Optimization on spheres and Stiefel manifolds

The real Stiefel manifold St(n,p) consists of n-by-p matrices with XᵀX=I, where 1≤p≤n. Its tangent matrices satisfy the skew-symmetry condition Xᵀη+ηᵀX=0. Decompose a tangent as XA+H, with A skew-symmetric and XᵀH=0. The first term rotates columns inside their current span; the second moves that span. The tangent dimension is np−p(p+1)/2. A sphere is the p=1 case, where the skew-symmetric one-by-one block vanishes.

For the induced metric, tangent inner products are Frobenius inner products, so an ambient objective gradient G becomes G−X sym(XᵀG). This is suitable for frame-sensitive objectives, such as matching labeled columns to specified reference vectors. An objective depending only on the span has an additional right-orthogonal symmetry and can instead be formulated on a Grassmann quotient. Conflating these problems can remove meaningful variables: a frame’s column labels matter for a general objective even though they are irrelevant to a subspace-only criterion.

The canonical Stiefel metric pairs ξ and η through I−XXᵀ/2 between their ambient factors. On the tangent decomposition, it gives one-half Frobenius weight to the skew frame-rotation block and full weight to the horizontal block. The induced metric gives full weight to both blocks. Thus they agree on horizontal directions but generally differ on frame rotations. Merely describing a Stiefel algorithm as Riemannian does not specify which of these geometries is intended.

For the canonical metric, the gradient corresponding to ambient G is G−XGᵀX. Its rotational component is twice the induced-gradient rotational component, matching the halved metric weight. Both satisfy the same differential identity when paired with their own metric. Their critical sets agree, but their norms and steps can differ. A Hessian or geodesic formula derived for one metric cannot be combined indiscriminately with the other gradient; this can change both the numerical method and the validity of a convergence argument.

For tangent η, the matrix X+η has Gram matrix I+ηᵀη, which is positive definite. Multiplying by its inverse square root gives the polar retraction Rₓ(η)=(X+η)(I+ηᵀη)⁻¹/². The output has orthonormal columns, and the first derivative at zero is the identity. Under the induced Frobenius geometry it is second order, because its initial ambient acceleration is −Xηᵀη, a normal matrix. This metric-qualified argument should not be silently recycled as a claim about a different connection.

A QR-based map with a consistent positive-diagonal convention also gives a local smooth retraction, but is generally only first order for the induced metric. Polar and QR updates can therefore have different pullback Hessians away from stationary points. In floating-point calculations, one should monitor XᵀX−I and the tangent constraint residual, while distinguishing these feasibility checks from a gradient or optimality certificate. An orthonormal matrix can be a saddle, a maximum, or far from any critical point of the objective.