Tensor-Centric Communication Mechanism

• Tensor-centric communication is defined as a system that organizes and minimizes tensor data movement by exploiting multidimensional structure and symmetry.
• It employs geometric partitioning, combinatorial designs, and tensor-aware data mapping to achieve near-optimal communication lower bounds.
• The approach enables predictable scaling in parallel tensor computations by ensuring balanced loads, efficient data reuse, and minimal runtime exchanges.

A tensor-centric communication mechanism is a system or algorithmic strategy that organizes and optimizes the movement of tensor data—and its associated computation—across distributed memory or processor architectures, explicitly exploiting tensor structure (e.g., symmetry, sparsity, or multidimensionality) to minimize communication. These mechanisms are central to large-scale parallel numerical linear algebra, high-dimensional data analysis, and modern deep learning workloads, where tensors frequently represent core data objects. Tensor-centric communication mechanisms differ from matrix- or vector-centric approaches by leveraging the multi-index iteration space, combinatorial symmetries, and geometric domain properties of tensors to achieve provably optimal or near-optimal data movement, often approaching information-theoretic lower bounds.

1. Fundamental Principles of Tensor-Centric Communication

The design of tensor-centric communication mechanisms is rooted in the recognition that naive parallel algorithms—such as those based on simple matrix multiplication analogues—fail to exploit tensor structure, leading to excessive or redundant data exchanges. Key principles include:

• Geometric Partitioning: The multi-dimensional iteration space of a tensor (e.g., the $(i,j,k)$ coordinates of a 3-way tensor) is partitioned into higher-dimensional blocks (e.g., tetrahedra for 3D, simplices for $d$-way tensors), as opposed to simple slices or slabs.
• Symmetry Exploitation: When the tensor is symmetric (invariant under index permutations), partitioning is performed so that each processor owns all arithmetic instances of each unique tensor entry, maximizing local reuse and preventing redundant communication of the same data.
• Tensor-Aware Input/Output Mapping: Associated vectors or matrices (e.g., for tensor-times-vector, tensor-times-matrix operations) are split and distributed in tight alignment with the geometric partitioning, ensuring that fragments of these operands flow only to processors that need them.
• Combinatorial Designs: Partitioning schemes often employ combinatorial objects such as block designs or Steiner systems to ensure balanced, non-overlapping coverage of the computation space, particularly in symmetric cases.
• Information-Theoretic Lower Bounds: Communication volume is bounded using geometric inequalities (e.g., Loomis–Whitney-type) that quantify the minimal number of unique data elements a processor must touch, given its assigned portion of the computation.

These principles allow the design of communication-optimal algorithms whose cost matches derived lower bounds up to a small constant and whose layout generalizes to arbitrary tensor orders and symmetric structures.

2. Communication Lower Bounds in Parallel Tensor Computations

The minimal communication necessary for parallel tensor operations is established using combinatorial geometry, discrete inequalities, and convex optimization. For symmetric third-order tensors (see (Daas et al., 18 Jun 2025)), using a set $V$ of assigned $(i, j, k)$ points in the strictly lower-tetrahedral index space, the following inequality holds:

$$6 |V| \leq \bigl| \phi_i(V) \cup \phi_j(V) \cup \phi_k(V) \bigr|^3$$

where $\phi_i(V)$ denotes the set of unique $i$ indices in $V$. This quantifies that the number of unique vector elements (i.e., $v_j$, $v_k$ needed for the contraction) grows as the cube root of the total arithmetic assigned.

For a global tensor of size $n^3$ divided across $P$ processors, the per-processor bandwidth lower bound for the symmetric "tensor-times-same-vector" (STTSV) operation is

$\geq 2 \frac{n}{P^{1/3}}$

asymptotically, i.e., the leading term for the minimal number of words communicated by any processor.

Such bounds are tight, in the sense that the constructive algorithms provided subsequently attain them in all regimes.

