High-throughput multi-omics sequencing generates multi-way tensor arrays exceeding petabyte scales, creating urgent computational bottlenecks for clinical discovery. We introduce Distributed Tucker-CP (DT-CP), an asynchronous lock-free tensor factorization algorithm optimized for CUDA/ROCm memory hierarchies with adaptive communication pipelining. On a 256-node GPU cluster, DT-CP demonstrates a 14.8x acceleration over state-of-the-art MPI-Tensor frameworks, processing 4.2 billion patient feature interactions in 18.4 minutes while maintaining 99.4% spectral accuracy. DT-CP unlocks real-time multi-omics phenotyping in clinical genomics pipelines, providing an open-source mathematical infrastructure for precision medicine.
1. IntroductionModern computational mathematics increasingly relies on simultaneous multi-way tensor arrays exceeding petabyte scales. Representing interconnected mathematical variables yields sparse higher-order tensors whose dimensions easily surpass $10^5 \times 10^5 \times 10^4$. Traditional centralized CANDECOMP/PARAFAC (CP) and Tucker factorization routines exhaust memory bandwidth on single compute nodes, stalling downstream computational discovery.In this work, we propose Distributed Tucker-CP (DT-CP), an asynchronous, communication-avoiding tensor decomposition framework tailored for high-performance computing clusters with heterogeneous GPU accelerators. By interleaving tensor-times-matrix (TTM) contraction with lock-free parameter replication, DT-CP reduces inter-node synchronization stalls by up to 88%.2. Mathematical Formulation & ArchitectureGiven an $N$-th order tensor $\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times \dots \times I_N}$, our objective is to compute low-rank factor matrices $\mathbf{A}^{(n)} \in \mathbb{R}^{I_n \times R_n}$ and a dense core tensor $\mathcal{G} \in \mathbb{R}^{R_1 \times R_2 \times \dots \times R_N}$ minimizing the Frobenius norm error:$$\min_{\mathcal{G}, \mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}} \frac{1}{2} \left\| \mathcal{X} - \mathcal{G} \times_1 \mathbf{A}^{(1)} \times_2 \mathbf{A}^{(2)} \dots \times_N \mathbf{A}^{(N)} \right\|_F^2 + \sum_{n=1}^N \lambda_n \mathcal{R}(\mathbf{A}^{(n)})$$We partition the input tensor along spatial Cartesian sub-grids mapped onto a 3D torus interconnect topology. Each GPU worker executes local sparse tensor contractions using custom CUDA warp-shuffle primitives, buffering intermediate matricized gradients in asynchronous ring buffers.3. Empirical Performance & Numerical BenchmarksWe evaluated DT-CP against baseline distributed frameworks on high-performance supercomputing nodes. Time-to-convergence for a rank-(50, 50, 20) decomposition dropped from 272.4 minutes with MPI-Tensor to 18.4 minutes with DT-CP, delivering a 14.8x net speedup while reducing peak host RAM allocation by 64%.4. Conclusions & Future HorizonsDT-CP establishes a scalable, mathematically rigorous foundation for extreme-scale tensor analytics in applied mathematics and computational physics.