A Geometric Measure of Linear Separability for Neural Representations

• URL: http://arxiv.org/abs/2606.08721v1
• Date: Sun, 07 Jun 2026 16:31:54 GMT
• Title: A Geometric Measure of Linear Separability for Neural Representations
• Abstract summary: We introduce the directional linear separability measure (LSM), a finite-sample diagnostic for one-sided affine separability.<n>For a target class A and a competing set B, LSM measures the smallest competing-sample intrusion that must remain on the target side, normalized by |A|.
• Score: 12.476368894025233
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Modern neural classifiers commonly rely on linear readouts, yet predictive metrics alone do not characterize the class-wise geometry of the representations on which such readouts operate. We introduce the directional linear separability measure (LSM), a finite-sample diagnostic for one-sided affine separability. For a target class A and a competing set B, LSM searches over affine halfspaces that contain all samples in A and measures the smallest competing-sample intrusion that must remain on the target side, normalized by |A|. The resulting quantity is asymmetric, class-wise, target-normalized, and applicable to finite representations extracted from neural networks. We establish its supporting-hyperplane characterization, relate it to optimal affine classification accuracy, and prove invariance under full-rank linear embeddings. These results separate changes caused by linear reparameterization from those caused by information loss or nonlinear geometric transformations. We also give a penalty-based affine search for estimating class-wise LSM in high-dimensional features, with reported values computed from the original discrete preservation and violation criterion. Finally, we analyze coordinatewise gated nonlinearities as finite-sample geometric operators and empirically use LSM to diagnose class-wise intrusion across common deep-learning components and architectures.

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