Dimension Reduction for Curves

• Dimension reduction for curves is the process of mapping high-dimensional curve data onto lower-dimensional spaces while preserving intrinsic geometry and reconstructability.
• The methodologies range from invertible manifold encodings using moving frames to random projections that maintain continuous Fréchet distances and intrinsic principal curves on spheres.
• Applications span shape analysis, time series clustering, and geometric modeling, with evaluation metrics focused on curvature preservation and fidelity of reconstruction.

Dimension reduction for curves studies how a curve, a sequence of points, or a manifold-valued trajectory can be represented in lower dimension while preserving properties such as reconstructability, intrinsic geometry, or task-relevant dissimilarities. In the literature summarized here, the subject spans invertible encodings of shape sequences on manifolds via moving frames and parallel transport (Yi et al., 2011), random linear projections for polygonal curves that preserve continuous Fréchet and related distances up to a factor of $1\pm\varepsilon$ (Psarros et al., 2022, Ebbens et al., 3 Jul 2026), and intrinsic principal-curve constructions for spherical data (Kim et al., 2020). A complementary line of work treats evaluation itself as a geometric problem, using curvature-based criteria to assess whether a reduced representation preserves the structure of curved data (Lara-Cabrera et al., 2023, Beylier et al., 16 Sep 2025).

1. Problem classes and formal viewpoints

Dimension reduction for curves is not a single problem. The sources distinguish at least three settings.

First, a curve may be a manifold-valued temporal object, such as a shape sequence evolving on a nonlinear and high-dimensional manifold. In that regime, the reduction problem is to encode the entire trajectory in a low-dimensional Euclidean signal while retaining an invertible mapping back to the manifold (Yi et al., 2011).

Second, a curve may be a polygonal curve in $\mathbb{R}^d$, as in high-dimensional time series. Here the objective is typically not pointwise reconstruction of every ambient coordinate, but preservation of a curve distance such as the continuous Fréchet distance, dynamic time warping, or Hausdorff distance after projection into $\mathbb{R}^t$ (Psarros et al., 2022, Ebbens et al., 3 Jul 2026).

Third, a curve may be the reduced object itself: principal curves fitted intrinsically to data on a manifold such as a sphere. In that case, dimension reduction proceeds by projecting data onto a continuous or piecewise-geodesic curve that is adapted to the manifold geometry (Kim et al., 2020).

Setting	Representative formulation	Reported emphasis
Manifold-valued curve	$X_t$ encoded by $X_0$, a moving frame, and $Z_t$	Invertible and computationally more efficient
Polygonal curves in $\mathbb{R}^d$	Random linear map $f:\mathbb{R}^d\to\mathbb{R}^t$	$1\pm\varepsilon$ preservation of curve distances
Data on $S^d$	Projection onto a spherical principal curve $\mathbb{R}^d$0	Self-consistency and exact geodesic projection

A recurrent misconception is to treat curve reduction as a direct extension of point-cloud reduction. The curve setting is stricter. For continuous Fréchet distance, the relevant geometry depends on all points on the polygonal interpolation, not only on vertices (Psarros et al., 2022). For manifold-valued sequences, global linearizations or PCA on spline coefficients fail to adapt to the specific geometry of a given curve and assume global flatness (Yi et al., 2011). This suggests that the central issue is not only dimensionality but also the compatibility between the reduction mechanism and the geometric structure carried by the curve.

2. Invertible reduction of curves on manifolds

A direct treatment of manifold-valued curves is given by the framework of moving frames and parallel transport in "A Invertible Dimension Reduction of Curves on a Manifold" (Yi et al., 2011). The basic object is a curve $\mathbb{R}^d$1 on a manifold $\mathbb{R}^d$2, represented through an initial point, an initial moving frame, and a Euclidean development $\mathbb{R}^d$3: $\mathbb{R}^d$4 Given the initial conditions, the paper states that there is a one-to-one correspondence between $\mathbb{R}^d$5 and $\mathbb{R}^d$6.

