Source code for cr.sparse._src.cvx.focuss

# Copyright 2022 CR-Suite Development Team
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
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import math
from jax import lax
from typing import NamedTuple, List, Dict
import jax.numpy as jnp
from jax.numpy.linalg import norm
import cr.nimble as crn
import cr.nimble.dsp as crdsp


def step_noiseless(Phi, y, x, p=1.):
    """A step in the FOCUSS algorithm

    FOCUSS (FOcal Underdetermined System Solver)
    uses an iterated reweighted least squares method.
    The algorithm is described in :cite:`elad2010sparse` (section 3.2.1).

    Examples:
        - :ref:`gallery:focuss:1`
    """
    xp = jnp.abs(x) ** p
    xp2 = xp * xp
    A = crn.diag_premultiply(xp2, Phi.T)
    G = Phi @ A 
    x = A @ jnp.linalg.solve(G, y)
    return x

class FocussState(NamedTuple):
    """State of iterative reweighted least squares algorithm
    """
    # The non-zero values
    x: jnp.ndarray
    "Solution vector"
    r: jnp.ndarray
    "The residuals"
    r_norm_sqr: float
    "The residual norm squared"
    iterations: int
    "Number of iterations"

    @property
    def length(self):
        return self.x.size

    @property
    def I(self):
        signal, mask = crdsp.energy_threshold(self.x, 0.99)
        return jnp.where(mask)[0]

    @property
    def x_I(self):
        I = self.I
        return self.x[I]

    def __str__(self):
        """Returns the string representation
        """
        s = []
        r_norm = math.sqrt(float(self.r_norm_sqr))
        x_norm = float(norm(self.x))
        I = self.I
        for x in [
            f'iterations={self.iterations}',
            f"m={len(self.r)}, n={self.length}, k={len(I)}",
            f'r_norm={r_norm:e}',
            f'x_norm={x_norm:e}',
            ]:
            s.append(x.rstrip())
        return u'\n'.join(s)


def matrix_solve_noiseless(Phi, y, p=1., max_iters=20):
    """Solves the sparse recovery problem using FOCUSS.

    Args:
        Phi(jax.numpy.ndarray): A sensing matrix / dictionary
        y(jax.numpy.ndarray): Measurements
        max_iters(int): Maximum number of iterations
        p(float): norm type [ p <= 1.]

    FOCUSS (FOcal Underdetermined System Solver)
    uses an iterated reweighted least squares method.
    The algorithm is described in :cite:`elad2010sparse` (section 3.2.1).

    Examples:
        - :ref:`gallery:focuss:1`
    """
    m, n = Phi.shape
    # squared norm of the signal
    y_norm_sqr = y.T @ y

    def init_func():
        # initial solution
        x = jnp.ones(n)
        xp = x
        # initialize residual
        r = y - Phi @ x
        r_norm_sqr = r.T @ r
        return FocussState(x=x, r=r, 
            r_norm_sqr=r_norm_sqr, 
            iterations=0)

    def body_func(state):
        xp = jnp.abs(state.x) ** p
        xp2 = xp * xp
        A = crn.diag_premultiply(xp2, Phi.T)
        G = Phi @ A 
        x = A @ jnp.linalg.solve(G, y)
        r = y - Phi @ x
        r_norm_sqr = r.T @ r
        return FocussState(x=x, r=r, 
            r_norm_sqr=r_norm_sqr, 
            iterations=state.iterations +1)

    def cond_func(state):
        # limit on number of iterations
        b = state.iterations < max_iters
        return b
    # state = init_func()
    # while cond_func(state):
    #     state = body_func(state)
    state = lax.while_loop(cond_func, body_func, init_func())
    return state