Tiling and coalescing: matrix transpose

Tiling and coalescing: matrix transpose#

Matrix transpose is a fundamental linear algebra operation that serves as a building block for many tasks. At a high level, it does not involve calculations; it involves only memory movement. Because GPUs typically have significantly higher memory bandwidth than CPUs, they have a unique advantage when performing matrix transpose operations.

Like the previous examples, the matrix transpose operation can benefit from a significantly parallel implementation, in which each thread is responsible for moving a matrix element.

 1// Simple matrix transpose kernel
 2__global__ void transpose_naive_kernel(float* in, float* out, int width, int height)
 3{
 4    int x_index = blockIdx.x * tile_dim + threadIdx.x;
 5    int y_index = blockIdx.y * tile_dim + threadIdx.y;
 6
 7    int in_index  = y_index * width + x_index;
 8    int out_index = x_index * height + y_index;
 9
10    out[out_index] = in[in_index];
11}

Memory access patterns#

Matrix transpose operations present a unique challenge in terms of memory access patterns. Theoretically, the number of reads and writes should be approximately equal, as you write all the data that you read from the input matrix. However, in practice, the actual memory traffic can be significantly higher due to the nature of how threads access memory.

To understand this behavior, you need to examine the memory access pattern carefully. Unlike matrix multiplication where the same data might be accessed multiple times, in matrix transpose each element is typically accessed only once. The issue instead lies in the address pattern of the read and write operations.

In a typical matrix transpose implementation, adjacent threads read data horizontally (row-wise) and write data vertically (column-wise). Since matrices are stored in row-major order, adjacent threads read data from contiguous memory addresses, which allows the GPU to coalesce these read operations into fewer memory transactions. However, when writing data vertically, the threads access memory addresses with large strides (the distance between consecutive elements in a column). This prevents write coalescing, as each thread’s write operation typically requires a separate memory transaction.

As a result, the read operations are coalescable and efficient, while the write operations are non-coalescable and generate significantly more memory traffic. This fundamental characteristic of matrix transpose operations makes them an excellent example for understanding the importance of memory access patterns and coalescing in GPU programming.

Memory coalescing with LDS#

It is possible to convert non-coalescable memory access to coalescable ones using the Local Data Store (LDS). The optimization approach splits the matrix transpose process into two steps. First, you move the data from the input matrix to LDS memory. For this, you read the data horizontally, ensuring that the memory access is coalesced. When the LDS buffer is filled, you then move the data from the LDS to the main memory. In this step, you read the data vertically and write them horizontally. Since LDS memory has a significantly higher bandwidth than global memory, reading them horizontally and vertically does not impact overall performance. However, you write the data horizontally so that the writes to the main memory can also be coalesced. Thus, you use the LDS as an intermediate data transfer buffer between the input and output matrices, allowing you to make both read and write operations coalescable.

The LDS-based kernel implementation uses the same interface as the naive implementation. The kernel uses a shared memory tile to temporarily store data before writing it back to the output matrix. This approach ensures that both the read from global memory and the write to global memory are coalescable, which significantly improves performance.

 1// LDS-based matrix transpose kernel for better memory coalescing
 2__global__ void transpose_lds_kernel(float* in, float* out, int width, int height)
 3{
 4    __shared__ float tile[tile_dim][tile_dim];
 5
 6    int x_tile_index = blockIdx.x * tile_dim;
 7    int y_tile_index = blockIdx.y * tile_dim;
 8
 9    int in_index  = (y_tile_index + threadIdx.y) * width + (x_tile_index + threadIdx.x);
10    int out_index = (x_tile_index + threadIdx.y) * height + (y_tile_index + threadIdx.x);
11
12    tile[threadIdx.y][threadIdx.x] = in[in_index];
13
14    __syncthreads();
15
16    out[out_index] = tile[threadIdx.x][threadIdx.y];
17}

In the LDS implementation, each threadblock loads a tile of the input matrix into shared memory in a coalesced fashion. The threads then transpose the data within the shared memory tile and write it back to the output matrix in a coalesced manner. This approach effectively solves the non-coalescable write problem of the naive implementation.

The LDS-based approach provides significant performance benefits because both read and write operations to global memory are now coalescable, reducing the total number of memory transactions. By using LDS as an intermediate buffer, you reduce the pressure on the GPU’s memory system and make better use of the available bandwidth. The tile-based approach improves cache hit rates by reusing data within the shared memory before writing it back to global memory.

This optimization technique is particularly important for memory-bound kernels like matrix transpose, where the performance is limited by memory bandwidth rather than computational throughput.

Implementation guidelines and performance considerations#

When implementing LDS-based matrix transpose optimization, consider the following practical aspects:

  • Tile size selection: Choose tile dimensions that balance LDS memory usage and coalescing benefits. Common sizes include 16 × 16 or 32 × 32 tiles, but the optimal size depends on your specific GPU architecture

  • Bank conflict avoidance: Structure your data access patterns to minimize memory bank conflicts in the LDS, which can reduce the effectiveness of the optimization

  • Memory alignment: Ensure proper memory alignment for both global memory accesses to maximize coalescing benefits

  • Edge case handling: Account for matrix dimensions that are not perfectly divisible by your chosen tile size, implementing proper boundary checks

The performance impact of LDS-based optimization varies depending on matrix dimensions, GPU architecture, and memory subsystem characteristics. For larger matrices, the benefits are typically more pronounced as memory bandwidth becomes the dominant performance factor.

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