1 Introduction
Audio source separation is the act of isolating soundproducing sources in an auditory scene. Examples include separating singing voice from accompanying music, the voice of a single speaker at a crowded party, or the sound of a car backfiring in a loud urban soundscape. Recent deep learning techniques have rapidly advanced the performance of such source separation algorithms leading to state of the art performance in the separation of music mixtures [1], separation of speech from nonstationary background noise [2], and separation of the voices from simultaneous overlapping speakers [3], often using only a single audio channel as input, i.e., no spatial information.
In this work we are concerned with separation networks that take as input a timefrequency (TF) representation of a signal (e.g., magnitude spectrogram), and either predict the separated source value in each TF bin directly or via a TF mask that when multiplied with the input recovers the separated signal. An inverse transform is then used to obtain the separated audio. One approach for training such algorithms uses some type of signal reconstruction error, such as the mean square error between magnitude spectra [2, 4]. An alternative approach referred to as deep clustering [3, 5, 6]
uses affinitybased training by estimating a highdimensional embedding for each TF bin, training the network with a loss function such that the embeddings for TF bins dominated by the same source should be close to each other and those for bins dominated by different sources should be far apart. This affinitybased training is especially valuable in tasks such as speech separation, as it avoids the permutation problem during network training where there is no straightforward mapping between the order of targets and outputs.
Deep clustering for music source separation was previously investigated with Chimera networks for singing voice separation in [7]. Chimera networks [7, 6] have multiple parallel output heads which are trained simultaneously on different tasks. Specifically, in the case of singing voice separation, one output head is used to directly approximate the soft mask for extracting vocals, while the other head outputs an embedding space that optimizes the deep clustering loss. When both heads are trained together, results are better than using any single head alone. Another approach for combining deep clustering and maskbased techniques was presented in [5]
where a deep clustering network, unfolded kmeans layers, and a second stage enhancement network are trained endtoend. The deep attractor network
[8] computes an embedding for each TF bin similar to deep clustering, but creates a mask based on the distance of each TF bin to an attractor point for each source. The attractors can either be estimated via kmeans clustering, or learned as fixed points during training.In this work, we consider deep attractorlike networks for separating multiple instruments in music mixtures. While embedding networks have typically been used in speech separation where all sources in a mixture belong to the same class (human speakers), we extend the formulation to situations where sources in a mixture correspond to distinct classes (e.g., musical instruments). Specifically, our classconditional embeddings work as follows: first, we propose using an auxiliary network to estimate a Gaussian distribution (mean vector and covariance matrix) in an embedding space for each instrument class we are trying to separate. Then, another network computes an embedding for each TF bin in a mixture, akin to deep clustering. Finally, a mask is generated based on the posterior distribution over classes for each TF bin. The network can be trained using a signal reconstruction objective, or in a multitask (Chimeralike) fashion with an affinitybased deep clustering loss used as a regularizer.
Deep clustering and deep attractor networks typically focus on speakerindependent speech separation where the mapping of input speaker to output index is treated as a nuisance parameter handled via permutation free training [3, 5, 9]. Several recent works on speakerconditioned separation [10, 11] allow separation of a targeted speaker from a mixture in a manner similar to how we extract specific instruments. Learning an embedding space for speaker separation that could also be used for classification was explored in recent work [12]. However, their work did this by introducing a classificationbased loss function. Here, the conditioning is introduced as input into the network rather than output from the network. Regarding specific musical instrument extraction from mixtures, a majority of methods [13, 14, 15, 1] use an independent deep network model for each instrument, and then combine these instrument specific network outputs in postprocessing using a technique such as the multichannel Wiener filter [14, 15]. While the efficacy of independent instrument modeling for musical source separation was confirmed by the results of a recent challenge [16], the requirements both in terms of computational resources and training data can be large, and scaling up the number of possible instruments can be prohibitive.
Recently, the work in [17]
demonstrated that a common embedding space for musical instrument separation using various deep attractor networks could achieve competitive performance. Our system is similar to the anchored and/or expectationmaximization deep attractor networks in
[17], but we use an auxiliary network to estimate the mean and covariance parameters for each instrument. We also explore what type of covariance model is most effective for musical source separation (tied vs. untied across classes, diagonal vs. spherical). Furthermore, we discuss a simple modification of our pretrained embedding networks for querybyexample separation [18, 19, 20], where given an isolated example of a sound we want to separate, we can extract the portion of a mixture most like the query without supervision.2 Embedding Networks
Let be the complex spectrogram of the mixture of sources for . An embedding network computes
(1) 
where is the input feature representation (we use the logmagnitude spectrogram in this work), and contains a dimensional embedding for each TF bin in the spectrogram. The function
is typically a deep neural network composed of bidirectional long shortterm memory (BLSTM) layers, followed by a dense layer. We then create a mask for each source
, with(2) 
from . Deep clustering [3, 5] builds binary masks via kmeans clustering on (soft masks can also be obtained via soft kmeans), and is trained by minimizing the difference between the true and estimated affinity matrices,
(3) 
where indicates which of the sources dominates each TF bin. Deep attractor networks [8] use the distance between the embeddings and fixed attractor points in the embedding space to compute soft masks, and are typically trained with a signal reconstruction loss function, such as the loss between the estimated and ground truth magnitude spectrograms
(4) 
where are the flattened spectrogram of the mixture, and ground truth source, respectively. We can obtain the separated time domain signal from the estimated magnitude after an inverse STFT using the mixture phase.
