Dynamic Stiefel Routing: Adaptive Manifold Techniques

• Dynamic Stiefel Routing is a family of data-adaptive mechanisms that constrain transformation learning to the Stiefel manifold for both spectral filtering and subspace selection.
• In graph-based models, it dynamically learns a spectral basis to enable efficient graph convolution, reducing noise and computational overhead.
• For EEG decoding, it implements a sample-wise adaptive subspace selection using a pool of expert filters to optimally project SPD inputs.

Searching arXiv for the cited papers and closely related background on SPD manifold networks and Stiefel-manifold optimization. Dynamic Stiefel Routing denotes a class of data-adaptive mechanisms in which the learned transformation is constrained to the Stiefel manifold, but the term is currently used in two materially different senses in the arXiv literature. In spatio-temporal time series forecasting, the relevant construction is the Dynamic Spatio-temporal Stiefel Graph Neural Network (DST-SGNN), where a dynamic adjacency and a Stiefel-constrained graph Fourier basis are used for efficient spectral graph convolution rather than for token-to-expert assignment (Zheng et al., 1 Jun 2025). In cross-domain EEG decoding, the term names an explicit sample-wise adaptive subspace selection mechanism, implemented as a mixture-of-experts layer on $\mathrm{St}(n,k)$ for covariance inputs on $\mathrm{SPD}(n)$ (Maia et al., 29 May 2026). Taken together, these works suggest that “dynamic Stiefel routing” is best understood not as a single canonical algorithm, but as a family of manifold-constrained adaptive basis-selection schemes.

1. Terminological scope and conceptual distinction

The two current uses of the term differ at the level of mechanism. In DST-SGNN, the paper states that it does not propose routing in the transformer/MoE sense of assigning tokens to experts or paths. Its “dynamic routing” idea is instead dynamic graph/transform adaptation: the model learns a time-varying graph adjacency and then learns a spectral transform matrix constrained on the Stiefel manifold, so that graph convolution can be done efficiently and with a built-in filtering effect (Zheng et al., 1 Jun 2025).

By contrast, the EEG paper defines dynamic Stiefel routing as a sample-wise adaptive subspace selection mechanism on the Stiefel manifold. Instead of a single projection matrix, the model maintains a pool of expert filters and computes routing weights for each sample; these weights determine a sample-specific projection filter used in a BiMap layer (Maia et al., 29 May 2026).

Setting	Adapted object	Meaning of routing
DST-SGNN	Dynamic adjacency and SGFT basis	Information is routed through a learned graph spectrum / basis
DASP for EEG	Pool of Stiefel expert filters	Each sample is routed to the most appropriate subspace filter via cross-attention

A common misconception is therefore to treat all uses of “dynamic Stiefel routing” as attention-style expert assignment. That description is accurate for the EEG setting, but not for DST-SGNN. The first usage is more precisely described as dynamic graph learning, spectral filtering, and manifold-constrained transform learning; the second is an actual routing layer with per-sample expert selection.

2. Geometric and spectral foundations

Both formulations are built around the Stiefel manifold. In DST-SGNN, the transform matrix satisfies

$$St(n,d) \triangleq \{F \in \mathbb{R}^{n \times d}: F^T F = I_d\},$$

so the columns of $F$ are orthonormal (Zheng et al., 1 Jun 2025). In the EEG formulation, each expert projector satisfies

$$\mathrm{St}(n,k) = \{W \in \mathbb{R}^{n\times k} : W^\top W = I_k\},$$

again enforcing orthonormal columns (Maia et al., 29 May 2026).

The graph-based formulation introduces the Stiefel Graph Fourier Transform:

$S(x) = F^T x, \qquad S^{-1}(x) = Fx,$

and defines the Stiefel Graph Spectral Convolution for a signal $x \in \mathbb{R}^{n \times 1}$ and kernel $g \in \mathbb{R}^{n \times 1}$ as

$x *_{s} g = F(F^T x \odot F^T g).$

For multi-dimensional features $X \in \mathbb{R}^{n \times k}$ and kernel $\mathrm{SPD}(n)$0, this becomes

$\mathrm{SPD}(n)$1

The matrix $\mathrm{SPD}(n)$2 is obtained by solving

$\mathrm{SPD}(n)$3

with

$\mathrm{SPD}(n)$4

The paper also derives a related maximization form,

$\mathrm{SPD}(n)$5

which makes explicit that the learned basis is chosen from the leading eigenvectors of the normalized adjacency (Zheng et al., 1 Jun 2025).

