Variable-Invariant 2D State Space Model

• VI 2D SSM is a multivariate time series framework that enforces permutation-equivariance to honor variable exchangeability.
• It utilizes a self + pooled update mechanism to reduce variable-axis dependency from O(C) to O(1) and simplifies stability analysis to two scalar modes.
• The VI 2D Mamba architecture integrates multi-scale temporal branches, spectral analysis, and permutation-invariant aggregation for efficient forecasting, classification, and anomaly detection.

Variable-Invariant Two-Dimensional State Space Model (VI 2D SSM) is a multivariate time-series modeling framework defined by permutation-equivariance along the variable axis. It was introduced to address the fact that many multivariate time series have no canonical ordering over variables, so models that impose an ordered scan over variables violate exchangeability and the permutation symmetry principle. In the formulation of "Permutation-Equivariant 2D State Space Models: Theory and Canonical Architecture for Multivariate Time Series," VI 2D SSM is presented as the realization of the canonical permutation-equivariant linear 2D state-space form through permutation-invariant aggregation, eliminating sequential dependency chains along variables, reducing variable-axis dependency depth from $\mathcal{O}(C)$ to $\mathcal{O}(1)$, and simplifying stability analysis to two scalar modes (Jeong et al., 7 Mar 2026).

1. Permutation symmetry and variable-axis exchangeability

The starting point of VI 2D SSM is the claim that multivariate time series (MTS) modeling often imposes an artificial ordering over variables. The relevant symmetry condition is formalized as variable-axis exchangeability. Let $X\in\mathbb{R}^{C\times T}$ be a multivariate time series with $C$ variables. The data-generating distribution is exchangeable along the variable axis if, for every permutation $\pi\in S_C$,

$P\bigl(X=x\bigr)=P\bigl(X^\pi=x^\pi\bigr),$

where $X^\pi$ permutes the rows of $X$ by $\pi$ (Jeong et al., 7 Mar 2026).

Under this assumption, a valid multivariate dynamical model $f$ must satisfy permutation-equivariance along variables: $\mathcal{O}(1)$0 This means that relabeling variables before applying the model is equivalent to relabeling outputs afterward. In the VI 2D SSM framework, this is not an optional architectural preference but a symmetry principle that constrains admissible dynamics (Jeong et al., 7 Mar 2026).

The significance of this formulation is that it shifts the modeling problem from axis-dependent recurrence to symmetry-preserving state evolution. This suggests that the variable axis should not be treated as a sequence unless the data-generating process itself provides an ordered semantics. In systems without such semantics, ordered recurrence becomes misaligned with the assumed invariance structure.

2. Canonical linear form under permutation-equivariance

VI 2D SSM is derived from a characterization theorem for linear variable coupling in a 2D state-space system. At fixed time $\mathcal{O}(1)$1, the most general linear update on a vertical state $\mathcal{O}(1)$2 is written as

$\mathcal{O}(1)$3

Permutation-equivariance requires the coupling matrices $\mathcal{O}(1)$4 and $\mathcal{O}(1)$5 to commute with every permutation matrix $\mathcal{O}(1)$6 (Jeong et al., 7 Mar 2026).

The key theorem states that a matrix $\mathcal{O}(1)$7 satisfies

$\mathcal{O}(1)$8

if and only if

$\mathcal{O}(1)$9

Accordingly, the vertical-axis update must take the self + pooled form

$X\in\mathbb{R}^{C\times T}$0

The variable coupling therefore decomposes into local self-dynamics and a global pooled interaction (Jeong et al., 7 Mar 2026).

In continuous time, the full coupled 2D SSM with global summary field $X\in\mathbb{R}^{C\times T}$1 is

$X\in\mathbb{R}^{C\times T}$2

After ZOH discretization with step $X\in\mathbb{R}^{C\times T}$3, this becomes

$X\in\mathbb{R}^{C\times T}$4

In this formulation, VI 2D SSM realizes the canonical equivariant form through permutation-invariant aggregation rather than through an ordered recurrence over variables (Jeong et al., 7 Mar 2026).

