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N.Boumal's Lecture: Optimization on Manifolds

This website links you to the relevant book and lectures provided by N. Boumal.

Tutorial 01: Rayleigh Quotient

The problem of Tutorial 01 is the following:

--- Problem 01 - Rayleigh Quotient ---

Given a symmetric $A\in\mathbb{R}^{n\times n}$, we want:

$$\max_{x\in\mathbb{R}^n}x^TAx,\quad \text{s.t.}\quad |x|=1.$$

The manifold is the sphere: $\mathcal{M}=\mathbb{S}^{n-1}={x\in\mathbb{R}^n:|x|=1}$, and the cost function to minimize is $f(x)=-x^TAx$.

--- End ---

Tutorial 02: Riemannian Gradient Descent (RGD) on Product of Spheres

The problem of Tutorial 02 is the following:

--- Problem 02 - RGD on Product of Spheres ---

Let $\mathcal{M}=\mathbb{S}^{m-1}\times\mathbb{S}^{n-1}$, which is an embedded submanifold of $\mathcal{E}=\mathbb{R}^m\times\mathbb{R}^n$ with its usual Euclidean structure. We turn $\mathcal{M}$ into a Riemannian submanifold by using the Euclidean structure of the ambient space $\mathcal{E}=\mathbb{R}^m\times\mathbb{R}^n$. Let $\mathcal{M}\in\mathbb{R}^{m\times n}$ and

$$f:\mathcal{M}\rightarrow\mathbb{R},\quad f(x,y)=x^TMy.$$

In the exercise Product o spheres, you showed that $f:\mathcal{M}\rightarrow\mathbb{R}$ is smooth, and worked out an expression for the Riemannian gradient of $f$. Now, we want to solve

$$\max_{(x,y)\in\mathcal{M}}\ f(x,y).$$

--- End ---

Note that the maximum value of $f$ on $\mathcal{M}$ is the largest singular value of $M$. Since in the book and the following, the RGD iterates toward the negative of gradient (i.e. toward the minimum of $f$), we need to transform this maximization problem to a minimization problem by adding a negative sign in the objective function.

To perform gradient descent method, we need obtain the Riemannian gradient. We introduce the steps to approach the Riemannian gradient as the followings:

  1. Since $f$ is also a smooth extension to the Euclidean space, to obtain the Riemannian Gradient, we can first obtain the Euclidean gradient $\mathrm{grad}\bar{f}(x,y)$. The product manifolds gives a nice equality that $\text{T}_{(x,y)}\mathcal{M}=\text{T}_x\mathbb{S}^{m-1}\times\text{T}_y\mathbb{S}^{n-1}$. That means we can compute the tangent space at $x$ and $y$ independently. Obviously, $\bar{f}=x^TMy$ with $x,y\in\mathcal{E}$ is a smooth extension of $f$. The Euclidean gradient can be written independently as well (Ex. 3.67)

$$\begin{aligned} \mathrm{grad}\bar{f}(x,y)&=\left(\mathrm{grad}(x\mapsto\bar{f}(x,y))(x),\mathrm{grad}(y\mapsto\bar{f}(x,y)(y))\right)\\ &=(\partial\bar{f}(x,y)/\partial x,\partial\bar{f}(x,y)/\partial y)\\ &=(My,M^Tx). \end{aligned}$$

  1. Derive a formula for the orthogonal projection from $\mathcal{E}$ onto the tangent space $\text{T}{(x,y)}\mathcal{M}$. Taking the tangent space at $x$ as an example. We know the tangent space $\mathrm{T}x\mathbb{S}^{m-1}={v\ \vert\ v^Tx=0}$. Since the tangent space is a subspace of vector space $\mathcal{E}$, any vector $u\in\mathcal{E}$ can be decomposed into two parts - one is on the tangent space and the other is orthogonal to the tangent space, i.e. $u=u{|}+u{\perp}$, where $\mathrm{Proj}x(u)=u{|}$ is the orthogonal projection onto $\mathrm{T}x\mathbb{S}^{n-1}$. Note that $x$ is also orthogonal to the tangent space. It shows the fact that $u{\perp}$ is parallel to $x$. As a result, we need only project $u$ on $x$ and then subtract it from $u$ itself, the remaining part is $u_{|}$, the orthogonal projection of $u$ on the tangent space. We know that $u_{\perp}=\frac{u^Tx}{x^Tx}x=u^Txx$, since $x\in\mathbb{S}^{n-1}$, i.e., $x^Tx=1$. Eventually, we obtain $\mathrm{Proj}_x(u)=u-u^Txx=u-xx^Tu=(I-xx^T)u$, resulting in the orthogonal projector for the unit sphere $\mathrm{Proj}_x=I-xx^T$.

  2. Project the Euclidean gradient to the tangent space, from which we can reach the Riemannian gradient

$$\begin{align*} \mathrm{grad}f(x,y)&=\mathrm{Proj}_{(x,y)}\left(\mathrm{grad}\bar{f}(x,y)\right)\\ &=\left(\mathrm{Proj}_x\left(\mathrm{grad}(x\mapsto\bar{f}(x,y))(x)\right),\mathrm{Proj}_y\left(\mathrm{grad}(y\mapsto\bar{f}(x,y))(y)\right)\right)\\ &=\left((I-xx^T)My,(I-yy^T)M^Tx\right). \end{align*}$$

  1. Map the new iteration back to manifold by retraction. There are many retractions for $\mathcal{M}$. One possible and the simplest retraction is to normalize the new data

$$\mathrm{R}:T\mathcal{M}\rightarrow\mathcal{M}:((x,y),(u,v))\mapsto\left(\frac{x+u}{|x+u|},\frac{y+v}{|y+v|}\right).$$

In a conclusion, the Riemannian Gradient is given by

--- Algorithm Begin ---

INPUT: $(x_0,y_0)\in\mathcal{M},\epsilon>0$, step size $\alpha>0$.

OUTPUT: Final position $(x,y)\in\mathcal{M}$.

  1. Let $(x,y)=(x_0,y_0)$ and compute $\mathrm{grad}(-f(x,y))$,
  2. While $|\mathrm{grad}(-f(x,y))|>\epsilon$,
  3. Let $(x,y)=\mathrm{R}_{(x,y)}(-\alpha\cdot\mathrm{grad}(-f(x,y)))$,
  4. Compute $\mathrm{grad}(-f(x,y))$,
  5. End while.

--- Algorithm End ---

Tutorial 03: RGD on Stiefel Manifold

--- *Problem 02: RGD on Stiefel Manifold ---

For $p<n$, consider the Stiefel manifold

$$\mathcal{M}=\mathrm{St}(n,p)=\lbrace X\in\mathbb{R}^{n\times p}:X^TX=I_p\rbrace.$$

We consider $\mathcal{M}$ as a Riemannian submanifold of $\mathbb{R}^{n\times p}$ endowed with usual the inner product $\langle X,Y\rangle=\mathrm{Tr}(X^TY)$. Let

$$f:\mathcal{M}\rightarrow\mathbb{R},\qquad f(X)=\mathrm{Tr}(X^TAX),$$

where $A$ is a real symmetric $n\times n$ matrix.

--- End ---