The EEG formulation instead starts from covariance inputs

$\mathrm{SPD}(n)$6

and uses the standard BiMap projection

$\mathrm{SPD}(n)$7

The paper’s premise is that a single shared $\mathrm{SPD}(n)$8 is geometrically mismatched under cross-domain shift, because covariance matrices from different subjects occupy systematically distinct regions of the SPD manifold (Maia et al., 29 May 2026).

This shared geometry explains why the two works can be discussed under the same label. In both cases, orthonormality is not merely a regularizer: it determines the admissible basis or projection family through which information may flow.

3. Dynamic graph/transform adaptation in DST-SGNN

DST-SGNN is defined by four components: patch-based decomposition of the input time series, seasonal/trend decomposition to build a hyperpatch graph, LDGOSM to dynamically learn the graph transform matrix on the Stiefel manifold, and SGSC/MSGSC to perform efficient spectral graph convolution (Zheng et al., 1 Jun 2025).

The paper’s “dynamic” aspect begins with an instance-dependent adjacency:

$\mathrm{SPD}(n)$9

This makes the graph adaptive to the current input features rather than fixed. The transform matrix is then parameterized linearly as

$$St(n,d) \triangleq \{F \in \mathbb{R}^{n \times d}: F^T F = I_d\},$$0

Substituting this into the Stiefel objective yields

$$St(n,d) \triangleq \{F \in \mathbb{R}^{n \times d}: F^T F = I_d\},$$1

which is the optimization solved by Linear Dynamic Graph Optimization on Stiefel Manifold (LDGOSM).

The LDGOSM procedure is explicitly given as follows:

1. Compute $$St(n,d) \triangleq \{F \in \mathbb{R}^{n \times d}: F^T F = I_d\},$$2.
2. Eigendecompose $$St(n,d) \triangleq \{F \in \mathbb{R}^{n \times d}: F^T F = I_d\},$$3.
3. Set

$$St(n,d) \triangleq \{F \in \mathbb{R}^{n \times d}: F^T F = I_d\},$$4

1. Compute

$$St(n,d) \triangleq \{F \in \mathbb{R}^{n \times d}: F^T F = I_d\},$$5

2. Eigendecompose

$$St(n,d) \triangleq \{F \in \mathbb{R}^{n \times d}: F^T F = I_d\},$$6

to get eigenvector matrix $$St(n,d) \triangleq \{F \in \mathbb{R}^{n \times d}: F^T F = I_d\},$$7.

3. Return

$$St(n,d) \triangleq \{F \in \mathbb{R}^{n \times d}: F^T F = I_d\},$$8

The motivation is computational. Direct eigen-decomposition of the Laplacian costs $$St(n,d) \triangleq \{F \in \mathbb{R}^{n \times d}: F^T F = I_d\},$$9, whereas LDGOSM reduces the total complexity to

$F$0

which is approximately linear in the number of nodes $F$1 when $F$2 (Zheng et al., 1 Jun 2025).

A further theoretical point is that SGSC is stated to be equivalent to a filtered graph spectral convolution:

$F$3

where

$F$4

$F$5 contains the corresponding eigenvectors, and the filter is diagonal,

$F$6

with

$F$7

The lower-eigenvalue components are therefore zeroed out. The authors interpret this as both filtering/noise reduction and complexity reduction.

The model extends SGSC to Multi-layer SGSC (MSGSC):

$F$8

with equivalent spectral form

$F$9

The implementation further specifies

$$\mathrm{St}(n,k) = \{W \in \mathbb{R}^{n\times k} : W^\top W = I_k\},$$0

where each $$\mathrm{St}(n,k) = \{W \in \mathbb{R}^{n\times k} : W^\top W = I_k\},$$1 is learned through

$$\mathrm{St}(n,k) = \{W \in \mathbb{R}^{n\times k} : W^\top W = I_k\},$$2

The stated effect is to capture short-range and deeper spatio-temporal interactions, multi-hop dependencies, and more complex correlations than a single spectral layer.

In this setting, “routing” is therefore only a high-level interpretation. There is no explicit routing module such as token-to-expert routing, hard node assignment, attention-based path selection, or cluster routing. The paper-specific meaning is closer to routing signal energy through selected spectral subspaces of a dynamically learned graph.

