竹杖芒鞋轻胜马,一蓑烟雨任平生

快速矩阵乘法

矩阵乘法

矩阵乘法是高性能计算以及深度学习的基石之一,矩阵乘法的优化一直是相关业界的关注重点。
矩阵乘法的定义非常简单,定义见:wiki

最简单的矩阵乘法实现

最简单的矩阵乘法实现,时间复杂度为O^3。

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static void mm_generate(float* matA,float* matB,float* matC,const int M,const int N,const int K,const int strideA,const int strideB,const int strideC)
{
for (int i = 0; i < M;i++)
{
for (int j = 0; j < N;j++)
{
float sum = 0.0f;
for (int k = 0; k < K;k++)
{
sum += matA[i*strideA + k] * matB[k*strideB + j];
}
matC[i*strideC + j] = sum;
}
}
}

转化为分块计算

矩阵转换为分块实现,时间复杂度仍然是O^3。

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static void mm_split(float* matA, float* matB, float* matC, const int M, const int N, const int K, const int strideA, const int strideB, const int strideC)
{
memset(matC, 0, M*strideC*sizeof(float));
//C11 = A11xB11 + A12XB21
for (int i = 0; i < M/2; i++)
{
for (int j = 0; j < N/2; j++)
{
float sum;
//A11XB11
sum = 0.0f;
for (int k = 0; k < K/2; k++)
{
sum += matA[i*strideA + k] * matB[k*strideB + j];
}
matC[i*strideC + j] += sum;
//A12XB21
sum = 0.0f;
for (int k = K / 2; k < K; k++)
{
sum += matA[i*strideA + k] * matB[k*strideB + j];
}
matC[i*strideC + j] += sum;
}
}
//C12 = A11XB12 + A12XB22
for (int i = 0; i < M / 2; i++)
{
for (int j = N/2; j < N; j++)
{
float sum;
//A11XB11
sum = 0.0f;
for (int k = 0; k < K / 2; k++)
{
sum += matA[i*strideA + k] * matB[k*strideB + j];
}
matC[i*strideC + j] += sum;
//A12XB21
sum = 0.0f;
for (int k = K / 2; k < K; k++)
{
sum += matA[i*strideA + k] * matB[k*strideB + j];
}
matC[i*strideC + j] += sum;
}
}
//C21 = A21XB11 + A22XB21
for (int i = M/2; i < M; i++)
{
for (int j = 0; j < N / 2; j++)
{
float sum;
//A11XB11
sum = 0.0f;
for (int k = 0; k < K / 2; k++)
{
sum += matA[i*strideA + k] * matB[k*strideB + j];
}
matC[i*strideC + j] += sum;
//A12XB21
sum = 0.0f;
for (int k = K / 2; k < K; k++)
{
sum += matA[i*strideA + k] * matB[k*strideB + j];
}
matC[i*strideC + j] += sum;
}
}
//C22 = A21XB12 + A22XB22
for (int i = M/2; i < M; i++)
{
for (int j = N/2; j < N; j++)
{
float sum;
//A11XB11
sum = 0.0f;
for (int k = 0; k < K / 2; k++)
{
sum += matA[i*strideA + k] * matB[k*strideB + j];
}
matC[i*strideC + j] += sum;
//A12XB21
sum = 0.0f;
for (int k = K / 2; k < K; k++)
{
sum += matA[i*strideA + k] * matB[k*strideB + j];
}
matC[i*strideC + j] += sum;
}
}
}

Strassen算法

Strassen算法,时间复杂度是O^2.81。定义见:wiki
c++实现:

