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 Sinkhorn (Hungarian/OT) loss with CUDA optimizations:

Two-stage kernel design - Separate kernels for cost matrix computation and Sinkhorn iteration.

Cost matrix parallelism - Each thread computes one element of the N×N pairwise distance matrix.

Shared memory Sinkhorn solver - Stores dual potentials (f, g) in shared memory for fast iterative updates.

Log-domain Sinkhorn - Uses log-sum-exp with max subtraction for numerical stability in exponentiation.

Batch parallelism in solver - Each thread processes one row/column in alternating Sinkhorn updates.

Vectorized distance computation - Efficient Euclidean distance calculation.

Warp reduction - Uses warp shuffle for final loss accumulation.

Entropy-regularized optimal transport - Solves Sinkhorn iterations for differentiable assignment.

Fixed-point iteration - Alternates between row and column normalization updates (5 iterations).

Efficient memory layout - Shared memory allocated for dual vectors and reduction buffer.

Regularization parameter - Uses ε=0.1 for entropy regularization strength.



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


class Model(nn.Module):
    def __init__(self):
        super(Model, self).__init__()
        self.epsilon = 0.1
        self.num_iters = 5

    def forward(self, x: torch.Tensor, target: torch.Tensor) -> torch.Tensor:
        x_sq = torch.sum(x ** 2, dim=1, keepdim=True)
        t_sq = torch.sum(target ** 2, dim=1, keepdim=True)
        dist_sq = x_sq + t_sq.t() - 2 * torch.matmul(x, target.t())

        C = dist_sq
        M = -C / self.epsilon

        f = torch.zeros(x.size(0), 1, device=x.device)
        g = torch.zeros(1, x.size(0), device=x.device)

        for _ in range(self.num_iters):
            f = -torch.logsumexp(M + g, dim=1, keepdim=True)
            g = -torch.logsumexp(M + f, dim=0, keepdim=True)

        log_P = f + M + g
        P = torch.exp(log_P)

        loss = torch.sum(P * C) / x.size(0)
        return loss


batch_size = 128
input_dim = 1024


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


def get_init_inputs():
    return []