4. Sample-wise expert routing on SPD manifolds

The EEG formulation makes routing explicit. The proposed layer, termed the Domain-Adaptive Stiefel Pool (DASP), maintains a pool of $$\mathrm{St}(n,k) = \{W \in \mathbb{R}^{n\times k} : W^\top W = I_k\},$$3 expert projection filters

$$\mathrm{St}(n,k) = \{W \in \mathbb{R}^{n\times k} : W^\top W = I_k\},$$4

computes routing weights

$$\mathrm{St}(n,k) = \{W \in \mathbb{R}^{n\times k} : W^\top W = I_k\},$$5

for each input covariance $$\mathrm{St}(n,k) = \{W \in \mathbb{R}^{n\times k} : W^\top W = I_k\},$$6, and uses them to form a sample-specific projection filter $$\mathrm{St}(n,k) = \{W \in \mathbb{R}^{n\times k} : W^\top W = I_k\},$$7, which is applied as

$$\mathrm{St}(n,k) = \{W \in \mathbb{R}^{n\times k} : W^\top W = I_k\},$$8

(Maia et al., 29 May 2026).

The routing weights are obtained by dot-product attention:

$$\mathrm{St}(n,k) = \{W \in \mathbb{R}^{n\times k} : W^\top W = I_k\},$$9

where $S(x) = F^T x, \qquad S^{-1}(x) = Fx,$0 is the matrix of keys and $S(x) = F^T x, \qquad S^{-1}(x) = Fx,$1 is the query. The query is built from two sources: a tangent-space representation of the covariance and a learnable domain embedding. The covariance is mapped to the tangent space at identity via

$S(x) = F^T x, \qquad S^{-1}(x) = Fx,$2

where $S(x) = F^T x, \qquad S^{-1}(x) = Fx,$3 extracts the upper-triangular vectorization with the usual $S(x) = F^T x, \qquad S^{-1}(x) = Fx,$4-scaled off-diagonals.

The paper introduces an optional fixed domain-discriminative projection:

$S(x) = F^T x, \qquad S^{-1}(x) = Fx,$5

with

$S(x) = F^T x, \qquad S^{-1}(x) = Fx,$6

The query is then

$S(x) = F^T x, \qquad S^{-1}(x) = Fx,$7

The adaptive filter is constructed by Riemannian interpolation around a dedicated anchor $S(x) = F^T x, \qquad S^{-1}(x) = Fx,$8:

$S(x) = F^T x, \qquad S^{-1}(x) = Fx,$9

where the tangent-space projection is

$x \in \mathbb{R}^{n \times 1}$0

the retraction $x \in \mathbb{R}^{n \times 1}$1 is a QR retraction back to $x \in \mathbb{R}^{n \times 1}$2, and

$x \in \mathbb{R}^{n \times 1}$3

Operationally, the layer encodes the sample and domain into a query, computes attention over expert keys, barycentrically interpolates the expert filters in tangent space around $x \in \mathbb{R}^{n \times 1}$4, retracts the result to the manifold, and applies the resulting subspace in the BiMap projection.

This is routing in the stricter sense: the network chooses, per sample, which orthonormal subspace should shape the projection. Unlike the DST-SGNN usage, the adaptive mechanism is not only spectral or graph-structural; it is a genuine expert-selection layer.

5. Degeneracy, committed routing, and structural remedies

A central theoretical claim of the EEG paper is that naive adaptive routing can collapse to ensemble averaging. If the routing weights are uniform,

$x \in \mathbb{R}^{n \times 1}$5

then

$x \in \mathbb{R}^{n \times 1}$6

which is constant across samples. Every input is then projected through the same effective filter, and the adaptive layer becomes indistinguishable from an equal-contribution combination of experts. The paper refers to this as the $x \in \mathbb{R}^{n \times 1}$7 proxy, and uses

$x \in \mathbb{R}^{n \times 1}$8

as the diagnostic for genuine routing (Maia et al., 29 May 2026).

The collapse is described as self-reinforcing. Uniform routing implies no specialization; no specialization weakens routing gradients because experts remain similar; weak routing gradients then keep routing close to uniform. The paper’s position is that adaptive subspace selection helps only when routing is genuinely committed rather than effectively averaged.

Three structural properties are introduced to break this degeneracy.

First, a symmetric anchor

$x \in \mathbb{R}^{n \times 1}$9

removes geometric proximity bias among experts. Without such an anchor, one expert can become privileged by the reference geometry, which suppresses fair competition and specialization.

Second, the query is made domain-discriminative through a frozen DSP projection

$g \in \mathbb{R}^{n \times 1}$0

whose columns are the top eigenvectors of the between-domain scatter

$g \in \mathbb{R}^{n \times 1}$1

This projection is computed once from training tangent vectors and then frozen. The stated purpose is to keep the query aligned with domain identity rather than letting it drift toward class-discriminative features.

