch element computes the same R. But because all threads in a warp are in the same batch element (since N is typically much larger than warp size), they all follow the same path. Good.
Wait, in my 1D grid formulation, consecutive threads in a warp might be in different batch elements if N is small. For example, with N=3, thread 0 processes batch 0 point 0, thread 1 processes batch 0 point 1, thread 2 processes batch 0 point 2, thread 3 processes batch 1 point 0. Then threads 0-2 load quat for batch 0, thread 3 loads quat for batch 1. This causes warp divergence!
To avoid this, I could use a 2D grid where blockIdx.x is batch and threads in a block only process one batch element. But then for small N, blocks are small.
Actually, let me think about whether warp divergence matters. If threads in a warp load different quaternions, they diverge on the quaternion values but follow the same code path. Since there's no branching based on quaternion values (except the normalization clamp, which all threads do), the only divergence is in memory addresses. On modern GPUs, divergent loads are handled fine - they just issue separate load instructions. There's no actual control flow divergence.
So the 1D grid is fine! No control flow divergence at all, just different data for different threads.
Wait, but what about the for loop? With `stride = gridDim.x * blockDim.x`, and total elements = B*N, if total elements is not a multiple of stride, some threads might have different numbers of iterations. Actually no, the grid-stride loop handles that naturally: some threads might do one extra iteration, but since the condition is the same for all threads in the warp (they check `linear < B * N` where `linear` differs by `stride`), they might diverge if `stride` causes some to be done and others not.
But `stride` is the same for all threads. The last warp might have some threads with `linear >= B*N` and others with `linear < B*N`. In the first iteration, threads with `linear >= B*N` don't enter the loop. The others do. Wait, but in CUDA, all threads in a warp execute the same instructions. If some threads need to loop and others don't, there IS control flow divergence because inactive threads are masked off.
For a grid-stride loop where total elements isn't a multiple of blockDim.x, the last block has some inactive threads. But in my loop, even the last block might have all threads active in the first few iterations. Let me trace:
Say B=1, N=260, threads_per_block=256, num_blocks=2.
stride = 2 * 256 = 512.
Thread 0: idx=0, linear=0, 512, ... Only linear=0 is < 260. So 1 iteration.
Thread 255: idx=255, linear=255, 767, ... Only linear=255 is < 260. So 1 iteration.
Thread 256: idx=256, linear=256, 768, ... linear=256 >= 260? No, 256 < 260. So 1 iteration.
Thread 257: idx=257, linear=257. 257 < 260. 1 iteration.
Thread 259: idx=259, linear=259. 259 < 260. 1 iteration.
Thread 260: idx=260, linear=260. 260 >= 260. 0 iterations.
So threads 0-259 do 1 iteration, threads 260-511 do 0 iterations. All threads in the first warp (0-31) do 1 iteration. But what about the second warp in the second block? Threads 256-287 are in warp 8. Threads 256-259 do 1 iteration, threads 260-287 do 0 iterations.
But wait, a warp executes in lockstep. Warp 8 has threads 256-287. In the first check of the for loop, threads 256-259 enter, threads 260-287 don't. The warp executes the loop body with a partial mask, which is fine but slightly less efficient. After the body, the loop condition is checked again. For threads 256-259, `linear += stride` gives 768, which is >= 260, so they exit too. The remaining threads in warp 8 also exit. So only one partially-active iteration per warp at the boundary.
This is standard and completely fine. The performance impact is negligible for large N.
For small N, like N=3 with 256 threads, most threads do 0 iterations. Very inefficient, but for tiny N all approaches are inefficient.
Let me also consider: should I use a dynamic block count? For H100, to maximize occupancy, a common heuristic is to launch enough blocks to fill the GPU. But for small problems, this doesn't help.
I'll stick with the 1D grid approach. It's clean and handles all sizes.
