Haar-tSVD: Haar-based Tensor SVD

• Haar-tSVD is a tensor singular value decomposition method that replaces Fourier-based transforms with Haar-based approaches to decouple tensor algebra and capture both global and local correlations.
• It employs discrete Haar wavelets and Haar-distributed orthogonal matrices to achieve effective low-rank approximations, justified by generalized Eckart–Young theorems.
• This technique is applied in imaging denoising and high-dimensional data analysis, offering computational efficiency and real-valued operations while integrating adaptive refinements.

Haar-tSVD is a specialization of tensor singular value decomposition in which the transform used to decouple the tensor algebra is Haar-based. In the algebraic literature, this includes t-SVD under a discrete Haar wavelet transform along the tube dimension and generalized t-SVD under any real orthogonal transform, including a Haar-distributed orthogonal matrix $Q$. In imaging, the term more specifically denotes a unified t-SVD projection combined with Haar transform for patch-group denoising, where global patch-level bases are coupled with a Haar basis on the grouping dimension to capture global and local correlations (Liao et al., 2020, Qi et al., 2021, Kong et al., 29 Dec 2025, Kong et al., 14 Aug 2025).

1. Terminological scope and conceptual variants

A third-order tensor $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$ admits a t-SVD

$\mathcal A=\mathcal U * \mathcal S * \mathcal V^T,$

where $\mathcal U$ and $\mathcal V$ are orthogonal tensors in the t-product sense and $\mathcal S$ is f-diagonal. Haar-tSVD arises when the transform underlying the t-product is chosen to be Haar-related rather than the normalized DFT. One form uses the discrete Haar wavelet as the invertible linear map $\mathcal L_H$ acting along the third mode. Another form replaces the DFT by an arbitrary real orthogonal matrix $Q\in\mathbb R^{p\times p}$, with a Haar-distributed orthogonal $Q$ interpreted as a random draw from $O(p)$. A third and more application-specific form uses the standard orthonormal Haar matrix on the patch-group axis while retaining a t-SVD projection at the patch level (Liao et al., 2020, Qi et al., 2021, Kong et al., 14 Aug 2025).

A common source of confusion is that “Haar” refers to two distinct objects in this literature. In one case it denotes the discrete Haar transform or Haar matrix used as a deterministic orthonormal basis. In the other it denotes Haar measure on the orthogonal group, so that a Haar-distributed orthogonal $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$0 is random. The cited works treat both constructions, but they are not identical specializations of t-SVD (Qi et al., 2021).

2. Tubal and tensorial algebra underlying Haar-tSVD

The generalized framework fixes a positive integer $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$1 and an invertible $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$2 matrix $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$3. Two $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$4-length vectors $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$5, viewed as tubal scalars, are multiplied by

$\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$6

where $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$7 is the Hadamard product. The set of length-$\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$8 vectors with this multiplication is denoted $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$9. An $\mathcal A=\mathcal U * \mathcal S * \mathcal V^T,$0-tubal vector is $\mathcal A=\mathcal U * \mathcal S * \mathcal V^T,$1, and an $\mathcal A=\mathcal U * \mathcal S * \mathcal V^T,$2 tubal matrix is $\mathcal A=\mathcal U * \mathcal S * \mathcal V^T,$3 with each $\mathcal A=\mathcal U * \mathcal S * \mathcal V^T,$4. Such an $\mathcal A=\mathcal U * \mathcal S * \mathcal V^T,$5 may be viewed equivalently as a third-order tensor in $\mathcal A=\mathcal U * \mathcal S * \mathcal V^T,$6. If $\mathcal A=\mathcal U * \mathcal S * \mathcal V^T,$7 and $\mathcal A=\mathcal U * \mathcal S * \mathcal V^T,$8, then the t-product is

$\mathcal A=\mathcal U * \mathcal S * \mathcal V^T,$9

Orthogonality means $\mathcal U$0, with $\mathcal U$1 the diagonal tubal identity (Qi et al., 2021).

A parallel formulation uses t-scalars as elements of $\mathcal U$2 equipped with circular convolution $\mathcal U$3 as the scalar product. A t-matrix is then a matrix whose entries are such t-scalars, and an invertible linear map $\mathcal U$4 is required to turn convolution into pointwise multiplication: $\mathcal U$5 In the $\mathcal U$6-domain, t-matrix multiplication decouples into ordinary matrix products on each slice, which is the structural reason that slice-wise SVD reconstructs a tensor SVD (Liao et al., 2020).

Within the tubal-matrix language, an $\mathcal U$7-diagonal tubal matrix $\mathcal U$8 has zero off-diagonal tubal-scalar entries. It is $\mathcal U$9-diagonal if each diagonal tubal scalar is symmetric, each is positive-semidefinite in the sense that its transform-domain entries are $\mathcal V$0, and the tubal singular values satisfy

$\mathcal V$1

The paper on tubal matrices identifies such $\mathcal V$2-diagonal tensors as the correct analogue of the singular-value block in generalized T-SVD (Qi et al., 2021).

