A Parametric Linear Conic Projection for Structured Sparse Attention

Abstract

In this paper, we introduce the Cone Alignment Index (CAI), a scale-invariant sparsity measure associated with a Lorentz-cone geometry. We provide a probabilistic analysis of the distribution of the CAI score, including a general theorem and explicit corollaries for Gaussian and uniform distributions. As a consequence, we show that, in the high-sparsity regime, the probability that the samples lie within the convex cone tends to one. Building upon this formulation, we derive two complementary projections: the \emph{Parametric Linear Conic Projection} (PLCP), an efficient closed-form approximation, and the \emph{Parametric Euclidean Conic Projection} (PECP), the exact Euclidean projection onto the cone boundary. Our analysis establishes several fundamental theoretical properties. We prove that PLCP and PECP are collinear, differ only by a positive radial scaling factor, and induce exactly the same sparsity threshold, thereby preserving an identical active support. Furthermore, we derive a support-identification theorem showing that, for any prescribed sparsity level $l<n$, the number of coefficients discarded at each iteration is theoretically controlled. Building on this theoretical foundation, we develop a bilevel PLCP framework for structured matrix sparsification. The proposed method naturally generates hardware-friendly structured sparsity patterns, including row-wise, column-wise, and diagonal masks, making it particularly well suited for accelerating Transformer attention mechanisms.