Let me finalize the code:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline
CUDA_SOURCE = r"""
#include <torch/extension.h>
#include <cuda_runtime.h>
__global__ void fused_quat_transform_kernel(
const float* __restrict__ points,
const float* __restrict__ quat,
const float* __restrict__ trans,
float* __restrict__ out,
int B, int N) {
int idx = blockIdx.x * blockDim.x + threadIdx.x;
int stride = gridDim.x * blockDim.x;
for (int linear = idx; linear < B * N; linear += stride) {
int b = linear / N;
int n = linear % N;
float qw = quat[b * 4 + 0];
float qx = quat[b * 4 + 1];
float qy = quat[b * 4 + 2];
float qz = quat[b * 4 + 3];
float norm = sqrtf(qw*qw + qx*qx + qy*qy + qz*qz);
float inv_norm = 1.0f / fmaxf(norm, 1e-8f);
qw *= inv_norm;
qx *= inv_norm;
qy *= inv_norm;
qz *= inv_norm;
float xx = qx * qx;
float yy = qy * qy;
float zz = qz * qz;
float xy = qx * qy;
float xz = qx * qz;
float yz = qy * qz;
float wx = qw * qx;
float wy = qw * qy;
float wz = qw * qz;
float m00 = 1.0f - 2.0f * (yy + zz);
float m01 = 2.0f * (xy - wz);
float m02 = 2.0f * (xz + wy);
float m10 = 2.0f * (xy + wz);
float m11 = 1.0f - 2.0f * (xx + zz);
float m12 = 2.0f * (yz - wx);
float m20 = 2.0f * (xz - wy);
float m21 = 2.0f * (yz + wx);
float m22 = 1.0f - 2.0f * (xx + yy);
int base = (b * N + n) * 3;
float px = points[base];
float py = points[base + 1];
float pz = points[base + 2];
float tx = trans[b * 3 + 0];
float ty = trans[b * 3 + 1];
float tz = trans[b * 3 + 2];
out[base] = px * m00 + py * m01 + pz * m02 + tx;
out[base + 1] = px * m10 + py * m11 + pz * m12 + ty;
out[base + 2] = px * m20 + py * m21 + pz * m22 + tz;
}
}
torch::Tensor fused_quat_transform(
torch::Tensor points,
torch::Tensor quat,
torch::Tensor trans) {
int B = points.size(0);
int N = points.size(1);
auto out = torch::empty_like(points);
const int threads_per_block = 256;
int total_elements = B * N;
int num_blocks = (total_elements + threads_per_block - 1) / threads_per_block;
num_blocks = max(num_blocks, 1);
fused_quat_transform_kernel<<<num_blocks, threads_per_block>>>(
points.data_ptr<float>(),
quat.data_ptr<float>(),
trans.data_ptr<float>(),
out.data_ptr<float>(),
B, N);
return out;
}
"""
fused_ext = load_inline(
name="fused_quat_transform",
cpp_sources="",
cuda_sources=CUDA_SOURCE,
functions=["fused_quat_transform"],
extra_cuda_cflags=["-O3"],
verbose=False,
)
class ModelNew(nn.Module):
def __init__(self):
super().__init__()
def forward(
self,
points: torch.Tensor,
quat: torch.Tensor,
trans: torch.Tensor,
) -> torch.Tensor:
return fused_ext.fused_quat_transform(points.contiguous(), quat.contiguous(), trans.contiguous())
```
I added `.contiguous()` to ensure inputs are contiguous. The outputs should also be contiguous if inputs are, but `empty_like` preserves strides. For contiguous inputs from `get_inputs()`, this is fine.
Wait, `load_inline` should happen at module load time, outside of the class definition. That's what I have. Good.
One concern: if the module is loaded multiple times or if there's a name collision with `fused_quat_transform`, it might fail. But `load_inline` uses a cache based on the name and source.
Actually, for evaluation, the script is executed with `exec()`, so `load_inline` will run each time. The cache should prevent recompilation if the source doesn't change.
Let me test this kernel.
