Quadratic Manifold Dimensionality Reduction

• Quadratic Manifold dimensionality reduction is a method that augments linear models with a quadratic term to capture curvature and nonlinear couplings in complex systems.
• It reconstructs high-dimensional states using a linear term plus a quadratic correction, improving accuracy for transport-dominated and structural dynamics problems.
• Advanced implementations use greedy basis selection, operator inference, and kernel methods to significantly outperform traditional linear dimensionality reduction techniques.

Quadratic Manifold (QM) dimensionality reduction denotes a family of low-dimensional modeling techniques in which quadratic structure is built directly into the reduction mechanism. In the canonical reduced-order modeling setting, a high-dimensional state is reconstructed from latent coordinates by a linear term plus a quadratic correction. In related supervised formulations, the same label is used for objectives that optimize sums of quadratic forms under manifold constraints or for projections that maximize class separability through first- and second-order statistics on statistical manifolds. Taken together, these formulations suggest a common theme: purely linear subspaces are replaced or augmented by curvature, quadratic couplings, or quadratic discriminability so that low-dimensional coordinates better capture the structure of the original problem (Jain et al., 2016, Geelen et al., 2022, Smith et al., 2016, Herrera-Esposito et al., 31 Jan 2025).

1. Terminological scope and core formulations

The literature uses “Quadratic Manifold” in several closely related but not identical senses. In one line of work, the manifold is a curved reconstruction set in the full state space. In another, the relevant manifold is the constraint set—typically a Stiefel manifold—over which a quadratic objective is optimized. In a third, the manifold is statistical, such as the SPD or Gaussian manifold used to encode class-conditional second-order structure. The unifying feature is not a single universal formula, but the explicit role of quadratic structure in the dimensionality-reduction map or criterion.

Formulation	Canonical expression	Representative papers
State-space quadratic manifold	$u \approx V z + \frac{1}{2}\Theta z z$, $$s \approx s_{\mathrm{ref}} + V\hat{s} + \bar V(\hat{s}\otimes \hat{s})$$	(Jain et al., 2016, Rutzmoser et al., 2016, Geelen et al., 2022)
Quadratic objective on a constraint manifold	$\min -\frac{1}{2}\sum_k w_k^\top R_k w_k$ subject to $W^\top W = I$	(Smith et al., 2016)
Statistical-manifold supervised reduction	$z=F^\top x$, maximize Fisher–Rao or QMI-based class separation	(Herrera-Esposito et al., 31 Jan 2025, Tangkaratt et al., 2015)

In the model-reduction literature, QM methods are motivated by the failure of linear subspaces when the solution manifold has slowly decaying Kolmogorov $n$-width, especially in transport-dominated or geometrically nonlinear problems. In the supervised literature, the quadratic component appears through covariance-aware discrimination, SPD-manifold geometry, or quadratic dependence measures rather than through a quadratic decoder (Ji et al., 26 Mar 2026, Herrera-Esposito et al., 31 Jan 2025).

2. Canonical state-space quadratic manifolds

The canonical QM ansatz in nonlinear structural dynamics writes the full displacement as a quadratic function of reduced coordinates: $$\mathbf{u} = \boldsymbol{\Gamma}(\mathbf{z}) = \mathbf{V}\mathbf{z} + \frac{1}{2}\boldsymbol{\Theta}\,\mathbf{z}\mathbf{z},$$ or equivalently

$$\mathbf{u} \approx \boldsymbol{\Phi}\mathbf{q} + \frac{1}{2}(\boldsymbol{\Omega}\cdot \mathbf{q})\cdot \mathbf{q}.$$

Here the linear part is spanned by a small set of vibration modes, while the curvature tensor is supplied by modal derivatives or static modal derivatives. The manifold is tangent at equilibrium to the modal subspace, and its second derivative determines the curvature of that subspace as modal amplitudes grow (Jain et al., 2016, Rutzmoser et al., 2016).

