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Projected Euclidean gradients are the Stiefel and Grassmann gradients

Statement

Let $F$ be smooth on an open neighborhood of $St (n, r)$ , and write $G = \nabla F (X)$ for its Euclidean gradient. Its restriction to Stiefel has Riemannian gradient $G - X sym (X^{T} G)$ . If $F (X Q) = F (X)$ for every $Q \in O (r)$ , it descends to a smooth Grassmann objective and its gradient has horizontal representative $(I - X X^{T}) G$ . In either metric, gradient zero is equivalent to a critical point of the restricted objective.

Facts & Assumptions

Given: $1 \leq r \leq n$ , a smooth ambient $F$ , $X \in St (n, r)$ , and, for the Grassmann conclusion, right-orthogonal invariance.

[F1]

The Frobenius projections to Stiefel and Grassmann tangent representatives are the formulas in Tangent spaces and Frobenius projections on Stiefel and Grassmann.

[F2]

A Riemannian gradient is the metric dual of the differential (The Riemannian gradient is the metric dual of the differential).

[F3]

Right multiplication by an orthogonal matrix changes the frame representative but not its subspace (Stiefel spaces, Grassmannians, and tautological bundles).

[F4]

The Jacobian definition gives the coordinate partial derivatives of $F$ as the entries of $G = \nabla F (X)$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case); for a smooth $F$ , its differential is linear and its value on a coordinate-basis matrix is the corresponding partial derivative.

[F5]

The Riemannian gradient vanishes exactly at critical points of a smooth function (The Riemannian gradient vanishes exactly at the critical points).

[F6]

The Frobenius pairing is the Euclidean coordinate inner product on matrix entries (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

Proof

technique · identify the metric dual by orthogonal projection

1.1F1F2F4F6givenalgebra

Let $E_{i j}$ be the coordinate-basis matrices. Smoothness of $F$ makes $d F_{X}$ linear, and by the definition in [F4], $d F_{X} [E_{i j}] = \partial_{i j} F (X) = G_{i j}$ . Expanding $Z = \sum_{i, j} Z_{i j} E_{i j}$ therefore gives $d F_{X} [Z] = \sum_{i, j} G_{i j} Z_{i j} = ⟨ G, Z ⟩_{F}$ . Restriction to Stiefel has this same differential on tangent vectors. By [F1], $Π_{X}^{S t} G$ is tangent and $G - Π_{X}^{S t} G$ is normal, hence $⟨ Π_{X}^{S t} G, Z ⟩_{F} = ⟨ G, Z ⟩_{F}$ . The metric-dual characterization [F2] proves the Stiefel formula.

1.2F3givenalgebra

Invariance and [F3] make $f ([X]) = F (X)$ well-defined. The graph charts for the quotient in [F3] provide smooth local frame sections; in each chart, $f$ is the composition of $F$ with such a section and is therefore smooth. For each skew-symmetric $Ω$ , the Cayley curve $Q (t) = (I_{r} - \frac{t}{2} Ω)^{- 1} (I_{r} + \frac{t}{2} Ω)$ satisfies $Q (t)^{T} Q (t) = I_{r}$ near zero by direct multiplication and has $Q (t) = I_{r} + t Ω + O (t^{2})$ . Invariance makes $F (X Q (t))$ constant, so $d F_{X} [X Ω] = 0$ for every vertical tangent; consequently the differential descends to horizontal representatives.

2.1F1F2F4F5F6givenstep 1.2∎

For a horizontal $Z$ , the descended differential is $⟨ G, Z ⟩_{F}$ by [F4] and [F6]. By [F1], $(I - X X^{T}) G$ is horizontal and its difference from $G$ is orthogonal to every horizontal $Z$ , so it is the metric dual on the quotient. [F5] gives the asserted equivalence between a zero gradient and criticality on each manifold.

Depends on

Used by

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Sources

P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Chapter 3 (2008) (standard reference, not scraped)
Nicolas Boumal, An Introduction to Optimization on Smooth Manifolds (2023 prepublication) (standard reference, not scraped)