One more thought: should I use `__ldg` for read-only loads? On modern GPUs, `__ldg` uses the read-only data cache (texture cache). For point data that is only read once, `__ldg` might help slightly. But for data that is accessed with strides or patterns that don't perfectly coalesce, the regular L1 cache is usually fine.
For the quat data, each thread in a warp might read the same quat (if they're in the same batch). `__ldg` could help here because it has broadcast capability. But L1 also broadcasts. The benefit is minimal.
Let me just use `const float* __restrict__` and let the compiler optimize.
Actually, I'll add `__ldg` for the quat and trans loads since they're broadcast within warps:
```cuda
float qw = __ldg(&quat[b * 4 + 0]);
...
```
But on some architectures, `__ldg` only works with `const float* __restrict__`. It should be fine.
Let me not overcomplicate it and just test the basic version first. We can optimize later if needed.
Oh wait, I need to be careful with `max`. In C++ `max` from `<algorithm>` works with integers, but CUDA also has `max` for floats. With `1` and `num_blocks` both being ints, `max(num_blocks, 1)` should work. But to be safe, I could write `num_blocks > 1 ? num_blocks : 1` or just rely on the fact that `(total + 255) / 256` is always >= 0, and = 0 only if total < 0, which can't happen. So `num_blocks` is always >= 1 for non-negative totals. Since B and N are sizes, they're non-negative. So `num_blocks` is at least 1 even when B=0 or N=0 (well, when B*N=0, num_blocks could be 0...).
Actually, when B=0 or N=0, `total_elements = 0`, and `(0 + 255) / 256 = 0`. Then we launch with 0 blocks, which might be problematic. But the reference kernel might also have issues with empty tensors. Let me add `num_blocks = max(num_blocks, 1)` to be safe. Or better, handle the empty case by returning early:
```cuda
if (B == 0 || N == 0) return out;
```
In the wrapper function.
Let me add that.
Also, I should double-check if the division `linear / N` and `linear % N` are expensive. CUDA has integer division instructions that are relatively slow (tens of cycles). For a grid-stride loop where each iteration does 2 integer divisions, this might be noticeable.
Alternative: I can restructure to avoid division in the inner loop. For example, using a 2D grid where blockIdx.x maps to batch and the y/threadIdx dimension maps to points within batch. But for a 1D grid with grid-stride loop over all elements, we'd need division.
I could precompute the batch and point index for the first iteration, then increment batch when n reaches N:
```cuda
int b = idx / N;
int n = idx % N;
for (int linear = idx; linear < B * N; linear += stride) {
// process (b, n)
n += stride;
b += n / N;
n = n % N;
}
```
But this still has division in the loop (n / N and n % N), though fewer calls. Actually, since stride might be large relative to N, `n` could increment by a large amount, and we'd need to divide by N each time.
Another optimization: if stride is a multiple of N, then the batch index never changes within a thread's loop. But stride = gridDim.x * blockDim.x, and we don't control it to be a multiple of N.
For large N (say N=100000), the division cost is amortized over many FLOPs. For small N (N=512), the arithmetic is so light that division might be a significant fraction.
Actually, let me count: for each point, we have ~10 operations for norm, ~20 for R construction, ~15 for the matmul+add. Total ~45 FLOPs. Integer division is ~20-30 cycles on H100. At ~2 GHz, that's ~10-15 ns. In that time, a float ALU can do ~20-30 FLOPs. So division is roughly comparable to ~20-30 FLOPs. For 45 FLOPs total, division could be 30-70% overhead per point. That's significant!
For small N, this matters more. For large N, memory latency dominates.
To avoid division, I can use a 2D grid:
```cuda
int b = blockIdx.x;
int n = blockIdx.y * blockDim.x + threadIdx.x;
if (b < B && n < N) {
// process (b, n)
}
```
No integer division! Let me use this approach.
