Simplicial covering dimension of extremal concept classes

TL;DR

We introduce the simplicial covering dimension $\mathop{\mathrm{sc}}(\mathcal{C})$ for binary concept classes, defined via the topology of the space of realizable distributions and refined by zero-loss sets. In the finite-domain setting, we prove $\mathop{\mathrm{lr}}(\mathcal{C}) = \mathop{\mathrm{sc}}(\mathcal{C}) + 1$, linking a topological invariant to list replicability. We classify the exact $\mathop{\mathrm{sc}}$-values for extremal classes: if $\mathcal{E} \neq \{\pm 1\}^{\mathcal{X}}$, then $\mathop{\mathrm{sc}}(\mathcal{E}) = \mathop{\mathrm{vc}}(\mathcal{E})$ and $\mathop{\mathrm{lr}}(\mathcal{E}) = \mathop{\mathrm{vc}}(\mathcal{E}) + 1$; for the full binary cube, $\mathop{\mathrm{sc}} = |\mathcal{X}| - 1$ and $\mathop{\mathrm{lr}} = |\mathcal{X}|$. The analysis leverages a deformation retraction between the Δ-structure of realizable distributions and the cubical complex of strongly shattered sets, together with Lebesgue’s covering theory, to derive tight bounds. The results unify and extend known values for several natural classes via a topological framework, and open questions remain about VC-based bounds and dual-VC interactions with $\mathop{\mathrm{sc}}(\cdot)$.

Abstract

Dimension theory is a branch of topology concerned with defining and analyzing dimensions of geometric and topological spaces in purely topological terms. In this work, we adapt the classical notion of topological dimension (Lebesgue covering) to binary concept classes. The topological space naturally associated with a concept class is its space of realizable distributions. The loss function and the class itself induce a simplicial structure on this space, with respect to which we define a simplicial covering dimension. We prove that for finite concept classes, this simplicial covering dimension exactly characterizes the list replicability number (equivalently, global stability) in PAC learning. This connection allows us to apply tools from classical dimension theory to compute the exact list replicability number of the broad family of extremal concept classes.

Figures (5)

• Figure 1: The illustrated open cover $\mathcal{A}$ of the square $[-1,1]^2$ has order $3$, but it admits a shrinkage $\mathcal{B}$ of order $2$, implying $\mathop{\mathrm{L}}\nolimits(\mathcal{A}) \le 2$. Moreover, since no set in $\mathcal{A}$ contains points from opposite faces, the Lebesgue covering theorem shows $\mathop{\mathrm{L}}\nolimits(\mathcal{A}) \ge 2$.
• Figure 2: The simplicial complexes $\Delta_\mathcal{C}$ and $\Delta_{\mathcal{C},1}$ for $\mathcal{C} = \{\texttt{++-}, \texttt{+++}, \texttt{+-+}, \texttt{--+}, \texttt{-++}\}$. Here, $\texttt{+}$ and $\texttt{-}$ are shorthand for $+1$ and $-1$, respectively.
• Figure 3: The cubical complex $\Gamma_{\mathcal{C}}$ and the simplicial complex $\Gamma_{\mathcal{C},1}$ for $\mathcal{C} = \{\texttt{++-}, \texttt{+++}, \texttt{+-+}, \texttt{--+}, \texttt{-++}\}$.
• Figure 4: The embeddings $\Gamma_{\mathcal{C},1}\hookrightarrow \Delta_{\mathcal{C},1}$ and $\Gamma_{\mathcal{C}} \hookrightarrow \Delta_{\mathcal{C}}$ for the concept class $\mathcal{C} = \{\texttt{++-}, \texttt{+++}, \texttt{+-+}, \texttt{--+}, \texttt{-++}\}$.
• Figure 5: The subcomplexes $\mathrm{St}_W(w)$ and $\mathrm{Lk}_W(w)$ for $w =$+**. Since $w$ is minimal in $W \setminus V(Q_{\mathcal{E}, 1})$, its link and star are independent of $W$.

Theorems & Definitions (65)

• Definition 1.1: List replicability
• Example 1.2: Sign patterns of convex sets MR683734
• Example 1.3: Homogeneous half-spaces
• Example 1.4: Axis-parallel boxes
• Example 1.5: Median classes
• Definition 1.7: Topological dimension
• Remark 1.8
• Theorem 1.8: Lebesgue covering theorem
• Theorem 1.9: munkres2000topology
• Definition 1.10: Simplicial covering dimension of a binary concept class