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Level surface

As explained in Math and Architectures of Deep Learning

A level surface in an arbitrary number of dimensions, generalizing constant-value contours and surfaces.

A level surface collects points in a multidimensional domain where a function has the same value. In two dimensions, these sets are level contours; in higher dimensions, they become level surfaces or hypersurfaces. section 3.3.2

For L(w₀,w₁)=√(w₀²+w₁²), the sets w₀²+w₁²=1 and w₀²+w₁²=25 are circles with constant values 1 and 5. Moving along a contour produces no change in L, whereas the gradient is normal to it and points toward greatest change. section 3.3.2

How Math and Architectures of Deep Learning covers it

How deep it goes
Defines contours and hypersurfaces, explains tangents, normals, gradients, and directional change, but stops short of implementation.
Examples
Uses concentric circles, concentric spheres, coaxial cylinders, heat maps, and radial gradient fields.
What it assumes
Assumes familiarity with functions, domains, gradients, and loss minimization while grounding the geometry in simple quadratic examples.
What you can do after
Evaluate how movement along or across a level surface changes a function and interpret the gradient geometrically.
How it connects
Ties level surfaces to gradients, directional derivatives, tangent directions, normal directions, and loss functions.
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This explainer was generated by Manning's knowledge graph from the text of Math and Architectures of Deep Learning. The section links point to the parts of the book it draws on.