This immediately changes the geometry of reduction. A linear ROM assumes that admissible states lie in a fixed flat space $\{Vz\}$; a QM assumes that admissible states lie on a curved set $\{Vz + \frac12 \Theta zz\}$. The tangent matrix becomes configuration-dependent,

$$s \approx s_{\mathrm{ref}} + V\hat{s} + \bar V(\hat{s}\otimes \hat{s})$$0

so the projection space evolves with the state instead of remaining fixed. In structural dynamics this captures configuration-dependent kinematics such as axial stretching induced by transverse motions, which linear modal truncation misses (Rutzmoser et al., 2016).

A data-driven, nonintrusive variant uses the same geometry in state-space form: $$s \approx s_{\mathrm{ref}} + V\hat{s} + \bar V(\hat{s}\otimes \hat{s})$$1 The quadratic term is learned from snapshot data and lies in the orthogonal complement of the POD subspace, so it acts as a data-driven closure term without introducing additional reduced coordinates. This is a central reason QMs are used against slowly decaying linear approximation error in transport-dominated systems (Geelen et al., 2022).

3. Construction and learning of the manifold

A standard construction begins with POD or PCA. Given snapshots, one computes a linear basis $$s \approx s_{\mathrm{ref}} + V\hat{s} + \bar V(\hat{s}\otimes \hat{s})$$2, encodes states by $$s \approx s_{\mathrm{ref}} + V\hat{s} + \bar V(\hat{s}\otimes \hat{s})$$3, and fits the quadratic correction by regularized least squares. In the feature-map form,

$$s \approx s_{\mathrm{ref}} + V\hat{s} + \bar V(\hat{s}\otimes \hat{s})$$4

the training problem is to minimize the reconstruction error of the nonlinear correction term, often with Tikhonov regularization. In explicit quadratic models, $$s \approx s_{\mathrm{ref}} + V\hat{s} + \bar V(\hat{s}\otimes \hat{s})$$5 collects all monomials $$s \approx s_{\mathrm{ref}} + V\hat{s} + \bar V(\hat{s}\otimes \hat{s})$$6 with $$s \approx s_{\mathrm{ref}} + V\hat{s} + \bar V(\hat{s}\otimes \hat{s})$$7; in nonintrusive ROM this appears as fitting $$s \approx s_{\mathrm{ref}} + V\hat{s} + \bar V(\hat{s}\otimes \hat{s})$$8 against linear projection residuals (Geelen et al., 2022, Schwerdtner et al., 2024).

A recurring difficulty is that leading principal components alone need not provide the best latent coordinates for a quadratic correction. “Greedy construction of quadratic manifolds for nonlinear dimensionality reduction and nonlinear model reduction” (Schwerdtner et al., 2024) shows that subspaces spanned only by the leading principal components can miss information necessary for the quadratic correction terms to be efficient. Its greedy method selects from leading as well as later principal components so that the corresponding linear approximations can be corrected most efficiently with quadratic terms, and the reported result is orders of magnitude higher accuracy than manifolds based on the leading principal components alone.

Later work extended this from batch to streaming settings. “Online learning of quadratic manifolds from streaming data for nonlinear dimensionality reduction and nonlinear model reduction” (Schwerdtner et al., 2024) introduces an online greedy method that updates quadratic manifolds in one pass as data points are received, using incremental truncated SVD. The stated purpose is in-situ analysis of numerical simulation data on the Petabyte scale, while eliminating expensive disk input/output operations as well as storing and loading data points once they have been processed.

Another development is task-aware training. “Operator Inference Aware Quadratic Manifolds with Isotropic Reduced Coordinates for Nonintrusive Model Reduction” (Schwerdtner et al., 28 Jul 2025) argues that training only for snapshot reconstruction can produce oscillatory or other non-smooth embeddings that hinder learning accurate reduced models. Its greedy procedure includes both reconstruction error and the prediction error of reduced models fitted to the embedded data, and introduces isotropic reduced coordinates so that each reduced coordinate trajectory has unit $$s \approx s_{\mathrm{ref}} + V\hat{s} + \bar V(\hat{s}\otimes \hat{s})$$9 norm over the snapshots. The paper reports reduced models with up to two orders of magnitude higher accuracy than QMs trained with respect to reconstruction error alone. A related stabilization strategy appears in “Nonlinear Model Order Reduction on Quadratic Manifolds via Greedy Algorithms with Dimension-Dependent Regularization” (Ji et al., 26 Mar 2026), which introduces a double-greedy algorithm on regularization parameters coupled with a standard greedy algorithm on the physical parameter.

