---
title: 'Haar-tSVD: Haar-based Tensor SVD'
url: https://www.emergentmind.com/topics/haar-tsvd
type: topic
---

# Haar-tSVD: Haar-based Tensor SVD

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 [2001.11708] [2105.00793] [2512.23569] [2508.10307].

## 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 [2001.11708] [2105.00793] [2508.10307].

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 \(Q\) is random. The cited works treat both constructions, but they are not identical specializations of t-SVD [2105.00793].

## 2. Tubal and tensorial algebra underlying Haar-tSVD

The generalized framework fixes a positive integer \(p\) and an invertible \(p\times p\) matrix \(L\). Two \(p\)-length vectors \(a,b\in\mathbb C^p\), viewed as tubal scalars, are multiplied by
\[
a *_L b := L^{-1}\bigl[(La)\circ(Lb)\bigr],
\]
where \(\circ\) is the Hadamard product. The set of length-\(p\) vectors with this multiplication is denoted \(K_p(L)\). An \(m\)-tubal vector is \(X\in K_p(L)^m\), and an \(m\times n\) tubal matrix is \(A=(a_{ij})\) with each \(a_{ij}\in K_p(L)\). Such an \(A\) may be viewed equivalently as a third-order tensor in \(\mathbb R^{m\times n\times p}\). If \(A\in K_p(L)^{m\times r}\) and \(B\in K_p(L)^{r\times n}\), then the t-product is
\[
(A *_L B)_{ij} = \sum_{\ell=1}^r a_{i\ell} *_L b_{\ell j}.
\]
Orthogonality means \(A *_L A^T = I\), with \(I_m(L)\) the diagonal tubal identity [2105.00793].

A parallel formulation uses t-scalars as elements of \(\mathbb C^{I_1\times\cdots\times I_N}\) equipped with circular convolution \(\,\boldsymbol{\ast}\,\) as the scalar product. A t-matrix is then a matrix whose entries are such t-scalars, and an invertible linear map \(\mathcal L\) is required to turn convolution into pointwise multiplication:
\[
\mathcal L(X_T \,\boldsymbol{\ast}\, Y_T)=\mathcal L(X_T)\odot \mathcal L(Y_T).
\]
In the \(\mathcal L\)-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 [2001.11708].

Within the tubal-matrix language, an \(f\)-diagonal tubal matrix \(S\in K_p(L)^{m\times n}\) has zero off-diagonal tubal-scalar entries. It is \(s\)-diagonal if each diagonal tubal scalar is symmetric, each is positive-semidefinite in the sense that its transform-domain entries are \(\ge 0\), and the tubal singular values satisfy
\[
\sigma_1(k)\ge \sigma_2(k)\ge \cdots \ge \sigma_{\min(m,n)}(k), \qquad k=1,\dots,p.
\]
The paper on tubal matrices identifies such \(s\)-diagonal tensors as the correct analogue of the singular-value block in generalized T-SVD [2105.00793].

## 3. Haar specializations of the t-SVD construction

When the discrete Haar wavelet is used as the transform, the mode-3 Haar transform \(\mathcal L_H\) acts on a 3D array \(X\in\mathbb C^{D_1\times D_2\times I_3}\) by
\[
\widetilde X=\mathcal L_H(X),\qquad
\widetilde X(:,:,i)=\sum_{k=1}^{I_3}X(:,:,k)\,H_{k,i}.
\]
For a tube of length \(L=I_3\), the one-step Haar matrix is
\[
H_2=\frac{1}{\sqrt 2}\begin{pmatrix}1&1\\1&-1\end{pmatrix},
\]
and a multilevel Haar transform \(H\) of size \(L\times L\) 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 \(r\), reassembling the slice factors, and applying the inverse Haar transform to recover \(U\), \(S\), and \(V\) [2001.11708].