3. Ordered recurrence, dependency depth, and structural consequences

A central theoretical claim is that ordered recurrence along variables is unnecessary and structurally suboptimal under permutation symmetry. Ordered-scan 2D SSMs define updates of the form

$X\in\mathbb{R}^{C\times T}$5

so, at fixed $X\in\mathbb{R}^{C\times T}$6, the computation graph along $X\in\mathbb{R}^{C\times T}$7 is a chain of length $X\in\mathbb{R}^{C\times T}$8. The resulting dependency depth is $X\in\mathbb{R}^{C\times T}$9 (Jeong et al., 7 Mar 2026).

VI 2D SSM instead computes a global summary

$C$0

through a single parallel reduction. After $C$1 is obtained, each $C$2 depends only on $C$3, local history, and $C$4, and the $C$5 variable updates are independent and can proceed in parallel. Under ideal parallelism, the reduction has $C$6 work but $C$7 span, so the overall variable-axis depth is $C$8 (Jeong et al., 7 Mar 2026).

This distinction is structurally important. The theory in (Jeong et al., 7 Mar 2026) does not merely recommend a more efficient implementation; it identifies a symmetry-constrained canonical form in which global pooling replaces variable-axis scans. A plausible implication is that the removal of ordered variable recurrence is not only computationally beneficial but also better aligned with exchangeable MTS data-generating assumptions.

4. Stability reduction to two scalar modes

Under the canonical form

$C$9

the vertical-state coupling acts on two invariant subspaces. On the zero-sum subspace $\pi\in S_C$0, one has

$\pi\in S_C$1

On the mean subspace $\pi\in S_C$2, one has

$\pi\in S_C$3

The corresponding eigenvalues are therefore

$\pi\in S_C$4

Discrete-time stability, expressed as $\pi\in S_C$5, reduces to the two scalar constraints

$\pi\in S_C$6

In continuous time, if $\pi\in S_C$7 is Hurwitz, meaning $\pi\in S_C$8, then $\pi\in S_C$9 has spectral radius $P\bigl(X=x\bigr)=P\bigl(X^\pi=x^\pi\bigr),$0 for all $P\bigl(X=x\bigr)=P\bigl(X^\pi=x^\pi\bigr),$1 (Jeong et al., 7 Mar 2026).

This stability reduction is one of the most distinctive analytical consequences of the symmetry constraint. Rather than requiring a full high-dimensional coupling analysis over the variable axis, the permutation-equivariant structure isolates a difference mode and a mean mode. This suggests that stability questions for the vertical interaction become substantially more tractable than in generic 2D state-space couplings.

5. VI 2D Mamba architecture

The architectural instantiation associated with VI 2D SSM is VI 2D Mamba, described as a unified architecture integrating multi-scale temporal dynamics and spectral representations (Jeong et al., 7 Mar 2026).

Its first component is permutation-invariant aggregation: $P\bigl(X=x\bigr)=P\bigl(X^\pi=x^\pi\bigr),$2 where $P\bigl(X=x\bigr)=P\bigl(X^\pi=x^\pi\bigr),$3 may be mean, sum, or attention pooling. This aggregation realizes the canonical pooled interaction required by permutation-equivariance (Jeong et al., 7 Mar 2026).

The second component is a set of multi-scale temporal branches. A long-term branch uses coarse step $P\bigl(X=x\bigr)=P\bigl(X^\pi=x^\pi\bigr),$4, and a short-term branch uses fine step $P\bigl(X=x\bigr)=P\bigl(X^\pi=x^\pi\bigr),$5. Each branch is a VI 2D SSM block with its own $P\bigl(X=x\bigr)=P\bigl(X^\pi=x^\pi\bigr),$6 parameters (Jeong et al., 7 Mar 2026).

The third component is a spectral branch. The input $P\bigl(X=x\bigr)=P\bigl(X^\pi=x^\pi\bigr),$7 is transformed to the frequency domain via DFT along $P\bigl(X=x\bigr)=P\bigl(X^\pi=x^\pi\bigr),$8, non-redundant frequencies are retained, and a VI 2D SSM is applied along the frequency axis with step $P\bigl(X=x\bigr)=P\bigl(X^\pi=x^\pi\bigr),$9 (Jeong et al., 7 Mar 2026).

The final component is adaptive gating/fusion: $X^\pi$0 where Gate is a small learned network that produces weights for each branch (Jeong et al., 7 Mar 2026).

