You write custom CUDA kernels to replace the PyTorch operators in the given EvoNorm 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 normalization+affine_transform+nonlinear_gating), or algorithmic changes (such as optimized memory access patterns). You are only limited by your imagination.

This document presents a CUDA-accelerated PyTorch module for computing Pearson correlation coefficients between two input tensors in batches. The implementation features a highly optimized GPU kernel that leverages several advanced techniques for maximum performance.
Key technical features include:
Memory Access Optimization: Uses vectorized float4 loads to maximize memory bandwidth utilization and employs Instruction-Level Parallelism (ILP=4) to hide memory latency.
Hierarchical Reduction: Implements a two-stage reduction strategy with warp-level reduction using __shfl_down_syncfollowed by block-level reduction via shared memory.
Dynamic Kernel Configuration: Automatically determines optimal kernel configuration based on GPU specifications (SM count) and problem size, splitting work across multiple blocks to maximize occupancy.
Numerical Stability: Incorporates epsilon term to prevent division by zero when computing the final correlation coefficient.
The kernel computes five statistical moments (sum_x, sum_y, sum_xx, sum_yy, sum_xy) in a single pass, then calculates the Pearson correlation using the formula:
cov_xy / sqrt(var_x * var_y)where cov_xy = sum_xy - D*mean_x*mean_yand var_x = sum_xx - D*mean_x*mean_x.
The module handles 4D tensors (NCHW format) and is particularly efficient for large channel dimensions where the vectorized approach provides significant speedups over naive implementations.

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

N, C, H, W = 32, 64, 56, 56
EPS = 1e-8

class PearsonCorrelation(nn.Module):
    def __init__(self, eps=1e-8):
        super().__init__()
        self.eps = eps

    def forward(self, x, y):
        x_flat = x.view(x.size(0), -1)
        y_flat = y.view(y.size(0), -1)

        x_mean = x_flat.mean(dim=1, keepdim=True)
        y_mean = y_flat.mean(dim=1, keepdim=True)

        x_centered = x_flat - x_mean
        y_centered = y_flat - y_mean

        cov = (x_centered * y_centered).sum(dim=1)
        x_var = (x_centered ** 2).sum(dim=1)
        y_var = (y_centered ** 2).sum(dim=1)

        denom = torch.sqrt(x_var * y_var)
        return cov / (denom + self.eps)

class Model(nn.Module):
    def __init__(self):
        super().__init__()
        self.op = PearsonCorrelation(EPS)

    def forward(self, x, y):
        return self.op(x, y)

def get_inputs():
    x = torch.randn(N, C, H, W, dtype=torch.float32)
    y = torch.randn(N, C, H, W, dtype=torch.float32)
    return [x, y]

def get_init_inputs():
    return []