Tensor Dynamic Mode Decomposition (TDMD)

• TDMD is a family of tensor-based DMD techniques that extend classical methods to naturally handle multidimensional data.
• The third-order T-product formulation leverages FFT-based slice-wise SVD and block-circulant operations to efficiently compute reduced-order models.
• Alternative formulations like TT-based and star_M-product methods offer significant runtime and memory improvements over traditional matrix DMD.

Tensor Dynamic Mode Decomposition (TDMD) denotes tensorial extensions of dynamic mode decomposition for spatiotemporal data that are naturally represented as tensors rather than as flattened matrices. In the 2025 third-order T-product formulation, TDMD identifies a third-order transition tensor $\mathscr A$ satisfying $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$ and extracts tensorial dynamic modes and temporal dynamics from that operator; related literature also includes tensor-based DMD in Tensor-Train (TT) form and tensor-based DMD in the $\star_M$-product framework, all motivated by the observation that conventional DMD is matrix-based and may be inefficient or inadequate for images, videos, higher-order networks, and other inherently multidimensional data (He et al., 4 Aug 2025, Klus et al., 2016, Saibaba et al., 13 Aug 2025).

1. Position within dynamic mode decomposition

Classical DMD starts from two snapshot arrays,

$$X_1=\bigl[x(0)\;\;x(\Delta t)\;\cdots\;x((M-1)\Delta t)\bigr],\qquad X_2=\bigl[x(\Delta t)\;\;x(2\Delta t)\;\cdots\;x(M\Delta t)\bigr],$$

and seeks a best-fit linear map $A$ with

$A\,X_1\approx X_2,\qquad A=X_2X_1^+.$

After a truncated SVD $X_1=U\Sigma V^*$, one forms the reduced operator

$\tilde A=U^*X_2V\Sigma^{-1},$

diagonalizes it, and reconstructs modes and temporal coefficients (Środa et al., 26 Sep 2025).

Tensor formulations retain that DMD logic while replacing matrix storage and matrix-only linear algebra by tensor representations, tensor pseudoinverses, and reduced operators defined in tensor algebras or compressed tensor networks. The literature contains several non-equivalent realizations of this idea.

Formulation	Data representation	Reduced or fitted operator
Third-order T-product TDMD	$\mathscr X_t\in\mathbb R^{n\times h\times m}$	$\mathscr A=\mathscr X_+\star\mathscr X_-^\dagger$
TT-based tensor DMD	$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$0	$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$1
$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$2-based tensor DMD	$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$3	$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$4

A common misconception is that TDMD names a single canonical algorithm. The literature instead presents distinct constructions based on different tensor models and operator algebras. This suggests that TDMD is best understood as a family of tensorial DMD schemes rather than a unique procedure.

2. Tensor models and algebraic foundations

The third-order T-product formulation is built on block-circulant unfolding and FFT-based slice-wise linear algebra. For $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$5, the T-product with $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$6 is

$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$7

The framework includes a T-identity, T-transpose, T-pseudoinverse, and T-orthogonality. Tensor SVD and tensor eigenvalue decomposition generalize matrix SVD and EVD: $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$8 with $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$9 and $\star_M$0 F-diagonal. Both TSVD and TEVD are implemented efficiently by applying FFT along the third mode, reducing to $\star_M$1 independent matrix SVDs or EVDs (He et al., 4 Aug 2025).

The 2016 tensor-based DMD uses the Tensor-Train format. A $\star_M$2-way tensor $\star_M$3 is represented as

$\star_M$4

with TT-ranks $\star_M$5, $\star_M$6, and cores $\star_M$7. The memory requirement is

$\star_M$8

The exposition explicitly attributes the TT format to Oseledets ’11, and under the standard successive-SVD construction the first $\star_M$9 cores become left-orthonormal (Klus et al., 2016).

