---
title: Matrix-Multiply-Assist (MMA) in Accelerators
url: https://www.emergentmind.com/topics/matrix-multiply-assist-mma
type: topic
---

# Matrix-Multiply-Assist (MMA) in Accelerators

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 [2307.01415, 2104.03142, 2305.18236, 2202.06819, 2511.10909].

## 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" [2307.01415], 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 [2307.01415].

## 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 [2104.03142, 2305.18236]. 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 [2104.03142].

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) [2511.10909, 2202.06819].

## 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 [2305.18236].
- 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 [2104.03142].
- 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 [2305.18236].

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 [2305.18236].

## 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 [2511.10909].
- 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 [2511.10909].
- 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 [2202.06819]).

## 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 [2307.01415].
- 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 [2104.03142]. Table below summarizes selected speedups (from [2305.18236]):

| 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 [2202.06819].

## 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 [2307.01415].
- **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 [2511.10909]. 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 [2307.01415].
- **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 [2305.18236].

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 [2511.10909].

## 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 [2307.01415].
- Compiler frameworks aim to incorporate auto-tuning of tile sizes and apply MMA abstractions to broader BLAS kernels (e.g., TRSM, SYR2K) [2305.18236].
- MMA behavioral models (e.g., MMA-Sim) expose subtle arithmetic differences between hardware platforms, enabling quantitative error tracking and architectural validation [2511.10909].
- 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.

Source: https://www.emergentmind.com/topics/matrix-multiply-assist-mma