Trainable Rank Decomposition Matrices

• Trainable rank decomposition matrices are structured, trainable approximations that decompose large weight matrices into low-rank factors for adaptive deep learning.
• They enable dynamic adaptation and efficient fine-tuning by optimizing low-dimensional subspaces using gradient-based methods and regularization techniques.
• Empirical results show techniques like LoRA, EDoRA, and LoRA-Mini achieve significant parameter reductions while maintaining or improving performance in model compression and tuning.

Trainable rank decomposition matrices provide a parameter-efficient and structured approach to representing or adapting large matrices and tensors, particularly in deep learning models. Unlike classical low-rank approximations calculated post hoc (e.g., via SVD), these methods endow the decomposition—often the low-rank factors themselves, or associated update rules—with trainable parameters, enabling dynamic adaptation, compression, or fine-tuning under explicit rank constraints. Major variants include trainable low-rank adapters for neural network fine-tuning, adaptive or mask-driven rank selection via gradient-based optimization, structured variational decompositions with rank/conditioning control, and IRLS-inspired quadratic regularization schemes for prescribed-rank training. These strategies underpin the latest advances in parameter-efficient tuning of LLMs, network compression, low-rank training, and task-specific adaptation.

1. Foundational Concepts and Motivation

Trainable rank decomposition matrices generalize static or analytical low-rank approximations by introducing learnable structures into one or more factors of the decomposition. Let $W\in\mathbb{R}^{m\times n}$ denote a weight matrix from a neural network layer (e.g., transformer attention or MLP layer). The central idea is to express $W$ (or its adaptation/fine-tuning update) as the product of low-rank factors parameterized to admit trainable degrees of freedom, thereby:

• Restricting updates/adaptation to a low-dimensional subspace to control parameter count.
• Incorporating learned or data-driven structure (as opposed to a pure SVD).
• Enabling adaptive selection of rank or support for layer-wise heterogeneity.
• Providing compatibility with existing architectures via differentiable modules.

Key motivations include memory- and compute-constrained fine-tuning of large models (e.g., LLMs), fast layer-wise adaptation, network compression and acceleration, and explicit control over the trade-off between representation capacity, generalization, and efficiency.

2. Parameter-Efficient Tuning via Trainable Low-Rank Factors

A primary application domain is parameter-efficient fine-tuning (PEFT) of pre-trained networks, where trainable rank decomposition matrices replace full matrix adaptation with structured, low-footprint updates. Representative methods include LoRA, LoRA-Mini, DoRA, and EDoRA:

LoRA (Low-Rank Adaptation):

A frozen weight $W$ receives a trainable rank-$r$ update parameterized as $\Delta W = AB$, with $A\in\mathbb{R}^{m \times r}$ and $B\in\mathbb{R}^{r \times n}$ (Singh et al., 24 Nov 2024). The fine-tuned weight is $W' = W + AB$. Trainable parameter count is $P_{\rm LoRA} = r(m+n)$.

EDoRA (Efficient Weight-Decomposed Low-Rank Adaptation):

A matrix $W$ undergoes truncated SVD, $W = U\Sigma V^\top$, $U\in\mathbb{R}^{m\times r}$, $V\in\mathbb{R}^{n\times r}$, $\Sigma \in \mathbb{R}^{r\times r}$.

• $U$ and $V$ are frozen (the principal subspaces), and a small trainable core $M\in\mathbb{R}^{r\times r}$ is injected: $W' = U \Sigma M V^\top$.
• For diagonal $M$, only $r$ parameters are trainable; for full $M$, $r^2$ parameters. This reduces the adaptation footprint dramatically compared to LoRA, e.g., for GPT-3 with $m=n=12,288$, $r=16$: $P_{\rm LoRA}\approx393,216$ vs $P_{\rm EDoRA}=256$ (Nasiri et al., 21 Jan 2025).
• Initialization sets $M=I + \epsilon$ to preserve initial performance and facilitate stable optimization.