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static void mm_strassen(float* matA, float* matB, float* matC, const int M, const int N, const int K, const int strideA, const int strideB, const int strideC)
{
if ((M <= 64) || (M%2 != 0 ||N%2 != 0 ||K%2!=0))
{
return mm_generate(matA, matB, matC, M, N, K, strideA, strideB, strideC);
}
memset(matC, 0, M*strideC*sizeof(float));
int offset = 0;
//M1 = (A11+A22)*(B11+B22)
std::vector<float> M1((M / 2) * (N / 2));
{
memset(&M1[0], 0, M1.size()*sizeof(float));
//M1_0 = (A11+A22)
std::vector<float> M1_0((M / 2) * (K / 2));
offset = M*strideA / 2 + K / 2;
for (int i = 0; i < M / 2; i++)
{
for (int j = 0; j < K/2; j++)
{
const int baseIdx = i*strideA + j;
M1_0[i*K/2+j] = matA[baseIdx] + matA[baseIdx + offset];
}
}
//M1_1 = (B11+B22)
std::vector<float> M1_1((K / 2) * (N / 2));
offset = K*strideB / 2 + N / 2;
for (int i = 0; i < K / 2; i++)
{
for (int j = 0; j < N / 2; j++)
{
const int baseIdx = i*strideB + j;
M1_1[i*N/2+j] = matB[baseIdx] + matB[baseIdx + offset];
}
}
mm_strassen(&M1_0[0], &M1_1[0], &M1[0], M / 2, N / 2, K / 2,
K/2,N/2,N/2);
}
//M2 = (A21+A22)*B11
std::vector<float> M2((M / 2) * (N / 2));
{
memset(&M2[0], 0, M2.size()*sizeof(float));
//M2_0 = (A21+A22)
std::vector<float> M2_0((M / 2) * (K / 2));
offset = K / 2;
for (int i = M / 2; i < M; i++)
{
for (int j = 0; j < K / 2; j++)
{
const int baseIdx = i*strideA + j;
M2_0[(i-M/2)*K/2+j] = matA[baseIdx] + matA[baseIdx + offset];
}
}
//M2_2 = B11
mm_strassen(&M2_0[0], &matB[N / 2], &M2[0], M / 2, N / 2, K / 2,
K / 2, strideB, N / 2);
}
//M3 = A11*(B12-B22)
std::vector<float> M3((M / 2) * (N / 2));
{
memset(&M3[0], 0, M3.size()*sizeof(float));
//M3_0 = A11
//M3_1 = (B12-B22)
std::vector<float> M3_1((K / 2) * (N / 2));
offset = K*strideB / 2;
for (int i = 0; i < K/2; i++)
{
for (int j = N/2; j < N; j++)
{
const int baseIdx = i*strideB + j;
M3_1[i*N/2+j-N/2] = matB[baseIdx] - matB[baseIdx + offset];
}
}
mm_strassen(matA, &M3_1[0], &M3[0], M / 2, N / 2, K / 2,
strideA, N / 2, N / 2);
}
//M4 = A22*(B21-B11)
std::vector<float> M4((M / 2) * (N / 2));
{
memset(&M4[0], 0, M4.size()*sizeof(float));
//M4_0 = A22
//M4_1 = (B12-B22)
std::vector<float> M4_1((K / 2) * (N / 2));
offset = K*strideB / 2;
for (int i = 0; i < K / 2; i++)
{
for (int j = N / 2; j < N; j++)
{
const int baseIdx = i*strideB + j;
M4_1[i*N/2+j-N/2] = matB[baseIdx + offset] - matB[baseIdx];
