Convexity-Driven Projection (CDP)

• CDP is a linear dimensionality reduction technique that retains detour-induced non-convexity by analyzing local graph structures from point clouds.
• It constructs a non-convexity structure matrix from weighted k-NN graph edges and selects projection directions through eigendecomposition, differing from variance-centric methods.
• The method provides rigorous guarantees with pairwise a-posteriori certificates and average-case spectral bounds, ensuring robust, verifiable projections.

Convexity-Driven Projection (CDP) is not a single standardized construct across the arXiv literature. In the explicit usage introduced for point clouds, it is a boundary-free linear method for dimensionality reduction that targets preserving detour-induced local non-convexity (Sanyal, 26 Sep 2025). In other literatures, the same initials can denote something else entirely: in low-$x$ deep inelastic scattering, “CDP” consistently means the Color Dipole Picture, not a separate convex-projection formalism (Boroun, 2022). Several convex optimization papers, meanwhile, do not introduce CDP as a formal name but develop projection-centered methodologies whose organizing principle is the use of convex structure, convex duality, or quasi-concavity to define or approximate projections (Kováčová et al., 2021, Kováčová et al., 2023). This suggests that the term functions less as a universally fixed formalism than as a label attached to several projection mechanisms whose defining feature is that geometry, convexity, or detour structure determines the projection itself.

1. Terminological scope and principal usages

In "Convexity-Driven Projection for Point Cloud Dimensionality Reduction" (Sanyal, 26 Sep 2025), CDP is a named algorithm. Given a point cloud $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$, it constructs a linear projection $V^\top:\mathbb{R}^d\to\mathbb{R}^k$ that preserves local detours measured on a $k$-NN graph. Its central signal is local non-convexity, quantified by discrepancies between Euclidean distances and graph shortest-path distances.

In "One-to-One correspondence of soft and hard Pomeron with the CDP of the gluon density at low $x$" (Boroun, 2022), by contrast, “CDP” is explicitly the Color Dipole Picture. The paper further states that “Convexity-Driven Projection” is not a standard term there; what is reconstructed from the paper is a projection of the measured proton structure function $F_2(x,Q^2)$ onto the gluon distribution $xg(x,Q^2)$ using the convex $W^2$-dependence implied by the dipole framework.

A third usage appears only implicitly. "Convex Projection and Convex Multi-Objective Optimization" (Kováčová et al., 2021) treats convex projection as the projection of a convex feasible set onto a subspace and proves an exact correspondence with an associated multi-objective convex optimization problem. "Computing the recession cone of a convex upper image via convex projection" (Kováčová et al., 2023) uses convex or polyhedral projection to compute or approximate recession cones for unbounded convex vector optimization problems. "Convex Optimization with an Interpolation-based Projection and its Application to Deep Learning" (Akrour et al., 2020) does not name CDP explicitly, but its interpolation-based feasibility map is entirely constructed from convexity of a domain-defining function.

The resulting picture is unavoidably plural. A common misconception is to treat CDP as a single standard algorithm. The literature instead contains one explicit point-cloud method, one unrelated acronym in DIS, and several projection-centered constructions that are “convexity-driven” only in a broader methodological sense (Sanyal, 26 Sep 2025, Boroun, 2022).

2. CDP as a point-cloud dimensionality reduction method

In its explicit arXiv definition, CDP begins with standardized points $\{p_i\}\subset\mathbb{R}^d$, builds a mutual $k$-NN graph $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$0, and assigns edge weights

$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$1

All-pairs shortest-path distances are then computed on $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$2: $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$3 For each pair, CDP defines the convexity ratio

$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$4

Pairs with $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$5 are locally convex in the graph-geometric sense; pairs with $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$6 exhibit strong detours and hence local non-convexity (Sanyal, 26 Sep 2025).