For grid configuration:
- Block size: 256 threads
- Grid: (B, (N + 255) / 256)
But what if B is large and N is small? Then we have many blocks in x but few in y. Each block in x processes one batch element with up to 256 threads. If N=100, then only 100 threads per block are active, and the remaining 156 are idle. This is fine.
What if B is small and N is large? We have few blocks in x but many in y. The warp scheduler distributes them across SMs.
What if B is small and N is very small? E.g., B=1, N=10. We have 1 block with 10 active threads. Very low occupancy, but very little work.
The 2D approach avoids integer division and is conceptually clean. Let me use it.
One more issue: for large N, say N=1000000, we'd have grid.y = 3907 blocks. grid.x = B. For B=64, total blocks = 64 * 3907 = 250,048. That's a lot of blocks but well within CUDA limits (65535 per dimension, but total blocks can be huge).
Actually, CUDA limits gridDim.y to 65535 in older architectures, but modern GPUs support much larger. Let me check: on H100, the maximum grid dimension is 2^31-1 for x, y, z. So we're fine.
Wait, but each block in the 2D grid can only cover one batch element. If N is not a multiple of 256, the last block for each batch has some inactive threads. This is fine.
Let me rewrite with 2D grid:
```cuda
__global__ void fused_quat_transform_kernel(
const float* __restrict__ points,
const float* __restrict__ quat,
const float* __restrict__ trans,
float* __restrict__ out,
int B, int N) {
int b = blockIdx.x;
int n = blockIdx.y * blockDim.x + threadIdx.x;
if (b >= B || n >= N) return;
// ... same computation ...
int base = (b * N + n) * 3;
// ... load, compute, store ...
}
```
And launch:
```cuda
dim3 grid(B, (N + threads_per_block - 1) / threads_per_block);
dim3 block(threads_per_block);
```
Wait, but what if B*N is huge and we want to use grid-stride to process multiple points per thread? That's incompatible with the 2D approach. For the first version, I'll just have each thread process one point.
For larger problems, we could process multiple points per thread by adding a loop:
```cuda
for (int n = start; n < N; n += blockDim.x * gridDim.y) {
if (n < N) {
// process
}
}
```
This is a grid-stride within the point dimension. Each block still handles one batch element, but threads loop over chunks of points. This doesn't add integer division overhead beyond the initial computation.
Let me use this for better occupancy with large N:
```cuda
__global__ void fused_quat_transform_kernel(...) {
int b = blockIdx.x;
int n_start = blockIdx.y * blockDim.x + threadIdx.x;
int n_stride = blockDim.x * gridDim.y;
if (b >= B) return;
// Load and normalize quaternion once
float qw = quat[b * 4 + 0];
float qx = quat[b * 4 + 1];
float qy = quat[b * 4 + 2];
float qz = quat[b * 4 + 3];
float norm = sqrtf(qw*qw + qx*qx + qy*qy + qz*qz);
float inv_norm = 1.0f / fmaxf(norm, 1e-8f);
qw *= inv_norm;
qx *= inv_norm;
qy *= inv_norm;
qz *= inv_norm;
float xx = qx * qx;
float yy = qy * qy;
float zz = qz * qz;
float xy = qx * qy;
float xz = qx * qz;
float yz = qy * qz;
float wx = qw * qx;
float wy = qw * qy;
float wz = qw * qz;
float m00 = 1.0f - 2.0f * (yy + zz);
float m01 = 2.0f * (xy - wz);
float m02 = 2.0f * (xz + wy);
float m10 = 2.0f * (xy + wz);
float m11 = 1.0f - 2.0f * (xx + zz);
float m12 = 2.0f * (yz - wx);
float m20 = 2.0f * (xz - wy);
float m21 = 2.0f * (yz + wx);
float m22 = 1.0f - 2.0f * (xx + yy);
float tx = trans[b * 3 + 0];
float ty = trans[b * 3 + 1];
float tz = trans[b * 3 + 2];
for (int n = n_start; n < N; n += n_stride) {
int base = (b * N + n) * 3;
float px = points[base];
float py = points[base + 1];
float pz = points[base + 2];
out[base] = px * m00 + py * m01 + pz * m02 + tx;
out[base + 1] = px * m10 + py * m11 + pz * m12 + ty;
out[base + 2] = px * m20 + py * m21 + pz * m22 + tz;
}
}
```
For the grid configuration:
- Block size: 256
- Grid: (B, num_point_blocks)
- num_point_blocks = (N + 255) / 256
With this, each batch element has `num_point_blocks` blocks. Each thread in a block processes points at stride `256 * num_point_blocks = 256 * ceil(N/256) >= N`. So each thread processes exactly 1 or 0 points! The loop doesn't help unless we deliberately launch fewer point blocks.
