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.

Technologies Used in This Code
Core Libraries & Frameworks
PyTorch: Deep learning framework

CUDA: NVIDIA's parallel computing platform for GPU acceleration

C++: For high-performance kernel implementation

PyTorch Specific Components
torch.nn.Module: Base class for neural network modules

torch.utils.cpp_extension.load_inline: For inline compilation of CUDA/C++ extensions

PyTorch Tensors: Multi-dimensional arrays with automatic differentiation

CUDA/C++ Implementation Details
CUDA Kernels: Custom GPU kernel (mahalanobis_kernel)

CUDA Math Functions: sqrtf() for square root computation

Parallel Reduction: Multi-level reduction with warp shuffles

Shared Memory: Using __shared__ for difference vector storage

Warp-Level Primitives: __shfl_down_sync() for warp reduction

Block-Level Parallelism: One CUDA block per batch element

Mathematical Components
Mahalanobis Distance: Statistical distance accounting for covariance

Quadratic Form: (x-μ)ᵀΣ⁻¹(x-μ) computation

Matrix-Vector Product: Efficient computation of quadratic form

Square Root: Final distance as sqrt(quadratic form)

Numerical Stability: Small epsilon (1e-6f) added before sqrt

Memory & Parallelism Patterns
Per-Batch Block Assignment: One CUDA block processes one batch element

Shared Memory Vector: Stores (x - mean) difference vector

Two-Level Reduction: Warp-level then block-level reduction

Parallel Matrix Processing: Threads process covariance matrix elements in parallel

Optimization Techniques
Warp Shuffle Operations: Efficient warp-level reduction without shared memory

Shared Memory Reuse: Difference vector stored once, used many times

Grid-Stride Loops: Threads process multiple matrix elements

Hierarchical Reduction: Two-level reduction for better parallelism

Numerical Safety: Epsilon prevents sqrt(0) issues

Performance Features
Massive Parallelization: GPU acceleration for distance computation

Memory Efficiency: Shared memory for frequently accessed data

Warp-Level Optimization: Leverages GPU warp architecture

Coalesced Memory Access: Sequential access patterns for covariance matrix

Batch Independence: Parallel processing of batch elements

Unique Implementation Aspects
Matrix-Based Distance: Unlike other divergences, uses covariance matrix

Two-Input Reduction: Warp shuffle and shared memory combined reduction

Quadratic Form Computation: Efficient parallel computation of xᵀAx

Statistical Normalization: Accounts for feature correlations via covariance

Multivariate Distance: Suitable for multivariate data distributions

Statistical/Machine Learning Applications
Anomaly Detection: Mahalanobis distance for outlier detection

Feature Correlation: Accounts for correlations between features

Covariance-Aware: Unlike Euclidean distance, considers feature relationships

Multivariate Analysis: Suitable for high-dimensional data

Numerical Considerations
Inverse Covariance Input: Requires pre-computed inverse covariance matrix

Positive Definite: Assumes covariance matrix is positive definite

Dimensionality: Performance scales with dimension squared (matrix elements)

Batch Processing: Efficient for multiple distance computations





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__()

    def forward(self, x, mean, inv_covariance):
        diff = x - mean
        temp = torch.mm(diff, inv_covariance)
        dist_sq = torch.sum(temp * diff, dim=1)
        dist = torch.sqrt(dist_sq + 1e-6)
        return dist.mean()

batch_size = 32
dim = 64

def get_inputs():
    x = torch.randn(batch_size, dim, requires_grad=True)
    mean = torch.randn(dim)
    # Generate positive definite matrix
    temp = torch.randn(dim, dim)
    inv_cov = torch.mm(temp, temp.t()) + torch.eye(dim)
    return [x, mean, inv_cov]

def get_init_inputs():
    return []