Tensor-Decomposition Local Block Multiplication

• Tensor-decomposition-based local block multiplication is a method that uses low-rank tensor factorizations to accelerate, compress, and structure matrix/tensor computations.
• It leverages various tensor formats, including CP, Tucker, and Kronecker, to reduce arithmetic complexity and enhance parallelism in scientific algorithms.
• Practical implementations benefit applications in neural network compression, secure computation, and multidimensional data analysis with significant efficiency improvements.

Tensor-decomposition-based local block multiplication refers to the use of low-rank tensor factorizations to accelerate, compress, or structure the multiplication of matrix or tensor blocks. Instead of performing dense block multiplications in the inner loops of algorithms for large-scale matrix computation, deep learning, or scientific computing, these approaches represent the bilinear map underlying matrix or tensor multiplication as a low-rank tensor (often a CP/Canonical Polyadic, Tucker, Kronecker, or hierarchical format), thereby reducing arithmetic complexity and enabling structural and parallel advantages. This paradigm is now central to modern fast matrix/tensor multiplication, communication-avoiding algorithms, privacy-preserving computation, neural network compression, and multidimensional data analysis.

1. Mathematical Foundations of Tensor-Decomposition-Based Block Multiplication

The core of these methods is the modeling of the block multiplication map as a structured multi-way tensor and seeking its low-rank decomposition. For two block matrices $X \in \mathbb{R}^{P \times Q}$ and $Y \in \mathbb{R}^{Q \times S}$, the bilinear map $Z = X Y$ can be encoded as a tensor $$T_{PQS} \in \mathbb{R}^{(PQ) \times (QS) \times (PS)}$$, where multiplication is realized through the contraction: $z = T_{PQS} \times_1 x \times_2 y$ with $x = \operatorname{vec}(X^T)$, $y = \operatorname{vec}(Y^T)$, $z = \operatorname{vec}(Z)$. A rank-$R$ CP decomposition of this tensor,

$T_{PQS} = \sum_{r=1}^R a_r \otimes b_r \otimes c_r$

with $a_r \in \mathbb{R}^{PQ}$, $b_r \in \mathbb{R}^{QS}$, $c_r \in \mathbb{R}^{PS}$, enables the computation of $z$ by $2R$ linear forms, $R$ multiplications, and a final assembly via $C w$ where $w=u \circ v$ is the Hadamard product of $u = A^T x$, $v = B^T y$ and $C$ collects $c_r$ as columns (Tichavsky, 2021, Tichavsky et al., 2016). Extensions to higher-order tensor convolutions or to block/local products are achieved by suitable definition of the multiplication tensor and its decomposition.

2. Classical, Structured, and Learned Low-Rank Multiplication Schemes

Tensor-decomposition-based block multiplication often leverages both classical and learned bilinear schemes:

• Classical Schemes: Algorithms such as Strassen (rank-7 for $2 \times 2$), Laderman (rank-23 for $3 \times 3$), and variants exploit symmetries and algebraic identities to minimize multiplication count in their respective block sizes, which can be naturally encoded in CP form (He et al., 14 Jan 2026, Khoruzhii et al., 13 Nov 2025). For structured matrices (symmetric, skew-symmetric, upper/lower-triangular), further reduction in rank is achievable; for instance, a $2\times2$ symmetric-symmetric multiplication with rank $5$ over $\mathbb{Q}$ (Khoruzhii et al., 13 Nov 2025).
• Numerical and Learned Schemes: Modern approaches, including reinforcement-learning-guided search (e.g., AlphaTensor-style) or constrained numerical optimization (e.g., Levenberg-Marquardt), discover decompositions with potentially lower rank or improved addition structure. Typical learned decompositions yield $T_\ell \ll s^3$ for a block of size $s \times s$ (where $T_\ell$ is the learned rank), allowing substantial acceleration (He et al., 14 Jan 2026, Tichavsky et al., 2016).
• Canonical Polyadic and Variants: CP remains the principal model, but block term, hierarchical Tucker, tensor train, Kronecker CP, and custom Kronecker-based CP (KCP) decompositions are also operational in diverse settings (Wang et al., 2020).

