Tensor Volumetric Operator (TenVOO) in 3D MRI Models

• The paper introduces TenVOO as a PEFT method that uses tensor network contractions to update 3D convolution kernels while preserving volumetric spatial structure.
• It achieves state-of-the-art MS-SSIM and competitive FID/MMD on various MRI datasets, reducing trainable parameters by over 300× compared to full fine-tuning.
• TenVOO’s design explicitly separates spatial indices across tensor cores, effectively modeling complex 3D anatomical dependencies for improved domain adaptation.

Searching arXiv for the specified paper to ground the article in the current record. Tensor Volumetric Operator (TenVOO) is a parameter-efficient fine-tuning (PEFT) method for three-dimensional U-Net-based denoising diffusion probabilistic models (DDPMs) applied to magnetic resonance imaging (MRI) generation. It was introduced to address the limited availability of PEFT schemes specifically tailored to 3D convolutional backbones, where preserving volumetric spatial structure is central to anatomical fidelity. TenVOO represents updates to 3D convolution kernels through tensor networks (TNs), replacing direct optimization of full 5D kernels with contractions of lower-order core tensors. In the reported setting, a DDPM pretrained on 59,830 T1-weighted UK Biobank brain MRI scans is adapted to ADNI, PPMI, and BraTS2021, and TenVOO is reported to achieve state-of-the-art performance in multi-scale structural similarity index measure (MS-SSIM) while requiring only $0.3\%$ of the trainable parameters of the original model (Li et al., 24 Jul 2025).

1. Conceptual basis and problem setting

TenVOO was proposed for PEFT of a large 3D U-Net-based DDPM whose noise predictor is a 3D U-Net, $\epsilon_\theta(x_t, t)$. The underlying motivation is the cost of adapting large volumetric generative models to multiple downstream datasets when full fine-tuning requires updating all convolutional parameters. For the model studied, full fine-tuning involves 166.67M trainable parameters, which is expensive in memory and storage if separate adapted models must be maintained for multiple domains. The same setting also increases training time and GPU memory usage because gradients and optimizer state must be tracked for the full parameter set (Li et al., 24 Jul 2025).

The central technical obstacle is the structure of 3D convolution. A 3D convolution layer has weight tensor

$$\mathcal{W} \in \mathbb{R}^{C_{out} \times C_{in} \times k_d \times k_h \times k_w},$$

with $C_{out} C_{in} k_d k_h k_w$ parameters. In volumetric MRI synthesis, these kernels must preserve complex 3D spatial dependencies that govern cortical geometry, ventricular topology, tissue boundaries, and lesion morphology. Existing PEFT methods such as LoRA, LoKr, and LoHa were described as being mainly designed for 2D convolutions or Transformers. When extended to 3D convolution, they still rely on simple low-rank factorizations across channels and kernel dimensions, and the spatial dimensions $(k_d, k_h, k_w)$ are not explicitly structured to model 3D volumetric dependencies. The reported experiments associate these methods with weaker MS-SSIM and structural distortions in generated 3D MRI (Li et al., 24 Jul 2025).

TenVOO addresses this limitation by parameterizing the kernel update $\Delta \mathcal{W}$ with a tensor network. Rather than learning a full dense residual kernel, it learns a compact network of tensor cores whose contraction is reshaped into the 3D convolution kernel. The stated design objective is not merely compression, but structured representation: spatial indices are separated across different TN cores so that volumetric dependencies are modeled explicitly through contraction topology rather than being absorbed into a generic low-rank matrix factorization.

2. Tensor-network formulation

The baseline 3D convolutional layer is written as

$\mathcal{Y} = \mathcal{W} * \mathcal{X} + b,$

where $\mathcal{X}$ is the input feature map, $b$ is the bias term, and $*$ denotes 3D convolution. The PEFT form freezes $\epsilon_\theta(x_t, t)$0 and learns only an update: $\epsilon_\theta(x_t, t)$1 In TenVOO, the update is generated by tensor contraction: $\epsilon_\theta(x_t, t)$2 where $\epsilon_\theta(x_t, t)$3 are trainable TN cores and $\epsilon_\theta(x_t, t)$4 denotes their contraction (Li et al., 24 Jul 2025).

