Linking Over Cones: Theory & Applications

• Linking over cones is a framework that uses geometric, topological, and algebraic approaches to construct and compare conic structures in various mathematical contexts.
• The methodology leverages variational schemes and topological linking to establish solution existence and quantify geometric alignment in spectral and activation spaces.
• Applications span nonlinear PDE analysis, representation learning via sparse autoencoders, and classical geometry through dual relationships in conic sections.

A cone, in mathematics, refers to a set closed under positive scalar multiplication, often arising as the nonnegative span of a collection of vectors. “Linking over cones” designates a family of geometric, topological, and algebraic methods for connecting, comparing, or constructing structures—optimizations, concept spaces, or solution sets—that are conic in nature, either in continuous activation spaces, functional analysis, or geometric domains. This framework is especially central in nonlinear analysis, concept learning, and classical algebraic geometry.

1. Formal Structure of Cones

A cone, formally, is the set

$$\mathrm{cone}(D_1,\dots,D_c) = \left\{v \in \mathbb{R}^d : v = D^{T}\alpha,\;\alpha \in \mathbb{R}_+^c\right\}$$

where the $D_i$ are linearly independent atoms (or directions) in activation, functional, or vector space. The cone is pointed and convex by construction—the nonnegativity constraint induces a geometric basis from the “extreme rays” $D_i$. In concept learning, cones parameterize subspaces of activations, admitting only positive combinations, thus capturing interpretable structures while restricting to semantically meaningful directions (Rocchi--Henry et al., 8 Dec 2025). In nonlinear spectral theory, cones structure function spaces indexed by nonlinear eigenvalue bounds (Mugnai et al., 2020).

2. Linking Over Cones: Topological and Variational Frameworks

Classically, variational analysis and nonlinear PDE existence proofs rely on finding critical points of energy functionals via topological linking (e.g., Rabinowitz saddle-point theorem). The “linking over cones” principle generalizes this by employing closed, symmetric cones rather than linear subspaces—crucial when operators lack linear eigenspaces or complementary subspaces. For instance, with the nonlocal, nonlinear fractional $p$-Laplacian

$$(-\Delta)_p^s\,u(x) = \mathrm{P.V.}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}\,dy,$$

spectral cones $C_m$, $C^m$ are defined by

$$C_m = \left\{u : \iint_Q \frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dxdy \leq \mu_m\int_\Omega |u|^p\,dx \right\},$$

$$C^m = \left\{u : \iint_Q \frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dxdy \geq \mu_{m+1}\int_\Omega |u|^p\,dx \right\}.$$

The linking framework then deploys minimax constructions over these cones, leveraging their cohomological indices rather than vector space dimensionality (Mugnai et al., 2020).

3. Linking and Containment Metrics in Concept Geometry

In the context of representation learning, linking over cones is interpreted as quantifying how well one cone (e.g., learned by a sparse autoencoder) captures or contains another (e.g., prescribed by a concept bottleneck model). The degree of geometric alignment is assessed by reconstructing supervised concept directions via sparse, nonnegative combinations of unsupervised atoms. Letting $V_i$ denote human-annotated concept directions and $D$ an unsupervised dictionary, one solves

$$\alpha_i^* = \arg\min_{\alpha \geq 0} \|V_i - \alpha^T D\|^2_2 + \lambda \|\alpha\|_1$$

and measures linkage by normalized reconstruction error $\delta_i = \|V_i - {\alpha_i^*}^T D\|_2/\|V_i\|_2$ and coverage

$$\mathrm{Cov} = \frac{\sum_i \| {\alpha_i^*}^T D\|_2^2 }{\sum_i \|V_i\|_2^2}.$$

A coverage near 1 indicates the unsupervised cone nearly subsumes the supervised cone, providing quantitative assessment of conic linkage (Rocchi--Henry et al., 8 Dec 2025).

4. Topological Index, Variational Schemes, and Nonlinear PDEs

In nonlinear PDEs, cones defined via nonlinear spectral bounds are indexed by the $\mathbb{Z}_2$-cohomological genus. The linking theorem states: if cones $C_-, C_+$ are closed, symmetric, and fulfill $C_- \cap C_+ = \{0\}$, $i(C_- \setminus \{0\}) = i(X \setminus C_+) = m$, then certain spheres and balls in $C_-, C_+$ link each other in the sense of Alexander–Spanier cohomology, ensuring existence of nontrivial critical points under Palais–Smale or Cerami compactness conditions. This method elegantly sidesteps the lack of linear decomposability in fractional $p$-Laplacian and similar operators, permitting weak solutions even without Ambrosetti–Rabinowitz conditions (Mugnai et al., 2020).

5. Geometric Linkages: Conics and Classical Conjunctions

In algebraic geometry, cones are central to the conic sections—ellipse and hyperbola—where “linking” describes dual geometric relationships. If an ellipse is obtained by intersecting a cone with a plane, the locus of all cone apexes generating the ellipse is a hyperbola, whose foci coincide with the ellipse’s vertices. Conversely, fixing a hyperbola, the locus of cone apexes is an ellipse. This two-way conjunction illustrates deep geometric “linking” via cones, with apex-loci equations

$$x^2/a^2 + y^2/b^2 = 1\quad \text{(ellipse)}, \qquad x^2/c^2 - y^2/b^2 = 1 \quad \text{(apex-locus, hyperbola)}$$

with focal relations $c^2 = a^2 - b^2$ (Kobiera, 2018).

6. Applications and Implications

Linking-over-cones facilitates existence proofs in nonlinear PDEs—fractional $p$-Laplacian problems with nonlocal Neumann boundary conditions admit solutions absent classical growth conditions, provided linking criteria over cones are established. In representation learning, linking quantifies model interpretability and the emergence of plausible concepts; principled metrics of conic containment and alignment elevate the study of sparse autoencoder discovery to quantitative rigor (Rocchi--Henry et al., 8 Dec 2025). In classical geometry, linking via cones exposes elegant dualities among conic sections (Kobiera, 2018).

7. Summary Table: Linking over Cones Across Domains

Domain	Cone Definition/Role	Linking Mechanism
Nonlinear PDEs (Mugnai et al., 2020)	Spectral cones $C_m, C^m$ in Banach space	Topological minimax linking
Concept Learning (Rocchi--Henry et al., 8 Dec 2025)	Nonnegative span of concept/activation directions	Containment/reconstruction
Classical Geometry (Kobiera, 2018)	Geometric locus of cone apexes generating conic section	Dual conic-locus linkages

Linking over cones thus constitutes a fundamental methodology for unifying spectral theory, topological analysis, representation learning, and classical geometry, providing structural insights, solution existence, and quantification of alignment or containment in diverse mathematical contexts.