}
}
mm_strassen(matA + M*strideA / 2 + K / 2, &M4_1[0], &M4[0], M / 2, K / 2, N / 2,
strideA, N / 2, N / 2);
}
//M5 = (A11+A12)*B22
std::vector<float> M5((M / 2) * (N / 2));
{
memset(&M5[0], 0, M5.size()*sizeof(float));
//M5_0 = (A11+A12)
std::vector<float> M5_0((M / 2) * (K / 2));
offset = K / 2;
for (int i = 0; i < M/2; i++)
{
for (int j = 0; j < K / 2; j++)
{
const int baseIdx = i*strideA + j;
M5_0[i*K / 2 + j] = matA[baseIdx] + matA[baseIdx + offset];
}
}
//M5_1 = B22
mm_strassen(&M5_0[0], &matB[K*strideB / 2 + N / 2], &M5[0], M / 2, N / 2, K / 2,
K / 2, strideB, N / 2);
}
//M6 = (A21-A11)*(B11+B12)
std::vector<float> M6((M / 2) * (N / 2));
{
memset(&M6[0], 0, M6.size()*sizeof(float));
//M6_0 = (A21-A11)
std::vector<float> M6_0((M / 2) * (K / 2));
offset = K*N / 2;
for (int i = 0; i < M / 2; i++)
{
for (int j = 0; j < K/2; j++)
{
const int baseIdx = i*strideA + j;
M6_0[i*K/2+j] = matA[baseIdx + offset] - matA[baseIdx];
}
}
//M6_1 = (B11+B12)
std::vector<float> M6_1((K / 2) * (N / 2));
offset = N / 2;
for (int i = 0; i < K / 2; i++)
{
for (int j = 0; j < N/2; j++)
{
const int baseIdx = i*strideB + j;
M6_1[i*N/2+j] = matB[baseIdx] + matB[baseIdx + offset];
}
}
mm_strassen(&M6_0[0], &M6_1[0], &M6[0], M / 2, N / 2, K / 2,
K / 2, N / 2, N / 2);
}
//M7 = (A12-A22)*(B21+B22)
std::vector<float> M7((M / 2) * (N / 2));
{
memset(&M7[0], 0, M7.size()*sizeof(float));
//M7_0 = (A12-A22)
std::vector<float> M7_0((M / 2) * (K / 2));
offset = M*strideA / 2;
for (int i = 0; i < M / 2; i++)
{
for (int j = K/2; j < K; j++)
{
const int baseIdx = i*strideA + j;
M7_0[i*K / 2 + j - K / 2] = matA[baseIdx] - matA[baseIdx + offset];
}
}
//M7_1 = (B21+B22)
std::vector<float> M7_1((K / 2) * (N / 2));
offset = N / 2;
for (int i = K/2; i < K; i++)
{
for (int j = 0; j < N / 2; j++)
{
const int baseIdx = i*strideB + j;
M7_1[(i-K/2)*N / 2 + j] = matB[baseIdx] + matB[baseIdx + offset];
}
}
mm_strassen(&M7_0[0], &M7_1[0], &M7[0], M / 2, N / 2, K / 2,
K / 2, N / 2, N / 2);
}
for (int i = 0; i < M / 2;i++)
{
for (int j = 0; j < N / 2;j++)
{
const int idx = i*N / 2 + j;
//C11 = M1+M4-M5+M7
matC[i*strideC + j] = M1[idx] + M4[idx] - M5[idx] + M7[idx];
//C12 = M3+M5
matC[i*strideC + j + N/2] = M3[idx] + M5[idx];
//C21 = M2+M4
matC[(i+M/2)*strideC + j] = M2[idx] + M4[idx];
//C22 = M1-M2+M3+M6
matC[(i+M/2)*strideC + j + N/2] = M1[idx] - M2[idx] + M3[idx] + M6[idx];
}
}
}

