Vivek Bharadwaj

Vivek Bharadwaj, Aydin Buluç, James Demmel

Sampled Dense Times Dense Matrix Multiplication (SDDMM) and Sparse Times Dense Matrix Multiplication (SpMM) appear in diverse settings, such as collaborative filtering, document clustering, and graph embedding. Frequently, the SDDMM output becomes the input sparse matrix for a subsequent SpMM operation. Existing work has focused on shared memory parallelization of these primitives. While there has been extensive analysis of communication-minimizing distributed 1.5D algorithms for SpMM, no such analysis exists for SDDMM or the back-to-back sequence of SDDMM and SpMM, termed FusedMM. We show that distributed memory 1.5D and 2.5D algorithms for SpMM can be converted to algorithms for SDDMM with identical communication costs and input / output data layouts. Further, we give two communication-eliding strategies to reduce costs further for FusedMM kernels: either reusing the replication of an input dense matrix for the SDDMM and SpMM in sequence, or fusing the local SDDMM and SpMM kernels. We benchmark FusedMM algorithms on Cori, a Cray XC40 at LBNL, using Erdos-Renyi random matrices and large real-world sparse matrices. On 256 nodes with 68 cores each, 1.5D FusedMM algorithms using either communication eliding approach can save at least 30% of time spent exclusively in communication compared to executing a distributed-memory SpMM and SDDMM kernel in sequence. On real-world matrices with hundreds of millions of edges, all of our algorithms exhibit at least a 10x speedup over the SpMM algorithm in PETSc. On these matrices, our communication-eliding techniques exhibit runtimes up to 1.6 times faster than an unoptimized sequence of SDDMM and SpMM. We embed and test the scaling of our algorithms in real-world applications, including collaborative filtering via alternating-least-squares and inference for attention-based graph neural networks.

Osman Asif Malik, Vivek Bharadwaj, Riley Murray

We show how to develop sampling-based alternating least squares (ALS) algorithms for decomposition of tensors into any tensor network (TN) format. Provided the TN format satisfies certain mild assumptions, resulting algorithms will have input sublinear per-iteration cost. Unlike most previous works on sampling-based ALS methods for tensor decomposition, the sampling in our framework is done according to the exact leverage score distribution of the design matrices in the ALS subproblems. We implement and test two tensor decomposition algorithms that use our sampling framework in a feature extraction experiment where we compare them against a number of other decomposition algorithms.

Vivek Bharadwaj, Austin Glover, Aydin Buluc et al.

Rotation equivariant graph neural networks, i.e. networks designed to guarantee certain geometric relations between their inputs and outputs, yield state of the art performance on spatial deep learning tasks. They exhibit high data efficiency during training and significantly reduced inference time for interatomic potential calculations compared to classical approaches. Key to these models is the Clebsch-Gordon (CG) tensor product, a kernel that contracts two dense feature vectors with a highly-structured sparse tensor to produce a dense output vector. The operation, which may be repeated millions of times for typical equivariant models, is a costly and inefficient bottleneck. We introduce a GPU sparse kernel generator for the CG tensor product that provides significant speedups over the best existing open and closed-source implementations. Our implementation achieves high performance by carefully managing the limited GPU shared memory through static analysis at model compile-time, minimizing reads and writes to global memory. We break the tensor product into a series of smaller kernels with operands that fit entirely into registers, enabling us to emit long arithmetic instruction streams that maximize instruction-level parallelism. By fusing the CG tensor product with a subsequent graph convolution, we reduce both intermediate storage and global memory traffic over naive approaches that duplicate input data. We also provide optimized kernels for the gradient of the CG tensor product and a novel identity for the higher partial derivatives required to predict interatomic forces. Our kernels offer up to 1.3x speedup over NVIDIA's closed-source cuEquivariance package, as well as 10x speedup over the widely-used e3nn package. In FP64 precision, we offer up to 6.2x inference-time speedup for the MACE chemistry foundation model over the original unoptimized version.