3. Haar specializations of the t-SVD construction

When the discrete Haar wavelet is used as the transform, the mode-3 Haar transform $\mathcal V$3 acts on a 3D array $\mathcal V$4 by

$\mathcal V$5

For a tube of length $\mathcal V$6, the one-step Haar matrix is

$\mathcal V$7

and a multilevel Haar transform $\mathcal V$8 of size $\mathcal V$9 is built by successive dyadic averaging and differencing. Haar-tSVD then proceeds by Haar-transforming along mode 3, computing an ordinary SVD on each frontal slice, truncating to target tubal-rank $\mathcal S$0, reassembling the slice factors, and applying the inverse Haar transform to recover $\mathcal S$1, $\mathcal S$2, and $\mathcal S$3 (Liao et al., 2020).

A second specialization replaces the DFT by an arbitrary real orthogonal $\mathcal S$4. Any real orthogonal $\mathcal S$5 is unitary and real-preserving in the paper’s sense, hence doubly real-preserving. Drawing $\mathcal S$6 from the Haar measure means choosing $\mathcal S$7 uniformly at random from $\mathcal S$8. Once $\mathcal S$9 is fixed, one builds a new t-product $\mathcal L_H$0 and the corresponding tubal scalar module $\mathcal L_H$1. The associated generalized T-SVD is obtained by the mode-3 transform

$\mathcal L_H$2

followed by ordinary matrix SVDs

$\mathcal L_H$3

assembly of $\mathcal L_H$4, $\mathcal L_H$5, and $\mathcal L_H$6 in the transform domain, and inverse transformation with $\mathcal L_H$7, yielding

$\mathcal L_H$8

This formulation retains the exact slice-wise SVD mechanism while changing the transform-domain basis (Qi et al., 2021).

These two constructions share the same decoupling principle but use different Haar objects. The discrete Haar wavelet is deterministic and structured; a Haar-distributed orthogonal $\mathcal L_H$9 is random and generic. This suggests that the phrase “Haar-tSVD” should be interpreted from context rather than treated as a single fixed algorithm.

4. Approximation theory and optimality results

The algebraic justification for Haar-tSVD is supplied by generalized Eckart–Young-type theorems. In the tubal-matrix framework, the Frobenius norm equals the sum of squares of all entries in the transform domain. The tubal rank is defined as

$Q\in\mathbb R^{p\times p}$0

where $Q\in\mathbb R^{p\times p}$1 are the diagonal tubal singular values from the generalized T-SVD. If $Q\in\mathbb R^{p\times p}$2 and $Q\in\mathbb R^{p\times p}$3, then among all $Q\in\mathbb R^{p\times p}$4 with $Q\in\mathbb R^{p\times p}$5, $Q\in\mathbb R^{p\times p}$6 is the unique minimizer of $Q\in\mathbb R^{p\times p}$7, and

$Q\in\mathbb R^{p\times p}$8

A second Eckart–Young theorem is obtained by linearizing the $Q\in\mathbb R^{p\times p}$9 frontal SVDs into a block-diagonal matrix of size $Q$0 and defining the $Q$1-rank through that linearization. Truncating the top $Q$2 entries in the sorted list of slice-wise singular values yields the best approximation of $Q$3-rank $Q$4 in Frobenius norm (Qi et al., 2021).

In the more general t-matrix formulation, any invertible $Q$5 that block-diagonalizes the t-scalar ring may be used, and if $Q$6 is unitary, then the Frobenius norm is preserved in the $Q$7-domain. Because the Haar transform is orthonormal, $Q$8, the factors satisfy $Q$9 and $O(p)$0, and the assembled Haar-tSVD gives the best tubal-rank-$O(p)$1 approximation in the Frobenius norm (Liao et al., 2020).

For the generalized orthogonal-transform setting, the cited paper states that the full characterization of $O(p)$2-diagonal tensors appears in Theorem 2.13, the two Eckart–Young analogues in Theorems 3.5 and 3.6, and stability and norm preservation under any doubly real-preserving unitary transform in Propositions 3.1–3.3 (Qi et al., 2021).

5. Haar-tSVD as a denoising method based on global and local circulant representation

In the denoising literature, Haar-tSVD denotes a computationally simple algorithm that exploits a unified t-SVD projection combined with Haar transform to capture global and local patch correlations. Each color patch $O(p)$3 is represented by a block-circulant matrix $O(p)$4, and multiplication by a block-circulant matrix is equivalent to circular convolution along modes. A group of $O(p)$5 matched patches is organized into a group-circulant matrix $O(p)$6, whose row-Gram matrix is also circulant. Its largest eigenvector is

$O(p)$7

and when $O(p)$8 is a power of two, $O(p)$9 coincides with the first row of the $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$00 Haar matrix. In the more detailed account, the second eigenpair for even $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$01 is

$\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$02

which corresponds to the second Haar basis. The practical consequence is that the Haar basis is used as a predefined surrogate for local PCA basis learning under the group circulant prior (Kong et al., 29 Dec 2025, Kong et al., 14 Aug 2025).