Ah, that's right. If `gridDim.y = ceil(N/256)`, then `n_stride = 256 * ceil(N/256) >= N`, so most threads only process 1 point. To have the loop be useful, we'd need `gridDim.y < ceil(N/256)`.
We could fix `gridDim.y` to a small constant and let the loop handle the rest:
```cuda
const int point_blocks = 4; // or some heuristic
dim3 grid(B, point_blocks);
```
Then `n_stride = 256 * 4 = 1024`. For N=100000, each thread processes ~100 points. For N=512, each thread processes ~1 point (since 512/1024 < 1, but some threads process 1).
But with fixed `point_blocks`, for small N we'd have many blocks where most threads are inactive. For example, with N=100 and point_blocks=4, we'd launch 4 blocks per batch, each with 256 threads, but only threads 0-24 in the first block are active. 768/1024 threads are idle. That's bad.
A better heuristic:
```cuda
int point_blocks = min((N + threads_per_block - 1) / threads_per_block, some_max);
```
Where `some_max` limits the blocks per batch. For example, `some_max = 32`. If N=1000000, we'd use 32 blocks per batch instead of 3907. Each thread then processes 1000000/(256*32) = 122 points.
But having more blocks isn't necessarily bad. In fact, having more blocks allows better SM distribution. The loop just adds overhead. For this kernel, since each point is independent and memory bandwidth is the bottleneck, having more blocks is usually better because the hardware can schedule them well.
So maybe the simple version without the loop is best. Let me just keep it simple:
```cuda
int b = blockIdx.x;
int n = blockIdx.y * blockDim.x + threadIdx.x;
if (b >= B || n >= N) return;
// compute R and t
// load point n
// compute and store
```
This is clean, has no division, and no loop overhead. Good.
For the reference case B=8, N=512:
- Grid: (8, 2)
- Total blocks: 16
- Each block: 256 threads
For B=64, N=65536:
- Grid: (64, 256)
- Total blocks: 16384
- This fills the H100 well.
For B=1, N=10000:
- Grid: (1, 40)
- Total blocks: 40
- Only 1 SM worth of work? Actually, 40 blocks can be distributed across SMs, so we use ~40/32 = 1-2 SMs at a time. But each block has 256 threads. Still somewhat occupancy-limited.
If we want better occupancy for small B but large N, we could use the 1D approach with grid-stride loop. But then we have integer division.
Actually, for small B and large N, we can use a mixed approach: each block processes a chunk of N points, and we have enough blocks. With grid=(1, 40) for B=1, N=10000, block_size=256:
- Total threads: 40 * 256 = 10240
- Warps: 320
- The H100 has 132 SMs, each with 2048 threads max = 64 warps max. Total theoretical warps: 132 * 64 = 8448. So 320/8448 = 3.8% theoretical occupancy.
But wait, at 32 blocks per SM max, with 256 threads each block, that's 8192 threads per SM, exceeding the 2048 limit. So max is 8 blocks of 256 threads = 2048 threads = 64 warps per SM. With 40 total blocks, we can have at most 40/132 ≈ 0.3 blocks per SM on average. So most SMs are idle. But each active SM has 64 warps, and with memory-bound kernels, each warp is often waiting for memory. Having more SMs active would help.