The key geometric device is the choice of a parallel moving frame along the curve, transported by the Levi-Civita connection. The encoding is therefore path-dependent and locally adaptive. Instead of searching for a global flat subspace for curve embedding, the method deploys a sequence of local flat subspaces adaptive to the geometry of both the curve and the manifold it lies on (Yi et al., 2011).

The reduction algorithm has two stages. In the embedding stage, an arbitrary initial frame at $\mathbb{R}^d$7 is parallel transported along the curve; all curve tangents are transported back to $\mathbb{R}^d$8; PCA is applied to these transported tangents; and the top $\mathbb{R}^d$9 orthonormal directions $\mathbb{R}^t$0 are used to define a new initial frame. The reduced curve is then written as

$\mathbb{R}^t$1

In the decoding stage, given $\mathbb{R}^t$2, the reduced initial frame, and $\mathbb{R}^t$3, the curve is reconstructed iteratively by

$\mathbb{R}^t$4

where $\mathbb{R}^t$5 is the projection to the manifold to counter drift in the ambient space (Yi et al., 2011).

The reported advantages are invertibility, computational efficiency, and locality. The computational cost is stated to be linear in the length of the curve and the dimension of the shape space, with efficient projection and back-projection due to known ambient and normal spaces of the shape manifold. Experiments on shape sequences, including Running, Jumping, Skipping, Siding, and Waving, embed the data into $\mathbb{R}^t$6 and reconstruct near-identical sequences from the low-dimensional representation (Yi et al., 2011). The same framework is said to enable efficient generative modeling of manifold-valued curves from low-dimensional latent trajectories. A plausible implication is that invertibility is not merely a reconstruction convenience; it turns the reduced trajectory into a valid generative coordinate system for the original curve class.

3. Random projections and distance preservation for polygonal curves

A second major line of work studies oblivious random projection for high-dimensional polygonal curves. The 2022 result on continuous Fréchet distance shows that one can reduce the dimension to $\mathbb{R}^t$7, where $\mathbb{R}^t$8 is the total number of input points, while preserving the continuous Fréchet distance between any two determined polygonal curves within a factor of $\mathbb{R}^t$9 and without any additive error (Psarros et al., 2022). For a set $X_t$0 of $X_t$1 polygonal curves, each of complexity at most $X_t$2, this is also stated as a target dimension $X_t$3, with

$X_t$4

for all $X_t$5, where $X_t$6 projects each vertex by a Johnson-Lindenstrauss type map (Psarros et al., 2022).

The technical difficulty is that continuous Fréchet distance depends on arbitrary points along segments, not only on vertices. The proof therefore uses a finite collection of auxiliary points associated with the relevant decision predicates, so that preserving distances among those points suffices to preserve the continuous decision structure of the free-space formulation (Psarros et al., 2022). The paper also reports applications to $X_t$7-center and $X_t$8-median clustering.

The 2026 generalization revisits this result through sparse oblivious subspace embeddings and extends it beyond Fréchet distance (Ebbens et al., 3 Jul 2026). For $X_t$9 piecewise linear surfaces $X_0$0 with at most $X_0$1 pieces, there exists a random linear map $X_0$2 with

$X_0$3

such that, with probability at least $X_0$4,

$X_0$5

for every pair $X_0$6. The generalized dissimilarity measure is

$X_0$7

with $X_0$8. The paper identifies Fréchet as the case $X_0$9, continuous DTW as $Z_t$0, and states that Hausdorff distance can be handled similarly (Ebbens et al., 3 Jul 2026).

For polygonal curves, corresponding to $Z_t$1, the bound simplifies to

$Z_t$2

described as an optimal JL-type bound up to constants (Ebbens et al., 3 Jul 2026). The proof relies on the observation that any point on a segment is a convex combination of a small number of vertices, so point differences lie in unions of sparse, low-dimensional subspaces. This generalization also extends to piecewise linear surfaces. A common misunderstanding is that such guarantees are limited to discrete curve distances. The papers separate the easy discrete case from the continuous case and emphasize that the continuous setting requires controlling all point differences, not merely vertex-to-vertex distances (Psarros et al., 2022, Ebbens et al., 3 Jul 2026).

