Cheap Tensor Partitioning Techniques

• Cheap tensor partitioning is a collection of methods to decompose tensors with minimal computational time, memory, and communication costs.
• It leverages techniques like mode-wise clustering, spectral relaxation, and operator-level partitioning to tackle challenges in deep neural networks and large-scale computations.
• Advanced strategies including PTAS, greedy algorithms, and metaheuristics such as simulated annealing provide scalable solutions with theoretical guarantees on performance.

Cheap tensor partitioning refers to algorithmic, computational, and system strategies for decomposing, clustering, or distributing tensors such that the partitioning cost—be it computational time, memory usage, communication overhead, or engineering labor—is minimized without sacrificing solution accuracy or scalability. The approach encompasses a variety of technical domains, including Boolean factorizations, hypergraph partitioning, operator-level graph partitioning for deep neural network training, distributed memory tensor computations, and optimization-of-contraction order in tensor networks. The following sections elaborate on salient methodologies, theoretical guarantees, and practical implementations driving the rapid and cost-effective partitioning of tensor data structures.

1. Complexity Reduction via Mode-wise Partitioning and Clustering

Partitioning a tensor by constraining one mode (dimension) to be clustered or non-overlapping is a powerful strategy to circumvent the inherent NP-hardness of general tensor factorization. In the binary setting—e.g., Boolean tensor clustering (Metzler et al., 2015)—the algorithm SaBoTeur unfolds the tensor in the mode to be partitioned and performs k-medoids style clustering, restricting centroids to rank-1 binary matrices. This restriction effectively regularizes the decomposition, dramatically reducing the model’s degrees of freedom and sidestepping the combinatorial optimization pitfalls associated with overlapping components. Two approximation schemes are central: a polynomial-time approximation scheme (PTAS) and a deterministic algorithm that achieves a 0.828‑approximation for rank-1 factorizations (by exploring a search restricted to sampled rows), both enabling scalable clustering with strong similarity guarantees.

A similar reduction in hypergraph partitioning is achieved via “mode collapse” and spectral relaxation (Ghoshdastidar et al., 2016). Here, the m-way affinity tensor is collapsed into an n×n matrix via summation over the last m–2 modes, and classical spectral methods are invoked, transforming an otherwise intractable high-order partitioning into manageable quadratic problems. The central innovation is further realized in sampling-based algorithms such as TTM and its iterative “Tetris” variant, which intelligently select edges or tuples for evaluation, yielding substantial cost reductions in memory and computational resources—often with high-probability guarantees on partitioning accuracy.

2. Operator- and Dataflow-level Partitioning for Deep Learning Systems

For deep neural networks that are bottlenecked by GPU memory limitations, operator-level graph partitioning systems such as Tofu (Wang et al., 2018), ParDNN (Qararyah et al., 2020), and TOAST (Alabed et al., 20 Aug 2025) deliver fully automated, cost-effective tensor partitioning at scale. Tofu utilizes a Halide-inspired Tensor Description Language (TDL) and symbolic interval analysis to identify valid partition strategies for each operator, while a dynamic programming search recursively optimizes the partition plan to minimize the overall communication volume. Its methodology enables training of DNNs that exceed a single GPU’s memory footprint, achieving up to 400% speedup versus alternatives.

ParDNN approaches the problem via computational-graph slicing, clustering critical paths onto primary devices, then mapping secondary operations with memory-aware heuristic adjustments to avoid memory overflows. The core strength is its static, framework-independent placement file strategy, enabling partitioning of models with billions of parameters (hundreds of thousands of operations) in seconds or minutes.

TOAST advances partitioning through principled static analysis (Named Dimension Analysis), which condenses the decision space by tracking tensor index identities and conflicts, thus exposing efficient high-level sharding actions. Coupled with Monte Carlo Tree Search (MCTS), TOAST navigates the exponential search space, discovering previously unknown partitioning solutions that outperform industrial benchmarks across diverse hardware and architectures.

3. Optimization Techniques: Sampling, Greedy, and Metaheuristics

Sampling-based partitioning (uniform and weighted) is a recurring technique for managing computational expense in high-order hypergraph partitioning (Ghoshdastidar et al., 2016). Weighted sampling concentrates effort on informative hyperedges, achieving weak consistency of partitioning with sample sizes orders of magnitude below the tensor’s total size. Iterative sampling, e.g., Tetris, refines edge selection adaptively, focusing partitioning effort where true cluster structure is most pronounced.

Greedy strategies are leveraged in geometric and graph-based partitioning software (Sasidharan, 4 Mar 2025), where hierarchical kd-tree decomposition and space-filling curve ordering (Morton, Hilbert) precede parallel greedy knapsack assignments. This combination preserves spatial locality, load balance, and minimizes inter-partition communication—all at a computational cost comparable to parallel sorting.

