Power Iteration

Power Iteration is a Linear Algebra method for approximating the dominant Eigenvalues and Eigenvectors of a matrix

Suppose $A$ is symmetric

  • then Eigendecomposition of $A$ is $A = Q \Lambda Q^T$
  • and $A^k = Q \Lambda^k Q^T$
  • let $\mathbf q_i$ be the columns of $Q$

Dominant eigenvalues and eigenvectors

  • $\lambda_1$ is the dominant eigenvalue of $A$ if $|\lambda_1| > |\lambda_i|$ for all $i = 2, ...$
  • The corresponding eigenvector $\mathbf q_1$ is also called dominant


The Power Method

Algorithm

  • initial approximation - random unit vector $x_0$
  • $x_1 = A x_0$
  • $x_2 = A A x_0 = A^2 x_0$
  • $x_3 = A A A x_0 = A^3 x_0$
  • ...
  • until converges

For large powers of $k$, we will obrain a good approximation of the dominant eigenvector


Finding the eigenvalue:

  • if $v$ is an eigenvector of $A$, then its eigenvalue is
  • $\lambda = \cfrac{v^T A^T v}{v^T v} = \cfrac{(A v)^T v}{v^T v}$
  • This is called Rayleigh Quotent
  • proof: suppose $\lambda$ is eigenvalue of $A$ and $v$ is its eigenvector
  • $\cfrac{(A v)^T v}{v^T v} = \cfrac{\lambda \, v^T v}{v^T v} = \lambda$


Performance:

  • computing $A \cdot A^{k-1}$ is costly
  • go the other direction:
  • use recurrent relation $x_{k} = A x_{k-1}$


if the power method generates a good approximation of the eigenvector, the approximation of the eigenvalue is also good


Normalization

The values produced by the power method are quite large:

  • better if we scale them down
  • should scale the values at each iteartion
  • for example, can find the largest component of $x_k$ and divide by it
  • more commonly, we just unit-normalize it


Implementation

Python with Numpy:

def eigenvalue(A, v):
    Av = A.dot(v)
    return v.dot(Av)

def power_iteration(A):
    n, d = A.shape

    v = np.ones(d) / np.sqrt(d)
    ev = eigenvalue(A, v)

    while True:
        Av = A.dot(v)
        v_new = Av / np.linalg.norm(Av)

        ev_new = eigenvalue(A, v_new)
        if np.abs(ev - ev_new) < 0.01:
            break

        v = v_new
        ev = ev_new

   return ev_new, v_new


Convergence of the Power Method

Theorem:

  • if $A$ is diagonalizable and has dominant eigenvalue,
  • then power iteration sequence $Ax, A^2 x, A^3 x, ...$ converges to the dominant eigenvector (scaled)

Proof:

  • since $A$ is diagonalizable, it has $n$ linearly independent eigenvectors $v_1, ..., v_n$ and corresponding eigenvalues $\lambda_1, ..., \lambda_n$.
  • let $v_1$ and $\lambda_1$ be dominant.
  • because the eigenvectors are independent, they form a basis
    • so any vector $x_0$ can be represented as $x_0 = c_1 v_1 + ... + c_n v_n$
    • and $c_1$ must be non zero - otherwise $x_0$ is not a good choice, and we need to choose another initial vector
  • let's multiply both sides by $A$:
    • $A x_0 = c_1 A v_1 + ... + c_n A v_n = c_1 \lambda_1 v_1 + ... + c_n \lambda_n v_n$
  • repeat that $k$ times:
    • $A^k x_0 = c_1 \lambda_1^k v_1 + ... + c_n \lambda_n^k v_n$
    • or $A^k x_0 = \lambda_1^k \left[ c_1 v_1 + c_2 \left(\cfrac{\lambda_2}{\lambda_1} \right)^k v_2 ... + c_n \left(\cfrac{\lambda_n}{\lambda_1} \right)^k v_n \right]$
  • since $\lambda_1$ is dominating, the ratios $\left(\cfrac{\lambda_i}{\lambda_1} \right)^k \to 0$ as $k \to \infty$ for all $i$
  • so $A^k x_0 = \lambda_1^k c_1 v_1$ and it gets better as $k$ grows

$\square$


From the proof we can also see that if the ratio $\lambda_2 / \lambda_1$ is small, it will converge quickly, but if it's not - it will take many iterations

Examples:

  • A = [4 5; 6 5] - needs only 6 steps, ratio is 0.1
  • A = [-4 10; 7 5] - needs 68 steps, ratio is 0.9


Finding Other Eigenvectors

Naive Method

We can just remove the dominant direction from the matrix and repeat

So:

  • $A = Q \Lambda Q^T$, so $A = \sum_{i = 1}^n \lambda_i \mathbf q_i \mathbf q_i^T$
  • use power iteration to find $\mathbf q_1$ and $\lambda_1$
  • then let $A_2 \leftarrow A - \lambda_1 \mathbf q_1 \mathbf q_1^T$
  • repeat power iteration on $A_2$ to find $\mathbf q_2$ and $\lambda_2$
  • continue like this for $\lambda_3, \ ... \ , \lambda_n$


Inverse Iteration

Inverse iteration - to find the smallest eigenvalue http://ranger.uta.edu/~huber/cse4345/Notes/Eigenvalues.pdf


Shifts

We can focus at any other eigenvalue by shifting our matrix

  • Shifting is the procedure of adding some value $s$ to the main diagonal: $A + s I$
  • let $\mu$ be the smallest ev of the shifted $A$
    • i.e. $(A + s I) \mathbf x = \mu \mathbf x$
    • or $(A + s I) \mathbf x - \mu \mathbf x = (A + s I - \mu I) \mathbf x = 0$
    • and the matrix $A + s I - \mu I$ is singular
  • since $A + s I - \mu I$ is singular, $\lambda = \mu - s$ is the eigenvalue of $A$ and it's the value closest to $s$
  • So shifting allows to focus on eigenvalues closest to any particular value


Eigenvectors are not affected by shifts:

  • suppose $A \mathbf x = \lambda \mathbf x$
  • let's shift by $m$: $(A + m I) \mathbf x = A \mathbf x + m \mathbf x = \lambda \mathbf x + m \mathbf x = (\lambda + m) \mathbf x$
  • for the shifted $A$, the eigenvalue is $\lambda + m$, and the eigenvector stays the same


Orthogonal Iteration

Idea:

  • we know that other eigenvectors are orthogonal to the dominant one
  • so we can use the power method, and force that the second vector is orthogonal to the first one
  • this way we guarante that they will converge to two different eigenvectors
  • we can do this for many vectors, not just two
  • this is called "Orthogonal Iteration"

Orthogonal Iteration

  • Orthogonal Iteration is a block version of the Power Method, also sometimes called "Simultaneous (Power) Interation"
  • Instead of multiplying $A$ by just one vector, we multiply by multiple vectors $\{ q_1, ... q_r \}$, which we put in a matrix $Q$.
  • At each step we re-normalize the vectors, e.g. with QR Decomposition


Algorithm:

  • choose $Q_0$ such that $Q_0^T Q_0 = I$
  • for $k = 1, 2, ...$:
    • $Z_k = A Q_{k-1}$
    • $Q_k R_k = Z_k$ (QR decomposition)


Let's look at the sequence of $\{ R_k \}$:

  • $R_k = Q_k^T Z_k = Q_k^T A Q_{k - 1}$
  • if $\{ Q_k \}$ converges to some $Q$ then $Q^T A Q = R$ is upper triangular
  • This is a Schur Decomposition of $A$
  • Thus, the eigenvalues of $A$ are located on the main diagonal of $R$
  • And the columns of $Q$ are the eigenvectors


Implementation

Python with numpy:

def simultaneous_power_iteration(A, k):
    n, m = A.shape
    Q = np.random.rand(n, k)
    Q, _ = np.linalg.qr(Q)
    Q_prev = Q
 
    for i in range(1000):
        Z = A.dot(Q)
        Q, R = np.linalg.qr(Z)

        # can use other stopping criteria as well 
        err = ((Q - Q_prev) ** 2).sum()
        if i % 10 == 0:
            print(i, err)

        Q_prev = Q
        if err < 1e-3:
            break

    return np.diag(R), Q


Algorithms based on Power Iteration

QR Algorithm

Simultaneous Iteration typically takes a while to converge

  • Another, better way to do it is QR algorithm.
  • It's based on Simultaneous Iteration algorithm


Lanczos Algorithm

Lanczos algorithm is used for tridiagonalization of a matrix $A$

  • It's based on the power iteration algorithm
  • And can be used to find eigenvalues and eigenvectors


Applications


Sources

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