4. Projection of dynamics, force compensation, and structure preservation

Once a quadratic manifold is defined, the reduced dynamics are obtained by projecting the full equations onto the tangent space of the manifold. In structural dynamics, substituting

$\min -\frac{1}{2}\sum_k w_k^\top R_k w_k$0

into the full equations and testing with the tangent basis yields a reduced model with configuration-dependent mass and damping matrices, reduced internal forces, and an additional convective-type term proportional to $\min -\frac{1}{2}\sum_k w_k^\top R_k w_k$1. This is the essential Galerkin mechanism behind QM ROMs (Jain et al., 2016, Rutzmoser et al., 2016).

The 2016 generalization “Generalization of Quadratic Manifolds for Reduced Order Modeling of Nonlinear Structural Dynamics” (Rutzmoser et al., 2016) rederives the curvature tensor from a force-compensation condition,

$\min -\frac{1}{2}\sum_k w_k^\top R_k w_k$2

which shifts the quadratic behavior from the internal force to the manifold. This yields static derivatives as natural extensions of static modal derivatives for arbitrary linear bases, including Krylov vectors. The paper’s QM variants include QM-MD, QM-SMD, QM-KrySD, and mixed Krylov–mode constructions, together with orthogonalized variants that stabilize the tangent space.

For Hamiltonian systems, preserving geometric structure becomes central. “Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds” (Sharma et al., 2023) introduces two structure-aware variants. SMG-QMCL-ROM uses a Quadratic Manifold Cotangent Lift and symplectic manifold Galerkin projection to obtain a nonlinear Hamiltonian ROM with a symplectic nonlinear lift. Galerkin-BQ-ROM uses a blockwise-quadratic map together with a standard Galerkin projection, producing an approximately Hamiltonian model. The reported finding is that both approaches are effective for issuing predictions in settings well outside the range of their training data while providing more accurate solutions than linear symplectic reduced-order models.

A further generalization replaces explicit quadratic features by reproducing-kernel structure. “Kernel manifolds: nonlinear-augmentation dimensionality reduction using reproducing kernel Hilbert spaces” (Diaz et al., 29 Aug 2025) keeps the same POD-style linear encoder but learns the nonlinear correction from a user-defined RKHS: $\min -\frac{1}{2}\sum_k w_k^\top R_k w_k$3 This includes polynomial structure, feature-map corrections, and radial basis function-based corrections as special cases, with relatively low training cost and monotonically decreasing error as the latent space dimension increases. In that sense, QMs appear as one particular point in a broader nonlinear-augmentation family.

5. Supervised and information-geometric variants

In supervised dimensionality reduction, QM language is used differently but still centers on quadratic structure. “A Category Space Approach to Supervised Dimensionality Reduction” (Smith et al., 2016) represents each class as a one-dimensional subspace and learns orthonormal class axes $\min -\frac{1}{2}\sum_k w_k^\top R_k w_k$4 so that

$\min -\frac{1}{2}\sum_k w_k^\top R_k w_k$5

encodes class membership. After eliminating auxiliary variables, the method minimizes

$\min -\frac{1}{2}\sum_k w_k^\top R_k w_k$6

subject to $\min -\frac{1}{2}\sum_k w_k^\top R_k w_k$7, where $\min -\frac{1}{2}\sum_k w_k^\top R_k w_k$8 is the class-specific covariance matrix. This is a heterogeneous sum of quadratic forms on a Stiefel manifold, and the paper explicitly notes that the method can be interpreted as a Quadratic Manifold dimensionality reduction scheme because it couples a quadratic objective with manifold constraints.

