DEP: Dimensionally-Efficient Parametrization

• DEP is a parametrization strategy that represents constrained objects by matching the intrinsic degrees of freedom, ensuring explicit backmapping to valid design parameters.
• It employs methods such as logarithm-based mapping, PME, and NLPME to reduce ambient dimensionality while maintaining geometric accuracy and reconstructability.
• DEP facilitates efficient compression and seamless integration with engineering workflows, including CAD modeling, simulation, and transformer parameter optimization.

Searching arXiv for papers on "Dimensionally-Efficient Parametrization" and closely related parametric shape-design methods. Dimensionally-Efficient Parametrization (DEP) denotes a class of representations that encode an object, model, or design family with a number of coordinates aligned with the intrinsic degrees of freedom of the underlying constrained set, rather than with a naive ambient representation. In the literature surveyed here, DEP appears most explicitly as a real-coordinate parametrization of unitary matrices using exactly the real dimension of the unitary group (Alegría, 6 Oct 2025), and more broadly as a design principle for reduced-dimensional parametrizations that preserve admissibility, reconstructability, and compatibility with established engineering workflows. In parametric shape design, this principle is instantiated by Parametric Model Embedding (PME) and its nonlinear extension NLPME, which reduce dimensionality while preserving an explicit backmapping to admissible design parameters (Serani et al., 2022, Serani et al., 12 May 2026). Related work on CAD-integrated PGD, adaptive FFD parameterization, kernel morphing, and structured transformer parametrizations further situates DEP within a wider program of replacing redundant coordinates by compact, structured representations [(Sevilla et al., 2019); (Majd, 2015); (Łaniewski-Wołłk, 2013); (Lou et al., 6 Jan 2026)].

1. Conceptual scope and definition

DEP is used most explicitly in "Dimensionally-Efficient Transmission and Storage of Unitary Matrices" to describe a representation of $U\in\mathcal{U}(N)$ by exactly $N^2$ real numbers, matching the real dimension of the unitary group (Alegría, 6 Oct 2025). The paper’s motivation is that a unitary matrix has $N^2$ complex entries, that is, $2N^2$ real numbers if stored naively, whereas $\mathcal{U}(N)$ is an $N^2$-dimensional real manifold. In that setting, DEP is not merely compression in an empirical sense; it is an explicit coordinate system aligned with manifold dimension.

A broader interpretation emerges across parametric design papers that do not always use the DEP label, but describe closely related aims: reducing dimension, preserving variability, and maintaining a path back to the original CAD or parametric representation. PME is presented as a method that embeds the original design variables into a reduced-dimensionality representation of the design space (Serani et al., 2022). NLPME extends that logic by replacing the linear reduced subspace with a nonlinear latent representation while preserving explicit backmapping to admissible design parameters (Serani et al., 12 May 2026).

Within this broader usage, DEP is best understood as a structural requirement on reduced representations. The reduced coordinates must remain operationally meaningful: they should correspond to admissible objects, support reconstruction into the original parameter space, and fit downstream workflows such as optimization, surrogate modeling, meshing, simulation, or transmission. This distinguishes DEP from generic dimensionality reduction that reconstructs only ambient data and leaves the pre-image problem unresolved (Serani et al., 2022, Serani et al., 12 May 2026).

2. Geometric and algebraic foundations

The most explicit mathematical DEP construction in the supplied literature is the logarithm-based parametrization of the unitary group. The unitary group is

$$\mathcal{U}(N)=\{U\in\mathbb{C}^{N\times N}:U^HU=I_N\},$$

and the paper constructs an injective map

$f:\mathcal{U}(N)\to \mathbb{R}^{N^2},$

as well as a bounded version

$$f:\mathcal{U}(N)\to [-\sqrt{N}\pi,\sqrt{N}\pi]^{N^2}$$

(Alegría, 6 Oct 2025). The construction proceeds by mapping $U$ to its skew-Hermitian logarithm $N^2$0, then expanding that matrix in a fixed orthonormal basis of $N^2$1 under the Hilbert–Schmidt inner product. The explicit DEP map is

$N^2$2

with inverse

$N^2$3

(Alegría, 6 Oct 2025).

