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Spectral Norm by Power Iteration

JAX

Medium

Estimate the largest singular value of a matrix with a fixed number of power iterations from a given start vector.

The matrix w has shape (M, N) and the start vector v0 has shape (N,), and $\lVert \cdot \rVert$ is the Euclidean length. Start from $v = v_0 / \lVert v_0 \rVert$ . Each of the num_iters iterations performs

u = \frac{w v}{\lVert w v \rVert}, \qquad v = \frac{w^\top u}{\lVert w^\top u \rVert}.

After the last iteration, using the final v, set $\sigma = \lVert w v \rVert$ and $u = w v / \sigma$ . With num_iters equal to 0, v is the normalized start vector.

Your function is compiled with jax.jit, and num_iters arrives as a traced int32 scalar, so Python's range cannot read it.

Return a dict with the key sigma, a float32 JAX array of shape (), the key u, a float32 JAX array of shape (M,), and the key v, a float32 JAX array of shape (N,).

Examples

Input: w = [[3.0,0.0],[0.0,1.0]], v0 = [1.0,1.0], num_iters = 2

Output: {"sigma":2.999797,"u":[0.999992,0.004115],"v":[0.999924,0.012345]}

Explanation: The unit start vector is [0.707107,0.707107]. Each iteration multiplies the ratio of the first component of v to the second by 9, so after two iterations v is close to [1,0] and sigma is close to the true spectral norm of 3.

Input: w = [[1.0,2.0,0.0],[2.0,1.0,1.0]], v0 = [1.0,0.0,0.0], num_iters = 0

Output: {"sigma":2.236068,"u":[0.447214,0.894427],"v":[1.0,0.0,0.0]}

Input: w = [[2.0,1.0],[1.0,2.0],[0.0,1.0]], v0 = [0.0,2.0], num_iters = 5

Output: {"sigma":3.087253,"u":[0.671542,0.700041,0.242847],"v":[0.661745,0.749729]}

Hint 1

jax.lax.fori_loop(0, num_iters, body, v0) with body(i, v) -> new_v.

Hint 2

jnp.linalg.norm(v) gives a vector's length as a scalar array.

Hint 3

W @ v and W.T @ u are the two halves of one iteration.

Requirements

Run the iterations with jax.lax.fori_loop so that the compiled program contains a while loop, which is what fori_loop produces when its upper bound is traced. A program without a while loop is rejected.
Keep the loop value at the same shape and dtype on every iteration.

Constraints

w is a float32 JAX array of shape (M, N) with M and N from 1 to 5.
v0 is a float32 JAX array of shape (N,) with a nonzero length.
num_iters is an int32 JAX scalar of shape () with a value from 0 to 20.
w v and the transpose of w times u never become the zero vector during the iteration.

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