4. Intrinsic principal curves on spheres

When the ambient manifold is the sphere $Z_t$3, dimension reduction can proceed by fitting a principal curve directly on the manifold rather than by first flattening the data. "Spherical Principal Curves" defines a self-consistent principal curve $Z_t$4 for a random variable $Z_t$5 on $Z_t$6 by

$Z_t$7

where $Z_t$8 is the projection index of $Z_t$9 onto the curve (Kim et al., 2020).

The curve is represented as a piecewise-geodesic curve, given by points $\mathbb{R}^d$0 connected by geodesics. Projection is defined by minimizing geodesic distance, and the paper stresses that projection is exact: each data point is projected onto every geodesic segment, the closest projected point is chosen, and if the orthogonal projection falls outside a segment, the nearest endpoint is used (Kim et al., 2020). The update step admits an extrinsic and an intrinsic version. The extrinsic mean is

$\mathbb{R}^d$1

while the intrinsic mean is

$\mathbb{R}^d$2

The algorithm initializes with a principal circle estimated via gradient descent, alternates projection and expectation steps, and reparameterizes by unit speed until convergence (Kim et al., 2020).

The paper further gives stationarity conditions adapted to the sphere. Because vector addition is not defined, perturbations are taken along geodesics. The extrinsic and intrinsic critical-point conditions are formulated respectively through derivatives of $\mathbb{R}^d$3 and $\mathbb{R}^d$4 at $\mathbb{R}^d$5 (Kim et al., 2020). Empirically, the method is reported to yield lower reconstruction error and better projection continuity than the earlier Riemannian principal curves of Hauberg, with examples on earthquake data, motion capture data, and synthetic circles and waves on $\mathbb{R}^d$6 and $\mathbb{R}^d$7 (Kim et al., 2020).

The limitations are also explicit. Initialization sensitivity arises from non-convex optimization; $\mathbb{R}^d$8 and $\mathbb{R}^d$9 are chosen heuristically; and the top-down construction may not capture piecewise or intersecting structures well. The paper states that intrinsic approaches on generic manifolds beyond spheres remain an open computational challenge (Kim et al., 2020). This identifies a broader methodological tension in curve reduction on manifolds: exact intrinsic geometry improves fidelity, but it may also increase computational and optimization difficulty.

5. Ribaucour transformations and geometric coordinates

A more classical differential-geometric approach to dimension reduction appears in the theory of Ribaucour transformations. "Ribaucour coordinates" studies smooth and discrete curves, as well as higher-dimensional submanifolds, and proves a result for reduction of the ambient dimension (Burstall et al., 2017). For a smooth curve $f:\mathbb{R}^d\to\mathbb{R}^t$0, a fixed sphere $f:\mathbb{R}^d\to\mathbb{R}^t$1 not intersecting the curve, and a parallel normal field $f:\mathbb{R}^d\to\mathbb{R}^t$2, one constructs a Ribaucour partner curve $f:\mathbb{R}^d\to\mathbb{R}^t$3 that lies on the given sphere $f:\mathbb{R}^d\to\mathbb{R}^t$4. The formula is

$f:\mathbb{R}^d\to\mathbb{R}^t$5

where $f:\mathbb{R}^d\to\mathbb{R}^t$6 and $f:\mathbb{R}^d\to\mathbb{R}^t$7 are Möbius geometric lifts of $f:\mathbb{R}^d\to\mathbb{R}^t$8 and the unit normal $f:\mathbb{R}^d\to\mathbb{R}^t$9 (Burstall et al., 2017).

The same paper states that any curve in $1\pm\varepsilon$0 can locally be obtained from a circular arc by two commuting Ribaucour transformations, and the resulting coordinates are called Ribaucour coordinates (Burstall et al., 2017). The commutativity is supplied by Bianchi permutability: successive Ribaucour reductions can be applied in any order and yield the same result. Analogous constructions are given for discrete curves and for $1\pm\varepsilon$1-dimensional submanifolds, with smooth and discrete versions of the transformation onto fixed hyperspheres (Burstall et al., 2017).