Third, the paper introduces a decoupled key alignment loss

$g \in \mathbb{R}^{n \times 1}$2

Only the winning key is updated for each sample, and the query is stop-grad’ed. The total loss is

$g \in \mathbb{R}^{n \times 1}$3

The paper interprets this as training expert keys toward stable domain attractors rather than allowing them to chase a moving target induced by the classification objective.

The work further gives an automatic rule based on

$g \in \mathbb{R}^{n \times 1}$4

If $g \in \mathbb{R}^{n \times 1}$5, DSP is enabled, $g \in \mathbb{R}^{n \times 1}$6, and keys are decoupled from cross-entropy; if $g \in \mathbb{R}^{n \times 1}$7, DSP is disabled and keys are allowed to receive classification gradients. This is presented as a single data-driven rule that removes dataset-specific tuning.

A plausible synthesis is that “dynamic Stiefel routing” becomes substantive only when the geometry of the manifold, the statistics of domain separation, and the optimization of expert competition are all aligned. Without that alignment, adaptation can remain formally dynamic but functionally static.

6. Empirical profile, ablations, and recurring misconceptions

The empirical claims differ by application domain but converge on the importance of the Stiefel constraint. For DST-SGNN, extensive experiments on seven spatio-temporal datasets are reported to show that the method outperforms state-of-the-art methods while maintaining relatively low computational costs; the paper further states that it has especially strong results on CSI300 and exchange_rate, performs well on both periodic traffic data and non-periodic financial/power data, and maintains efficiency with relatively low parameter counts (Zheng et al., 1 Jun 2025).

The DST-SGNN ablation compares three variants: Rep-StdGSC, which replaces SGSC with standard graph spectral convolution using an orthogonal $g \in \mathbb{R}^{n \times 1}$8; Rep-SpatConv, which replaces SGSC with standard spatial graph convolution; and w/o-Stiefel, which removes the Stiefel constraint and replaces LDGOSM with an MLP. All three variants perform worse than DST-SGNN. The interpretation given by the authors is that the Stiefel manifold discards low-information eigenvalues, reduces noise, improves the ability to capture complex spatio-temporal structure, and does so efficiently.

For the EEG setting, the reported benchmark is balanced accuracy under binary classification of right-hand versus feet motor imagery, using 5-fold cross-validation with 70/15/15 trial splits, Adam with learning rate $g \in \mathbb{R}^{n \times 1}$9, and batch size $x *_{s} g = F(F^T x \odot F^T g).$0 (Maia et al., 29 May 2026). Relative to a standard single-stage SPDNet with fixed BiMap, DASP improves balanced accuracy from $x *_{s} g = F(F^T x \odot F^T g).$1 on Weibo2014, from $x *_{s} g = F(F^T x \odot F^T g).$2 on BNCI2015001, and from $x *_{s} g = F(F^T x \odot F^T g).$3 on BNCI2014001.

The EEG ablations are particularly diagnostic. On Weibo2014, the naive adaptive model reaches $x *_{s} g = F(F^T x \odot F^T g).$4 but has $x *_{s} g = F(F^T x \odot F^T g).$5, which the paper interprets as evidence that the gain came from a better ensemble mean rather than true routing. DASP attains $x *_{s} g = F(F^T x \odot F^T g).$6 with $x *_{s} g = F(F^T x \odot F^T g).$7 and lower entropy, consistent with committed routing. Additional ablations show that load-balancing loss increases entropy to $x *_{s} g = F(F^T x \odot F^T g).$8 but does not improve $x *_{s} g = F(F^T x \odot F^T g).$9; entropy loss similarly does not solve the problem; $X \in \mathbb{R}^{n \times k}$0 alone gives the biggest single improvement among structural fixes, with $X \in \mathbb{R}^{n \times k}$1; $X \in \mathbb{R}^{n \times k}$2 DSP improves domain alignment substantially; alignment loss alone is insufficient; and the full DASP gives the best overall performance.

These findings clarify two recurrent misconceptions. The first is that any adaptive mixture of Stiefel filters necessarily yields sample-wise specialization. The EEG results explicitly reject that view: naive routing can collapse to averaging. The second is that dynamic Stiefel routing must always involve attention. The STTS formulation shows the opposite: a model may dynamically route information through a learned spectral basis and still contain no explicit attention score matrix over neighbors, no softmax normalization of pairwise affinities, and no learned token-to-token routing policy.

Across both usages, the consistent theme is that the Stiefel constraint is not incidental. It structures the admissible basis, enables orthogonality-preserving adaptation, and, in the formulations presented so far, is tied either to spectral filtering on dynamic graphs or to adaptive subspace projection for SPD-valued inputs.