The $$X_1=\bigl[x(0)\;\;x(\Delta t)\;\cdots\;x((M-1)\Delta t)\bigr],\qquad X_2=\bigl[x(\Delta t)\;\;x(2\Delta t)\;\cdots\;x(M\Delta t)\bigr],$$0-product framework defines tensor multiplication through a unitary transform $$X_1=\bigl[x(0)\;\;x(\Delta t)\;\cdots\;x((M-1)\Delta t)\bigr],\qquad X_2=\bigl[x(\Delta t)\;\;x(2\Delta t)\;\cdots\;x(M\Delta t)\bigr],$$1. For $$X_1=\bigl[x(0)\;\;x(\Delta t)\;\cdots\;x((M-1)\Delta t)\bigr],\qquad X_2=\bigl[x(\Delta t)\;\;x(2\Delta t)\;\cdots\;x(M\Delta t)\bigr],$$2, one first forms $$X_1=\bigl[x(0)\;\;x(\Delta t)\;\cdots\;x((M-1)\Delta t)\bigr],\qquad X_2=\bigl[x(\Delta t)\;\;x(2\Delta t)\;\cdots\;x(M\Delta t)\bigr],$$3, multiplies frontal slices in the transform domain, and returns by $$X_1=\bigl[x(0)\;\;x(\Delta t)\;\cdots\;x((M-1)\Delta t)\bigr],\qquad X_2=\bigl[x(\Delta t)\;\;x(2\Delta t)\;\cdots\;x(M\Delta t)\bigr],$$4. The product is associative, has an identity tensor, and supports a slice-wise conjugate transpose, pseudoinverse, and $$X_1=\bigl[x(0)\;\;x(\Delta t)\;\cdots\;x((M-1)\Delta t)\bigr],\qquad X_2=\bigl[x(\Delta t)\;\;x(2\Delta t)\;\cdots\;x(M\Delta t)\bigr],$$5-unitary tensors. The same exposition states, following Kilmer–Kilmer–Kernfeld ’21, that one obtains a full $$X_1=\bigl[x(0)\;\;x(\Delta t)\;\cdots\;x((M-1)\Delta t)\bigr],\qquad X_2=\bigl[x(\Delta t)\;\;x(2\Delta t)\;\cdots\;x(M\Delta t)\bigr],$$6-SVD

$$X_1=\bigl[x(0)\;\;x(\Delta t)\;\cdots\;x((M-1)\Delta t)\bigr],\qquad X_2=\bigl[x(\Delta t)\;\;x(2\Delta t)\;\cdots\;x(M\Delta t)\bigr],$$7

and that truncation to the largest $$X_1=\bigl[x(0)\;\;x(\Delta t)\;\cdots\;x((M-1)\Delta t)\bigr],\qquad X_2=\bigl[x(\Delta t)\;\;x(2\Delta t)\;\cdots\;x(M\Delta t)\bigr],$$8 nonzero tubes gives the best Frobenius-norm t-rank-$$X_1=\bigl[x(0)\;\;x(\Delta t)\;\cdots\;x((M-1)\Delta t)\bigr],\qquad X_2=\bigl[x(\Delta t)\;\;x(2\Delta t)\;\cdots\;x(M\Delta t)\bigr],$$9 approximation (Saibaba et al., 13 Aug 2025).

3. Algorithmic constructions

In the third-order T-product formulation, one starts from a sequence of tensor snapshots $A$0, each in $A$1, and forms

$A$2

The goal is to identify $A$3 such that

$A$4

and to extract tensorial modes $A$5 and temporal dynamics $A$6. The reduced-order construction is: $A$7

$A$8

$A$9

The corresponding pseudocode performs FFT along mode 3, computes slice-wise economy SVDs, builds reduced matrices $A\,X_1\approx X_2,\qquad A=X_2X_1^+.$0, diagonalizes them, and assembles $A\,X_1\approx X_2,\qquad A=X_2X_1^+.$1 and $A\,X_1\approx X_2,\qquad A=X_2X_1^+.$2 by inverse FFT and un-bcirc operations (He et al., 4 Aug 2025).