This is a different notion of dimension reduction from random projection or PCA-based curve encoding. The objective is not a low-dimensional Euclidean latent signal with a norm-preservation bound, but a constructive reduction of the ambient dimension while preserving geometric structures such as curvature lines, normal bundles, and sphere congruences. A plausible implication is that Ribaucour coordinates occupy a bridge between geometric modeling and curve reduction: they reduce ambient complexity through transformation theory rather than through metric embedding.

6. Evaluation criteria, applications, and methodological limits

Evaluation of dimension reduction for curves has itself become a geometric topic. "An evaluation framework for dimensionality reduction through sectional curvature" introduces a metric based on sectional curvature behavior arising from Riemannian geometry (Lara-Cabrera et al., 2023). The setting uses a generator makegen that produces smooth immersions

$1\pm\varepsilon$2

with controllable curvature along each axis, including logistic, polyroll, sine, circle, and flat components. For planar curves, a curve with prescribed curvature function $1\pm\varepsilon$3 is reconstructed by

$1\pm\varepsilon$4

Given a dimensionality reduction map $1\pm\varepsilon$5, the composite map $1\pm\varepsilon$6 induces a pull-back metric, and the curvature distortion metric is

$1\pm\varepsilon$7

The paper argues that this metric captures both local and global geometric fidelity and properly penalizes self-intersections, severe flattening, and bending not matching the original, which NPR can miss (Lara-Cabrera et al., 2023).

A related 2025 paper constructs a curvature-based geometric profile for discrete metric spaces and compares original and embedded datasets via the 1-Wasserstein distance $1\pm\varepsilon$8 between curvature profiles (Beylier et al., 16 Sep 2025). The basic curvature quantity for a triple $1\pm\varepsilon$9 is

$S^d$0

where the $S^d$1 satisfy $S^d$2. For comparability and robustness, the profile is built using equilateral triangles binned by scale. The paper states that plotting $S^d$3 against target embedding dimension can indicate the optimal or critical dimension for faithful geometry preservation and can be used to estimate intrinsic dimensionality (Beylier et al., 16 Sep 2025).

These evaluation papers formalize an important correction to a common practice: neighbor preservation alone is not a sufficient proxy for curve faithfulness. The sources explicitly state that local metrics can be blind to global or topological distortions, whereas curvature-based metrics respond to the shape of curved data at multiple scales (Lara-Cabrera et al., 2023, Beylier et al., 16 Sep 2025).

The application range reported across the literature includes shape analysis, object tracking, activity recognition, high-dimensional time series, clustering, motion capture, earthquake analysis, and geometric modeling [(Yi et al., 2011); (Psarros et al., 2022); (Kim et al., 2020); (Burstall et al., 2017)]. The methodological limits are equally clear. Global PCA on spline coefficients ignores local geometry (Yi et al., 2011); non-invertible curve embeddings may lack reconstruction (Yi et al., 2011); intrinsic spherical methods are sensitive to initialization and parameter choice (Kim et al., 2020); continuous curve-distance embeddings are not terminal embeddings and depend on the total number of vertices (Psarros et al., 2022); and evaluation by curvature requires either explicit maps or careful estimation of derivatives or metric structure from discrete data (Lara-Cabrera et al., 2023, Beylier et al., 16 Sep 2025).

Taken together, these works show that dimension reduction for curves has developed along several non-equivalent axes: invertible local-frame encodings for manifold trajectories, metric-preserving random projections for polygonal curves, intrinsic principal-curve fitting on curved spaces, and geometry-aware evaluation based on curvature. This suggests that the decisive modeling choice is not simply the target dimension, but which curve property is to be preserved: reconstruction, geodesic self-consistency, pairwise dissimilarity, or large-scale geometry.