A key practical technique is the De Groote transformation, enabling adjustment of factor matrices in CP decompositions to yield sparser, integer-factor schemes with reduced additive complexity and potential stability improvements (Tichavsky, 2021).

3. Fast Matrix and Tensor Multiplication: Algorithmic Realizations

Two prototypical algorithmic frameworks illustrate tensor-decomposition-based block multiplication:

• Blockwise Bilinear Mapping: For each block pair, one computes the relevant linear forms in $X$ and $Y$ as dictated by the factor matrices, multiplies resulting scalars, and assembles the block product via linear combination of $w_r$ vectors, as in the workflow:
  1. For each $r = 1..R$, compute $a_r^T x$, $b_r^T y$.
  2. Compute product $a_r^T x \cdot b_r^T y = \alpha_r$.
  3. Aggregate $$Z = \sum_{r=1}^R \alpha_r\, \operatorname{mat}_{P \times S}(c_r)$$ (Tichavsky et al., 2016, Tichavsky, 2021).

• Tensor-Product or Convolutional Multiplication: For block-local or convolutional scenarios (as in t-SVD or block convolutional products), the product of two tensors $A, X$ is effected via a local block convolution:

  $$\mathcal{Y} = \langle \mathsf{TH}(\hat{\mathcal{A}}), \mathcal{X} \rangle$$

  where $\mathsf{TH}$ is a Toeplitz + Hankel tensor encoding the reflective/padded local structure, and fast diagonalization (e.g., DCT) is used (Molavi et al., 2023, Xu et al., 2019). Locality ensures each output block depends only on nearby input blocks.

• Kronecker-CP Decomposition in Neural Networks: In compressing RNNs, a Kronecker CP format stores weights as sums of Kronecker products of small CPs, enabling strictly/relaxed block-parallel algorithms that avoid materializing the full unfolding (Wang et al., 2020).

Pseudocode for each of the above follows directly from tensor contraction and the specific decomposition.

4. Complexity, Parallelism, and Practical Implementation

Tensor-decomposition-based block multiplication achieves:

• Arithmetic reduction: Lower multiplication count—e.g., $R=15$ for $3\times3$ by $3\times2$ blocks (vs. $18$ naively) (Tichavsky et al., 2016); significant for large block hierarchies or recursive algorithms (Khoruzhii et al., 13 Nov 2025).
• Additive complexity: While additions may increase, hardware acceleration and integer-friendly schemes can ameliorate the cost (Tichavsky, 2021).
• Space efficiency: KCP and related decompositions offer $O(dr(m+n)K)$ storage for order-$d$ tensors with $K$ blocks and CP-rank $r$ (Wang et al., 2020).
• Parallelism: Blockwise structure, e.g., the $K$ Kronecker branches in KCP, enables natural thread/process assignment: bulk numerics are local, with only minimal reductions needed (Wang et al., 2020).
• Exactness in Secure Protocols: In MPC, learning-augmented PSMM using tensor-decomposition-based local block multiplication certifiably preserves privacy/recovery thresholds, while yielding up to $80\%$ per-agent savings—since the protocol's information-theoretic properties depend only on implementation as a bilinear map (He et al., 14 Jan 2026).

