Basis Decomposition Module (BDM)

• Basis Decomposition Module is a framework that expresses complex structures as linear combinations of basis elements, enhancing compression and interpretability.
• In quantum simulation, BDM enables exact operator decomposition via spectral methods to reduce computational complexity and leverage symmetry.
• For neural architectures and vision, BDM refactors projection matrices and feature tensors to lower computational load while preserving essential information.

A Basis Decomposition Module (BDM) refers to a mathematical or computational mechanism that decomposes complex structures—such as operators, matrices, tensors, features, or even entire modules—into linear (or, in some contexts, algorithmically minimal) combinations of basis elements. The specific construction of BDM may differ significantly between fields including algorithmic complexity, quantum simulation, neural attention, visual representation learning, and persistence modules, but the unifying thread is the use of bases to factor, compress, or interpret high-dimensional structures via modular decomposition.

1. Formal Definitions and Mathematical Frameworks

A BDM is characterized by the principle of expressing a target object $X$ (e.g., a vector, matrix, tensor, function, or module) as a combination of basis elements $\{b_i\}$ with coefficients $\{\alpha_i\}$ that are computed via task- and field-specific procedures. In the setting of neural networks and feature maps, a BDM typically performs

$X \approx \sum_{i=1}^c \alpha_i(P_i) P_i$

where $P_i$ are learned or fixed basis feature representations and $\alpha_i$ are decomposition coefficients computed by inner products or linear projections. In matrix factorization scenarios, e.g., attention or quantum simulation, BDM exploits exact or low-rank factorizations such as $W = [I_r; C]B$ to achieve compression and accelerated computation (Zhao, 2 Oct 2025, Kaicher, 2021). In persistence module decomposition, BDM identifies interval bases that permit direct sum decomposition into indecomposable modules (Gregorio et al., 2021). For algorithmic complexity, BDM approximates the complexity of $X$ by partitioning it into sub-blocks and combining the code lengths of their minimal programs (Zenil et al., 2016).

2. BDM in Algorithmic Complexity Estimation

Within the context of algorithmic complexity, BDM extends the Coding Theorem Method (CTM) by partitioning an object $X$ into small, overlapping blocks and accumulating their precomputed CTM complexities, together with the bits needed to code their multiplicities. The key formula is

$$\mathrm{BDM}(X) = \sum_i \left[ \mathrm{CTM}(r_i) + \log_2 n_i \right]$$

where $\{r_i\}$ are the unique sub-blocks under chosen size/overlap parameters and $n_i$ are their counts. This approach makes BDM a hybrid estimator that interpolates between exact CTM for small blocks and a block-wise Shannon entropy measure when CTM becomes degenerate or unavailable for large structures. Bounds relate the BDM estimate to true algorithmic complexity $K(X)$: $$K(X) \leq \mathrm{BDM}(X) + O(\log^2|A|) + \epsilon$$ (Zenil et al., 2016).

BDM provides efficient, polynomial-time complexity estimation for multi-dimensional data (arrays, tensors, graphs) by using a w-dimensional CTM table. Asymptotically, BDM converges to the Shannon entropy for maximal-entropy objects, but detects algorithmic regularities that entropy and standard compressors (e.g., LZ77, BZip2) systematically miss.

3. Basis Decomposition Modules in Quantum Simulation

In quantum simulations, particularly of electronic structure Hamiltonians, BDM refers to an exact sum-of-squares, pairwise decomposition of the two-body operator. The process involves flattening the four-index two-electron tensor into a $N_f^2 \times N_f^2$ matrix $H$, spectral decomposition $H = O\Sigma O^T$, followed by reshaping and splitting each eigenmode into symmetric and antisymmetric parts. Each block is further diagonalized, yielding a sum of pairwise number-operator products:

$$H_2 = \sum_{\ell \in S} \frac{1}{2}\sigma_\ell \sum_{a,b} \mu_{\ell,a}\mu_{\ell,b} n_{\ell,a} n_{\ell,b} + \sum_{\ell \in A} \frac{1}{2}\sigma_\ell \sum_{a,b} (i\nu_{\ell,a})(i\nu_{\ell,b}) n_{\ell,a} n_{\ell,b}$$

This construction reduces computational cost, exploits sparsity or truncation, and is valid for complex-valued basis sets (Kaicher, 2021). The corresponding BDM pseudocode takes antisymmetrized integrals, forms and spectrally decomposes $H$, processes each mode, and finally returns a compact list of modes for simulation backends.