The architecture can be summarized as follows.

Component	Definition	Role
Permutation-invariant aggregation	$X^\pi$1	Global summary over variables
Multi-scale temporal branches	Long-term branch with $X^\pi$2; short-term branch with $X^\pi$3	Temporal dynamics at different scales
Spectral branch	DFT along $X^\pi$4, keep non-redundant frequencies, apply VI 2D SSM with $X^\pi$5	Frequency-domain dynamics
Adaptive gating/fusion	$X^\pi$6	Learned branch weighting

Taken together, these components instantiate the theoretical claim that symmetry-preserving variable interaction can be combined with temporal and spectral modeling in a single 2D SSM architecture.

6. Benchmarks, efficiency, and empirical position

The reported experiments cover forecasting, classification, anomaly detection, and scaling behavior. For long-term forecasting on ETT, ECL, Traffic, Weather, and Exchange with metrics MSE/MAE, VI 2D Mamba achieves lowest average MSE on 4/8 datasets and lowest MAE on 3/8, outperforming Chimera (2D SSM), TimePro, Simba, TCN, and various Transformers (Jeong et al., 7 Mar 2026).

For short-term forecasting on M4 with metrics SMAPE/MASE/OWA, it is reported as second-best overall, just behind Chimera, while remaining better than most other baselines (Jeong et al., 7 Mar 2026). For classification on the UEA/MTS archive using accuracy, the average is approximately $X^\pi$7, compared with Chimera at $X^\pi$8, with the explicit note that the model retains strong performance given much lower computational cost (Jeong et al., 7 Mar 2026). For anomaly detection on SMD, MSL, SMAP, SWaT, and PSM using F1, VI 2D Mamba achieves best average F1 of approximately $X^\pi$9, compared with Chimera at $X$0 and the next best at approximately $X$1 (Jeong et al., 7 Mar 2026).

The scaling results emphasize structural efficiency. Per-epoch training time is described as nearly flat as $X$2 grows, reaching 6 sec/epoch even at $X$3, compared to Chimera’s approximately 90 sec at $X$4. FLOPs and peak GPU memory are reported as comparable to 1D SSM (Mamba) and far below 2D scans (Jeong et al., 7 Mar 2026).

These results position VI 2D SSM as a symmetry-constrained alternative to scan-based 2D state-space models. The empirical picture is not one of universal dominance on every benchmark, since Chimera remains ahead on M4 and slightly ahead on average classification accuracy, but the reported combination of anomaly-detection performance, forecasting strength, and variable-axis scaling is consistent with the theoretical emphasis on structural scalability.

7. Related uses of “variable-invariant” and interpretive boundaries

The phrase “variable-invariant” also appears in a different state-space context in "Disentangled State Space Representations" (Miladinović et al., 2019). There, a variable-invariant SSM is described for a 2D bouncing-ball setting in which the transition map is decomposed into domain-invariant dynamics and domain-specific effects governed by a domain variable such as gravity. The formulation writes

$X$5

or, in a gating-style implementation,

$X$6

In that setting, invariance concerns separation of domain-invariant and domain-specific dynamics rather than permutation-equivariance along a variable axis (Miladinović et al., 2019).

This distinction is important for avoiding a common misconception. VI 2D SSM in (Jeong et al., 7 Mar 2026) is not primarily a disentanglement model over latent domain variables; it is a permutation-equivariant 2D state-space model for multivariate time series whose central invariance concerns exchangeability across variables. By contrast, the DSSM formulation uses invariant/specific decomposition to support transfer across domains such as different gravity regimes, with a VAE-style posterior over latent state and domain variables (Miladinović et al., 2019).

A plausible implication is that the two lines of work address different symmetry structures. One treats permutation symmetry over observed variables in MTS; the other treats invariance to domain-specific variation in latent dynamics. They are therefore related at the level of symmetry-aware state-space modeling, but they are not interchangeable formulations of the same problem.

VI 2D SSM is consequently best understood as the canonical linear 2D state-space realization consistent with variable-axis permutation symmetry, with global pooled interaction replacing ordered variable scans, stability collapsing to two scalar conditions, and VI 2D Mamba providing the principal architectural instantiation for forecasting, classification, and anomaly detection tasks (Jeong et al., 7 Mar 2026).