Matrix-Multiply-Assist (MMA) in Accelerators

• Matrix-Multiply-Assist (MMA) is a paradigm that optimizes general matrix multiplication by replacing traditional multipliers with efficient addition chains and on-chip memory operations.
• It leverages specialized hardware instructions, low-precision arithmetic, and domain-specific data layouts to significantly improve throughput, energy efficiency, and latency.
• MMA is integrated into modern processors like IBM POWER10, NVIDIA GPUs, and AMD accelerators, enabling scalable deep learning and scientific computing applications.

Matrix-Multiply-Assist (MMA) refers to a class of accelerator architectures, hardware instructions, and algorithms that improve the throughput and efficiency of matrix multiplication, a fundamental computational kernel in machine learning and scientific computing. Under the MMA paradigm, the core task of $\mathtt{GEMM}$—general matrix–matrix multiplication—is accelerated via architectural features that (i) replace general-purpose multiplication hardware with specialized low-cost operations, (ii) encode the multiply-accumulate pattern in compact micro-operations, and (iii) allow exploitation of low-precision arithmetic and domain-specific data layouts for maximal hardware utilization. MMA is prominent in both classical and novel chip architectures, including addition-only accelerators, IBM POWER10, NVIDIA and AMD GPUs, and emerging compiler and software abstractions (Cussen et al., 2023, Moreira et al., 2021, Kuzma et al., 2023, Choi et al., 2022, Xie et al., 14 Nov 2025).

1. MMA Algorithms and the Addition-Only Paradigm

Traditional hardware matrix multiplication relies on resource-intensive multipliers coupled with adders for each $a_{ik} b_{kj}$ term. Matrix-Multiply-Assist (MMA) algorithms, such as described in "Matrix Multiplication Using Only Addition" (Cussen et al., 2023), eliminate scalar multipliers altogether, substituting each multiplication with a constant-depth sequence of additions and on-chip copy (pointer-follow) operations. The core insight leverages structural sparsity and value redundancy in large, fixed-precision vectors, enabling encoding of scalar products as preprocessed addition chains:

• Each column vector $V = [v_1, \dots, v_n]^\top$ is preprocessed: align (right-shift trailing zeros), sort, deduplicate (unique residues $S$), and difference encode ($D$).
• For each scalar $c$, compute $[c d_1, \dots, c d_m]$ using shift-add (“Russian-Peasant” or recursive) methods; reconstruct $c S$ by prefix sum; retrieve each $c v_i$ via indexed lookup and apply bit shift $h_i$ if needed.

This design reduces the number of addition operations per multiply on random data to under one on average (e.g., for 24-bit mantissas, $C(n,2^{24})/n \approx 1$ at $n \gtrsim 10^5$), offering substantial area, energy, and latency savings as multipliers are completely eliminated from the architecture (Cussen et al., 2023).

2. MMA Architectures and Instruction Set Facilities

MMA is implemented at the hardware level in several architectures. A prominent example is the MMA facility in Power ISA v3.1, deployed in IBM POWER10 processors (Moreira et al., 2021, Kuzma et al., 2023). Key characteristics include:

• Accumulator and Vector-Scalar Register Organization: 64 × 128-bit VSRs and 8 × 512-bit accumulators, each holding a $4 \times 4$ or $4 \times 2$ (fp64) matrix.
• Execution Pipeline: Two parallel pipelines (MU2, MU3) instantiate a matrix math engine that delivers up to two MMA instructions per cycle, sustained.
• Instruction Catalog: Support for integer (int4/int8/int16) and floating-point (fp16/bf16/fp32/fp64) outer-product updates, with row/col/product masks for fine-grained control.
• Mathematical Semantics: Facility for 4×4, 4×2, and masked rank-$k$ outer-products, with direct mapping to C/C++ compiler built-ins.

MMA instructions implement:

$$A_{ij} \leftarrow \sum_{k=0}^{K-1} X_i^{(k)} Y_j^{(k)} + A_{ij}$$

for $4\times 4$ blocks, with variants for different arithmetic types and masking strategies (Moreira et al., 2021).

On NVIDIA and AMD architectures, MMA units known as Tensor Cores and Matrix Cores perform block matrix multiply-accumulate operations in a single instruction (e.g., WMMA on NVIDIA: $\mathtt{mma.m16n16k16.f32.f16.f16.f32}$) with support for a variety of low-precision (FP16, INT4, FP8, BF16) formats and systematic accumulator promotion (e.g., INT4 $\to$ INT32) (Xie et al., 14 Nov 2025, Choi et al., 2022).

3. MMA Software Integration: Compilers and Intrinsics

Compiler-level support allows software frameworks to exploit MMA instructions directly, bypassing the need for hand-crafted microkernels:

• LLVM provides the @llvm.matrix.multiply intrinsic, parameterized by operand types and tile sizes, abstracting the interface for matrix blocks. Lowerings translate this high-level operation into hardware-specific microkernels or vectorized IR (Kuzma et al., 2023).
• In C/C++, built-in functions such as __builtin_mma_xvbf16ger2pp and related intrinsics map one-to-one to the underlying hardware instructions for POWER10 MMA (Moreira et al., 2021).
• For machine learning workloads, frameworks automatically partition computation into matrix tiles matching hardware-specific MMA block sizes, handling register allocation and accumulator management. Optimizations cover tiling, packing, and data layout transformations to maximize memory locality and cache efficiency (Kuzma et al., 2023).