A key preparatory step is channel tensorization. The input and output channels are factorized as

$\epsilon_\theta(x_t, t)$5

These factors, together with the spatial indices $\epsilon_\theta(x_t, t)$6, are distributed across different TN cores. This placement is central to the method: the paper emphasizes that each spatial dimension is handled separately within the network rather than merged into a single matrix-like dimension.

Two variants are defined.

Variant	Structural characterization	Parameter count
TenVOO-L	LoRA-like TN design; chain-like TN with separated spatial dimensions	$\epsilon_\theta(x_t, t)$7
TenVOO-Q	QuanTA-like TN design; more internal rank-only cores and high effective rank	$\epsilon_\theta(x_t, t)$8

TenVOO-L is described as a chain-like TN whose cores correspond to combinations of input-channel factors, output-channel factors, and spatial dimensions. Its principal design property is spatial separation: each of $\epsilon_\theta(x_t, t)$9, $$\mathcal{W} \in \mathbb{R}^{C_{out} \times C_{in} \times k_d \times k_h \times k_w},$$0, and $$\mathcal{W} \in \mathbb{R}^{C_{out} \times C_{in} \times k_d \times k_h \times k_w},$$1 is attached to a different core. The paper also states that if one kernel dimension is removed, TenVOO-L degenerates to a 2D-convolution TN.

TenVOO-Q extends QuanTA to the 5D convolution kernel. Its defining distinction is the presence of more internal $$\mathcal{W} \in \mathbb{R}^{C_{out} \times C_{in} \times k_d \times k_h \times k_w},$$2 cores, consistent with a quantum-circuit-inspired TN that maintains high effective rank from small cores. The difference between $$\mathcal{W} \in \mathbb{R}^{C_{out} \times C_{in} \times k_d \times k_h \times k_w},$$3 and $$\mathcal{W} \in \mathbb{R}^{C_{out} \times C_{in} \times k_d \times k_h \times k_w},$$4 reflects a different allocation of the physical indices across the network, including the appearance of $$\mathcal{W} \in \mathbb{R}^{C_{out} \times C_{in} \times k_d \times k_h \times k_w},$$5 in the $$\mathcal{W} \in \mathbb{R}^{C_{out} \times C_{in} \times k_d \times k_h \times k_w},$$6 term for TenVOO-Q (Li et al., 24 Jul 2025).

3. Reparameterization and efficiency

The paper reports that naively imposing $$\mathcal{W} \in \mathbb{R}^{C_{out} \times C_{in} \times k_d \times k_h \times k_w},$$7 at initialization can hurt performance because of the TN structure. To address this, TenVOO uses a frozen-copy reparameterization inspired by QuanTA. The TN cores are initialized as $$\mathcal{W} \in \mathbb{R}^{C_{out} \times C_{in} \times k_d \times k_h \times k_w},$$8, and a frozen copy of the initial TN contraction is stored. The convolution is then defined by subtracting the frozen initial TN output and adding the current trainable TN output: $$\mathcal{W} \in \mathbb{R}^{C_{out} \times C_{in} \times k_d \times k_h \times k_w},$$9 with $C_{out} C_{in} k_d k_h k_w$0 still given by the reshaped contraction of the current trainable cores (Li et al., 24 Jul 2025).

This reparameterization ensures that at initialization the effective kernel is approximately $C_{out} C_{in} k_d k_h k_w$1, because the frozen TN output and the initial trainable TN output coincide. During optimization, only the TN cores are updated while the frozen copy remains fixed. The method therefore behaves as a residual adaptation around the pretrained 3D convolution kernel, but with the residual itself constrained by a tensor-network parameterization.

