%matplotlib inline
import seaborn
import numpy, scipy, matplotlib.pyplot as plt, sklearn, IPython.display as ipd
import librosa, librosa.display
plt.rcParams['figure.figsize'] = (14, 4)

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Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) is an algorithm that factorizes a nonnegative matrix, $X$, into a product of two nonnegative matrices, $W$ and $H$. It is an unsupervised iterative algorithm that minimizes a distance between $X$ and the product $WH$:

$$ \min_{W, H} d(X, WH) $$

If $X$ has dimensions $M$ by $N$, then $W$ will have dimensions $M$ by $R$, and $H$ will have dimensions $R$ by $N$, where inner dimension $R$ is the rank or number of components of the decomposition.

When applied to a musical signal, we find that NMF can decompose the signal into separate note events. Therefore, NMF is quite useful and popular for tasks such as transcription and source separation.

The input, $X$, is often a magnitude spectrogram. In such a case, we find that the columns of $W$ represent spectra of note events, and the rows of $H$ represent temporal envelopes of the same note events.

Let's load a signal:

x, sr = librosa.load('audio/conga_groove.wav')
print sr

22050

Play the audio signal:

ipd.Audio(x, rate=sr)

Compute the STFT:

S = librosa.stft(x)
print S.shape

(1025, 188)

Display the spectrogram:

Smag = librosa.amplitude_to_db(S)
librosa.display.specshow(Smag, sr=sr, x_axis='time', y_axis='log')
plt.colorbar()

<matplotlib.colorbar.Colorbar at 0x106de1210>

librosa.decompose.decompose

We will use librosa.decompose.decompose to perform our factorization. librosa uses sklearn.decomposition.NMF by default as its factorization method.

X = numpy.absolute(S)
n_components = 6
W, H = librosa.decompose.decompose(X, n_components=n_components, sort=True)
print W.shape
print H.shape

(1025, 6)
(6, 188)

Let's display the spectra:

plt.figure(figsize=(12, 6))
logW = numpy.log10(W)
for n in range(n_components):
    plt.subplot(numpy.ceil(n_components/2.0), 2, n+1)
    plt.plot(logW[:,n])
    plt.ylim(-2, logW.max())
    plt.xlim(0, W.shape[0])
    plt.ylabel('Component %d' % n)

/Users/steve/miniconda2/lib/python2.7/site-packages/ipykernel_launcher.py:2: RuntimeWarning: divide by zero encountered in log10
  

Let's display the temporal activations:

plt.figure(figsize=(12, 6))
for n in range(n_components):
    plt.subplot(numpy.ceil(n_components/2.0), 2, n+1)
    plt.plot(H[n])
    plt.ylim(0, H.max())
    plt.xlim(0, H.shape[1])
    plt.ylabel('Component %d' % n)

Finally, re-create the individual components, and listen to them. To do this, we will reconstruct the magnitude spectrogram from the NMF outputs and use the phase spectrogram from the original signal.

reconstructed_signal = scipy.zeros(len(x))
for n in range(n_components):
    Y = scipy.outer(W[:,n], H[n])*numpy.exp(1j*numpy.angle(S))
    y = librosa.istft(Y)
    reconstructed_signal[:len(y)] += y
    ipd.display( ipd.Audio(y, rate=sr) )

Listen to the reconstructed full mix:

ipd.Audio(reconstructed_signal, rate=sr)

Listen to the residual:

residual = x - reconstructed_signal
residual[0] = 1 # hack to prevent automatic gain scaling
ipd.Audio(residual, rate=sr)

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