The TT-based construction begins with snapshots $A\,X_1\approx X_2,\qquad A=X_2X_1^+.$3, $A\,X_1\approx X_2,\qquad A=X_2X_1^+.$4, gathered into $A\,X_1\approx X_2,\qquad A=X_2X_1^+.$5-mode tensors $A\,X_1\approx X_2,\qquad A=X_2X_1^+.$6 and $A\,X_1\approx X_2,\qquad A=X_2X_1^+.$7. Standard DMD would vectorize these to $A\,X_1\approx X_2,\qquad A=X_2X_1^+.$8, $A\,X_1\approx X_2,\qquad A=X_2X_1^+.$9, then form $X_1=U\Sigma V^*$0. The TT method instead computes the pseudoinverse of a tensor unfolding directly from TT cores. After left-orthonormalizing early cores, right-orthonormalizing later cores, and applying an SVD to the middle core, one obtains

$X_1=U\Sigma V^*$1

This is the main theoretical result stated in the paper: no large-scale optimization is required, and the result is exact up to TT-truncation error. The reduced DMD matrix is then assembled from small factors,

$X_1=U\Sigma V^*$2

and standard or exact modes are recovered by replacing the final TT core by the relevant low-dimensional vector or coefficient block (Klus et al., 2016).

In the $X_1=U\Sigma V^*$3-product formulation, the tensor-regression viewpoint is explicit: $X_1=U\Sigma V^*$4 After a truncated $X_1=U\Sigma V^*$5-SVD,

$X_1=U\Sigma V^*$6

one forms the reduced tensor

$X_1=U\Sigma V^*$7

computes a $X_1=U\Sigma V^*$8-Schur decomposition

$X_1=U\Sigma V^*$9

and defines the tensor of modes by

$\tilde A=U^*X_2V\Sigma^{-1},$0

State reconstruction is then written as

$\tilde A=U^*X_2V\Sigma^{-1},$1

The same work also gives a randomized streaming algorithm that maintains two sketches, updates them batchwise with Gaussian sketch-tensors, and then invokes a static $\tilde A=U^*X_2V\Sigma^{-1},$2-DMD on the low-rank approximation (Saibaba et al., 13 Aug 2025).

A further extension integrates tensor-train DMD into a predictor-corrector method for nonequilibrium Green’s function calculations in quantics tensor-train form. There, $\tilde A=U^*X_2V\Sigma^{-1},$3 is orthogonalized and decomposed about a central bond as

$\tilde A=U^*X_2V\Sigma^{-1},$4

After eigendecomposition, the predictor takes the form

$\tilde A=U^*X_2V\Sigma^{-1},$5

and is coupled to a causality-preserving block-time-stepping corrector that iterates only on newly added time blocks (Środa et al., 26 Sep 2025).

4. Computational scaling and compression

The third-order T-product paper provides an explicit complexity comparison. Let $\tilde A=U^*X_2V\Sigma^{-1},$6 be the spatial dimension, $\tilde A=U^*X_2V\Sigma^{-1},$7 the number of snapshots, $\tilde A=U^*X_2V\Sigma^{-1},$8 the lateral mode size, $\tilde A=U^*X_2V\Sigma^{-1},$9 the number of tensor channels, and $\mathscr X_t\in\mathbb R^{n\times h\times m}$0. TDMD slice-wise performs an SVD of an $\mathscr X_t\in\mathbb R^{n\times h\times m}$1 matrix per frontal slice, giving

$\mathscr X_t\in\mathbb R^{n\times h\times m}$2

Standard flatten-then-DMD instead requires the SVD of an $\mathscr X_t\in\mathbb R^{n\times h\times m}$3 matrix,

$\mathscr X_t\in\mathbb R^{n\times h\times m}$4

Memory-wise, TDMD stores $\mathscr X_t\in\mathbb R^{n\times h\times m}$5 parameters in $\mathscr X_t\in\mathbb R^{n\times h\times m}$6 plus $\mathscr X_t\in\mathbb R^{n\times h\times m}$7 in $\mathscr X_t\in\mathbb R^{n\times h\times m}$8, versus $\mathscr X_t\in\mathbb R^{n\times h\times m}$9 for full DMD (He et al., 4 Aug 2025).

