cuFastTuckerPlus: A Stochastic Parallel Sparse FastTucker Decomposition Using GPU Tensor Cores (2404.10087v2)

Abstract: Sparse tensors are prevalent in real-world applications, often characterized by their large-scale, high-order, and high-dimensional nature. Directly handling raw tensors is impractical due to the significant memory and computational overhead involved. The current mainstream approach involves compressing or decomposing the original tensor. One popular tensor decomposition algorithm is the Tucker decomposition. However, existing state-of-the-art algorithms for large-scale Tucker decomposition typically relax the original optimization problem into multiple convex optimization problems to ensure polynomial convergence. Unfortunately, these algorithms tend to converge slowly. In contrast, tensor decomposition exhibits a simple optimization landscape, making local search algorithms capable of converging to a global (approximate) optimum much faster. In this paper, we propose the FastTuckerPlus algorithm, which decomposes the original optimization problem into two non-convex optimization problems and solves them alternately using the Stochastic Gradient Descent method. Furthermore, we introduce cuFastTuckerPlus, a fine-grained parallel algorithm designed for GPU platforms, leveraging the performance of tensor cores. This algorithm minimizes memory access overhead and computational costs, surpassing the state-of-the-art algorithms. Our experimental results demonstrate that our method achieves a speedup of $3X$ to $5X$ compared to state-of-the-art algorithms.

• The paper introduces FastTuckerPlus, a stochastic non-convex optimization method that accelerates sparse tensor decomposition.
• It reformulates tensor factorization into two interleaved subproblems solved via stochastic gradient descent for faster convergence.
• Leveraging GPU Tensor Cores, cuFastTuckerPlus distributes computation in parallel, achieving up to a 5X speedup over conventional methods.

Enhancing Tensor Decomposition: Introducing the FastTuckerPlus Algorithm for HHLST

GPU-Accelerated Sparse Tensor Decomposition

Sparse tensor decomposition is a fundamental operation in high-dimensional data analysis, enabling the extraction of simpler, interpretable data structures from complex datasets. In particular, the Tucker decomposition method has gained traction for its ability to uncover latent structures within tensors. However, traditional Tucker decomposition algorithms struggle with large-scale, high-order, high-dimensional sparse tensors (HHLST), often requiring impractical computational resources. Recent developments in fast Tucker decomposition algorithms have sought to address these challenges, yet the need for further improvements in efficiency and scalability remains.

FastTuckerPlus: A Stochastic Non-Convex Optimization Approach

In response to the limitations of existing algorithms, this paper introduces FastTuckerPlus, a novel approach to sparse tensor decomposition. FastTuckerPlus redefines the optimization problem underlying tensor decomposition into two non-convex subproblems, solved alternately using a Stochastic Gradient Descent (SGD) strategy. This innovation enables FastTuckerPlus to converge to an optimal solution faster than state-of-the-art techniques that rely on convex optimization methods.

A key advantage of FastTuckerPlus is its ability to effectively navigate the simple optimization landscape presented by tensor factorization problems. Empirical results demonstrate that FastTuckerPlus converges more rapidly to a global optimum, showcasing the potential of non-convex optimization in tensor decomposition tasks.

cuFastTuckerPlus: Leveraging GPU Tensor Cores for Paralleled Efficiency

The paper further extends the FastTuckerPlus algorithm to cuFastTuckerPlus, which is tailored for parallel execution on GPU platforms equipped with Tensor Cores. cuFastTuckerPlus distributes the computation efficiently across the GPU's tens of thousands of Tensor Cores, minimizing memory access overhead and significantly speeding up computations.

Experimental results reveal that cuFastTuckerPlus achieves a speedup of up to $5X$ over existing algorithms in single iteration time. This remarkable improvement underscores the algorithm's capability to handle HHLST more efficiently than any current method, blending advanced mathematical strategies with cutting-edge computational technology.

Implications and Future Directions

The introduction of FastTuckerPlus and its GPU-accelerated variant, cuFastTuckerPlus, marks a significant advancement in sparse tensor decomposition. Beyond the theoretical contributions, these algorithms offer practical tools for analyzing large-scale datasets across various domains, from social network analysis to neuroscience.

Looking ahead, the success of FastTuckerPlus opens new avenues for exploring non-convex optimization strategies in tensor decomposition and beyond. As computational architectures continue to evolve, the integration of such algorithms with hardware innovations will likely unveil even more efficient data analysis methodologies.

Moreover, the adaptability of cuFastTuckerPlus to leverage GPU Tensor Cores invites further research into algorithm-hardware co-design, promising to enhance the performance of data-intensive applications dramatically. In conclusion, FastTuckerPlus not only represents a notable achievement in tensor decomposition but also sets the stage for future discoveries in high-dimensional data analysis and computational optimization strategies.