3. Optimal Tensor Block Partitioning Schemes

Communication-optimal algorithms partition the symmetric index space into blocks that are higher-dimensional analogues of matrix triangular blocks. In the 3-way case:

• Tetrahedral Blocks: Each processor is assigned a subset $R_p$ (of size $q+1$ from a total of $q^2+1$) using a Steiner $(q^2+1, q+1, 3)$ system, for a suitably chosen prime power $q$.
• Assignment: Each $R_p$ identifies which indices $(i, j, k)$ a processor owns, ensuring that every unordered triple appears in precisely one processor's block. This achieves perfect load balance and maximal reuse.
• Tensor Storage: Each processor stores only those entries it owns, removing the need for runtime communication of tensor elements.
• Vector Distribution: The input vector $v$ is cut into blocks matched to $R_p$, and each block is held initially by processors that need it, requiring only a small all-to-all communication.
• Local Computation and Output Gathering: Local ternary multiplications are performed, partial results are accumulated, and only the minimal vector portions necessary are redistributed to assemble the final output.

This framework generalizes naturally to higher-order symmetric tensors, with simplex block partitions and analogous geometric inequalities.

4. Communication-Optimality and Algorithmic Workflow

The optimal algorithm for the STTSV problem follows a four-stage workflow:

1. Local Packing: Processors pack their portion of the input vector for exchange.
2. All-to-All Exchange: Each processor gathers precisely the vector blocks it needs (one per mode index in its block).
3. Local Computation: All required ternary multiplications are performed locally, eliminating further need for tensor or vector communication.
4. Result Redistribution: Partial sums of the output vector are gathered across the appropriate processors and summed.

The communication volume for both vector gather and output redistribution steps is:

$$2 \left( \frac{n}{P^{1/3}} - \mathcal{O} \left( \frac{n}{P} \right) \right)$$

This matches exactly the lower bound, confirming tight optimality of the tensor-centric communication design.

5. Generalization to Other Tensor Operations

The tensor-centric mechanism is not limited to third-order symmetric tensor contractions. The combinatorial partitioning, data alignment, and lower-bound reasoning generalize to:

• Higher-Order Tensor Contractions: Using higher-simplex blocks and high-dimensional analogues of the Loomis–Whitney inequality.
• Multiple-Mode Matrix and Vector Products: Algorithmic templates adapt to contractions with more modes or mixed symmetry, after suitable adjustment of block design.
• Other Symmetric or Structured Tensor Classes: Variants (e.g., anti-symmetric, partially symmetric) can modify the block partitioning and communication paths to preserve optimality.
• Related Work on Symmetric Tensor Contractions: The bilinear-algorithm framework (Solomonik et al., 2017) reveals arithmetic-communication trade-offs; symmetry reduces arithmetic but can increase communication unless partitioning is chosen judiciously.

This approach thus provides a unified route for constructing minimal communication layouts for a broad class of tensor computations.

6. Implications for Parallel and Distributed System Design

The tensor-centric communication paradigm offers architectural and practical insights:

• Separation of Data and Compute Movement: By aligning data layout with computation needs, expensive runtime redistributions of large tensor volumes are avoided; only small, targeted exchanges of vector/matrix slices remain on critical paths.
• Predictable Scaling: Communication cost scales as a sublinear power of $P$ (specifically $n / P^{1/3}$ in 3-way symmetric case), enabling strong scaling and efficiency on modern supercomputers.
• Block Designs Inform Load Balancing: Use of combinatorial block designs guarantees per-processor load balance both in memory and arithmetic.
• Applicability Beyond Tensors: The geometric and combinatorial lower-bounding arguments underpinning these mechanisms are applicable to other high-dimensional algebraic structures, potentially enabling similar in-memory computation for structured graphs, hypermatrices, or multi-modal data.

A plausible implication is that future distributed frameworks for large-scale symmetric tensor computations will increasingly adopt tensor-centric communication mechanisms as primitives, exposing block-partitioning parameters and supporting hardware–software co-design to maximize overall throughput and minimize network contention.