With 1D grid and 10240 total elements, using 10240/256 = 40 blocks, same issue.
To get more blocks, we need to launch more than the minimum. For example, we could launch `num_blocks = max(40, 1000)` or something. But then many blocks would be very small (only 1 warp or less active). This is fine on modern GPUs - the scheduler handles it.
For the 2D approach, we can't have more than ceil(N/256) blocks in the y direction without having completely empty blocks. So for B=1, N=10000, we're stuck with ~40 blocks.
An alternative: use a 1D grid with enough blocks to fill the GPU, and each block processes a contiguous chunk of points. To avoid division, we can pass the point offset as part of the linear index calculation.
Actually, if we restructure:
```cuda
int linear = blockIdx.x * blockDim.x + threadIdx.x;
int b = linear / N;
```
Division is unavoidable in a fully linearized mapping unless we encode batch in block index and point in thread index. But that's what the 2D approach does.
Let me think about another approach: have each block process a tile of points, and the tiles are assigned round-robin across all blocks.
```cuda
int total_point_tiles = (B * N + tile_size - 1) / tile_size;
int tile_id = blockIdx.x;
int start = tile_id * tile_size;
for (int i = threadIdx.x; i < tile_size && (start + i) < B * N; i += blockDim.x) {
int linear = start + i;
int b = linear / N;
int n = linear % N;
// process
}
```
Still has division. But each warp processes a contiguous range of `linear` values, so `b` is the same for most of the warp if `tile_size` is small compared to N, or if the tile is aligned.
Hmm, what if we process points in a batched manner where each block handles one batch, but we launch many blocks per batch as before? With B=1, we can't get more than ceil(N/256) blocks.
For better occupancy with small B, we could process multiple batches per block:
```cuda
int batches_per_block = max(1, target_batches / gridDim.x);
```
But this is getting complicated. For the evaluation, I suspect they'll use reasonable sizes where either B or N is large enough to provide sufficient parallelism.
Let me just go with the 2D approach. It's the cleanest and avoids integer division.
Actually, one more optimization: we could use warp-level primitives to share the quaternion computation. Since all threads in a warp for the same batch compute the same R, we could have one thread compute it and broadcast to others using `__shfl_sync`. But the computation is 20 FLOPs, which is cheaper than a shuffle instruction latency. Not worth it.
Similarly for shared memory: storing R in shared memory would require a syncthreads and a load. The computation is cheap enough that recomputing is fine.
OK, I'll finalize with the 2D approach. Let me also think about compiler optimizations. I should use `-O3`. And maybe `--use_fast_math` to speed up `sqrtf`. But `fast_math` changes `sqrtf` to an approximate version. With atol=1e-4, it should be fine. Let me test without it first, then try with it.
Actually, one more thing. Does the reference code use double precision for any intermediate computations? Let me look at the reference:
```python
q = F.normalize(q, dim=-1, eps=1e-8)
```
In PyTorch, `F.normalize` for float32 inputs does float32 computation. So we match.
OK, let me write the kernel code string properly and test it.
Wait, I need to make sure all the tensors are float32. The problem says "fp32 precision." So I can just use float.
Let me also consider: should I check that the tensors are contiguous? `empty_like` preserves the layout of `points`. If `points` is not contiguous but has strides that make it logically (B, N, 3), `empty_like` creates an output with the same strides, and our kernel assumes contiguous layout.
To be safe, I could make everything contiguous. But the reference `Model` also uses `torch.matmul(points, R.transpose(...))`, and if `points` is non-contiguous, PyTorch handles it correctly. So should I support non-contiguous?
For simplicity, let me handle this in the wrapper:
```python
def forward(self, points, quat, trans):
# Ensure contiguous for our kernel
points = points.contiguous()
quat = quat.contiguous()
trans = trans.contiguous()
return fused_ext.fused_quat_transform(points, quat, trans)
```
This is safe and doesn't affect the common case where inputs are already contiguous.
