Enterprise AI Analysis
Space Filling Curves Revolutionize Matrix Multiplication
Unlocking SOTA performance with minimal code and platform-oblivious efficiency.
Executive Impact Summary
This work introduces SFC-CA GEMM, a novel algorithm leveraging Space Filling Curves (SFC) to dramatically improve General Matrix Multiplication (GEMM) performance. By converting multi-dimensional coordinates to 1D, SFC-CA ensures inherent data locality and minimizes communication across memory hierarchies. It achieves state-of-the-art results on multiple CPU platforms, outperforming vendor libraries by up to 2x, with a remarkably compact code footprint (~30 LOC). The approach is platform-oblivious and shape-oblivious, addressing the 'glass jaws' performance issues seen in current vendor-optimized libraries.
0x
Performance Boost
0
LOC Code Compactness
0-15%
Roofline Proximity
Deep Analysis & Enterprise Applications
Select a topic to dive deeper, then explore the specific findings from the research, rebuilt as interactive, enterprise-focused modules.
SFC Fundamentals
CA Algorithms
Performance Results
Space Filling Curves (SFC) convert multi-dimensional coordinates (e.g., 2D) into a single dimension (1D), preserving locality. This work uses Generalized Hilbert Curves, which extend classical Hilbert curves to rectangles of arbitrary sizes. The recursive nature of SFC allows for locality-aware partitioning of computation space, ensuring that adjacent 1D SFC indices correspond to neighboring boxes in the 2D space, thus obviating the need for explicit cache-blocking and loop reordering.
Enterprise Process Flow
Multi-D Coordinates
→
SFC Mapping
→
1D Index Order
→
Locality Preservation
→
Efficient Data Access
1.34x
Less L2 Misses (SFC-CA 2D vs oneDNN)
Communication-Avoiding (CA) 2.5D GEMM algorithms replicate input tensors to provably minimize communication/data-movement. This extends classic 2D algorithms by organizing processors in a 3D logical grid, reducing bandwidth costs by a factor of √c where c is the replication factor. Our SFC-based partitioning seamlessly integrates these CA algorithms, enabling effective 2.5D and 3D GEMM decompositions with minimal code changes.
| Feature | SFC-CA GEMM | Vendor-Optimized (oneDNN) |
|---|---|---|
| Performance | Up to 2x faster (geom. mean) | Performance 'glass jaws' |
| Communication Cost | Provably minimized (critical path) | Suboptimal, high data movement |
| Tuning Effort | Platform-oblivious, shape-oblivious | Expensive, platform-specific auto-tuning needed |
| Code Compactness | ~30 LOC | Complex, intricate indexing |
2.9x
Less L2 Misses (SFC-CA 2.5D vs oneDNN for M=4096, N=8192, K=4096)
SFC-CA GEMM consistently tracks the roofline closely (typically within 5-15%) across various CPU platforms (x86 and Arm/Aarch64). Experimental results demonstrate substantial performance improvements, with geometric mean speedups of up to 2x over vendor-optimized libraries like oneDNN and ACL. The algorithm dynamically adapts to different matrix shapes and aspect ratios, mitigating performance 'glass jaws' often seen in highly tuned but brittle vendor implementations.
EMR (Intel Xeon Emerald Rapids) Performance
On a 64-core Intel Xeon Emerald Rapids platform with AMX, SFC-CA GEMM shows a 1.4x geometric mean speedup over oneDNN for BF16 GEMM. It consistently tracks the roofline, demonstrating superior cache locality and communication efficiency across diverse matrix configurations, especially for shapes with high operational intensity.
GNR (Intel Xeon Granite Rapids) Performance
On a 128-core Intel Xeon Granite Rapids platform with AMX, SFC-CA GEMM achieves an even more significant 2x geometric mean speedup over oneDNN. This highlights the algorithm's scalability and efficiency on newer, higher-core-count architectures, maintaining tight roofline proximity.
ZEN5 (AMD EPYC) Performance
On a 96-core AMD EPYC (ZEN5) server, SFC-CA GEMM delivers a 1.4x geometric mean speedup over oneDNN using AVX512 BF16 FMA instructions. The consistency across different x86 architectures underscores the portability and robustness of the SFC-CA approach.
GVT4 (Arm Graviton 4) Performance
On a 96-core Arm Graviton 4 server, SFC-CA GEMM outperforms ARM Compute Library (ACL) by 1.4x geometric mean speedup using BFMMLA instructions. This demonstrates the cross-architecture portability and effectiveness, even when compared to highly optimized Arm-specific libraries.
Advanced ROI Calculator
Estimate the potential performance and efficiency gains for your enterprise workloads with our AI solution.
Annual Savings Potential
$0
Hours Reclaimed Annually
0
Strategic Implementation Roadmap
Our proven phased approach ensures a smooth transition and rapid value realization.
Discovery & Planning
Thorough assessment of existing infrastructure and identification of key optimization targets for GEMM workloads. Define clear performance metrics and integration strategy.
SFC-CA Integration
Seamless integration of SFC-CA GEMM algorithm with existing tensor computation frameworks. Leverage TPPs for architecture-specific code generation while maintaining high-level abstraction.
Performance Validation
Extensive benchmarking and validation against current vendor libraries and target roofline performance. Fine-tune K_layers replication factor for optimal communication avoidance.
Scaling & Continuous Optimization
Scale across various CPU platforms and future architectures. Implement continuous monitoring and optimization for evolving workloads.
Ready to Transform Your AI Performance?
Schedule a personalized consultation to explore how Space Filling Curves can unlock new levels of efficiency for your enterprise.
Questions? Let's Talk.
Our experts are available to discuss your specific challenges and how SFC-CA can provide a competitive edge.
Book a CallReady to Get Started?
Book Your Free Consultation.
Let's Discuss Your AI Strategy!
Lets Discuss Your Needs
Select a Date
Sun
Mon
Tue
Wed
Thu
Fri
Sat