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 CUDA kernel implements optimized Variation of Information (VI) calculation with:

Memory Optimization:

Fixed 32×32 histogram in shared memory (4KB)

Vectorized memory access using float4 for 4x bandwidth

Shared memory for marginals, reduction buffers, and min/max values

Parallelization Strategy:

One block per batch sample with 256 threads

Warp-level reduction for min/max/sum operations

Parallel histogram binning with atomic operations

Concurrent computation of MI, H(X), and H(Y)

Computational Optimization:

Efficient min/max reduction using warp shuffles

Vectorized range calculation for normalization

Double precision for entropy summation

Epsilon stabilization for logarithmic operations

Fast math compilation flags

Work Distribution:

Threads process 4-element vectors via float4

Residual elements handled by thread 0

Parallel marginal probability computation

Separate entropy calculations for X and Y

Final VI calculation: H(X) + H(Y) - 2×MI

The implementation efficiently computes VI through shared memory histogram construction and parallel entropy calculations.

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, bins=32, eps=1e-12):
        super().__init__()
        self.bins = bins
        self.eps = eps

    def forward(self, x: torch.Tensor, y: torch.Tensor) -> torch.Tensor:
        b, n = x.shape

        min_x = x.min(dim=1, keepdim=True)[0]
        max_x = x.max(dim=1, keepdim=True)[0]
        min_y = y.min(dim=1, keepdim=True)[0]
        max_y = y.max(dim=1, keepdim=True)[0]

        x_norm = (x - min_x) / (max_x - min_x + 1e-6)
        y_norm = (y - min_y) / (max_y - min_y + 1e-6)

        x_bin = torch.clamp((x_norm * self.bins).long(), 0, self.bins - 1)
        y_bin = torch.clamp((y_norm * self.bins).long(), 0, self.bins - 1)

        joint_idx = x_bin * self.bins + y_bin
        hist = torch.zeros(b, self.bins * self.bins, device=x.device, dtype=torch.float32)
        ones = torch.ones_like(joint_idx, dtype=torch.float32)
        hist.scatter_add_(1, joint_idx, ones)

        p_xy = hist.view(b, self.bins, self.bins) / n

        p_x = p_xy.sum(dim=2)  # (B, bins)
        p_y = p_xy.sum(dim=1)  # (B, bins)

        h_x = -torch.sum(p_x * torch.log(p_x + self.eps), dim=1)
        h_y = -torch.sum(p_y * torch.log(p_y + self.eps), dim=1)

        p_x_p_y = torch.bmm(p_x.unsqueeze(2), p_y.unsqueeze(1))
        mi = torch.sum(p_xy * torch.log((p_xy + self.eps) / (p_x_p_y + self.eps)), dim=(1, 2))

        return h_x + h_y - 2 * mi


batch_size = 128
feature_dim = 512


def get_inputs():
    x = torch.randn(batch_size, feature_dim, dtype=torch.float32)
    y = torch.randn(batch_size, feature_dim, dtype=torch.float32)
    return [x, y]


def get_init_inputs():
    return [32]