At the model level, the reported trainable parameter counts are 0.60M for TenVOO-L and 0.58M for TenVOO-Q, compared with 166.67M for full fine-tuning. The corresponding ratios are approximately 0.36% and 0.35% of the full parameter count, consistent with the abstract’s rounded statement of “only $C_{out} C_{in} k_d k_h k_w$2 of the trainable parameters.” The paper characterizes this as about a 300× reduction in trainable parameters.

The work does not provide explicit FLOP formulas. It states qualitatively that inference cost does not significantly decrease if the TN is merged into the effective convolution kernel, because the forward convolution still uses the full kernel. The practical efficiency gain lies instead in training and storage: gradients are computed only for the TN cores, optimizer state is much smaller, and per-dataset PEFT modules are lightweight to store (Li et al., 24 Jul 2025).

4. Incorporation into the 3D DDPM

The DDPM follows a standard formulation. The forward diffusion process is

$C_{out} C_{in} k_d k_h k_w$3

and the reverse process is

$C_{out} C_{in} k_d k_h k_w$4

with the mean parameterized through the noise predictor $C_{out} C_{in} k_d k_h k_w$5 (Li et al., 24 Jul 2025).

The U-Net contains 3D convolutional ResNet blocks, attention blocks with value and query projections, time embedding and projection layers, and the usual downsampling, upsampling, and skip-connection structure. TenVOO is integrated as a PEFT mechanism at fine-tuning time rather than as part of pretraining. The pretrained convolution kernels are loaded and frozen, while newly initialized TN cores are introduced to parameterize the residual kernel updates.

The modules adapted in the main PEFT setting are:

• ResNet 3D convolutional layers
• Value and query projection layers in ResNet attention blocks
• Time embedding and time projection layers

For convolutional layers, the adaptation uses TenVOO-L or TenVOO-Q. For linear layers, the paper states that it “directly integrate[s] QuanTA,” tensorizing input and output feature dimensions into 3rd-order tensors and constructing a QuanTA TN analogously to the TenVOO-Q convolutional variant. Linear layers account for about 10% of the trainable parameters in the original model, and these are also parameter-reduced through QuanTA. In the main PEFT experiments, other components, including upsampling and downsampling convolutions and skip pathways, remain frozen. In the joint fine-tuning setting, these non-PEFT layers are also fully fine-tuned (Li et al., 24 Jul 2025).

A common misconception would be to treat TenVOO as a change to the pretrained generative architecture itself. The reported workflow is more specific: pretraining on UK Biobank is performed as a standard unconditional DDPM, and TenVOO is inserted only during downstream adaptation.

5. MRI datasets, training protocol, and reported results

The pretraining corpus consists of 59,830 T1-weighted brain MRI scans from UK Biobank (UKB). These scans are resampled to $C_{out} C_{in} k_d k_h k_w$6, skull-stripped, and registered to MNI152 space. The DDPM is implemented with MONAI and is unconditional. For downstream adaptation, three T1-weighted brain MRI datasets are used: PPMI with 979 scans, ADNI with 545 scans, and BraTS2021 with 327 T1-weighted scans selected from 1251 training samples based on visual quality, following Dorjsembe et al. PPMI and ADNI are processed with the CAT12 toolbox using resampling to $C_{out} C_{in} k_d k_h k_w$7, skull stripping, bias correction, tissue segmentation, registration to MNI152, and intensity normalization. BraTS2021 uses resampling to $C_{out} C_{in} k_d k_h k_w$8 and official skull-stripping preprocessing. UKB, PPMI, and ADNI are padded and cropped to $C_{out} C_{in} k_d k_h k_w$9; BraTS2021 is padded and cropped to $(k_d, k_h, k_w)$0; all are then resized to $(k_d, k_h, k_w)$1. Each downstream dataset uses a 90%/10% train/evaluation split (Li et al., 24 Jul 2025).

Fine-tuning uses MSE on predicted noise, Adam, a learning rate of $(k_d, k_h, k_w)$2, batch size 1, and 4 gradient accumulation steps, giving an effective batch size of 4. The TN rank is set to 4 in the main TenVOO experiments.

