OT-Augmented Feature Set

• OT-augmented feature sets are representations enriched using optimal transport theory to align features across modalities, resolutions, and symmetry groups.
• They leverage advanced methods like bispectral embeddings, cross-modal adapters, and OT-guided concatenation to enhance discriminative power.
• Empirical studies show improved semantic alignment and detection accuracy, though increased feature dimensionality and computational overhead are notable challenges.

An OT-augmented feature set generically refers to a representation or embedding constructed or aligned using optimal transport (OT) theory, either by directly optimizing for correspondences between sets or by systematically enriching features through OT-driven mechanisms. In contemporary machine learning, graphics, vision, and multimodal learning, such feature sets are used to robustify alignment, factor out nuisance modalities, and enhance discriminative power—especially across modalities, resolutions, or groups with nontrivial symmetry structure. Representative frameworks include symmetry-aware bispectral OT (Ma et al., 25 Sep 2025), cross-modal OT adapters (Ji et al., 19 Mar 2025), OT-guided feature intertwining (Li et al., 2019), domain-adaptive OT-based representation alignment (Yu et al., 2022), and in-model transformation via feature concatenation (Lyu et al., 1 Feb 2025).

1. Mathematical Formulation of OT-Augmented Feature Sets

OT-augmented feature construction centers on the empirical or entropy-regularized Kantorovich optimal transport problem. Given two empirical distributions $\mu=\sum_{i} a_i \delta_{x_i}$ and $\nu=\sum_{j} b_j\delta_{y_j}$ in feature space, with cost function $C_{ij}=d(x_i,y_j)$ for some metric $d$, the OT coupling plan $\Gamma^*$ is obtained as

$$\min_{\Gamma \in \Pi(\mu,\nu)} \langle \Gamma, C \rangle - \varepsilon H(\Gamma) \quad \text{where} \quad \Pi(\mu,\nu)=\{\Gamma\geq 0 | \Gamma 1 = a,\, \Gamma^T 1 = b\}$$

with entropic regularization $H(\Gamma)=-\sum_{i,j} \Gamma_{ij} \log \Gamma_{ij}$, solved efficiently by Sinkhorn-Knopp iterations. The resulting transport plan defines a correspondence metric, a feature transformation, or a soft/hard assignment between domains.

In bispectral OT (Ma et al., 25 Sep 2025), the cost matrix is defined between bispectrum-invariant feature tensors, yielding symmetry-invariant transport plans. In modality bridging (Ji et al., 19 Mar 2025), feature correlation is measured by cosine distance, and OT aligns the distributions of visual and textual tokens, producing harmonized cross-modal embeddings.

2. Methods for Constructing OT-Augmented Features

Key instantiations encompass:

• Bispectral Feature Embedding: For signals or images acted on by a (finite or compact) group $G$, the bispectrum $B(f)$ is calculated using the generalized Fourier transform over $G$’s irreducible representations. For commutative $G$, $$B_{i,j}(f) = \hat{f}_i \cdot \hat{f}_j \cdot \hat{f}^*_{i+j \, \mathrm{mod}\, n}$$, providing an invariant that factors out nuisance group actions but retains full structural information. This bispectrum is computed per image (e.g., by polar gridding and FFT over angular bins, followed by channel-wise feature vectorization), and then used as the input for OT matching (Ma et al., 25 Sep 2025).
• Modality-Adapter Construction via OT: In cross-modal problems, such as few-shot remote sensing scene classification, visual patch embeddings and text token embeddings are projected into a common space, then aligned via an OT adapter module incorporating cross-modal attention. The OT cost matrix is defined via cosine similarity, with Sinkhorn-regularized coupling driving feature updates. Sample-level entropy-aware weighting further stabilizes the learning (Ji et al., 19 Mar 2025).
• Feature Intertwining and Domain Alignment: In detection frameworks, higher-resolution (“reliable”) feature sets are upsampled and aligned with lower-resolution (“less reliable”) sets via OT. Feature tokens are mapped through a critic network, cosine-distance cost matrices are computed, and Sinkhorn’s algorithm yields the coupling used to define loss terms that backpropagate, pulling the two distributions closer (Li et al., 2019).
• Oblique Decision Tree Feature Augmentation: In tabular learning, OT-augmentation appears as sequential concatenation of oblique-projection scores along a decision path: for sample $x_i$, at each split node $t$ with projection vector $w_t$, the augmented feature is $[x_i^{(t)},\, w_t^T x_i]$ (Lyu et al., 1 Feb 2025). This carries discriminative splits forward, deepening the effective representational capacity.
• Domain Structure Alignment in Pose Estimation: RF-based pose features and ground-truth pose features (after respective encoders) are coupled via OT plans, using $L_1$ metric in cost matrix and updating encoder parameters to minimize the induced OT loss (Yu et al., 2022).