Decomposition	Rank Example	Multiplications (vs naive)	Parallelizability
Strassen (2x2)	$R=7$	$7$ vs $8$	Yes, along each branch
CP $3\times3,3\times2$	$R=15$	$15$ vs $18$	Yes
KCP (RNN, d=4)	$K$ blocks, $r$-CP	$O(d r^2 K)$	K-block parallel
Learned LA-PSMM	$T_\ell\ll s^3$	drastic reduction	block- and agent-parallel

5. Extensions: Block Convolutional and Cosine-transform Products

Several recent works generalize blockwise multiplication to tensor convolution structures under varied boundary conditions:

• t-Product and Block Convolutional Tensor Decomposition: t-Product (with circulant structure—periodic BCs) admits diagonalization by FFT, but the $\star_c$-product (reflective BCs) yields a Toeplitz+Hankel block structure, allowing DCT-based diagonalization and purely real arithmetic (Xu et al., 2019, Molavi et al., 2023). These products are local: each output block relates only to nearby input blocks, supporting cache- and GPU-friendly implementation.
• SVD-like Factorizations: Both t-SVD (FFT) and $\star_c$-SVD (DCT) decompose input tensors into orthogonal factors and diagonal tensors with efficient invertibility and optimal storage; experiments show $\star_c$-SVD achieves similar or better accuracy at lower cost for large 3D and multimodal data (Molavi et al., 2023).
• Practical Impact: In applications (compression, classification, clustering), DCT/block-local convolutional decompositions halve the runtime of standard t-SVD and reduce memory and arithmetic costs while matching or improving empirical performance (Molavi et al., 2023, Xu et al., 2019). The block-local view opens avenues for custom transforms (e.g., DST, DWT) and for integration of hierarchical or sparse structures.

6. Applications and Empirical Results

• Matrix Multiplication: Flip-graph search and learning-guided decomposition yield practical schemes for structured/local block multiplication, e.g., $4\times4$ SYRK with rank $34$ (Khoruzhii et al., 13 Nov 2025); these schemes are integrated recursively for improved asymptotic multiplicative factors $\gamma$.
• Secure Computation: Learning-augmented PSMM achieves up to $80\%$ reduction in per-agent computation for collaborative matrix multiplication, with no impact on threshold or privacy (He et al., 14 Jan 2026).
• Neural Network Compression: Kronecker-CP-based RNN layers realize compression ratios up to $2.8\times 10^5$ with significant reduction in forward/backward time, and block-parallelizability facilitates GPU and multi-core execution (Wang et al., 2020).
• Multidimensional Data Analysis: Local block multiplication via DCT/toeplitz-hankel structures improves SVD-based tensor methods in compression, clustering, and principal component extraction across diverse datasets (Molavi et al., 2023).

7. Challenges, Optimizations, and Future Directions

Key challenges tied to tensor-decomposition-based local block multiplication include:

• Optimal Rank Routine Discovery: For larger block sizes and more complex structures (e.g., $n\geq5$), discovery of minimum-rank bilinear schemes remains computationally demanding (Khoruzhii et al., 13 Nov 2025).
• Multi-objective Optimization: Existing searches focus on minimizing multiplicative rank. Simultaneous optimization of rank, addition count, and numerical stability remains largely unaddressed; extensions to the flip-graph and RL-guided pipelines are plausible research avenues (Khoruzhii et al., 13 Nov 2025).
• Field Lifting and Algebraic Constraints: Many low-rank schemes require denominators (e.g., $1/2$), complicating integer or fixed-point implementation; field-specific design (e.g., for cryptographic protocols) is ongoing (Khoruzhii et al., 13 Nov 2025, He et al., 14 Jan 2026).
• Hierarchical and Hybrid Structures: Nesting local block multipliers within global fast multiplication or compression algorithms is essential for achieving both theoretical and real-world gains (Tichavsky et al., 2016, Tichavsky, 2021).
• Exploration of Nonstandard Tensor Products: Extensions to wavelet- or STFT-local transforms, or low-rank schemes for convolution, polynomial, and hypercomplex multiplication, are proposed, with open questions regarding regularization and transform selection (Xu et al., 2019, Molavi et al., 2023).

This field, integrating algebraic complexity, tensor analysis, learning-based search, and high-performance numerics, continues to progress rapidly, driven by both theoretical limits and practical requirements of large-scale scientific and machine learning workloads.