LoRA-Mini:

Further decomposes the LoRA matrices $A$ and $B$ into four factors, training only the two innermost ("inner") matrices while keeping the "outer" matrices frozen. For $A=A_\mathrm{aux}A_\mathrm{train}$, $B=B_\mathrm{train}B_\mathrm{aux}$, with only $A_\mathrm{train}\in\mathbb{R}^{a\times r}$ and $B_\mathrm{train}\in\mathbb{R}^{r\times b}$ trainable, the update is $$\Delta W = A_\mathrm{aux}A_\mathrm{train}B_\mathrm{train}B_\mathrm{aux}$$. Trainable parameter count is $r(a+b)$, yielding up to $20\times$ reduction over standard LoRA while retaining performance (Singh et al., 24 Nov 2024).

Comparison Table: Parameter Counts

Method	Trainable Parameters	Expressivity Note
LoRA	$r(m+n)$	Standard low-rank update
EDoRA (full $M$)	$r^2$	Can mix principal subspaces
EDoRA (diag $M$)	$r$	Only scales singular directions
LoRA-Mini	$r(a+b)$	Only two compact inner matrices train

Empirical benchmarks indicate that these low-rank adaptation schemes (especially EDoRA) achieve competitive or superior downstream accuracy with orders-of-magnitude reduction in trainable parameters, as corroborated on GLUE and MT benchmarks (Nasiri et al., 21 Jan 2025, Singh et al., 24 Nov 2024).

3. Adaptive Rank Selection and Structured Decomposition

Adaptive mechanisms for selecting or optimizing ranks (rather than pre-specifying them) are implemented via binary masks, regularization, or data-driven ranking of basis elements:

MARS (Masked Automatic Ranks Selection):

Imposes learnable binary masks $m_k\in\{0,1\}^{r_k}$ on the cores of a tensor decomposition (e.g., Tensor-Train, Tucker). During training, mask variables $\phi_k(s)\in[0,1]$ are optimized by relaxed MAP via Continuous Concrete distributions and a Bernoulli prior favoring sparsity. The final effective rank is the number of mask values learned to be nonzero. This provides fully trainable, data-driven rank adaptation and can yield $>100\times$ compression with minimal accuracy loss on network benchmarks (Kodryan et al., 2020).

Maestro/LoD (Low-rank Ordered Decomposition):

Factorizes each weight as $W^i = U^i (V^i)^\top$, with columns ordered by importance via nested ordered dropout. Hierarchical group-lasso penalties drive unimportant columns to zero and enable progressive, per-layer rank shrinkage during training. The process probabilistically samples effective rank and prunes based on learned groupwise norm thresholds, resulting in empirical footprint reductions of $4\times$–$9\times$ (ResNet, VGG) and accuracy preservation (Horvath et al., 2023).

4. Variational and Regularized Low-Rank Training

Beyond analytical factorization, several frameworks directly impose trainable decompositions through variational or regularized formulations.

Structured Variational D-Decomposition:

Represents $A\approx PDQ$, $P\in\mathbb{R}^{n\times k}$, $D\in\mathbb{R}^{k\times k}$, $Q\in\mathbb{R}^{k\times n}$, with an augmented objective minimizing $\|A - PDQ\|_F^2 + \lambda \mathcal{R}(P,D,Q)$. The regularizer $\mathcal{R}$ accommodates squared norms, sparsity terms, and a log-condition-number penalty on $D$ for numerical stability. Optimization proceeds via block-coordinate descent, with each sweep costing $O(n^2k)$. This variational decomposition yields stable, efficient low-rank models competitive with (or outperforming) SVD, CUR, and sparse PCA, especially under noise or ill-conditioning (Katende, 10 Jun 2025).

Q3R (Quadratic Reweighted Rank Regularizer):

Facilitates prescribed-rank training by imposing a smoothed log-determinant surrogate for matrix rank:

$R_\epsilon(W) = \sum_k f_\epsilon(\sigma_k(W)) ,$

where $f_\epsilon$ is quadratic below threshold $\epsilon$ and logarithmic above. This is majorized at each iteration via IRLS, leading to a dynamically updated quadratic regularizer $Q^t(W) = \frac{1}{2}\mathrm{Tr}[W^\top A^t W]$, where $A^t$ is constructed from the partial SVD of the current weight. Integration with first-order optimizers is straightforward, and rank targets are enforced by shrinking $\epsilon$ toward the smallest unwanted singular value. Q3R achieves strong empirical sparsity-accuracy trade-offs for transformers, outperforming LoRA/LoRITa at high-compression regimes (Ghosh et al., 6 Nov 2025).