This formulation is dimensionally efficient because $N^2$4, exactly the number of real coordinates used. The same paper proves boundedness: $N^2$5 which facilitates quantization and transmission (Alegría, 6 Oct 2025). It also outlines lower-dimensional variants for constrained subclasses, including $N^2$6 dimensions for $N^2$7, $N^2$8 for symmetric unitary matrices, and $N^2$9 for rotation matrices in $N^2$0 (Alegría, 6 Oct 2025).

A related, though architecturally rather than manifold-theoretically framed, example appears in CRoPE. There the argument is that standard real-valued $N^2$1 projections in RoPE are not equivalent to a complex linear map, and that constraining them to the complex-linear subspace removes redundancy while retaining the essential structure of rotary embeddings (Lou et al., 6 Jan 2026). The proposed block-structured matrix

$N^2$2

is the real representation of a complex linear operator, saving about $N^2$3 of the parameters in the $N^2$4 projections (Lou et al., 6 Jan 2026). This suggests a general DEP principle: identify the true structured function class first, then parameterize that class directly rather than overparameterizing it in an ambient coordinate system.

3. DEP in parametric shape design: PME and NLPME

In shape design, DEP is closely associated with reduced-dimensional parametrization of CAD or parametric design spaces. PME was introduced to address the limitation of geometry-only reduction methods that compress geometry well but often lose the ability to map reduced coordinates back to the original design variables (Serani et al., 2022). PME constructs an augmented feature space combining the shape modification vector and the original design-variable vector: $N^2$5 The reduction is performed through a weighted generalized PCA in which the design-variable block is assigned zero weight, so the latent coordinates remain geometry-driven while the eigenvectors retain the information required for backmapping (Serani et al., 2022).

The importance of this construction is theoretical as well as practical. PME preserves the same nonzero eigenvalues for the geometry part as KLE, so it does not sacrifice the geometric variance captured by KLE; rather, it augments the eigenspace with parameterization information (Serani et al., 2022). This is why PME can be viewed as a DEP-style embedding: it reduces dimension without severing the connection to the original parametric model.

NLPME generalizes this philosophy to nonlinear latent spaces. In the notation of the paper, the original parametric design space is

$N^2$6

with forward parametric map

$N^2$7

(Serani et al., 12 May 2026). NLPME replaces the linear reduced subspace by a nonlinear encoder–decoder chain: $N^2$8 The critical distinction from a standard autoencoder is that NLPME does not decode directly to geometry. It decodes to design parameters, and only then is geometry recovered through the forward parametric map or a differentiable surrogate (Serani et al., 12 May 2026).

The geometry-consistency loss is

$N^2$9

and no explicit parameter-reconstruction term is included. Instead, the decoder output is constrained to the admissible range by a sigmoid activation, so $2N^2$0 in normalized parameter space (Serani et al., 12 May 2026). The resulting representation is therefore nonlinear in latent space but still explicit in the original parametric variables.

4. Reconstruction, admissibility, and the pre-image problem

A defining DEP criterion in engineering design is explicit backmapping. PME’s reduced basis is partitioned into geometric and parametric parts,

$2N^2$1

so that reduced coordinates $2N^2$2 yield the backmapping

$2N^2$3

(Serani et al., 12 May 2026). In the linear PME formulation, the same latent coefficients that reconstruct geometry also reconstruct the original design variables through the embedded modes (Serani et al., 2022).

This parameter-mediated reconstruction addresses the pre-image problem emphasized in both PME and NLPME. Geometry-only reduction methods such as KLE/POD/PCA reduce the dimensionality of the shape modification field but typically do not provide a direct inverse mapping to the original CAD parameterization (Serani et al., 2022). Generic nonlinear encoders may reconstruct geometry accurately, but without a reliable inverse map to design parameters they are hard to use in CAD-driven design workflows (Serani et al., 12 May 2026).

The CAD/forward-parametric reconstruction pipeline in NLPME makes this point explicit. Reconstruction occurs in two steps: first decode latent variables into admissible design parameters, then generate geometry from those parameters through $2N^2$4 or $2N^2$5 (Serani et al., 12 May 2026). Because the exact CAD generator may be non-differentiable, the paper uses a differentiable surrogate

$2N^2$6

trained by

$2N^2$7

(Serani et al., 12 May 2026). This preserves differentiability during training while keeping the representation tied to the original parametric structure.

The same admissibility issue appears in linear PME optimization. The paper penalizes reduced-space points whose reconstructed original variables fall outside bounds by

$2N^2$8

with $2N^2$9 (Serani et al., 2022). This indicates that reduced coordinates alone are not sufficient; DEP requires a feasible operational map back to the underlying constrained design space.