A second specialization replaces the DFT by an arbitrary real orthogonal \(Q\in\mathbb R^{p\times p}\). Any real orthogonal \(Q\) is unitary and real-preserving in the paper’s sense, hence doubly real-preserving. Drawing \(Q\) from the Haar measure means choosing \(Q\) uniformly at random from \(O(p)\). Once \(Q\) is fixed, one builds a new t-product \(*_Q\) and the corresponding tubal scalar module \(K_p(Q)\). The associated generalized T-SVD is obtained by the mode-3 transform
\[
\bar A(:,:,k):=Q\cdot A(:,:,k),\qquad k=1,\dots,p,
\]
followed by ordinary matrix SVDs
\[
\bar A(:,:,k)=U^{(k)}D^{(k)}[V^{(k)}]^H,
\]
assembly of \(\widetilde U\), \(\widetilde V\), and \(S\) in the transform domain, and inverse transformation with \(Q^T\), yielding
\[
A = U *_Q S *_Q V^T.
\]
This formulation retains the exact slice-wise SVD mechanism while changing the transform-domain basis [2105.00793].

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 \(Q\) 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
\[
\mathrm{Rank}_t(A)=\#\{\, i:\|s_{ii}\|>0\,\},
\]
where \(s_{ii}\) are the diagonal tubal singular values from the generalized T-SVD. If \(S_r=\mathrm{diag}(s_{11},\dots,s_{rr},0,\dots,0)\) and \(A_r=U *_L S_r *_L V^T\), then among all \(B\) with \(\mathrm{Rank}_t(B)\le r\), \(A_r\) is the unique minimizer of \(\|A-B\|_F\), and
\[
\|A-A_r\|_F^2=\sum_{i>r}\|s_{ii}\|^2.
\]
A second Eckart–Young theorem is obtained by linearizing the \(p\) frontal SVDs into a block-diagonal matrix of size \(mp\times np\) and defining the \(B\)-rank through that linearization. Truncating the top \(j\) entries in the sorted list of slice-wise singular values yields the best approximation of \(B\)-rank \(\le j\) in Frobenius norm [2105.00793].

In the more general t-matrix formulation, any invertible \(\mathcal L\) that block-diagonalizes the t-scalar ring may be used, and if \(\mathcal L\) is unitary, then the Frobenius norm is preserved in the \(\mathcal L\)-domain. Because the Haar transform is orthonormal, \(\|X\|_F=\|\mathcal L_H(X)\|_F\), the factors satisfy \(U^H\boldsymbol{\ast}U=I\) and \(V^H\boldsymbol{\ast}V=I\), and the assembled Haar-tSVD gives the best tubal-rank-\(r\) approximation in the Frobenius norm [2001.11708].

For the generalized orthogonal-transform setting, the cited paper states that the full characterization of \(s\)-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 [2105.00793].

## 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 \(\mathcal P\in\mathbb R^{ps\times ps\times c}\) is represented by a block-circulant matrix \(\mathrm{bcirc}(\mathcal P)\), and multiplication by a block-circulant matrix is equivalent to circular convolution along modes. A group of \(K\) matched patches is organized into a group-circulant matrix \(\mathrm{circ}(\mathcal G)\), whose row-Gram matrix is also circulant. Its largest eigenvector is
\[
\mathbf u_{\max}=\tfrac1{\sqrt K}(1,1,\dots,1)^T,
\]
and when \(K\) is a power of two, \(\mathbf u_{\max}\) coincides with the first row of the \(K\times K\) Haar matrix. In the more detailed account, the second eigenpair for even \(K\) is
\[
\hat\lambda=\sum_{i=1}^K(-1)^i\mathbf p_i^T\sum_{j=1}^K(-1)^j\mathbf p_j,\qquad
\hat u=\tfrac1{\sqrt K}[-1,1,-1,1,\dots]^T,
\]
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 [2512.23569] [2508.10307].

The algorithm separates global and local structure. Global patch-level bases \(\mathcal U\) and \(\mathcal V\) 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 \(\mathcal G_{\rm noisy}\in\mathbb R^{ps\times ps\times 3\times K}\), the forward transform is
\[
\mathcal S_{\rm noisy}
=\mathcal U^T * \mathcal G_{\rm noisy} * \mathcal V \times_4 U_{\rm Haar}^T,
\]
followed by hard thresholding
\[
\mathcal S_{\rm truncate}(i)=
\begin{cases}
\mathcal S_{\rm noisy}(i), & |\mathcal S_{\rm noisy}(i)|\ge \tau,\\
0, & |\mathcal S_{\rm noisy}(i)|< \tau,
\end{cases}
\qquad
\tau=\sigma\sqrt{2\ln(cKps^2)}.
\]
The reconstructed group is
\[
\mathcal G_{\rm estimate}
=\mathcal U * \mathcal S_{\rm truncate} * \mathcal V^T \times_4 U_{\rm Haar}^{-1},
\]
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 [2512.23569].