5. Data-Driven and Learning-Based Low-Rank Factorization

Data-driven low-rank approximations leverage learning to optimize reduced representations under structured constraints.

Learning-Based Low-Rank Approximations:

Given a collection of input matrices, instead of random sketch matrices $S$ (as in classical sketch-and-SVD), the nonzero elements of $S$ become trainable parameters, optimized by backpropagating through a differentiable SVD pipeline. This learned sketch minimizes Frobenius reconstruction loss over the training set and can be hybridized with random sketches for worst-case approximation guarantees. Experimental results demonstrate up to $20\times$ lower approximation error relative to random projections, with negligible additional computational cost (Indyk et al., 2019).

DeepTensor:

Extends the notion of trainable factors into the nonlinear regime by parameterizing the low-rank factors $U$, $V$ as outputs of small untrained deep networks, optimized to minimize mean-square error. DeepTensor leverages implicit regularization by network structure to achieve denoising, completion, and classification performance superior to classical SVD/PCA under non-Gaussian noise, and provides fast, separable decompositions for tensors (Saragadam et al., 2022).

6. Empirical Impact and Application Domains

Trainable rank decomposition matrices underpin several high-impact applications:

• LLM and transformer fine-tuning: EDoRA, LoRA-Mini, and Q3R enable sublinear memory/compute footprint, faster adaptation, and robust performance on benchmark NLP tasks (GLUE, MT, code generation) (Nasiri et al., 21 Jan 2025, Singh et al., 24 Nov 2024, Ghosh et al., 6 Nov 2025).
• Compression and acceleration: LoRD and Maestro deliver $15$–$25\%$ parameter compression and $15$–$22\%$ inference speedup for large LLMs and DNNs, with layerwise adaptability and drop-in PyTorch compatibility (Kaushal et al., 2023, Horvath et al., 2023).
• Adaptive model selection: MARS and Maestro dynamically select or prune ranks during training, obviating cumbersome post-hoc selection or iterative pruning (Kodryan et al., 2020, Horvath et al., 2023).
• Structured regularization: D-decomposition and Q3R allow explicit control of conditioning, spectral alignment, and rank, benefiting ill-conditioned, multiclass, or high-noise matrix factorization (Katende, 10 Jun 2025, Ghosh et al., 6 Nov 2025).

7. Limitations, Open Questions, and Future Outlook

Common limitations include:

• Dependence on the expressivity of the top-$r$ subspace (misses fine-grained directions if $r$ is too small).
• Hyperparameter sensitivity for rank, regularization strength, and initialization.
• Non-convexity and potential local minima in variational/lasso-regularized settings.
• Remaining gap in automated, scalable rank selection for large heterogeneous architectures.

Open directions include integration of nonlinear or attention-based “mixers” in place of linear trainable cores, blockwise or local low-rank parameterization for vision and multimodal processing, and seamless embedding of adaptive rank decomposition within emerging foundation model pipelines.

References:

• "EDoRA: Efficient Weight-Decomposed Low-Rank Adaptation via Singular Value Decomposition" (Nasiri et al., 21 Jan 2025)
• "LoRA-Mini : Adaptation Matrices Decomposition and Selective Training" (Singh et al., 24 Nov 2024)
• "MARS: Masked Automatic Ranks Selection in Tensor Decompositions" (Kodryan et al., 2020)
• "Maestro: Uncovering Low-Rank Structures via Trainable Decomposition" (Horvath et al., 2023)
• "Q3R: Quadratic Reweighted Rank Regularizer for Effective Low-Rank Training" (Ghosh et al., 6 Nov 2025)
• "Structured Variational $D$-Decomposition for Accurate and Stable Low-Rank Approximation" (Katende, 10 Jun 2025)
• "Learning-Based Low-Rank Approximations" (Indyk et al., 2019)
• "DeepTensor: Low-Rank Tensor Decomposition with Deep Network Priors" (Saragadam et al., 2022)
• "LORD: Low Rank Decomposition Of Monolingual Code LLMs For One-Shot Compression" (Kaushal et al., 2023)