5. Representative workflows and applications

In parametric shape design, PME and NLPME are demonstrated on concrete CAD-based design spaces. PME is shown on the reparameterization of 2D Bézier curves and 3D free-form deformation design spaces, together with simulation-driven optimization of a subsonic airfoil and a naval destroyer in calm water (Serani et al., 2022). For the airfoil case, a 14-variable Bézier-based parameterization is reduced to $\mathcal{U}(N)$0 variables at 95% retained variance, with dimensionality reduction about 70% and NMSE about 2.5% (Serani et al., 2022). For the DTMB 5415 destroyer, an FFD model with $\mathcal{U}(N)$1 design variables is reduced to $\mathcal{U}(N)$2 variables at 95% retained variance, again about 70% reduction and NMSE about 4% (Serani et al., 2022).

NLPME is validated on a bio-inspired autonomous underwater glider with a CAD-based parametric hull model (Serani et al., 12 May 2026). The half-body has $\mathcal{U}(N)$3 free design variables. The dataset is generated by Sobol sampling with $\mathcal{U}(N)$4 sampled configurations, of which $\mathcal{U}(N)$5 valid geometries are retained after filtering. Each valid geometry is represented by $\mathcal{U}(N)$6 geometric degrees of freedom, corresponding to the 3D coordinates of $\mathcal{U}(N)$7 aligned surface points (Serani et al., 12 May 2026). The CAD generation pipeline uses OpenCASCADE and includes trailing-edge regularization, spanwise tangency constraints, repeated tip sections, and trimming operations (Serani et al., 12 May 2026).

The normalized reconstruction error used for comparison is

$\mathcal{U}(N)$8

(Serani et al., 12 May 2026). At the 5% threshold, PME requires $\mathcal{U}(N)$9, DAE $N^2$0, and NLPME $N^2$1. At the 1% threshold, PME requires $N^2$2, DAE $N^2$3, and NLPME $N^2$4 (Serani et al., 12 May 2026). The paper’s interpretation is that DAE can be slightly more compact because it directly decodes geometry, whereas NLPME preserves explicit backmapping to admissible parameters and retains most of the nonlinear compression gain (Serani et al., 12 May 2026).

DEP-style ideas also appear in offline parametric PDE solution methods integrated with CAD. In the PGD framework, geometry parameters are treated as extra coordinates, a family of admissible shapes $N^2$5 is described through a mapping

$N^2$6

and a generalized solution $N^2$7 is computed off-line as a separated representation (Sevilla et al., 2019). The paper emphasizes seamless integration with CAD preprocessors by using NURBS control points as geometric parameters (Sevilla et al., 2019). This is DEP-like in the sense that the geometric degrees of freedom become explicit low-dimensional coordinates in a reusable computational vademecum.

6. Relation to adjacent parametrization strategies

DEP is related to, but distinct from, several nearby strategies.

Adaptive FFD parameterization seeks parameterization adaptivity rather than explicit latent embedding. In 3D aerodynamic shape optimization, the procedure is to optimize in a low-dimensional FFD space, monitor when the control lattice becomes irregular, rebuild the parameterization around the current shape, and continue optimization with the same geometry but a more regular parameterization (Majd, 2015). The tested coarse, medium, and fine parameterizations have 8, 20, and 32 degrees of freedom, respectively, and the adaptive method is reported to achieve a better final cost and improved convergence on a transonic business-aircraft wing case (Majd, 2015). This is DEP-like because it keeps the parameter space compact and well-conditioned, though it is not a latent-space embedding in the PME sense.

Morphing-based kernel parameterization provides another DEP-style mechanism. In "Automatic parametrization and mesh deformation for CFD optimization," the design variables are the displacements of a small set of automatically selected morphing nodes $N^2$8, while the full mesh deformation is reconstructed by

$N^2$9

and, after preprocessing,

$$\mathcal{U}(N)=\{U\in\mathbb{C}^{N\times N}:U^HU=I_N\},$$0

(Łaniewski-Wołłk, 2013). Node placement is driven by the posterior variance

$$\mathcal{U}(N)=\{U\in\mathbb{C}^{N\times N}:U^HU=I_N\},$$1

(Łaniewski-Wołłk, 2013). Here dimensional efficiency arises from adaptive sparse control-point selection rather than from PCA, manifold learning, or Lie-algebra coordinates.