But wait, if the reference works with non-contiguous inputs, our kernel should too. The specification says "Accept the same constructor arguments as Model" and "same signature". It doesn't explicitly say we need to support every possible input layout... but correctness-wise, if the evaluation uses non-contiguous inputs, we'd fail.
Let me look at `get_inputs()`:
```python
def get_inputs():
points = torch.randn(batch_size, num_points, 3)
quat = torch.randn(batch_size, 4)
trans = torch.randn(batch_size, 3)
return [points, quat, trans]
```
These are contiguous by default. So `.contiguous()` is harmless.
OK, let me write the final code and test it via the compile tool.
One more thing: I need to make sure the CUDA source doesn't have any C++11/14/17 features that might cause issues. But `torch::Tensor` and `torch::empty_like` are standard PyTorch C++ API.
For `load_inline`, the `cpp_sources` can be empty since all our code is in `cuda_sources`. The wrapper function needs to be in `cuda_sources` with `extern "C"` or just as a regular C++ function. `load_inline` will generate the pybind11 bindings.
Let me write it now.
Actually, for `load_inline` with `cuda_sources`, I need to be careful about the function naming. The function should be callable from Python. `load_inline` auto-generates the binding.
Let me use the API correctly:
```python
fused_ext = load_inline(
name="fused_quat_transform",
cpp_sources="", # optional, can be omitted
cuda_sources=CUDA_SOURCE,
functions=["fused_quat_transform"],
extra_cuda_cflags=["-O3"],
verbose=False,
)
```
This should work.
Actually, looking at the PyTorch docs, `cpp_sources` is for C++ source code. If we're only using CUDA, we can provide `cuda_sources`. But sometimes `load_inline` expects at least one of `cpp_sources` or `cuda_sources`.
OK, let me just provide both - `cpp_sources=""` might work, or I can put the wrapper in `cpp_sources` and the kernel in `cuda_sources`.
For `load_inline`, the kernel typically goes in `cuda_sources` and the host wrapper can also go in `cuda_sources` (it will be compiled as CUDA code). Actually, any code in `cuda_sources` is compiled with nvcc. Code in `cpp_sources` is compiled with the C++ compiler.
Let me put everything in `cuda_sources`:
```python
fused_ext = load_inline(
name="fused_quat_transform",
cuda_sources=CUDA_SOURCE,
functions=["fused_quat_transform"],
extra_cuda_cflags=["-O3"],
verbose=False,
)
```
This should work. Let me test compile.
Actually, I just realized `load_inline` might cache the compiled extension based on the source code. If the source changes but the name stays the same, it might use the old cached version. To avoid this during development, I should change the name or delete the cache. But since we're just submitting one version, this is fine.
Hmm, I should also verify the dtype handling. What if the inputs are not float32? The reference `Model` uses the default dtype. `get_inputs()` uses `torch.randn` which creates float32 tensors by default (unless the default dtype is changed). Our kernel specifically expects float. We should add a dtype check or cast:
```python
if points.dtype != torch.float32:
points = points.float()
```
Actually, the specification says "produce numerically equivalent outputs at fp32 precision." If the inputs are float64, we'd need to cast them. But `get_inputs()` creates float32. Let me just handle it by checking/casting in the forward method. Or, I can write a templated kernel...
For simplicity, let me just assume float32 and add a check:
```python
def forward(self, points, quat, trans):
return fused_ext.fused_quat_transform(
points.contiguous(), quat.contiguous(), trans.contiguous()
)
```
If inputs are not float, it will fail. But the test inputs are float.
Actually wait, I should check: does `fused_quat_transform` need to handle different tensor devices? The reference runs on the default device. Our kernel runs on CUDA. Let me make sure points/quat/trans are on CUDA. But `get_inputs()` doesn't specify device. In the evaluation, they probably move them to CUDA before calling forward.
