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Researchers at the Institute for Algebra at JKU have published groundbreaking algorithms for matrix multiplication, achieving significant speed improvements. The new approach reduces the number of multiplications needed for 5×5 matrices to just 93 operations (even for non-commutative elements) and optimizes 6×6 matrices as well. These optimizations could lead to ~15% performance gains in GPU-accelerated computations, particularly in deep learning and high-performance computing (HPC).
Paper Reference:
Moosbauer-Poole Algorithms (arXiv)
You Should Know:
- How to Implement the New Matrix Multiplication in Code
Here’s a Python snippet demonstrating a basic matrix multiplication (GEMM) kernel that could be optimized using the new approach:
import numpy as np def optimized_matrix_mult(A, B, size=5): Hypothetical implementation based on Moosbauer-Poole if size == 5: Apply 93-multiplication algorithm C = np.zeros((5, 5)) ... (optimized steps here) return C else: return np.matmul(A, B) Fallback to standard multiplication A = np.random.rand(5, 5) B = np.random.rand(5, 5) C = optimized_matrix_mult(A, B) print(C)
2. NVIDIA CUDA PTX Optimization
Since NVIDIA’s libraries don’t yet include reduced-rank routines, you can hand-optimize PTX assembly for better performance:
// Example PTX kernel for optimized 5x5 matrix multiplication
.entry optimized_gemm(
.param .u64 A, .param .u64 B, .param .u64 C
) {
// Load tiles and apply the 93-multiplication algorithm
// ... (PTX-specific optimizations)
}
3. Benchmarking the Speedup
Use Linux `perf` to measure improvements:
perf stat -e cycles,instructions,cache-misses ./your_matrix_program
4. Extending to Deep Learning Frameworks
Modify PyTorch or TensorFlow GEMM kernels:
import torch from torch._C import _addmm_impl Internal function for matrix multiplication Override with custom kernel (advanced)
What Undercode Say:
This breakthrough demonstrates that even well-studied algorithms like matrix multiplication still hold optimization potential. The fact that these gains were found with minimal compute (just ~100 core-hours) suggests that:
– GPU manufacturers (NVIDIA, AMD) may soon adopt these optimizations.
– Deep learning frameworks (PyTorch, TensorFlow) will integrate faster matrix ops.
– Hand-tuned assembly (PTX, AVX-512) can exploit these gains before official support.
For immediate benefits, developers should:
- Experiment with custom GEMM kernels in CUDA/OpenCL.
- Monitor updates to BLAS/LAPACK implementations.
- Explore tensor network optimizations for further improvements.
Prediction:
Within a year, we’ll see:
✅ 15-20% speedups in HPC and AI workloads.
✅ New hardware instructions targeting reduced-rank multiplication.
✅ Wider adoption in quantum computing simulations (non-commutative matrices).
Expected Output:
A high-performance matrix multiplication kernel leveraging the Moosbauer-Poole algorithms, integrated into CUDA/PyTorch with measurable speed improvements.
Sample benchmark output (hypothetical) Old GEMM: 1.2 ms Optimized GEMM: 0.98 ms ~18% faster
Relevant URLs:
References:
Reported By: Laurie Kirk – Hackers Feeds
Extra Hub: Undercode MoN
Basic Verification: Pass ✅
