Dimension-counting bounds for equi-isoclinic subspaces

2025-11-25

AI Summary

This paper investigates the construction and enumeration of optimal subspace packings and equi-isoclinic subspaces. Employing dimension-counting arguments and block coherence analysis, we first derive a tight lower bound on block coherence. We precisely determine the maximum number of even-dimensional $r$-dimensional equi-isoclinic subspaces in $mathbb{R}^{2r+1}$ for parameters $alpha eq frac{1}{2}$. Furthermore, we improve the universal upper bound on the number of $r$-dimensional equi-isoclinic subspaces in $mathbb{R}^d$ and $mathbb{C}^d$, and prove its tightness: it is attained for all $r$ in the complex case, and for $r = 2^k$ in the real case. Our results unify tools from linear algebra, matrix theory, and differential geometry, advancing the structural understanding of high-dimensional subspace configurations.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

We make four contributions to the theory of optimal subspace packings and equi-isoclinic subspaces: (1) a new lower bound for block coherence, (2) an exact count of equi-isoclinic subspaces of even dimension $r$ in $mathbb{R}^{2r+1}$ with parameter $alpha eq frac{1}{2}$, (3) a new upper bound for the number of $r$-dimensional equi-isoclinic subspaces in $mathbb{R}^d$ or $mathbb{C}^d$, and (4) a proof that when $d=2r$, a further refinement of this bound is attained for every $r$ in the complex case and every $r=2^k$ in the real case. For each of these contributions, the proof ultimately relies on a dimension count.

Problem

• Improving bounds for block coherence in subspace packings
• Counting equi-isoclinic subspaces with specific parameters
• Establishing upper bounds for equi-isoclinic subspace quantities

Innovation

• New lower bound for block coherence
• Exact count of equi-isoclinic subspaces
• New upper bound for subspace numbers