The admissible set is

$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$7

with user-chosen threshold $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$8. CDP also defines the admissible non-convexity index

$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$9

Smaller $V^\top:\mathbb{R}^d\to\mathbb{R}^k$0 means stronger average detours among selected pairs. For each admissible pair, the normalized direction is

$V^\top:\mathbb{R}^d\to\mathbb{R}^k$1

The central object is the non-convexity structure matrix

$V^\top:\mathbb{R}^d\to\mathbb{R}^k$2

Each $V^\top:\mathbb{R}^d\to\mathbb{R}^k$3 is a rank-1 projector, the weight $V^\top:\mathbb{R}^d\to\mathbb{R}^k$4 emphasizes strong detours, and the matrix is positive semidefinite. CDP then computes an eigendecomposition

$V^\top:\mathbb{R}^d\to\mathbb{R}^k$5

orders eigenvalues $V^\top:\mathbb{R}^d\to\mathbb{R}^k$6, and chooses the projection matrix

$V^\top:\mathbb{R}^d\to\mathbb{R}^k$7

The projected point cloud is

$V^\top:\mathbb{R}^d\to\mathbb{R}^k$8

This makes the method explicitly linear. Its criterion is not variance retention, neighborhood smoothness, or global geodesic preservation; it is the retention of ambient directions along which the graph geometry displays significant detour structure (Sanyal, 26 Sep 2025).

3. Guarantees, certificates, and evaluation protocol

CDP provides two kinds of guarantees. The first is a pairwise a-posteriori certificate. After projection, define the projected shortest-path distance $V^\top:\mathbb{R}^d\to\mathbb{R}^k$9 on the graph with the same edge set and projected edge weights

$k$0

The projected convexity ratio is

$k$1

For each admissible pair, CDP defines

$k$2

and

$k$3

where $k$4 is a projected shortest path between $k$5 and $k$6. The theorem is

$k$7

There is also a uniform graph-wise bound

$k$8

The lower side is controlled by how well the admissible pair direction is captured by the subspace; the upper side is controlled by the minimum captured cosine along the projected shortest path (Sanyal, 26 Sep 2025).

The second guarantee is an average-case spectral bound. If $k$9 is sampled from admissible directions with probability proportional to $x$0, then $x$1 up to normalization. For

$x$2

the expected captured energy is

$x$3

This is the fraction of total eigenvalue mass explained by the top-$x$4 eigenvectors, directly analogous to explained variance but for detour directions. The paper then gives the quantile statement

$x$5

When $x$6 is close to $x$7, most admissible directions are well captured.

The evaluation protocol mirrors these guarantees. It reports a fixed-pairs detour error

$x$8

where $x$9 averages $F_2(x,Q^2)$0 over the original admissible set $F_2(x,Q^2)$1, and a reselected-pairs detour error

$F_2(x,Q^2)$2

where $F_2(x,Q^2)$3 is computed on the reselected admissible set $F_2(x,Q^2)$4. It also reports quantiles of $F_2(x,Q^2)$5 and $F_2(x,Q^2)$6, together with $F_2(x,Q^2)$7 (Sanyal, 26 Sep 2025).

4. Relation to neighboring dimensionality reduction methods

CDP differs from PCA because PCA builds a covariance matrix of points and chooses directions of maximum variance, while CDP builds $F_2(x,Q^2)$8 from detour-weighted pairwise directions and chooses directions where non-convexity concentrates. A direction can therefore have low global variance and still be retained by CDP if it is crucial for obstacle- or curvature-induced detours (Sanyal, 26 Sep 2025).

It also differs from classical MDS and Isomap. Classical MDS tries to preserve pairwise Euclidean distances by spectral embedding of a centered distance matrix. Isomap builds a $F_2(x,Q^2)$9-NN graph, computes graph geodesic distances, and then performs MDS on those distances. CDP uses a $xg(x,Q^2)$0-NN graph and shortest-path distances as Isomap does, but it does not aim to preserve all geodesics globally; it focuses on the subset of pairs where detours are strong and then constructs a linear projection rather than a nonlinear embedding. Compared with Laplacian Eigenmaps, LPP, NPP, and OLPP, CDP does not minimize graph smoothness or keep neighbors close. Its structure matrix is not a Laplacian; it is a sum of projectors $xg(x,Q^2)$1 weighted by $xg(x,Q^2)$2 (Sanyal, 26 Sep 2025).

