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Source code for librosa.util._nnls

```#!/usr/bin/env python
# -*- coding: utf-8 -*-
"""Non-negative least squares"""

# The scipy library provides an nnls solver, but it does
# not generalize efficiently to matrix-valued problems.
# We therefore provide an alternate solver here.
#
# The vectorized solver uses the L-BFGS-B over blocks of
# data to efficiently solve the constrained least-squares problem.

import numpy as np
import scipy.optimize
from .utils import MAX_MEM_BLOCK

__all__ = ["nnls"]

def _nnls_obj(x, shape, A, B):
"""Compute the objective and gradient for NNLS"""

# Scipy's lbfgs flattens all arrays, so we first reshape
# the iterate x
x = x.reshape(shape)

# Compute the difference matrix
diff = np.einsum("mf,...ft->...mt", A, x, optimize=True) - B

# Compute the objective value
value = (1 / B.size) * 0.5 * np.sum(diff ** 2)

grad = (1 / B.size) * np.einsum("mf,...mt->...ft", A, diff, optimize=True)

def _nnls_lbfgs_block(A, B, x_init=None, **kwargs):
"""Solve the constrained problem over a single block

Parameters
----------
A : np.ndarray [shape=(m, d)]
The basis matrix
B : np.ndarray [shape=(m, N)]
The regression targets
x_init : np.ndarray [shape=(d, N)]
An initial guess
**kwargs

Returns
-------
x : np.ndarray [shape=(d, N)]
Non-negative matrix such that Ax ~= B
"""

# If we don't have an initial point, start at the projected
# least squares solution
if x_init is None:
x_init = np.einsum("fm,...mt->...ft", np.linalg.pinv(A), B, optimize=True)
np.clip(x_init, 0, None, out=x_init)

# Adapt the hessian approximation to the dimension of the problem
kwargs.setdefault("m", A.shape[1])

# Construct non-negative bounds
bounds = [(0, None)] * x_init.size
shape = x_init.shape

# optimize
x, obj_value, diagnostics = scipy.optimize.fmin_l_bfgs_b(
_nnls_obj, x_init, args=(shape, A, B), bounds=bounds, **kwargs
)
# reshape the solution
return x.reshape(shape)

[docs]def nnls(A, B, **kwargs):
"""Non-negative least squares.

Given two matrices A and B, find a non-negative matrix X
that minimizes the sum squared error::

err(X) = sum_i,j ((AX)[i,j] - B[i, j])^2

Parameters
----------
A : np.ndarray [shape=(m, n)]
The basis matrix
B : np.ndarray [shape=(..., m, N)]
**kwargs

Returns
-------
X : np.ndarray [shape=(..., n, N), non-negative]
A minimizing solution to ``|AX - B|^2``

--------
scipy.optimize.nnls
scipy.optimize.fmin_l_bfgs_b

Examples
--------
Approximate a magnitude spectrum from its mel spectrogram

>>> y, sr = librosa.load(librosa.ex('trumpet'), duration=3)
>>> S = np.abs(librosa.stft(y, n_fft=2048))
>>> M = librosa.feature.melspectrogram(S=S, sr=sr, power=1)
>>> mel_basis = librosa.filters.mel(sr=sr, n_fft=2048, n_mels=M.shape[0])
>>> S_recover = librosa.util.nnls(mel_basis, M)

Plot the results

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(nrows=3, sharex=True, sharey=True)
>>> librosa.display.specshow(librosa.amplitude_to_db(S, ref=np.max),
...                          y_axis='log', x_axis='time', ax=ax[2])
>>> ax[2].set(title='Original spectrogram (1025 bins)')
>>> ax[2].label_outer()
>>> librosa.display.specshow(librosa.amplitude_to_db(M, ref=np.max),
...                          y_axis='mel', x_axis='time', ax=ax[0])
>>> ax[0].set(title='Mel spectrogram (128 bins)')
>>> ax[0].label_outer()
>>> img = librosa.display.specshow(librosa.amplitude_to_db(S_recover, ref=np.max(S)),
...                          y_axis='log', x_axis='time', ax=ax[1])
>>> ax[1].set(title='Reconstructed spectrogram (1025 bins)')
>>> ax[1].label_outer()
>>> fig.colorbar(img, ax=ax, format="%+2.0f dB")
"""

# If B is a single vector, punt up to the scipy method
if B.ndim == 1:
return scipy.optimize.nnls(A, B)[0]

n_columns = MAX_MEM_BLOCK // (np.prod(B.shape[:-1]) * A.itemsize)
n_columns = max(n_columns, 1)

# Process in blocks:
if B.shape[-1] <= n_columns:
return _nnls_lbfgs_block(A, B, **kwargs).astype(A.dtype)

x = np.einsum("fm,...mt->...ft", np.linalg.pinv(A), B, optimize=True)
np.clip(x, 0, None, out=x)
x_init = x

for bl_s in range(0, x.shape[-1], n_columns):
bl_t = min(bl_s + n_columns, B.shape[-1])
x[..., bl_s:bl_t] = _nnls_lbfgs_block(
A, B[..., bl_s:bl_t], x_init=x_init[..., bl_s:bl_t], **kwargs
)
return x
```