The evaluation metrics are FID, MMD, and MS-SSIM, with FID and MMD scaled by $(k_d, k_h, k_w)$3. Med3D is used to compute feature representations for FID and MMD. In the PEFT-only setting, the reported results are as follows:

• On ADNI, TenVOO-L achieves the best PEFT MS-SSIM at 0.663, while TenVOO-Q gives the best PEFT FID and MMD at 13.475 and 9.502.
• On PPMI, TenVOO-Q achieves the best MS-SSIM at 0.756, exceeding full fine-tuning’s 0.646.
• On BraTS2021, TenVOO-L achieves the best FID, MMD, and MS-SSIM at 3.168, 0.908, and 0.581, outperforming full fine-tuning and all PEFT baselines.

The paper concludes from these results that TenVOO attains state-of-the-art MS-SSIM and competitive FID/MMD while using fewer parameters than LoRA, LoKr, and LoHa and less than $(k_d, k_h, k_w)$4 of the full-fine-tuning parameter count (Li et al., 24 Jul 2025).

In the joint fine-tuning setting, where non-PEFT layers are also updated, all methods use about 47M trainable parameters. TenVOO-L gives the best MS-SSIM on ADNI (0.804) and BraTS2021 (0.715), as well as markedly stronger BraTS2021 FID/MMD (1.256 / 0.190). TenVOO-Q gives the best MS-SSIM on PPMI (0.837). The reported ablation over $(k_d, k_h, k_w)$5 shows that MS-SSIM improves as rank increases, while parameter count also rises according to the polynomial dependence in the parameter-count expressions. The chosen $(k_d, k_h, k_w)$6 is described as a good trade-off between parameter efficiency and structural quality.

6. Interpretation, limitations, and research context

The qualitative results reported in the paper align with the metric-level emphasis on structural fidelity. Baseline PEFT methods are described as producing structural distortions in generated brain MRIs, including deformed cortical surfaces, blurred or inconsistent tissue boundaries, and artifacts around ventricles or lesions. TenVOO-based models are described as preserving global brain shape and local anatomical structures more faithfully, with especially strong behavior on BraTS2021, where tumor regions and surrounding tissues show sharper structure and more realistic spatial placement (Li et al., 24 Jul 2025).

This pattern is used in the paper to support the claim that TenVOO better captures complex 3D spatial dependencies. The strongest empirical indicators are the MS-SSIM improvements across datasets, the rank-ablation trend in which higher rank improves MS-SSIM, and the particularly large gains on BraTS2021, which is structurally distinct from UK Biobank pretraining data. A plausible implication is that explicit separation of spatial indices within the TN becomes most beneficial when the downstream domain shift is driven by volumetric anatomical reconfiguration rather than only by subtle distributional changes.

The work also states several limitations and assumptions. Performance is sensitive to the TN rank $(k_d, k_h, k_w)$7; small $(k_d, k_h, k_w)$8 may underfit, while large $(k_d, k_h, k_w)$9 increases parameter count and potentially contraction cost. Tensor-network contraction introduces overhead, although the paper describes this as modest relative to 3D convolution and notes that the effective kernel can be pre-merged for inference. The method is designed and evaluated on T1-weighted brain MRI, and its tensorization choices or ranks may require modification for other modalities or anatomical regions. The reported behavior on PPMI is more mixed in FID and MMD, suggesting that a PEFT method emphasizing spatial adaptation may not align equally well with every dataset shift. The underlying assumption is that the pretrained 3D DDPM is sufficiently general that small TN-based updates can adapt it to new domains (Li et al., 24 Jul 2025).

Within the paper’s own framing, TenVOO exemplifies the use of tensor networks as efficient parameterizations for deep models, but its direct contribution is narrower and more specific: a PEFT mechanism for 3D convolutional diffusion models in medical imaging. Its significance lies in showing that volumetric structure can be encoded in the adaptation mechanism itself, rather than left to generic low-rank residuals that do not explicitly organize spatial dimensions.