The TT formulation emphasizes that matrix DMD on ambient dimension $\mathscr A=\mathscr X_+\star\mathscr X_-^\dagger$0 requires $\mathscr A=\mathscr X_+\star\mathscr X_-^\dagger$1 memory for $\mathscr A=\mathscr X_+\star\mathscr X_-^\dagger$2 and $\mathscr A=\mathscr X_+\star\mathscr X_-^\dagger$3 if $\mathscr A=\mathscr X_+\star\mathscr X_-^\dagger$4 is formed, while the TT representation uses

$\mathscr A=\mathscr X_+\star\mathscr X_-^\dagger$5

memory and has a cost to build $\mathscr A=\mathscr X_+\star\mathscr X_-^\dagger$6 from small factors of $\mathscr A=\mathscr X_+\star\mathscr X_-^\dagger$7. The paper’s conclusion is that if the data admits low TT ranks $\mathscr A=\mathscr X_+\star\mathscr X_-^\dagger$8, one gains orders-of-magnitude in time and memory (Klus et al., 2016).

The $\mathscr A=\mathscr X_+\star\mathscr X_-^\dagger$9-product work gives a parallel cost comparison. For matrix DMD on $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$00 and $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$01, the dominant term is $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$02. For $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$03-DMD with t-rank $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$04, the t-SVD cost is

$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$05

the Schur step costs $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$06, and mode recombination costs $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$07, with $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$08 if $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$09 is FFT-based. It also states that transforms and per-slice operations are embarrassingly parallel (Saibaba et al., 13 Aug 2025).

For QTT-based predictor-corrector TDMD, QTT-SVD and pseudoinverse for DMD cost $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$10, contractions cost $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$11, and the block solver per iteration is also $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$12, where $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$13 is bond dimension and $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$14. The corresponding motivation is that direct matrix representations of two-time functions scale as $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$15, whereas QTT storage scales as $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$16 (Środa et al., 26 Sep 2025).

Across these variants, the shared computational principle is low-rank tensor structure. What differs is the algebra in which “rank” is defined: tubal rank in the T-product setting, TT-ranks in TT/QTT, and t-rank in the $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$17-product framework.

5. Empirical performance and application domains

The 2025 third-order T-product TDMD paper reports two benchmark categories. On synthetic data consisting of $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$18 tensors with 20 snapshots, TDMD achieves relative reconstruction errors on the order of $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$19 to $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$20 across rank truncations $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$21 down to $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$22, whereas standard DMD with flattening gives a fixed error of approximately $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$23, insensitive to truncation. Runtime is approximately $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$24–$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$25 s for TDMD versus $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$26–$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$27 s for DMD, and memory usage is $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$28–$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$29 MB versus $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$30–$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$31 MB. On video data with $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$32 frames, TDMD yields per-frame relative error of approximately $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$33–$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$34, while DMD yields approximately $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$35–$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$36; in dynamic component separation, TDMD cleanly extracts a static background and moving square, while DMD mixing persists (He et al., 4 Aug 2025).

The 2016 TT-based study illustrates tensor DMD on fluid-dynamics problems. For a two-dimensional von Kármán vortex street on a $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$37 grid with $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$38 snapshots, standard DMD uses $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$39, whereas the tensor formulation stores $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$40 in TT format with ranks $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$41 and $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$42. The reported wall-clock runtime is approximately $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$43 for standard DMD and $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$44 for TDMD, corresponding to a $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$45 speed-up, and visual DMD modes match to machine precision. For two merging vortices with $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$46 up to $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$47 and $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$48 time steps, the paper reports approximately $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$49 speed-up for exact TT and up to $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$50 for $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$51, with relative errors in eigenvalues and modes below $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$52 for small $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$53. For flow around a blunt body with data $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$54, TT truncation with $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$55 yields runtime $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$56 versus $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$57, first two eigenvalue error approximately $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$58, and mode-error $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$59. The paper’s summary states that TDMD reduced memory and runtime by factors of $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$60–$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$61 while producing the same dynamical spectra and spatial modes up to TT-truncation error (Klus et al., 2016).