Let me look at the reference Model. It uses PyTorch ops which work on whatever device the tensors are on. Our CUDA kernel requires CUDA tensors. But for the evaluation, they're probably testing on CUDA.
One more consideration: does `torch.utils.cpp_extension.load_inline` work inside the `exec()` call? The evaluation executes the module with `exec()`. `load_inline` compiles at import time. This should be fine, but compilation might take a few seconds. The timing evaluation probably doesn't include import time, only `forward()` time.
OK, I'm ready to write the code and compile it.
Let me draft it carefully:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline
CUDA_SOURCE = r"""
#include <torch/extension.h>
#include <cuda_runtime.h>
__global__ void fused_quat_transform_kernel(
const float* __restrict__ points,
const float* __restrict__ quat,
const float* __restrict__ trans,
float* __restrict__ out,
int B, int N) {
int b = blockIdx.x;
int n = blockIdx.y * blockDim.x + threadIdx.x;
if (b >= B || n >= N) return;
float qw = quat[b * 4 + 0];
float qx = quat[b * 4 + 1];
float qy = quat[b * 4 + 2];
float qz = quat[b * 4 + 3];
float norm = sqrtf(qw * qw + qx * qx + qy * qy + qz * qz);
float inv_norm = 1.0f / fmaxf(norm, 1e-8f);
qw *= inv_norm;
qx *= inv_norm;
qy *= inv_norm;
qz *= inv_norm;
float xx = qx * qx;
float yy = qy * qy;
float zz = qz * qz;
float xy = qx * qy;
float xz = qx * qz;
float yz = qy * qz;
float wx = qw * qx;
float wy = qw * qy;
float wz = qw * qz;
float m00 = 1.0f - 2.0f * (yy + zz);
float m01 = 2.0f * (xy - wz);
float m02 = 2.0f * (xz + wy);
float m10 = 2.0f * (xy + wz);
float m11 = 1.0f - 2.0f * (xx + zz);
float m12 = 2.0f * (yz - wx);
float m20 = 2.0f * (xz - wy);
float m21 = 2.0f * (yz + wx);
float m22 = 1.0f - 2.0f * (xx + yy);
float tx = trans[b * 3 + 0];
float ty = trans[b * 3 + 1];
float tz = trans[b * 3 + 2];
int base = (b * N + n) * 3;
float px = points[base];
float py = points[base + 1];
float pz = points[base + 2];
out[base] = px * m00 + py * m01 + pz * m02 + tx;
out[base + 1] = px * m10 + py * m11 + pz * m12 + ty;
out[base + 2] = px * m20 + py * m21 + pz * m22 + tz;
}
torch::Tensor fused_quat_transform(
torch::Tensor points,
torch::Tensor quat,
torch::Tensor trans) {
int B = points.size(0);
int N = points.size(1);
auto out = torch::empty_like(points);
if (B == 0 || N == 0) {
return out;
}
const int threads_per_block = 256;
int blocks_y = (N + threads_per_block - 1) / threads_per_block;
dim3 grid(B, blocks_y);
dim3 block(threads_per_block);
fused_quat_transform_kernel<<<grid, block>>>(
points.data_ptr<float>(),
quat.data_ptr<float>(),
trans.data_ptr<float>(),
out.data_ptr<float>(),
B, N);
return out;
}
"""
_fused_ext = load_inline(
name="fused_quat_transform_v1",
cuda_sources=CUDA_SOURCE,
functions=["fused_quat_transform"],
extra_cuda_cflags=["-O3"],
verbose=False,
)
class ModelNew(nn.Module):
def __init__(self):
super().__init__()
def forward(
self,
points: torch.Tensor,
quat: torch.Tensor,
trans: torch.Tensor,
) -> torch.Tensor:
points = points.contiguous()
quat = quat.contiguous()
trans = trans.contiguous()
return _fused_ext.fused_quat_transform(points, quat, trans)
```
I changed the name to `fused_quat_transform_v1` to avoid cache issues if I iterate.
Let me compile this first.