The algorithm separates global and local structure. Global patch-level bases $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$03 and $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$04 are estimated once from many reference patches. Local redundancy across a matched group is then decorrelated by the Haar transform along the fourth mode. For a noisy group $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$05, the forward transform is

$\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$06

followed by hard thresholding

$\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$07

The reconstructed group is

$\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$08

and overlapping group estimates are returned to the image grid by weighted averaging. The method is described as a one-step, parallelizable plug-and-play denoiser that eliminates the need for learning local bases (Kong et al., 29 Dec 2025).

The denoising papers explicitly motivate this construction by the approximation

$\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$09

which supports estimating global t-SVD bases once from many noisy patches, together with the circulant-Haar observation that the dominant PCA direction is already encoded by the Haar constant vector (Kong et al., 29 Dec 2025).

6. Adaptive variants, computational characteristics, applications, and limitations

An adaptive extension, A-Haar-tSVD, augments the basic pipeline with noise estimation. First, a lightweight CNN predicts $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$10 from a finite set of noise levels. Then the second-largest eigenvalue $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$11 of $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$12 is used as an indicator of inner-group similarity. Let $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$13 be the rank position of $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$14 among the sorted eigenvalues. The adjustment rule is

$\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$15

To reduce cost, one account computes $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$16 on a random subset of groups and fuses via majority voting; another describes sampling a few groups per subimage and selecting the final $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$17 by majority voting (Kong et al., 29 Dec 2025, Kong et al., 14 Aug 2025).

A further refinement, RA-Haar-tSVD, integrates deep neural networks through a mean-patch FCN. The FCN takes the noisy group-mean patch $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$18 and outputs a refined mean patch $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$19. During inference, the FCN replaces only the first row of $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$20, preserving other coefficients. The stated rationale is that under severe noise, patch matching and fixed Haar bases may fail, so learning-based refinement can help (Kong et al., 29 Dec 2025).

Reported computational costs depend on the level of abstraction. In the generalized tubal-matrix algorithm with a dense orthogonal $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$21, each mode-3 multiply costs $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$22, $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$23 SVDs of $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$24 cost $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$25, and the total is $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$26. In the image-denoising formulation, per-group complexity is reported as $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$27 via fast Haar in one summary and as $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$28 when $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$29 is treated explicitly in another. This suggests a distinction between fast-Haar implementation and dense Haar multiplication (Qi et al., 2021, Kong et al., 29 Dec 2025, Kong et al., 14 Aug 2025).

Empirical results in the denoising papers place Haar-tSVD between classical patch-based denoisers and larger learned models. One summary states that on synthetic Kodak at $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$30, Haar-tSVD matches CBM3D and MSt-SVD at approximately $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$31 and $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$32, while on DND and SIDD, A-Haar-tSVD yields approximately $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$33 and $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$34, improving approximately $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$35–$\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$36 dB over CBM3D and MSt-SVD. A more detailed table gives $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$37 on DND, $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$38 on SIDD-val, $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$39 on CC15, $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$40 on PolyU, $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$41 on HighISO, and $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$42 on IOCI for A-Haar-tSVD; it also reports $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$43 on CRVD and $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$44 on IOCV for video, and gives Real-HSI, fluorescence microscopy, and MRI results (Kong et al., 29 Dec 2025, Kong et al., 14 Aug 2025).

The same sources describe several practical advantages and limitations. Haar-tSVD on $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$45 is reported to run in approximately $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$46 s on CPU, comparable to CBM3D at $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$47 s and faster than PCA-tSVD methods at approximately $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$48–$\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$49 s, while the CNN noise estimator trains in approximately $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$50 min. In the more algebraic setting, using a real orthogonal transform avoids complex arithmetic; with the discrete Haar transform, computation is $\mathcal A\in\mathbb R^{n_1\times n_2\times n_3}$51 per tube and Haar matrices are well-conditioned and real for real data. Failure modes are reported when noise is non-stationary or patch recurrence is very low, and a potential future remedy is multi-scale downsampling (Kong et al., 29 Dec 2025, Liao et al., 2020).

Across these strands of work, Haar-tSVD denotes a family of tensor factorizations and tensor-processing algorithms that preserve the t-SVD principle while substituting a Haar-related basis for the usual Fourier basis. The common theme is transform-domain decoupling with retained orthogonality and low-rank optimality; the main divergence lies in whether “Haar” means a wavelet basis, a Haar-distributed orthogonal transform, or the predefined group-wise Haar matrix used in denoising.