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

Tensor.numel(): Method to get total number of elements

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

CUDA Math Functions: logf() for logarithmic computations

Parallel Reduction: Tree-based reduction using shared memory

Shared Memory: Using __shared__ for inter-thread communication

Atomic Operations: atomicAdd for thread-safe global updates

Grid-Stride Loops: Efficient memory access pattern

Mathematical Components
Itakura-Saito Distance: Spectral distance measure for signals/spectra

Ratio Computation: y/(x+eps) ratio calculation

Logarithmic Term: log(ratio+eps) component

Three-Term Formula: ratio - log(ratio) - 1 formulation

Numerical Stability: Epsilon (eps) to prevent division by zero and log(0)

Signal Processing/Spectral Analysis Components
Spectral Distance: Originally designed for power spectra comparison

Scale Invariance: Property of Itakura-Saito distance

Non-Negative Inputs: Typically used with power spectra (non-negative values)

Element-Wise Computation: Independent computation across frequency bins

Optimization Techniques
Shared Memory Reduction: Parallel tree reduction within thread blocks

Grid-Stride Loops: Efficient handling of arbitrary tensor sizes

Numerical Stability: Dual epsilon usage for division and log operations

Fused Computation: Complete distance calculation in single kernel

Element-Wise Parallelism: Massive parallelism across all tensor elements

Performance Features
Massive Parallelization: GPU acceleration for distance computation

Memory Efficiency: Shared memory for intermediate reduction results

Atomic Accumulation: Safe parallel sum across thread blocks

Scalable Design: Efficient for any tensor shape/size

Mean Normalization: Final division by total number of elements

Unique Implementation Aspects
Dual Epsilon Protection: Prevents both division by zero and log(0)

Ratio-Based Computation: Core of Itakura-Saito formulation

Element-Wise Metric: Unlike matrix-based distances, operates element-wise

Signal Processing Focus: Specialized for spectral/power distribution comparison

Scale Invariant: Important property preserved in implementation

Applications & Use Cases
Speech Processing: Originally for speech spectrum comparison

Audio Signal Analysis: Power spectrum distance measurement

Non-Negative Matrix Factorization: Common divergence measure in NMF

Spectral Data: Suitable for any non-negative spectral/power data

Numerical Considerations
Non-Negative Inputs: Expects non-negative values (typical for spectra)

Epsilon Selection: Small but non-zero to ensure numerical stability

Ratio Stability: Protected against both numerator and denominator extremes

Mean Computation: Averages across all elements (not batch mean)




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, eps=1e-8):
        super(Model, self).__init__()
        self.eps = eps

    def forward(self, input, target):
        ratio = target / (input + self.eps)
        loss = ratio - torch.log(ratio + self.eps) - 1.0
        return loss.mean()

batch_size = 32
input_dim = 1024

def get_inputs():
    input = torch.abs(torch.randn(batch_size, input_dim, requires_grad=True)) + 0.1
    target = torch.abs(torch.randn(batch_size, input_dim)) + 0.1
    return [input, target]

def get_init_inputs():
    return [1e-8]