New Geometric Measure Quantifies Linear Separability in Neural Network Representations
A team of researchers has introduced the Directional Linear Separability Measure (LSM), a new diagnostic tool for assessing how well neural network representations separate classes geometrically. The measure is asymmetric, class-specific, and designed to work on finite samples extracted from trained neural networks, filling a gap left by purely predictive accuracy metrics. It offers a principled way to distinguish changes caused by linear reparameterization from those caused by genuine information loss or nonlinear geometric transformations.
Researchers have proposed the Directional Linear Separability Measure (LSM), a finite-sample diagnostic aimed at characterizing the class-wise geometry of neural network representations. Unlike standard predictive metrics, LSM evaluates one-sided affine separability: for a target class A and a competing set B, it searches over affine halfspaces containing all samples in A and quantifies the minimum intrusion of competing samples that cannot be excluded, normalized by the size of A. The measure is proven to be invariant under full-rank linear embeddings, meaning it is insensitive to linear reparameterizations and responds only to information loss or nonlinear geometric changes. The authors establish a supporting-hyperplane characterization, relate LSM to optimal affine classification accuracy, and provide a penalty-based algorithm for estimating it in high-dimensional feature spaces. They also analyze coordinatewise gated nonlinearities as geometric operators and apply LSM empirically to diagnose class-wise intrusion patterns across common deep-learning components and architectures. The work was submitted to arXiv on June 7, 2026, and has not yet undergone formal peer review.
What's missing
As a preprint, this work has not yet been peer-reviewed. Key open questions include: how LSM scales computationally to very large datasets and high-dimensional representations, whether the penalty-based estimation introduces systematic bias, and how LSM compares empirically to existing separability metrics such as Fisher's discriminant ratio or centered kernel alignment across diverse architectures and tasks.
What different sources said
- arXiv cs.LGCenter
A Geometric Measure of Linear Separability for Neural Representations
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