Math Problem Statement

for (int k = 0; k < K; ++k) { for (int c = 0; c < C; ++c) { float *filters_ptr = filter + (k * C + c) * sizeF; sgemm(&G[0][0], filters_ptr, tmp_u, 4, 3, 3); sgemm(tmp_u, &G_T[0][0], u, 4, 3, 4); for (int xi = 0; xi < 4; ++xi) { int base_index = ((xi * 4) * K + k) * C + c; memcpy(&U[base_index], &u[xi * 4], 4 * sizeof(float)); } } }我们将U矩阵的存储方式改变，而后U矩阵的读取方式也要相应的改变最后V矩阵和U矩阵的计算结果要保持不变float tmp_v[16]; float d[16]; // d: [4 * 4]; float v[16]; // v: [4 * 4]; #pragma omp parallel for collapse(2) private(tmp_v, d, v) for (int n = 0; n < N; ++n) for (int c = 0; c < C; ++c) { for (int y = 0; y < outHeight / 2; ++y) { for (int x = 0; x < outWidth / 2; ++x) {

// Generate d_cb for (int iy = 0; iy < 4; ++iy) for (int ix = 0; ix < 4; ++ix) d[iy * 4 + ix] = image[(n * C + c) * sizeI + (y * 2 + iy) * inWidth + (x * 2 + ix)]; sgemm(&B_T[0][0], d, tmp_v, 4, 4, 4); sgemm(tmp_v, &B[0][0], v, 4, 4, 4); int b = ((n * outHeight / 2) + y) * outWidth / 2 + x; for (int xi = 0; xi < 4; ++xi) for (int nu = 0; nu < 4; ++nu) V[((long)(xi * 4 + nu) * C + c) * P + b] = v[xi * 4 + nu]; } } }

// M[xi, nu, :, :] = U[xi, nu, :, :] * V[xi, nu, :, :] for (int xi = 0; xi < 4; ++xi) { for (int nu = 0; nu < 4; ++nu) { float *M_ptr = M + (long)(xi * 4 + nu) * K * P; float *U_ptr = U + (long)(xi * 4 + nu) * K * C; float *V_ptr = V + (long)(xi * 4 + nu) * C * P; sgemm_parallel(U_ptr, V_ptr, M_ptr, K, C, P); } }

// Y = A_T * m * A float mm[16]; // 4 * 4 float tmp_m[8]; // 2 * 4 float temp_out[4]; // 2 * 2 for (int n = 0; n < N; ++n) for (int k = 0; k < K; ++k) { for (int y = 0; y < outHeight / 2; ++y) { for (int x = 0; x < outWidth / 2; ++x) { int b = (n * outHeight / 2 + y) * outWidth / 2 + x; for (long xi = 0; xi < 4; ++xi) { for (long nu = 0; nu < 4; ++nu) { mm[xi * 4 + nu] = M[((xi * 4 + nu) * K + k) * P + b]; } } sgemm(&A_T[0][0], mm, tmp_m, 2, 4, 4); sgemm(tmp_m, &A[0][0], temp_out, 2, 4, 2); for (int i = 0; i < 2; ++i) for (int j = 0; j < 2; ++j) out[(long)((n * K + k) * outHeight + y * 2 + i) * outWidth + x * 2 + j] = temp_out[i * 2 + j]; } } } }

Solution

The provided code is a complex nested loop structure for matrix manipulations using SGEMM (Single-Precision General Matrix Multiply) functions, which are optimized for parallel computations. Here is a detailed explanation of the key components and operations:

  1. Initialization and SGEMM Calls:

    • The first nested loops iterate over k and c to handle different filters and channels.
    • For each combination of k and c, the filters_ptr is set to point to the current filter.
    • SGEMM is called twice to perform matrix multiplications and store results in tmp_u and u.
    • The memcpy function copies the results to the U matrix with a specific indexing pattern.

  2. Parallel Processing with OpenMP:

    • The second set of nested loops involves parallel processing using OpenMP.
    • These loops iterate over n, c, y, and x to process image patches.
    • The d array is populated with image data for the current patch.
    • SGEMM is used to perform matrix multiplications and store results in tmp_v and v.
    • The results are stored in the V matrix using a specific indexing pattern.

  3. Matrix Multiplication and Storage:

    • The third section performs matrix multiplications for each combination of xi and nu and stores the results in the M matrix.
    • The U and V matrices are multiplied using sgemm_parallel.

  4. Final Output Computation:

    • The final nested loops iterate over n, k, y, and x to compute the final output matrix out.
    • Intermediate results are stored in mm, tmp_m, and temp_out arrays.
    • SGEMM is used to perform matrix multiplications with the A_T and A matrices to get the final output values.

Understanding the Changes:

  • The code suggests a change in how the U matrix is stored, requiring corresponding changes in how it is read.
  • The overall structure ensures the computation result remains unchanged by properly handling matrix operations and indexing.

Detailed Explanation of Specific Parts:

  • SGEMM Functions: These are highly optimized routines for matrix multiplications, crucial for performance in such intensive computations.
  • OpenMP Directives: Enable parallel processing, significantly speeding up the computations by distributing work across multiple threads.
  • Indexing Patterns: Specific patterns ensure data is correctly accessed and stored in the multi-dimensional matrices involved.

Would you like more detailed insights into any specific part of the code, or do you have any questions?

Potential Questions:

  1. How does SGEMM optimize matrix multiplications?
  2. What role does OpenMP play in this code?
  3. Can you explain the specific indexing patterns used for the U and V matrices?
  4. How are the changes to the U matrix storage handled in the code?
  5. What are the benefits of using matrix multiplication in this context?

Tip:

Understanding and optimizing matrix operations are crucial for performance in numerical computations and machine learning.

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Math Problem Analysis

Mathematical Concepts

Matrix Operations
SGEMM Functions
Parallel Processing

Formulas

Matrix multiplication

Theorems

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Suitable Grade Level

Advanced Level

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