New Geometric Measure Quantifies Linear Separability in Neural Network Representations
Researchers introduced the directional linear separability measure (LSM), a diagnostic tool that quantifies how well neural network representations can be separated by linear classifiers. LSM is asymmetric and class-wise, measuring the minimum intrusion of competing samples into target class regions, and is invariant under linear reparameterization. This work helps characterize the geometric properties of neural representations beyond predictive accuracy alone.
A new paper on arXiv proposes the directional linear separability measure (LSM) to diagnose one-sided affine separability in neural network representations. Unlike predictive metrics that only measure overall accuracy, LSM provides a class-wise, target-normalized measure of how cleanly a target class can be separated from competing classes using affine halfspaces. The authors establish theoretical properties including a supporting-hyperplane characterization, relate LSM to optimal affine classification accuracy, and prove invariance under full-rank linear embeddings—meaning the measure captures genuine geometric changes rather than mere reparameterization. They develop a penalty-based search algorithm for estimating LSM in high-dimensional features and empirically apply it to analyze common deep-learning components and architectures. This work bridges the gap between predictive performance and the underlying geometric structure of learned representations.
What's missing
The paper does not discuss computational complexity or scalability of the LSM estimation algorithm to very high-dimensional representations. Additionally, the practical implications for improving neural network design or training based on LSM diagnostics are not elaborated.
What different sources said
- arXiv cs.LGCenter
A Geometric Measure of Linear Separability for Neural Representations
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