The compiler-driven approach enables retargeting: introducing new intrinsic lowerings for accelerators (e.g., Intel AMX, Arm ME) and adjusting packing layouts allows performance portability across hardware generations (Kuzma et al., 2023).

4. MMA in Deep Learning Accelerators

MMAs are central to the performance of deep neural network training and inference, especially on GPUs:

• Each MMA operation computes a blockwise $\mathtt{D = A \times B + C}$, with variations in operand shapes and floating-point formats across architectures (e.g., HMMA.16816 for $16 \times 8 \times 16$ per warp).
• MMA units support mixed-precision computation (TensorFloat-32, FP16, BF16, FP8, INT4), and can deliver petascale FLOPS for GEMM workloads in DNN layers (Xie et al., 14 Nov 2025).
• Precision and rounding modes differ by design. AMD CDNA3 implements “fused-dot-round-down-add” (FDRDA) with asymmetric rounding; NVIDIA Tensor Cores use fused-dot-add (FDA) with precise exponent alignment and round-to-zero/truncate, affecting accumulation error and DNN training stability (Xie et al., 14 Nov 2025).
• In reduced-precision MMA (FP16/INT4), data reuse and packing overheads become dominant. Scheduling must account for operand grouping constraints, blocking/packing choices, and register file layout. Automatic search for optimal scheduling delivers substantial runtime gains in convolutional kernels (up to $3.85\times$ over baseline, as in (Choi et al., 2022)).

5. Performance, Complexity, and Energy Analysis

MMA architectures achieve substantial efficiency improvements:

• The addition-only MMA eliminates multipliers, shrinking area per PE by $3$–$6\times$ and reducing energy per multiply-accumulate accordingly. Overall, addition-only PEs provide $>3\times$ throughput per unit area compared to conventional designs (Cussen et al., 2023).
• POWER10 MMA achieves $2.6\times$ VSX throughput at $+8\%$ power or $5.8\times$ POWER9 at $-24\%$ power, with area overhead $\approx 2\%$ for the Matrix Math Engine (Moreira et al., 2021). Table below summarizes selected speedups (from (Kuzma et al., 2023)):

Matrix size	vs. OpenBLAS	vs. Eigen	MMA/VSX
16×16	+10%	+160%	–
32×32	+10%	+83%	–
2048×2048	0.96×	+83%	2.6×

• On NVIDIA GPUs, Tensor Core throughput scales to $\sim$1 PFLOPS (FP16) on Ada Lovelace and RTX Blackwell; MMA microkernel scheduling matches or exceeds hand-tuned code using compiler-level optimization (Choi et al., 2022).

6. Limitations, Scalability, and Interoperability

MMA algorithms and architectures, while highly efficient, are subject to several constraints:

• Worst-case behavior: Addition-only MMA shows overhead when vectors are adversarial or $n=O(b)$, but for $n\gg b$ (typical ML regime), savings dominate (Cussen et al., 2023).
• Precision and error control: Floating-point MMAs exhibit rounding, subnormal, and overflow behavior unique to each hardware vendor. Lower accumulation precision in FP8 may destabilize LLM training (Xie et al., 14 Nov 2025). Subnormal flushing imposes precision loss; asymmetric rounding induces bias.
• Integration with existing frameworks: MMA blocks are plug-compatible with tile-based and fast-matrix methods (tiling, Strassen), and apply to both square and non-square, dense or sparse, signed/int/fp and mixed-precision models (Cussen et al., 2023).
• Software and compiler limitations: For large $mr/kr/nr$ tile sizes, intrinsic unrolling is a bottleneck; complex data shuffling and register pressure can cause backend spilling (Kuzma et al., 2023).

A plausible implication is that increased standardization of accumulator formats, rounding modes, and compiler abstractions would further ease cross-platform reproducibility and algorithmic stability in MMA-enabled DNN workloads (Xie et al., 14 Nov 2025).

7. Extensions and Future Directions

MMA continues to be extended in breadth and depth:

• Research prototypes (e.g., addition-only accelerators) show that further minimizing individual compute unit complexity is possible for low- and medium-precision ML kernels (Cussen et al., 2023).
• Compiler frameworks aim to incorporate auto-tuning of tile sizes and apply MMA abstractions to broader BLAS kernels (e.g., TRSM, SYR2K) (Kuzma et al., 2023).
• MMA behavioral models (e.g., MMA-Sim) expose subtle arithmetic differences between hardware platforms, enabling quantitative error tracking and architectural validation (Xie et al., 14 Nov 2025).
• Increasing support for mixed-precision, masked operations, sparse formats, and non-square tiles extends MMA applicability to a diverse set of domains including scientific computing, signal processing, and advanced ML models.

MMA is positioned as a foundational component of high-performance computing platforms, driving both hardware and software innovation for efficient matrix computation at scale.