In Nvidia’s Ampere microarchitecture, Tensor Cores are specialized hardware units designed to accelerate matrix operations, which are fundamental to many machine learning and scientific computing tasks. The primary mathematical functions performed by Tensor Cores can be expressed algebraically as follows:
Matrix Multiplication and Accumulation (MMA)
The core operation of Tensor Cores is the fused multiply-add (FMA) operation applied to matrices. Specifically, for matrices \(A\), \(B\), and \(C\), the operation is:
\[ D = \alpha \cdot (A \times B) + \beta \cdot C \]
Here, \(\alpha\) and \(\beta\) are scaling factors (typically 1 in most machine learning applications), and \(A \times B\) represents matrix multiplication. The resulting matrix \(D\) is the output of the operation. This can be broken down into element-wise operations:
\[ D_{ij} = \alpha \sum_{k} A_{ik} B_{kj} + \beta C_{ij} \]
where \(D_{ij}\) is the element in the \(i\)-th row and \(j\)-th column of matrix \(D\).
Mixed-Precision Arithmetic
Tensor Cores in the Ampere architecture are designed to handle mixed-precision computations efficiently. They support operations where inputs are in half-precision (FP16) or integer (INT8) formats, and the accumulation is done in single-precision (FP32). This can be expressed as:
\[ D_{ij} = \sum_{k} \text{FP32}(\text{FP16}(A_{ik}) \cdot \text{FP16}(B_{kj})) + C_{ij} \]
or for integer arithmetic:
\[ D_{ij} = \sum_{k} \text{INT32}(\text{int8}(A_{ik}) \cdot \text{INT8}(B_{kj})) + C_{ij} \]
Tensor Contractions
Tensor Cores also perform tensor contractions, which generalize matrix multiplications to higher dimensions. For example, a contraction of a 3D tensor \(A\) and a 3D tensor \(B\) can be written as:
\[ D_{ijkl} = \sum_{m} A_{ijm} B_{mkl} \]
This operation is crucial in various applications, such as higher-order neural networks and certain types of scientific simulations.
In summary, the main mathematical functions performed by Tensor Cores in Nvidia’s Ampere microarchitecture can be expressed algebraically as variations of matrix multiplications and tensor contractions, with a focus on mixed-precision arithmetic to enhance computational efficiency and performance.