Metaheuristic partitioning via simulated annealing (Geiger et al., 28 Jul 2025) refines tensor network partitionings, escaping local minima through stochastic acceptance of higher-cost moves. By directly optimizing the contraction tree’s operation count and memory footprint—not just edge cuts—this approach achieves up to 8× reductions in cost over naive or general-purpose hypergraph algorithms, evidenced by high Pearson correlation between model estimates and real wallclock times across the MQT Bench suite.

4. Spectral and Low-Rank Tensor Approximation Frameworks

Spectral approaches generalize matrix-based partitioning to tensors using low-rank Tucker (best rank-$(2,2,\lambda)$) approximations (Eldén et al., 2020). Through constrained maximization on Grassmann manifolds, the factor matrices $U$ recover the block (indicator) structure in the data, revealing partitions analogous to those signaled by leading eigenvectors in standard graph spectral clustering. Importantly, if a tensor exhibits reducibility (block diagonal or approximately so), the resulting decomposition automatically inherits that structure. Perturbation theory analyses provide guarantees on recovery in the presence of noise.

Tensor spectral clustering methodologies extend the paradigm to higher-order motifs in networks (Benson et al., 2015), such as triangles, directed cycles, or feedback loops, by formulating transition probability tensors and applying multilinear PageRank. The clustering objective becomes that of minimizing “cuts” in the motif space rather than in individual edges, yielding partitions more structurally faithful and less disruptive to network dynamics.

Pseudo-PageRank frameworks for hypergraph partitioning (Chen et al., 2023) avoid expensive stochastic (“dangling”) corrections and leverage the sparsity of the undirected/directed adjacency Laplacian tensors. The tensor splitting algorithms converge linearly under mild conditions, allowing efficient, exact partitioning in regimes characterized by sparse motif connectivity and large data scales.

5. Cost and Memory Optimization in Dynamic and Distributed Settings

Memory-aware tensor partitioning plays a critical role in dynamic (rematerialization-enabled) neural network training (Zhang et al., 2023). Coop introduces a sliding window eviction strategy classified under “cheap tensor partitioning,” targeting contiguous blocks of low-cost tensors for eviction (DRL-informed cost density). By grouping tensors cheaply rematerializable at one end of the memory pool, Coop minimizes fragmentation, recomputation time, and achieves up to 2× reduction in total memory usage across benchmarks. Its approach contrasts with earlier rematerialization techniques that treat all free blocks as equivalent, thereby incurring unnecessary recomputation and fragmentation costs.

In distributed and many-core HPC environments (Sasidharan, 4 Mar 2025), geometric and statistical partitioning algorithms are specifically tailored for mesh, point location, and general graph partitioning, and are highly relevant for large sparse tensor workloads. By using kd-tree decomposition, space-filling curves, and parallel greedy knapsack across processors, these algorithms quickly adapt partitions to shifting load distributions, ensuring both minimal partitioning overhead and communication cost—demonstrated by experimental results on modern many-core architectures.

6. Applications and Implications

Cheap tensor partitioning methodologies are consistently demonstrated as effective in diverse domains:

• Dynamic networks: Mode-wise clustering reveals adaptive communities (Metzler et al., 2015), while higher-order motif-preserving partitioning identifies functional or anomalous sub-networks (Benson et al., 2015).
• Computer vision and subspace clustering: Sampling-based spectral partitioning (Ghoshdastidar et al., 2016), as well as multi-linear PageRank (Chen et al., 2023), enable efficient cluster recovery even when the tensor affinity graphs are vast and highly sparse.
• Distributed tensor decompositions: Schemes such as Lite (Chakaravarthy et al., 2018) directly address computational and SVD load balancing in block-sparse settings, outperforming both fine- and coarse-grained strategies in HOOI iterations.
• Large model training: Operator- and graph-level partitioners (Wang et al., 2018, Qararyah et al., 2020, Alabed et al., 20 Aug 2025, Zhang et al., 2023) facilitate full or partial model parallelism, avoiding manual code restructuring, OOM failures, and suboptimal device utilization—especially critical when batch sizes or parameter counts are increased for throughput or convergence improvements.
• Classical tensor network contraction and quantum simulation: Simulated annealing-based partition refinement (Geiger et al., 28 Jul 2025) achieves dramatic reductions in floating-point and memory operations, validated against HyperOptimizer and other partitioning baselines.

Collectively, these advances establish cheap tensor partitioning as a foundational technique for scalable, interpretable, and resource-efficient computation across scientific, engineering, and machine learning applications.