4. BDM in Neural Architectures and Attention Acceleration

In LLMs and vision architectures, BDM (also denoted BD or BD Attention) leverages an exact matrix identity to factor multi-head projection matrices into linear combinations of basis rows or columns. For a given $W \in \mathbb{R}^{m \times n}$ of rank $r$, the representation

$$W = \begin{bmatrix} I_r \ C \end{bmatrix} B$$

where $B$ collects linearly independent basis rows and $C$ encodes coefficient matrices, is utilized to rewrite multi-head Q/K/V projections, reducing both parameter count and floating-point operations by up to 25%, with negligible impact on perplexity and no retraining (Zhao, 2 Oct 2025). The same decomposition is applied to attention outputs (VO products), and the approach is compatible and orthogonal to engineering optimizations such as FlashAttention.

A key feature is the maintenance of lossless or numerically exact outputs, as the transformation preserves full-rank properties almost surely in practice. The BDM replacement is performed during an offline phase, requiring only a few seconds even for large-scale models, and inference employs fused kernel operations on the re-factored projections.

5. Feature-Space BDM for Compact Representation in Vision

For small target detection and other dense prediction tasks in vision, BDM projects feature tensors onto task-adaptive, typically high-frequency or motion-difference basis kernels. Let $T \in \mathbb{R}^{B\times G\times 1\times F}$ be the reshaped feature map, and $P=[P_1,\ldots,P_c]$ the stack of basis features. BDM computes

$S = T P^\top, \qquad O = S P$

reconstructing a selective approximation that amplifies features aligned with the basis (target-like patterns) and suppresses redundant or background structure. Bases in modules such as SD$^2$M (spatial difference decomposition) are designed to emphasize edge or difference patterns at multiple dilations, while temporal variants (TD$^2$M) exploit frame differences to extract motion cues (Hu et al., 3 Dec 2025).

Empirically, learned difference bases outperform hand-crafted and strictly orthogonal bases. BDM incurs minimal parameter and compute overhead, enabling real-time applications at hundreds of frames per second while achieving SOTA mean IoU on ISTD benchmarks.

6. BDM in Persistence Module and Topological Data Analysis

In algebraic topology and TDA, BDM refers to parallelized algorithms for computing interval bases of persistence modules, viewed as representations of type-$A_n$ quivers or graded modules over $\mathbb{F}[x]$. Each module $M$ decomposes as a direct sum of indecomposable intervals, and BDM computes a set of homogeneous generators whose cyclic submodules give a direct sum decomposition:

$M \cong \bigoplus_{m=1}^r I_{[b_m,d_m]}$

The parallel BDM algorithm tracks kernel flags of structure maps, identifies basis elements per filtration step, and assembles the global basis with minimal inter-processor communication. The algorithm achieves $O(\bar{m}^2 m)$ complexity, outperforming classical approaches based on Smith Normal Form, especially for high-step, small-width modules (Gregorio et al., 2021). It applies directly to persistent homology and can be extended to track harmonic generators via Hodge decomposition.

7. Comparative Performance and Implementation Considerations

Field/Context	BDM Role	Principal Gains
Algorithmic complexity	Complexity estimation	Detects nonstatistical regularities. Hybrid estimator interpolates between CTM and entropy (Zenil et al., 2016)
Quantum simulation	Operator decomposition	Reduces exponent scaling of two-body term simulation; exploits symmetry and truncation (Kaicher, 2021)
Neural architectures	Exact parameter reduction	25% weight compression, 30%+ speedup, lossless inference in MHA (Zhao, 2 Oct 2025)
Computer vision (ISTD, etc.)	Feature factorization	SOTA detection mIoU, interpretable basis selection, negligible params (Hu et al., 3 Dec 2025)
Persistence modules	Interval basis computation	Parallel, scalable, outperforms SNF (Gregorio et al., 2021)

Across domains, BDM modules combine theoretical compressibility, computational leverage, and adaptability. In learning architectures, bases may be fixed, hand-crafted, or learned end-to-end. In information-theoretic contexts, bases encode minimal programs or symbolic patterns. In operator-theoretic settings, BDM harmonizes low-rank and symmetry constraints with numerical and storage efficiency. In combinatorial or algebraic settings, BDM algorithms parallelize classical decomposition procedures with provable correctness and optimality.

8. Limitations and Extensions

BDM’s limitations are domain-specific. Large block sizes may degrade approximation quality if precomputed tables or basis sets become intractable. In quantum algorithms, spectral decomposition bottlenecks dominate for very large $N_f$. In deep models, BDM may require alignment strategies for pathological weight matrices, and in vision, strictly orthogonal bases are suboptimal and computationally expensive relative to non-orthogonal or task-adapted schemes. Open directions include the integration of BDM with quantization, pruning, dynamic basis selection, joint factorization across multiple network layers, and expansion to multimodal data decompositions (Zhao, 2 Oct 2025, Hu et al., 3 Dec 2025).

BDM modules, through their shared principle of basis-driven factorization, continue to deliver scalable gains and interpretability across the computational sciences.