You write custom CUDA kernels to replace the pytorch operators in the given GeGLU architecture to get speedups.

You have complete freedom to choose the set of operators you want to replace. You may make the decision to replace some operators with custom CUDA kernels and leave others unchanged. You may replace multiple operators with custom implementations, consider operator fusion opportunities (combining multiple operators into a single kernel, for example, combining chunk+gelu+elementwise_mul), or algorithmic changes (such as optimized memory access patterns). You are only limited by your imagination.

This code implements total correlation + ELU activation with CUDA optimizations:

Shared memory staging - Loads x and target into shared memory for reuse across multiple reduction phases.

Two-stage computation - First computes means, then uses them for covariance and variance calculations.

Triple parallel reduction - Warp shuffle for three sums: covariance, var_x, var_t.

Three shared memory buffers - Separate buffers for covariance, var_x, and var_t to avoid bank conflicts.

Broadcast means - Stores computed means in shared memory for all threads to access.

Numerical stability - Adds 1e-6 to denominator for safe division in correlation calculation.

ELU activation - Computes Exponential Linear Unit: max(0,x) + min(0,exp(x)-1).

Grid-stride loop - Threads process multiple elements for load balancing.

CUDA math functions - Uses sqrtf() and expf() for hardware acceleration.

Memory coalescing - Contiguous tensor access patterns.

Batch parallelism - One CUDA block per input row with dynamic shared memory allocation.



Here's an example to show you the syntax of inline embedding custom CUDA operators in torch: The example given architecture is:
import torch
import torch.nn as nn
import torch.nn.functional as F


class Model(nn.Module):
    def __init__(self, target):
        super(Model, self).__init__()
        self.target = nn.Parameter(target)

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        mean_x = x.mean(dim=-1, keepdim=True)
        mean_t = self.target.mean(dim=-1, keepdim=True)

        xm = x - mean_x
        tm = self.target - mean_t

        cov = torch.sum(xm * tm, dim=-1)
        sx = torch.sqrt(torch.sum(xm * xm, dim=-1))
        st = torch.sqrt(torch.sum(tm * tm, dim=-1))

        correlation = cov / (sx * st + 1e-6)

        return F.elu(correlation)


batch_size = 128
input_dim = 1024


def get_inputs():
    x = torch.randn(batch_size, input_dim)
    return [x]


def get_init_inputs():
    target = torch.randn(input_dim)
    return [target]