# Copyright 2021 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
# limitations under the License.
import jax
import jax.numpy as jnp
from jax import vmap, jit, lax
from .defs import RecoverySolution, SPState
from cr.nimble.dsp import largest_indices
import cr.sparse as crs
[docs]def matrix_solve(Phi, y, K, max_iters=None, res_norm_rtol=1e-4):
r"""Solves the sparse recovery problem :math:`y = \Phi x + e` using Subspace Pursuit for matrices
"""
## Initialize some constants for the algorithm
M, N = Phi.shape
# squared norm of the signal
y_norm_sqr = y.T @ y
max_r_norm_sqr = y_norm_sqr * (res_norm_rtol ** 2)
if max_iters is None:
max_iters = M
def init():
# compute the correlations of atoms with signal y
h = Phi.T @ y
# Pick largest K indices [this is first iteration]
I = largest_indices(h, K)
# Pick corresponding atoms to form the K wide subdictionary
Phi_I = Phi[:, I]
# Solve least squares over the selected indices
x_I, r_I_norms, rank_I, s_I = jnp.linalg.lstsq(Phi_I, y)
# Compute new residual
r = y - Phi_I @ x_I
# Compute residual norm squared
r_norm_sqr = r.T @ r
# Assemble the algorithm state at the end of first iteration
return RecoverySolution(x_I=x_I, I=I, r=r, r_norm_sqr=r_norm_sqr, iterations=1, length=Phi.shape[1])
def body(state):
# compute the correlations of dictionary atoms with the residual
h = Phi.T @ state.r
# Ignore the previously selected atoms
h = h.at[state.I].set(0)
# Pick largest K indices
I_new = largest_indices(h, K)
# Combine with previous K indices to form a set of 2K indices
I_2k = jnp.hstack((state.I, I_new))
# Pick corresponding atoms to form the 2K wide subdictionary
Phi_2I = Phi[:, I_2k]
# Solve least squares over the selected 2K indices
x_p, r_p_norms, rank_p, s_p = jnp.linalg.lstsq(Phi_2I, y)
# pick the K largest indices
Ia = largest_indices(x_p, K)
# Identify indices for corresponding atoms
I = I_2k[Ia]
# TODO consider how we can exploit the guess for x_I
# # Corresponding non-zero entries in the sparse approximation
# x_I = x_p[Ia]
# Form the subdictionary of corresponding atoms
Phi_I = Phi[:, I]
# Solve least squares over the selected K indices
x_I, r_I_norms, rank_I, s_I = jnp.linalg.lstsq(Phi_I, y)
# Compute new residual
r = y - Phi_I @ x_I
# Compute residual norm squared
r_norm_sqr = r.T @ r
return RecoverySolution(x_I=x_I, I=I, r=r, r_norm_sqr=r_norm_sqr, iterations=state.iterations+1, length=Phi.shape[1])
def cond(state):
# limit on residual norm
a = state.r_norm_sqr > max_r_norm_sqr
# limit on number of iterations
b = state.iterations < max_iters
c = jnp.logical_and(a, b)
return c
state = lax.while_loop(cond, body, init())
return state
matrix_solve_jit = jit(matrix_solve, static_argnums=(2), static_argnames=("max_iters", "res_norm_rtol"))
[docs]def operator_solve(Phi, y, K, max_iters=None, res_norm_rtol=1e-4,
tracker=crs.noop_tracker):
r"""Solves the sparse recovery problem :math:`y = \Phi x + e` using Subspace Pursuit for linear operators
Examples:
- :ref:`gallery:0001`
- :ref:`gallery:0002`
- :ref:`gallery:0003`
- :ref:`gallery:0004`
- :ref:`gallery:0006`
- :ref:`gallery:0007`
"""
trans = Phi.trans
## Initialize some constants for the algorithm
M = y.shape[0]
# squared norm of the signal
y_norm_sqr = jnp.abs(jnp.vdot(y, y))
y_norm = jnp.sqrt(y_norm_sqr)
# scale the signal down.
scale = 1.0 / y_norm
y = scale * y
dtype = jnp.float64 if Phi.real else jnp.complex128
max_r_norm_sqr = (res_norm_rtol ** 2)
if max_iters is None:
max_iters = M
min_iters = min(3*K, 20)
def init():
# Data for the previous approximation [r = y, x = 0]
I_prev = jnp.arange(0, K)
x_I_prev = jnp.zeros(K, dtype=dtype)
r_norm_sqr_prev = 1.