The denoising papers explicitly motivate this construction by the approximation
\[
\sum_i\mathbf p_{{\rm noisy},i}\mathbf p_{{\rm noisy},i}^T
\approx
\sum_i\mathbf p_{{\rm clean},i}\mathbf p_{{\rm clean},i}^T+\sigma^2I,
\]
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 [2512.23569].

## 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 \(\sigma_{\rm est}\) from a finite set of noise levels. Then the second-largest eigenvalue \(\hat\lambda\) of \(\mathrm{circ}(\mathcal G)\mathrm{circ}(\mathcal G)^T\) is used as an indicator of inner-group similarity. Let \(a\) be the rank position of \(\hat\lambda\) among the sorted eigenvalues. The adjustment rule is
\[
\hat\sigma=
\begin{cases}
\sigma_{\rm est}/\beta, & a\le \gamma,\\
\sigma_{\rm est}, & a>\gamma,
\end{cases}
\qquad
\beta=1.2,\ \gamma=13.
\]
To reduce cost, one account computes \(a\) on a random subset of groups and fuses via majority voting; another describes sampling a few groups per subimage and selecting the final \(\hat\sigma\) by majority voting [2512.23569] [2508.10307].

A further refinement, RA-Haar-tSVD, integrates deep neural networks through a mean-patch FCN. The FCN takes the noisy group-mean patch \(\overline{\mathbf p}_{\rm noisy}\in\mathbb R^{3ps^2}\) and outputs a refined mean patch \(\overline{\mathbf p}_{\rm est}\). During inference, the FCN replaces only the first row of \(\mathbf U_{\rm Haar}\), 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 [2512.23569].

Reported computational costs depend on the level of abstraction. In the generalized tubal-matrix algorithm with a dense orthogonal \(Q\), each mode-3 multiply costs \(O(mnp^2)\), \(p\) SVDs of \(m\times n\) cost \(O(p\cdot mn\cdot \min(m,n))\), and the total is \(O(mnp^2+p\cdot mn\cdot \min(m,n))\). In the image-denoising formulation, per-group complexity is reported as \(O(KW^2ps+K\,ps^3+K\log K)\) via fast Haar in one summary and as \(O(KW^2ps+K\,ps^3+K^3)\) when \(U_{\rm Haar}\in\mathbb R^{K\times K}\) is treated explicitly in another. This suggests a distinction between fast-Haar implementation and dense Haar multiplication [2105.00793] [2512.23569] [2508.10307].

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 \(\sigma=20/50\), Haar-tSVD matches CBM3D and MSt-SVD at approximately \(33.5/0.89\) and \(29.1/0.78\), while on DND and SIDD, A-Haar-tSVD yields approximately \(38.25/0.94\) and \(35.6/0.925\), improving approximately \(0.3\)–\(0.8\) dB over CBM3D and MSt-SVD. A more detailed table gives \(38.25/0.945\) on DND, \(35.58/0.925\) on SIDD-val, \(38.24/0.963\) on CC15, \(38.87/0.971\) on PolyU, \(40.63/0.974\) on HighISO, and \(41.52/0.978\) on IOCI for A-Haar-tSVD; it also reports \(37.16/0.963\) on CRVD and \(38.97/0.978\) on IOCV for video, and gives Real-HSI, fluorescence microscopy, and MRI results [2512.23569] [2508.10307].

The same sources describe several practical advantages and limitations. Haar-tSVD on \(512\times512\times3\) is reported to run in approximately \(4.5\) s on CPU, comparable to CBM3D at \(3.6\) s and faster than PCA-tSVD methods at approximately \(7\)–\(40\) s, while the CNN noise estimator trains in approximately \(23\) min. In the more algebraic setting, using a real orthogonal transform avoids complex arithmetic; with the discrete Haar transform, computation is \(O(I_3)\) 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 [2512.23569] [2001.11708].

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.

Source: https://www.emergentmind.com/topics/haar-tsvd