# Hessian

/ˈhɛʃən/ (US) · /hˈɛʃən/ (UK) · CEFR C2 · Noun

Source: https://englishreference.com/word/hessian
Updated: 2026-09-28

## Definition

a mathematical grid of second derivatives that shows the curvature or steepness of an optimization landscape.

For teachers: a square matrix of second-order partial derivatives of a scalar-valued function, describing the local curvature of the loss surface in optimization.

## Examples

- Checking the eigenvalues of the Hessian reveals whether a point is a valley bottom or a saddle.
- Second-order optimizers use the Hessian to take smarter steps toward the minimum of the loss function.
- Analyzing the eigenspectrum of the Hessian matrix reveals flat directions in parameter space that correlate strongly with superior test generalization.

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