Euclidean Gradient not converging for L(Q)=|h Q Qᵀ g|² with Q=I−2vvᴴ (complex sphere)

[Keyvan Aslansefat]

I have a real-valued loss function as follows:

L(Q)=|hQQTg|2,

where h∈C1×K, g∈CK×1, Q∈CK×K. I obtained the gradient of L wrt Q∗ as follows:

dL=2hQQTg(hHgH+g∗h∗)Q∗:dQ∗.

Moreover, the matrix Q itself can be written as Q=IK−2vvH, where v∈CK×1, and has a unit norm, i.e., spherecomplexfactory. I calculated the gradient of L with respect to v∗ as follows, assuming v is constant, but the optimization problem is not converging.

dL=−4h(IK−2vvH)(IK−2vvH)Tg(hHgH+g∗h∗)(IK−2vvH)∗v:dv∗.

Does anybody have an idea about the problem? I code is as follows:

M = spherecomplexfactory(K, 1);

P.M = M;

P.cost = @(v) -costFun(hr, ht, v);

P.egrad = @(v) -gradFun(P, hr, ht, v);

[v, ~] = conjugategradient(P, [], []);

checkgradient(P)

function c = costFun(hr, ht, v)

I = eye(length(hr));

Q = (I-2*v*v');

Theta = Q*Q.';

c = abs(hr*(Theta)*ht)^2;

end

function g = gradFun(P, hr, ht, v)

I = eye(length(hr));

Q = (I-2*v*v');

Theta = Q*Q.';

G = 2*hr*Theta*ht*((conj(ht)*conj(hr)).' + (conj(ht)*conj(hr)))*conj(Q);

g = -2*G*v;

end

[Nicolas Boumal]

Hello,

If you believe the issues may come from an incorrect derivation of the gradient because the variable is complex, then I suggest you write v = u + iw (separate v into real and imaginary parts, so that u and v are two real vectors), and compute the gradient with respect to u and v (as one normally would for a function of real variables). Then, you can recombine these into a complex vector (gradient wrt u + i gradient wrt w).

Best,

Nicolas