Coppersmith-Winograd算法

Coppersmith-Winograd算法,时间复杂度是O^2.38。定义见:wiki
c++实现:

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static void mm_winograd(float* matA, float* matB, float* matC, const int M, const int N, const int K, const int strideA, const int strideB, const int strideC)
{
if ((M <= 64) || (M % 2 != 0 || N % 2 != 0 || K % 2 != 0))
{
return mm_generate(matA, matB, matC, M, N, K, strideA, strideB, strideC);
}
memset(matC, 0, M*strideC*sizeof(float));
int offset = 0;
std::vector<float> S1((M / 2) * (K / 2));
std::vector<float> S2((M / 2) * (K / 2));
std::vector<float> S3((M / 2) * (K / 2));
std::vector<float> S4((M / 2) * (K / 2));
for (int i = 0; i < M / 2;i++)
{
for (int j = 0; j < K / 2;j++)
{
const int idx = i*K / 2 + j;
//S1 = A21 + A22
S1[idx] = matA[(i + M / 2)*strideA + j] + matA[(i + M / 2)*strideA + j + K / 2];
//S2 = S1 - A11
S2[idx] = S1[idx] - matA[i*strideA + j];
//S3 = A11 - A21
S3[idx] = matA[i*strideA + j] - matA[(i + M / 2)*strideA + j];
//S4 = A12 - S2
S4[idx] = matA[i*strideA + j + K / 2] - S2[idx];
}
}
std::vector<float> T1((K / 2) * (N / 2));
std::vector<float> T2((K / 2) * (N / 2));
std::vector<float> T3((K / 2) * (N / 2));
std::vector<float> T4((K / 2) * (N / 2));
for (int i = 0; i < K / 2; i++)
{
for (int j = 0; j < N / 2; j++)
{
const int idx = i*N / 2 + j;
//T1 = B21 - B11
T1[idx] = matB[(i + K / 2)*strideB + j] - matB[i*strideB + j];
//T2 = B22 - T1
T2[idx] = matB[(i + K / 2)*strideB + j + N / 2] - T1[idx];
//T3 = B22 - B12
T3[idx] = matB[(i + K / 2)*strideB + j + N / 2] - matB[i*strideB + j + N / 2];
//T4 = T2 - B21
T4[idx] = T2[idx] - matB[(i + K / 2)*strideB + j];
}
}
//M1 = A11*B11
std::vector<float> M1((M / 2) * (N / 2));
{
memset(&M1[0], 0, M1.size()*sizeof(float));
mm_winograd(matA, matB, &M1[0], M / 2, N / 2, K / 2,
strideA, strideB, N / 2);
}
//M2 = A12*B21
std::vector<float> M2((M / 2) * (N / 2));
{
memset(&M2[0], 0, M2.size()*sizeof(float));
mm_winograd(matA + K / 2, matB + K*strideB/2, &M2[0], M / 2, N / 2, K / 2,
strideA, strideB, N / 2);
}
//M3 = S4*B22
std::vector<float> M3((M / 2) * (N / 2));
{
memset(&M3[0], 0, M3.size()*sizeof(float));
mm_winograd(&S4[0], matB + K*strideB/2 + N / 2, &M3[0], M / 2, N / 2, K / 2,
K/2, strideB, N / 2);
}
//M4 = A22*T4
std::vector<float> M4((M / 2) * (N / 2));
{
memset(&M4[0], 0, M4.size()*sizeof(float));
mm_winograd(matA + M*strideA / 2 + K / 2, &T4[0], &M4[0], M / 2, N / 2, K / 2,
strideA, N / 2, N / 2);
}
//M5 = S1*T1
std::vector<float> M5((M / 2) * (N / 2));
{
memset(&M5[0], 0, M5.size()*sizeof(float));
mm_winograd(&S1[0], &T1[0], &M5[0], M / 2, N / 2, K / 2,
K / 2, N/2, N / 2);
}
//M6 = S2*T2
std::vector<float> M6((M / 2) * (N / 2));
{
memset(&M6[0], 0, M6.size()*sizeof(float));
mm_winograd(&S2[0], &T2[0], &M6[0], M / 2, N / 2, K / 2,
K / 2, N / 2, N / 2);
}
//M7 = S3*T3
std::vector<float> M7((M / 2) * (N / 2));
{
memset(&M7[0], 0, M7.size()*sizeof(float));
mm_winograd(&S3[0], &T3[0], &M7[0], M / 2, N / 2, K / 2,
K / 2, N / 2, N / 2);
}
for (int i = 0; i < M / 2; i++)
{
for (int j = 0; j < N / 2; j++)
{
const int idx = i*N / 2 + j;
//U1 = M1 + M2
const auto U1 = M1[idx] + M2[idx];
//U2 = M1 + M6
const auto U2 = M1[idx] + M6[idx];
//U3 = U2 + M7
const auto U3 = U2 + M7[idx];
//U4 = U2 + M5
const auto U4 = U2 + M5[idx];
//U5 = U4 + M3
const auto U5 = U4 + M3[idx];
//U6 = U3 - M4
const auto U6 = U3 - M4[idx];
//U7 = U3 + M5
const auto U7 = U3 + M5[idx];
//C11 = U1
matC[i*strideC + j] = U1;
//C12 = U5
matC[i*strideC + j + N / 2] = U5;
//C21 = U6
matC[(i + M / 2)*strideC + j] = U6;
//C22 = U7
matC[(i + M / 2)*strideC + j + N / 2] = U7;
}
}
}

其他优化方法

除了在时间复杂度上的优化以外,还可以通过多线程、SIMD指令、GPGPU等工程方法进行优化。

参考