Dimension Reduction for Curves: Simplified and Generalized

Abstract: We revisit random projections for reducing the dimension of high-dimensional polygonal curves. Drawing from the toolbox of randomized linear algebra, we give a considerably simplified proof of the known $O(\varepsilon^{-2}\log(nm))$ bound on the target dimension of a random projection that preserves the continuous Fréchet distance of polygonal curves up to a factor $(1\pm\varepsilon)$. Our proof is based on the concept of sparse oblivious subspace embeddings. While previous techniques were limited to the case of the Fréchet distance, our techniques are fairly general and extend to all possible distance measures that involve the maximum, a sum or an integral over Euclidean distances between pairs of points on both input curves. We define a generalized dissimilarity measure for curves that includes several popular measures such as Fréchet, $q$-DTW, Hausdorff, etc. as special cases and show that the same dimension reduction technique works for this generalized dissimilarity measure. Finally, we apply the same framework for dimension reduction to piecewise linear surfaces, after extending the distance measure suitably to such surfaces.

• The paper introduces a unified framework leveraging sparse oblivious subspace embeddings to reduce dimensions while preserving Fréchet, DTW, and Hausdorff distances.
• It simplifies prior proofs and extends techniques to high-dimensional surfaces, achieving controlled error bounds for piecewise linear structures.
• The method supports algorithmic improvements in clustering, classification, and proximity search, paving the way for dimension-independent coresets.

Dimension Reduction for Curves and Surfaces: Simplified and Generalized Framework

Overview and Motivation

The paper "Dimension Reduction for Curves: Simplified and Generalized" (2607.03112) presents a broadly applicable, theoretically grounded approach to dimension reduction for geometric objects such as polygonal curves and piecewise linear surfaces, under a wide class of dissimilarity measures including Fréchet, dynamic time warping (DTW), Hausdorff, and variants thereof. The principal advances lie in (i) simplifying the proofs for known bounds on dimension reduction for the Fréchet distance, (ii) extending the dimension reduction techniques to generalized dissimilarity measures via a unified framework, and (iii) providing a rigorous extension to higher-dimensional surfaces.

Dimension reduction transforms high-dimensional computational geometry problems into tractable instances with reduced dimensionality, retaining essential geometric structure up to a controlled multiplicative error. The methods here leverage random linear mappings based on oblivious subspace embeddings, producing significant algorithmic improvements for computational tasks whose complexity is dimension-dependent. The results are germane not only for theoretical geometry but also for practical applications such as scalable clustering, classification, and proximity search over curves/surfaces.

Technical Contributions

Simplified Dimension Reduction for the Fréchet Distance

The paper revisits the problem of dimension reduction for polygonal curves under the continuous Fréchet distance. Previous work established the necessity of embedding dimension $t=O(\varepsilon^{-2} \log(nm))$ to maintain $(1\pm\varepsilon)$ distortion for $n$ curves of complexity $m$ [PsarrosR23, PsarrosR25]. The authors introduce a succinct proof based on sparse oblivious subspace embeddings, eliminating the need for intricate combinatorial or logical predicate arguments. They show that every potential vector distance between points on two curves can be expressed as a linear combination of four vertices, enabling the direct application of $4$-sparse embeddings (see [MaiMM0SW23]). This yields the same target dimension, ensuring all pairwise (vertex-indexed) distances are preserved within the desired approximation bounds.

Generalization to a Unified Distance Framework

A salient contribution is the extension of the dimension reduction technique to a highly general dissimilarity measure:

$$d(\sigma_i, \sigma_j) = \inf_{(\alpha,\beta)\in T} \left( \int_{[0,1]^\gamma} \|\sigma_i(\alpha(r)) - \sigma_j(\beta(r))\|_2^q \, d\mu(r) \right)^{1/q}$$

This formulation encapsulates Fréchet (as $q\to\infty$), DTW ($q=1$), Hausdorff, and other relevant metrics. The framework is agnostic to whether the measure is defined via a maximum, sum, or integral over Euclidean distances, and thus supports discrete and continuous variants, as well as piecewise linear surfaces for arbitrary parameter $\gamma$.

Dimension Reduction for Piecewise Linear Surfaces

For piecewise $\gamma$-dimensional linear surfaces, every difference vector between points on two surfaces can be written as a linear combination of $(1\pm\varepsilon)$0 vertices. The authors refine the embedding count to avoid overcounting overlapping subspaces, achieving an embedding dimension

$(1\pm\varepsilon)$1

for $(1\pm\varepsilon)$2 surfaces with $(1\pm\varepsilon)$3 pieces, with $(1\pm\varepsilon)$4 probability. This result subsumes and generalizes the curve case ($(1\pm\varepsilon)$5), and is supported by geometric arguments leveraging Carathéodory's theorem and union bounds across surface pairings.

Rigorous Treatment of Discrete Traversals for Surfaces

Unlike curves with natural vertex sequences, surfaces lack canonical ordering for discrete traversals. The paper introduces a Voronoi-based construction of traversals induced by homeomorphisms, ensuring consistency with curve traversals and suitability for discrete Fréchet and DTW distances. Every discrete traversal is shown to correspond to some homeomorphism, and vice versa, thus unifying the continuous and discrete settings in the generalized distance framework.

Implications and Applications

Algorithmic Impact

Algorithms for continuous and discrete Fréchet, DTW, Hausdorff and related distances are frequently dimension-sensitive. By reducing the dimensionality to $(1\pm\varepsilon)$6 or $(1\pm\varepsilon)$7, computational tasks such as clustering, proximity search, and classification over curves/surfaces can be executed with substantially reduced complexity, often with linear or logarithmic dependence on the underlying quantity of interest. The approach also provides a foundation for constructing dimension-independent coresets and dimension-free bounds in learning contexts.

Extensions to $(1\pm\varepsilon)$8 Spaces and Other Metrics

The framework is extendable to normed spaces with $(1\pm\varepsilon)$9-stable distributions for $n$0-norms ($n$1), though median estimators replace the $n$2 norm in the reduced space [Indyk2006, MaiMM0SW23]. Algorithms specifically querying $n$3 distances remain compatible; however, more complex operations may require adaptation.

Limitations and Open Directions

While the dimension reduction guarantees are broad, efficient algorithms for many surface distance measures are currently available only in low dimensions or for special classes (e.g., simple polygons). The framework thus potentially supports future algorithmic advances once these tasks are generalized to higher dimensions. Extensions to terminal embeddings for preserving distances to arbitrary curves and truly dimension-free coreset constructions in clustering under curve/surface metrics are actively pursued research directions [munteanu2026terminal].

Conclusion

This paper presents a highly general and robust framework for dimension reduction under curve and surface similarity/dissimilarity measures. The use of sparse and oblivious subspace embeddings significantly simplifies proofs and broadens the applicability to arbitrary distances expressible as infima over integrals of Euclidean norms. The unified treatment of discrete and continuous traversals, extension to higher-dimensional surfaces, and compatibility with $n$4 spaces provide a compelling foundation for theoretical and practical advances in computational geometry and geometric data analysis. Anticipated future developments include efficient dimension-reduced algorithms for surfaces, core-set construction and learning-theoretic bounds independent of ambient dimension, and further exploration of dimension-free geometric embeddings in computational geometry and machine learning.