The $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$62-product study evaluates cylinder-wake flow at $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$63, NOAA sea-surface temperature data, and a streaming cylinder dataset. On the cylinder-wake dataset of 150 snapshots of vorticity on a $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$64 grid, $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$65-DMDII consistently attains lower reconstruction relative error at fixed storage, often half or less than either DMD or $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$66-DMD. On the first 200 of 1,727 weekly NOAA SST maps on a $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$67 grid, standard DMD slightly outperforms $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$68-DMDII at low storage, but the errors converge as storage grows. In a streaming test with 200 snapshots on a $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$69 grid split into 20 batches of 10, streaming $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$70-DMDII yields $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$71–$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$72 lower reconstruction error on each batch and overall, and end-to-end CPU time for $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$73-DMD is often $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$74–$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$75 faster at the same error level (Saibaba et al., 13 Aug 2025).

The QTT-NEGF predictor-corrector application extends tensor DMD into many-body nonequilibrium simulation. Applied to the Hubbard model on a $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$76 lattice within the nonequilibrium $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$77 approximation, it demonstrates stable propagation up to $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$78 inverse hoppings, exceeding both matrix-based implementations and previous QTT solvers. The predictor typically reduces the initial error on a new block to approximately $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$79, and the corrector then requires $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$80 iterations to reach approximately $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$81 (Środa et al., 26 Sep 2025).

6. Implementation practice and conceptual boundaries

The third-order T-product paper gives explicit implementation guidelines. Tubal rank $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$82 is chosen by inspecting the singular-tube Frobenius norms $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$83 and dropping those below a tolerance $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$84, with $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$85 given as an example. For separating persistent versus transient modes, it recommends $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$86 on $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$87. It also recommends the MATLAB “tensor-tensor-product-toolbox” available at github.com/lu-group, or Python implementations based on FFT with NumPy/SciPy for slice-wise SVD/EVD, together with three efficiency tips: precompute FFT along mode 3 once, use economy-size SVD with $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$88, and parallelize across slices $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$89 (He et al., 4 Aug 2025).

The $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$90-product paper places tensor DMD in direct relation to traditional DMD and physics-informed DMD. It states that

$\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$91

is equivalent to a constrained matrix regression over a structured subspace $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$92 of block-structured matrices, and interprets $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$93-DMD as the case in which the operator must lie in the $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$94-algebra defined by $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$95. This provides an explicit connection to piDMD, which also imposes subspace constraints on $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$96 (Saibaba et al., 13 Aug 2025).

A second misconception is that tensorization by itself guarantees exact recovery or superior results. The literature qualifies that point. In the TT framework, the pseudoinverse theorem is exact apart from TT-truncation error (Klus et al., 2016). In the third-order T-product formulation, the practical algorithm is a reduced-order approximation with chosen tubal rank (He et al., 4 Aug 2025). In the $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$97-product setting, accuracy depends on the selected transform $\mathscr X_{t+1}=\mathscr A\star\mathscr X_t$98, the truncation strategy, and whether one uses fixed-rank tr-tSVDM or slice-adaptive tr-tSVDMII (Saibaba et al., 13 Aug 2025). The numerical evidence consistently favors structure-preserving tensor formulations over flatten-then-DMD on the reported examples, but the reported comparisons are framework-specific rather than universal.

A third misconception is that tensor DMD is confined to fluid mechanics or image-like data. The cited applications do include vortex streets, merging vortices, blunt-body flow, cylinder wakes, sea-surface temperature fields, and video sequences, but the QTT predictor-corrector formulation is presented as a general framework not limited to nonequilibrium Green’s functions, requiring only that data at fixed times be stored in QTT form and that one can perform a self-consistent or nonlinear update on a small time block while holding the past fixed (Środa et al., 26 Sep 2025).

Taken together, these works define TDMD as a technically diverse but conceptually coherent research area: DMD-style Koopman-oriented regression and modal analysis carried out in tensor representations that preserve multiway structure, expose low-rank organization, and replace large dense matrix operations by tensor-algebraic or tensor-network computations.