Stiefel Manifold Integration

IntegrateUnitary.jl supports integration over the Stiefel manifold $V_k(\mathbb{C}^d)$, the set of all $d \times k$ matrices with orthonormal columns:

\[V_k(\mathbb{C}^d) = \{ V \in \mathbb{C}^{d \times k} \mid V^\dagger V = I_k \}.\]

Mathematical Background

Connection to the Unitary Group

The uniform (Haar) measure on $V_k(\mathbb{C}^d)$ is induced by the Haar measure on $U(d)$. A Haar-random Stiefel frame is obtained by taking the first $k$ columns of a Haar-random unitary:

\[V = U_{1:d,\; 1:k}, \quad U \sim \mathrm{Haar}(U(d)).\]

Consequently, every polynomial integral over the Stiefel manifold reduces to unitary Weingarten calculus:

\[\int_{V_k(\mathbb{C}^d)} f(V)\, dV = \int_{U(d)} f\!\left(U_{1:d,\, 1:k}\right) dU.\]

IntegrateUnitary.jl exploits this reduction internally — the full symbolic Weingarten engine is automatically available for Stiefel integrals.

Special case: $k = 1$ and pure states

When $k = 1$, the Stiefel manifold reduces to the complex unit sphere $S^{2d-1}$, and the Haar measure coincides with the Fubini–Study measure on Haar-random pure quantum states. In code, dStiefel(d, 1) and dPsi(d) are equivalent. See Pure States.

[!IMPORTANT] The rank $k$ must satisfy $k \le d$. Passing k > d raises an ArgumentError at measure construction time.

Usage

Use the dStiefel(d, k) measure. The @integrate macro automatically identifies V as the random Stiefel matrix.

Basic integration

using IntegrateUnitary, Symbolics
@variables d

# E[|V_{11}|^2] = 1/d
@integrate abs(V[1, 1])^2 dStiefel(d, 2)
# Output: 1/d

The result $1/d$ matches the unitary case: the first column of a Stiefel frame is a Haar-random unit vector, so each component has average squared magnitude $1/d$.

Off-diagonal element (same column, different rows)

# E[|V_{11}|^2 * |V_{21}|^2]: two entries from the same column
@integrate abs(V[1, 1])^2 * abs(V[2, 1])^2 dStiefel(d, 2)
# Output: 1 / (d*(d+1))

This is strictly less than $1/d^2$, reflecting the negative correlation (anti-correlation) between entries of the same column imposed by the unit-norm constraint.

Cross-column correlation

# E[|V_{11}|^2 * |V_{12}|^2]: entries from different columns
@integrate abs(V[1, 1])^2 * abs(V[1, 2])^2 dStiefel(d, 2)
# Output: 1 / (d*(d+1))

The same value as the within-column result above follows from the unitary invariance of the measure.

Second-moment matrix

Matrix integration confirms the isotropic structure $\mathbb{E}[V V^\dagger] = \frac{k}{d} I_d$:

# E[V * V'] for d=3, k=2 (concrete dimensions required for matrix integration)
@integrate V * V' dStiefel(3, 2)
# Output: (2/3) * I_3

Asymptotic expansion

Large-$d$ expansions are fully supported. Pass the exact result as a rational function of d to asymptotic, or integrate and expand in one step:

using IntegrateUnitary, Symbolics
@variables d

# Exact: 1/(d*(d+1)); leading large-d behaviour:
asymptotic(1 / (d * (d + 1)), d, 3)
# Output: 1/d^2 - 1/d^3 + 1/d^4

Manual integration

For more control, create the SymbolicMatrix explicitly:

using IntegrateUnitary, Symbolics
@variables d
V = SymbolicMatrix(:V, :V)
integrate(abs(V[1, 1])^2, dStiefel(d, 2))
# Output: 1/d

Potential Pitfalls

[!IMPORTANT]

Symbolic (d) Pitfalls

• Results are rational functions of $d$ with poles at small integer values (typically $d < k$ for degree-$k$ moments).
• Use evaluate to resolve removable singularities when substituting specific numeric dimensions.

See Also

• Pure States — the $k = 1$ special case (dPsi)
• Unitary Integration — underlying Weingarten engine
• Asymptotic Expansions — large-$d$ limit analysis

References

• Edelman, A., Arias, T. A., & Smith, S. T. (1998). The geometry of algorithms with orthogonality constraints. SIAM Journal on Matrix Analysis and Applications, 20(2), 303–353.
• Collins, B., & Śniady, P. (2006). Integration with respect to the Haar measure on unitary, orthogonal and symplectic groups. Communications in Mathematical Physics, 264(3), 773–795.

API Reference

StiefelMeasure(dim, k)

dStiefel(dim, k)