Compared with t-SNE and UMAP, CDP is linear, represented by an explicit matrix $xg(x,Q^2)$3, and comes with a-posteriori certificates and average-case spectral bounds. The trade-off is equally explicit: it is less flexible for complicated manifold unfolding, but more predictable and verifiable on the detour geometry it targets (Sanyal, 26 Sep 2025).

The method is correspondingly sensitive to graph and threshold choices. Too small $xg(x,Q^2)$4 can disconnect the graph or overemphasize detours; too large $xg(x,Q^2)$5 can make shortest-path distances approach Euclidean distances and suppress the very signal CDP is designed to preserve. Too small $xg(x,Q^2)$6 yields very few admissible pairs; too large $xg(x,Q^2)$7 dilutes the detour signal. The dominant computational cost is all-pairs shortest paths on a sparse graph, stated as $xg(x,Q^2)$8, followed by construction of $xg(x,Q^2)$9 at worst $W^2$0 and eigendecomposition of a $W^2$1 PSD matrix (Sanyal, 26 Sep 2025).

The paper’s examples are consistent with that design goal. In a toy $W^2$2 example, the eigenvalues of $W^2$3 are approximately $W^2$4, the spectral capture is $W^2$5, and the uniform upper bound is $W^2$6. On Swiss roll, torus, S-curve, helix, Möbius strip, Klein bottle, and annulus-with-obstacle benchmarks, CDP is reported to preserve “around-the-hole” structure better than PCA/LPP while remaining linear; relative to UMAP it is more constrained but provides explicit detour guarantees (Sanyal, 26 Sep 2025).

5. Convex projection lineages in optimization and discrete convex analysis

A second major line of work uses “projection” in the literal convex-analytic sense of projecting a feasible set onto a subspace or projecting a point onto a cone. "Convex Projection and Convex Multi-Objective Optimization" (Kováčová et al., 2021) defines the convex projection problem as

$W^2$7

for a nonempty convex set $W^2$8, and associates to it the multi-objective convex optimization problem

$W^2$9

The paper proves an exact equivalence between solutions of the convex projection and solutions of the associated multi-objective problem, and for approximate solutions derives sharp error multipliers $\{p_i\}\subset\mathbb{R}^d$0 and $\{p_i\}\subset\mathbb{R}^d$1 linking projection error and multi-objective error (Kováčová et al., 2021).

"Approximations of unbounded convex projections and unbounded convex sets" (Kováčová et al., 2023) extends this viewpoint from bounded convex sets to unbounded ones. It treats the target as a linear image

$\{p_i\}\subset\mathbb{R}^d$2

and introduces finite $\{p_i\}\subset\mathbb{R}^d$3-solutions that simultaneously approximate the set and its recession cone $\{p_i\}\subset\mathbb{R}^d$4. The algorithms use weighted sums, Pascoletti–Serafini scalarizations, and norm minimization to construct inner and outer polyhedral approximations. The core inclusions are

$\{p_i\}\subset\mathbb{R}^d$5

with Hausdorff control

$\{p_i\}\subset\mathbb{R}^d$6

This directly turns convex projection into a controlled approximation problem for both the finite and asymptotic geometry (Kováčová et al., 2023).

A related development appears in "Computing the recession cone of a convex upper image via convex projection" (Kováčová et al., 2023). There the main object is the recession cone $\{p_i\}\subset\mathbb{R}^d$7 of the upper image of an unbounded convex vector optimization problem. The paper identifies

$\{p_i\}\subset\mathbb{R}^d$8

where

$\{p_i\}\subset\mathbb{R}^d$9

and then expresses $k$0 or its bounded base $k$1 as a convex projection defined by dual feasibility conditions. Approximating $k$2 and dualizing yields a controlled outer approximation of $k$3. In this literature, projection is therefore the computational vehicle for extracting directions of unboundedness (Kováčová et al., 2023).

Discrete convex analysis supplies an additional closure theory. "Projection and Convolution Operations for Integrally Convex Functions" (Moriguchi et al., 2017) defines the projection of $k$4 by

$k$5

and proves that if $k$6 is integrally convex, then $k$7 is integrally convex as well. The same stability holds for globally and locally discrete midpoint convex functions. By contrast, convolution of two integrally convex functions may fail to preserve integral convexity, while convolution with a separable convex function does preserve it (Moriguchi et al., 2017).