A different supervised interpretation appears in “Supervised Quadratic Feature Analysis: Information Geometry Approach for Dimensionality Reduction” (Herrera-Esposito et al., 31 Jan 2025). SQFA learns a linear projection $\min -\frac{1}{2}\sum_k w_k^\top R_k w_k$9 so that class distributions in the projected space are maximally separated on the SPD/Gaussian statistical manifold via affine-invariant or Fisher–Rao distances. The projected class statistics are encoded through first- and second-order moments, and the objective maximizes pairwise manifold distances between class Gaussians. The paper frames this as a route to quadratic discriminability aligned with QDA, and states that, from the perspective of information geometry, SQFA is to QDA what drLDA is to cLDA.

A third supervised strand uses quadratic dependence rather than quadratic reconstruction. “Direct Estimation of the Derivative of Quadratic Mutual Information with Application in Supervised Dimension Reduction” (Tangkaratt et al., 2015) formulates supervised dimension reduction as maximizing Quadratic Mutual Information between projected inputs and outputs. QMI is based on an $W^\top W = I$0 distance rather than the KL divergence, so it is described as more robust against outliers than standard MI. The paper’s central methodological point is that directly estimating the derivative of QMI is more accurate than differentiating an estimated QMI, and this derivative is then used for subspace search.

6. Empirical behavior, applications, and limitations

QM dimensionality reduction has been applied to geometrically nonlinear structural dynamics, transport-dominated PDEs, Hamiltonian wave systems, turbulent flows, acoustic waves, Kelvin–Helmholtz instability, high-speed aerothermodynamics, and shock-dominated compressible flows. In structural mechanics, QMs can match or nearly match the accuracy of linear manifolds augmented with modal or static derivatives while retaining only the primary modal coordinates, as shown for flat plates and wing structures (Jain et al., 2016, Rutzmoser et al., 2016). In transport and wave problems, data-driven QMs reduce projection and state errors relative to purely linear POD or linear symplectic ROMs (Geelen et al., 2022, Sharma et al., 2023).

The most consistent empirical pattern is that the benefit of a QM depends strongly on how the latent coordinates and nonlinear correction are chosen. Greedy quadratic manifolds can deliver orders of magnitude higher accuracy than PCA-based QMs (Schwerdtner et al., 2024). Operator-inference-aware QMs can improve downstream reduced-model accuracy by up to two orders of magnitude relative to reconstruction-only training (Schwerdtner et al., 28 Jul 2025). Online greedy learning has been demonstrated on data that far exceed common disk input/output capabilities and volumes as well as main-memory sizes, including a Petabyte-scale Kelvin–Helmholtz example (Schwerdtner et al., 2024). Kernel manifolds often outperform POD and several recent QM approaches on several example problems, especially when the closure relation between retained and truncated modes is not well captured by a quadratic polynomial (Diaz et al., 29 Aug 2025).

The limitations are equally well documented. In nonlinear structural dynamics, the quadratic coupling assumption works best when the dominant nonlinearity is of cable or membrane type, so that axial or membrane effects can be statically condensed as quadratic functions of slower coordinates. When large rotations and fully coupled geometric nonlinearities dominate, the quadratic constraint can become unphysical; the cantilever example in the generalized structural QM paper is the canonical failure case, where linear-augmented bases outperform the QM (Rutzmoser et al., 2016). In data-driven settings, performance depends sensitively on regularization, feature scaling, and basis selection, which is why dimension-dependent regularization, isotropic reduced coordinates, and greedy selection of subspaces or kernels have become recurring themes (Ji et al., 26 Mar 2026, Schwerdtner et al., 28 Jul 2025). In streaming methods, the quality of the quadratic manifold depends on the quality of the incremental truncated SVD and therefore on the truncation rank $W^\top W = I$1 (Schwerdtner et al., 2024). In supervised Stiefel-manifold formulations, global optimality can be certified by a sufficient Rapcsák condition, but this remains a post-hoc criterion rather than a generic algorithmic guarantee (Smith et al., 2016).

Taken together, the literature indicates that QM dimensionality reduction is best understood not as a single algorithm, but as a design principle: introduce quadratic structure at the level of the decoder, the reduced dynamics, or the discriminative criterion so that low-dimensional coordinates can represent curvature, coupling, or covariance-aware class structure that linear methods systematically miss.