# compute the correlations of atoms with signal y
h = trans(y)
# Pick largest K indices [this is first iteration]
I = largest_indices(h, K)
# Pick corresponding atoms to form the K wide subdictionary
Phi_I = Phi.columns(I)
# Solve least squares over the selected indices
x_I, r_I_norms, rank_I, s_I = jnp.linalg.lstsq(Phi_I, y)
# Compute new residual
r = y - Phi_I @ x_I
# Compute residual norm squared
r_norm_sqr = jnp.abs(jnp.vdot(r, r))
# Assemble the algorithm state at the end of first iteration
return SPState(x_I=x_I, I=I, r=r, r_norm_sqr=r_norm_sqr,
iterations=1,
I_prev=I_prev, x_I_prev=x_I_prev, r_norm_sqr_prev=r_norm_sqr_prev)
def body(state):
I_prev = state.I
x_I_prev = state.x_I
r_norm_sqr_prev = state.r_norm_sqr
# compute the correlations of dictionary atoms with the residual
h = trans(state.r)
# Ignore the previously selected atoms
h = h.at[state.I].set(0)
# Pick largest K indices
I_new = largest_indices(h, K)
# Combine with previous K indices to form a set of 2K indices
I_2k = jnp.hstack((state.I, I_new))
# Pick corresponding atoms to form the 2K wide subdictionary
Phi_2I = Phi.columns(I_2k)
# Solve least squares over the selected 2K indices
x_p, r_p_norms, rank_p, s_p = jnp.linalg.lstsq(Phi_2I, y)
# pick the K largest indices
Ia = largest_indices(x_p, K)
# Identify indices for corresponding atoms
I = I_2k[Ia]
# TODO consider how we can exploit the guess for x_I
# # Corresponding non-zero entries in the sparse approximation
# x_I = x_p[Ia]
# Form the subdictionary of corresponding atoms
Phi_I = Phi.columns(I)
# Solve least squares over the selected K indices
x_I, r_I_norms, rank_I, s_I = jnp.linalg.lstsq(Phi_I, y)
# Compute new residual
r = y - Phi_I @ x_I
# Compute residual norm squared
r_norm_sqr = jnp.abs(jnp.vdot(r, r))
return SPState(x_I=x_I, I=I, r=r, r_norm_sqr=r_norm_sqr,
iterations=state.iterations+1,
I_prev=I_prev, x_I_prev=x_I_prev, r_norm_sqr_prev=r_norm_sqr_prev
)
def cond(state):
# limit on residual norm
a = state.r_norm_sqr > max_r_norm_sqr
# limit on number of iterations
b = state.iterations < max_iters
c = jnp.logical_and(a, b)
# checking if support is still changing
d = jnp.any(jnp.not_equal(state.I, state.I_prev))
# consider support change only after some iterations
d = jnp.logical_or(state.iterations < min_iters, d)
c = jnp.logical_and(c, d)
jax.debug.callback(tracker, state, more=c)
return c
state = lax.while_loop(cond, body, init())
# scale back the result
x_I = y_norm * state.x_I
r = y_norm * state.r
r_norm_sqr = state.r_norm_sqr * y_norm_sqr
return RecoverySolution(x_I=x_I, I=state.I, r=r, r_norm_sqr=r_norm_sqr,
iterations=state.iterations, length=Phi.shape[1])
operator_solve_jit = jit(operator_solve, static_argnames=("Phi", "K", "tracker"))
solve = operator_solve_jit