3. Integration with Learning Pipelines

OT-augmented feature sets are integrated into end-to-end pipelines in diverse ways:

• As feature representation: Bispectral embeddings, cross-modal OT-aligned vectors, and concatenated oblique features serve directly as input to downstream modules (classification, regression, or detection heads).
• As loss terms or regularizers: OT costs (usually entropy-regularized Sinkhorn divergences) are used as auxiliary losses to enforce alignment between sets (modalities, resolutions, domains). In object detection, OT loss is added to the RoI head losses at each feature pyramid level (Li et al., 2019).
• In iterative or phase-based training schemes: For example, RFPose-OT (Yu et al., 2022) uses a three-phase regime: encoder training on ground-truth, OT-based alignment of RF with pose features, and fine-tuning all heads; OT loss is used to minimize distributional discrepancy, leading to OT-augmented inference features.
• In attention-adapter blocks: Within multimodal transformers, cross-modal attention combined with OT cost drives repeated feature enrichment (e.g. in remote sensing OTAT: visual patch attention enhances textual token embeddings, whose distribution is then regularized using OT plans and EAW loss (Ji et al., 19 Mar 2025)).

4. Symmetry-Awareness and Robustness

A distinguishing property of OT-augmented features is robust invariance to specified nuisance transformations, especially group actions:

• Bispectral OT Feature Sets: By embedding signals/images in bispectrum-invariant spaces, all information other than group-induced variation (e.g. rotations, shifts) is preserved; empirical analysis confirmed near-perfect label recovery under arbitrary rotations, far outperforming raw pixel OT which simply matches by orientation (Ma et al., 25 Sep 2025).
• Multimodality and Structural Harmonization: OT adapters, by optimizing transport plans in common representational spaces (e.g. viewing CLIP’s visual and textual tokens as discrete empirical distributions), yield semantically harmonized cross-modal features, crucial in data-sparse regimes (Ji et al., 19 Mar 2025).
• Resolution/Domain Alignment: OT augments less reliable or ambiguous features by coupling them to their more informative or prototypical domain counterparts, driving semantic centroid compactness and substantially improving detection metrics (Li et al., 2019).

5. Empirical Benefits and Complexity Considerations

The primary empirical gains from OT-augmented feature sets include:

• Semantic Alignment: Near-block-diagonal coupling plans, improved class preservation under nuisance transformations (e.g. >80% label recovery in MNIST under random rotation with bispectral OT vs. 33% for raw feature OT) (Ma et al., 25 Sep 2025).
• Cross-Modal Transfer Performance: State-of-the-art accuracy in few-shot scene classification via cross-modal OT adapter tuning—even with severely mismatched modality density—proven through experiments on standard remote sensing benchmarks (Ji et al., 19 Mar 2025).
• Improved Generalization: In regression and tabular tasks, oblique decision trees with feature-concatenation exhibit quadratically decaying depth-bias ($O(1/K^2)$) and superior sample efficiency, dominating baselines under shallow tree constraints (Lyu et al., 1 Feb 2025).
• Resolution-Enhanced Detection: Object detectors exhibit significant AP boosts (+2.0 mAP overall, especially on small/ambiguous objects), directly attributable to the OT-leveraged intertwiner module (Li et al., 2019).

Computational drawbacks include feature dimension explosion (e.g. bispectral MNIST features reach $D=22,400$), quadratic cost matrix construction, and GPU-bounded Sinkhorn iterations ($O(\varepsilon^{-1} n m)$). Dimensionality reduction and approximate OT solvers are recommended for large-scale settings.

6. Limitations and Guidelines

Limitations of OT-augmented feature set methodologies include:

• Assumption of Known Group Structure: Symmetry-aware OT requires knowledge of acting group and precise nuisance definition; for more complex, approximate, or compositional groups, learned equivariant encoders may be necessary (Ma et al., 25 Sep 2025).
• High Feature Dimensionality: Aggressive O($R K^2$) growth in bispectral methods, expanding feature-concatenation in ODTs, and token-level OT in cross-modal tasks may impede scalability. PCA or selective invariant extraction is recommended where completeness is less crucial.
• Computational Overhead: Cost matrix and Sinkhorn iterations are bottlenecks, especially for large $n, m$; batching and GPU acceleration are recommended.
• Applicability Restricted to Data Domains: While OT and its augmentations perform robustly in images, time-series, and certain structured data, extension to non-image modalities or to datasets lacking clear ground-truth alignment remains challenging.

Normalization of feature vectors and proper regularization tuning ($\varepsilon$, $d(\cdot,\cdot)$ metric selection) are essential to avoid scale artifacts.

7. Representative Table: OT-Augmented Feature Set Construction

Method	Primary Augmentation Mechanism	Domain
Bispectral OT (Ma et al., 25 Sep 2025)	Bispectrum invariant embedding	Vision/Signals
OT Adapter (Ji et al., 19 Mar 2025)	Cross-modal OT with attention	Multimodal/RS
Feature Intertwiner (Li et al., 2019)	OT-aligned feature sets	Object detection
FC-ODT (Lyu et al., 1 Feb 2025)	Oblique projection concatenation	Tabular
RFPose-OT (Yu et al., 2022)	RF-to-pose feature domain transport	Pose estimation

Each approach leverages OT-based coupling to generate feature sets that inherit invariance, semantic compactness, and cross-domain harmonization crucial for robust learning across diverse modalities and symmetries.