Embedded constraints and tangent projections

Let h map an open part of Rⁿ smoothly into Rᵐ, and suppose its derivative has full row rank at every point of the zero set under consideration. The regular level-set theorem gives an embedded manifold of dimension n−m. Differentiating h along any feasible smooth curve shows that its velocity lies in the kernel of Dh. Regularity supplies the converse locally: every vector in that kernel is a tangent velocity. Merely counting equations without checking their independent derivatives can give the wrong dimension.

The rank assumption belongs to the chosen representation as well as to the feasible set. For h(t)=t² at zero, the feasible set is a smooth zero-dimensional singleton, but Dh vanishes and its kernel is the whole line. The kernel formula therefore fails for that singular description. For h(x,y)=xy, the zero set has a crossing and is not even a manifold near the origin. These examples explain why a failed constraint qualification is a warning to inspect the geometry rather than a license to treat every derivative-kernel direction as feasible.

With the Euclidean inner product, the normal space of a regular level set is the range of Dh(x) transposed. Minimizing the squared distance from an ambient vector z to the derivative kernel gives a small normal equation. Full row rank makes Dh times its transpose invertible, producing the displayed orthogonal projector. This is a linear projection at one point, not a projection of the point itself onto the nonlinear feasible set. Its output and the removed component are orthogonal in the declared Euclidean metric.

For the unit sphere, use h(x)=(∥x∥²−1)/2. Its derivative is x transposed, so projection subtracts the radial component. For the Stiefel constraint XᵀX=I, tangent matrices satisfy Xᵀη+ηᵀX=0, and the Frobenius-orthogonal projection subtracts X times the symmetric part of XᵀZ. These formulas exploit the constraints’ structure and avoid forming a large explicit Jacobian. Their metric qualification matters: weighted or canonical metrics generally require different gradient or projection formulas.

For a tangent η, the straight ambient point x+tη satisfies the constraints only to first order. On a sphere its squared norm becomes one plus t²∥η∥², so every nonzero finite step leaves the sphere. A retraction will repair this by mapping a tangent displacement back to the manifold while preserving its initial velocity. Projecting an ambient gradient and then adding it without such a feasible update is therefore not yet a complete manifold algorithm. The derivative constraint controls velocity rather than all higher-order terms.

Computationally, tangent projection should be checked by its defining identities: the output satisfies the differentiated constraint, projection is idempotent, and the residual is normal. These tests reveal transposes, missing symmetric factors and inappropriate metrics. They do not certify a global algorithmic rate, because convergence additionally depends on step choice, objective regularity and the update map. Starting with the correct tangent geometry isolates those later assumptions instead of mixing a geometric error with a line-search or curvature issue.