Structured architectural reparameterization, exemplified by CRoPE, belongs to the same family at a more abstract level. CRoPE constrains RoPE $$\mathcal{U}(N)=\{U\in\mathbb{C}^{N\times N}:U^HU=I_N\},$$2 projections to the complex-linear subspace, saving about 50% of the parameters in those layers while leaving RoPE and CRoPE “essentially indistinguishable” in validation loss across PG-19, WikiText-2, and Penn Treebank in the reported experiments (Lou et al., 6 Jan 2026). A plausible implication is that DEP can refer not only to coordinate systems over constrained geometric objects, but also to architectural parameter spaces whose naive formulation contains symmetrically redundant degrees of freedom.

A notable source of ambiguity is the acronym “DEP” itself. In "DEP-RL: Embodied Exploration for Reinforcement Learning in Overactuated and Musculoskeletal Systems," DEP means Differential Extrinsic Plasticity, a self-organizing exploration controller defined by

$$\mathcal{U}(N)=\{U\in\mathbb{C}^{N\times N}:U^HU=I_N\},$$3

and an online update rule for $$\mathcal{U}(N)=\{U\in\mathbb{C}^{N\times N}:U^HU=I_N\},$$4 (Schumacher et al., 2022). That usage is unrelated to Dimensionally-Efficient Parametrization. The shared acronym should therefore not be taken to indicate conceptual continuity.

7. Significance, limitations, and open issues

Across the surveyed literature, DEP is significant because it targets a specific failure mode of many reduced representations: they compress data but detach it from the constrained object or workflow of interest. PME and NLPME were developed precisely because engineers need reduced variables that can be turned back into valid CAD parameters for meshing, simulation, optimization, and exploration (Serani et al., 2022, Serani et al., 12 May 2026). In wireless communications, the logarithm-based DEP of unitary matrices is significant because it reduces storage and feedback from $$\mathcal{U}(N)=\{U\in\mathbb{C}^{N\times N}:U^HU=I_N\},$$5 real numbers to $$\mathcal{U}(N)=\{U\in\mathbb{C}^{N\times N}:U^HU=I_N\},$$6 while providing an explicit inverse map and bounded coordinate range (Alegría, 6 Oct 2025). In transformer architectures, CRoPE is significant because it shows that a mathematically structured subspace can remove redundancy with negligible out-of-sample impact (Lou et al., 6 Jan 2026).

The limitations depend on the DEP instantiation. PME is intrinsically linear and may become inefficient when the sampled design space is governed by nonlinear geometric variability, such as variability induced by lofting, section coupling, and other CAD construction operations (Serani et al., 12 May 2026). NLPME addresses this, but comparison with a deep autoencoder shows that direct geometry decoding can still be slightly more compact at strict error thresholds (Serani et al., 12 May 2026). In PME optimization, latent coordinates can map outside the original parameter bounds, requiring feasibility handling via penalties (Serani et al., 2022).

In explicit manifold DEP for unitary matrices, the dominant computational cost is eigendecomposition for the matrix logarithm and exponential, with time complexity $$\mathcal{U}(N)=\{U\in\mathbb{C}^{N\times N}:U^HU=I_N\},$$7 assuming enough hardware parallelism (Alegría, 6 Oct 2025). The bounded interval used for quantization is useful but not necessarily tight, and the paper notes that nonuniform or vector quantization could improve performance (Alegría, 6 Oct 2025).

More broadly, the collected papers suggest two persistent tensions. The first is between compactness and operational admissibility: direct geometric or unconstrained latent decoding may yield the smallest latent dimension, while parameter-mediated reconstruction better preserves engineering meaning (Serani et al., 12 May 2026). The second is between global explicitness and local practicality: logarithm-based or basis-based DEP offers exact manifold coordinates (Alegría, 6 Oct 2025), whereas kernel morphing, FFD adaptation, and PGD-based geometric parametrization emphasize workflow integration and deployability [(Łaniewski-Wołłk, 2013); (Majd, 2015); (Sevilla et al., 2019)].

Taken together, these works portray DEP not as a single universal algorithm but as a methodological criterion: represent constrained objects or design families with coordinates commensurate with their true degrees of freedom, and do so in a way that preserves reconstruction, admissibility, and utility in the target computational pipeline (Serani et al., 2022, Serani et al., 12 May 2026, Alegría, 6 Oct 2025).