Still another projection tradition concerns cones. "Replacing projection on finitely generated convex cones with projection on bounded polytopes" (Nurminski, 2020) reformulates projection onto a finitely generated cone $k$8 as projection onto the bounded polytope

$k$9

provided $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$00 is chosen using the least-norm element of the convex hull of generators. The paper states the equivalence

$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$01

and implements it with the Cone Truncated to Polytope algorithm (Nurminski, 2020).

Finally, "A Projection Framework for Testing Shape Restrictions That Form Convex Cones" (Fang et al., 2019) uses the distance to a convex cone,

$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$02

as a test statistic for shape restrictions such as monotonicity, concavity, supermodularity, or negative semidefiniteness. The test’s bootstrap validity hinges on a cone-specific monotonicity property of the projection distance, which allows the procedure to avoid estimating local parameter spaces (Fang et al., 2019). Across these papers, projection is not a visualization device but a structural operator that preserves convex classes, recovers recession directions, or defines inferential distances.

6. Domain-specific adaptations, proximal variants, and the low-$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$03 DIS acronym

Several newer applications reinterpret projection through convexity or quasi-concavity while remaining outside the point-cloud definition. In "Convex Optimization with an Interpolation-based Projection and its Application to Deep Learning" (Akrour et al., 2020), the feasible set is

$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$04

with convex differentiable $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$05, and the projection map is

$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$06

where $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$07 is strictly feasible. Rather than using projected gradient descent, the paper performs gradient descent on $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$08 and proves convergence for linear objectives and arbitrary convex and Lipschitz domain-defining inequality constraints. The same projection is then embedded as a differentiable layer in reinforcement learning and supervised learning settings (Akrour et al., 2020).

"Projection onto cones generated by epigraphs of perspective functions" (Briceño-Arias et al., 2024) studies orthogonal projection onto

$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$09

where $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$10 is the perspective of a proper convex lower semicontinuous function. The paper states that the projection can be computed efficiently using only two scalar equations involving the proximity operator of the underlying function. This recovers projections onto exponential and power cones and extends to the hyperbolic cone (Briceño-Arias et al., 2024).

A functional convexity version appears in "D-Convexity: A Unified Differentiable Convex Shape Prior via Quasi-Concavity for Data-driven Image Segmentation" (Chen et al., 19 May 2026). There the network output $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$11 is required to be quasi-concave, equivalently all super-level sets

$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$12

must be convex. The paper derives zero-, first-, and second-order quasi-concavity conditions, including the 2D second-order quadratic form

$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$13

and uses them in a Convex Gradient Projection Module that solves

$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$14

This is not named CDP, but it is explicitly a projection-like proximal step toward a convexity-constrained function class (Chen et al., 19 May 2026).

The acronym’s most divergent use remains low-$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$15 DIS. In "One-to-One correspondence of soft and hard Pomeron with the CDP of the gluon density at low $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$16" (Boroun, 2022), CDP means Color Dipole Picture. The central projection-like relation is

$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$17

combined with the dipole-based asymptotic form

$\{p_i\}_{i=1}^N \subset \mathbb{R}^d$18

The paper emphasizes that the hard-Pomeron trajectory alone does not converge to the CDP asymptotic behavior for $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$19, whereas the soft+hard Pomeron model does converge for $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$20 (Boroun, 2022). Here “convexity-driven projection” is an interpretive reconstruction, not a named formalism.

Taken together, these literatures support a precise but limited synthesis. The explicit, standardized meaning of Convexity-Driven Projection is presently the point-cloud method of detour-preserving linear dimensionality reduction (Sanyal, 26 Sep 2025). Beyond that, the phrase describes a wider methodological family in which projection operators are defined, approximated, or regularized by convexity, quasi-concavity, recession geometry, or cone structure rather than by variance or Euclidean distance alone (Kováčová et al., 2021, Kováčová et al., 2023, Akrour et al., 2020, Briceño-Arias et al., 2024).