Chimera networks [7, 6] combine signal reconstruction and deep clustering losses, using two heads stemming from the same underlying network (stacked BLSTM layers). In this work, we also combine the loss functions from (3) and (4) (with equal weighting), but the gradients from both propagate into the same embedding space, rather than separate heads.
3 Conditioning embeddings on class
When we are interested in separating sources that belong to distinctly different groups, i.e., classes, each source has an associated class label , and we assume here that each mixture contains at most one isolated source per class label. Estimating the mask in (2) for source (class) and TF bin is then equivalent to estimating the posterior over classes given the corresponding dimensional network embedding. For simplicity we use a Gaussian model of the embedding space and obtain the mask from
(5) 
The Gaussian parameters and class prior
for each class are learned endtoend along with the embedding network. The generation of the parameters of each Gaussian from the auxiliary classconditional network is the maximization step in the expectationmaximization (EM) algorithm (trained through stochastic gradient descent), and the generation of the mask is the expectation step. Rather than unfolding a clustering algorithm as in
[5], we instead can learn the parameters of the clustering algorithm efficiently via gradient descent. Further, the soft mask is generated directly from the posteriors of the Gaussians, rather than through a secondstage enhancement network as in [5]. A diagram of our system can be seen in Fig. 1.We also draw a connection between the classconditional masks of (5) and the adaptive pooling layers for sound event detection in [21], which are also conditioned on class label. In [21]
, an activation function that is a variant of softmax with learnable parameter
is introduced. Ifis very high, the function reduces to a max function, heavily emphasizing the most likely class. If it is low, energy is spread more evenly across classes approximating an average. Our work uses a similar idea for source separation. A softmax nonlinearity is comparable to the posterior probability computation used in the expectation step of the EM algorithm in our Gaussian mixture model (GMM). For a GMM with tied spherical covariance,
is the inverse of the variance. A similar formulation of softmax was also used in
[5], where kmeans was unfolded on an embedding space. In that work was set manually to a high value for good results. In our work, we effectively learn the optimal (the inverse of the covariance matrix) for signal reconstruction rather than setting it manually, but still conditioning it on class as in [21] for source separation.4 Experiments
Our experiment is designed to investigate whether our proposed classconditional model outperforms a baseline mask inference model. We also explore which covariance type is most suitable for music source separation. We do this by evaluating the SDR of separated estimates of vocals, drums, bass, and other in the MUSDB [22] corpus using the museval package^{1}^{1}1https://github.com/sigsep/sigsepmuseval. Finally, we show the potential of our system to perform querying tasks with isolated sources.
4.1 Dataset and training procedure
We extend Scaper [23], a library for soundscape mixing designed for sound event detection, to create large synthetic datasets for source separation. We apply our variant of Scaper to the MUSDB training data, which consists of 100 songs with vocals, drums, bass, and other stems to create training mixtures and validation mixtures, all of length seconds at a sampling rate of kHz. Of the songs, we use for training and for validation. The remaining songs in the MUSDB testing set are used for testing. The training and validation set mixtures are musically incoherent (randomly created using stems from different songs) and each contains a random second excerpt from a stem audio file in MUSDB (vocals, drums, bass, and other). All four sources are present in every training and validation mixture.
Time domain stereo audio is summed to mono and transformed to a singlechannel logmagnitude spectrogram with a window size of samples ( ms) and a hop size of samples. Our network consists of a stack of 4 BLSTMs layers with units in each direction for a total of . Before the BLSTM stack, we project the logmagnitude spectrogram to a melspectrogram with mel bins. The mel spectrogram frames are fed to the BLSTM stack which projects every timemel bin to an embedding with dimensions.
The auxiliary classconditional network takes as input a onehot vector of size , one for each musical source class in our dataset. It maps the onehot vector to the parameters of a Gaussian in the embedding space. For an embedding space of size , and diagonal covariance matrix, the onehot vector is mapped to a vector of size : for the mean, for the variance, and for the prior. After the parameters of all Gaussians are generated, we compute the mask from the posteriors across the GMM using Eq. (5). The resultant mask is then put through an inverse mel transform to project it back to the linear frequency domain, clamped between and , and applied to the mixture spectrogram. The system is trained end to end with
loss and the embedding space is regularized using the deep clustering loss function. To compute the deep clustering loss, we need the affinity matrix for the melspectrogram space. This is computed by projecting the ideal binary masks for each source into mel space and clamping between
and . The deep clustering loss is only applied on bins that have a log magnitude louder than db, following [3].We evaluate the performance of multiple variations of classconditional embedding networks on the MUSDB18 [22] dataset using sourcetodistortion ratio (SDR)^{2}^{2}2https://github.com/sigsep/sigsepmuseval. At test time, we apply our network to both stereo channels independently and mask the two channels of complex stereo spectrogram. We explore several variants of our system, specifically focusing on the possible covariance shapes of the learned Gaussians. We compare these models to a baseline model that is simply a mask inference network (the same BLSTM stack) with outputs (one mask per class) followed by a sigmoid activation. All networks start from the same initialization and are trained on the same data.
4.2 Results
Table 1 shows SDR results for the baseline model and the four covariance model variants. We find that all four of our models that use an embedding space improve significantly on the baseline for vocals and other sources. The best performing model is a GMM with tied spherical covariance, which reduces to soft kmeans, as used in [5]. The difference here is that the value for the covariance is learned rather than set manually. The covariance learned was , or an value of , close to the value of found in [5]. The embedding space for this model on a sample mixture is visualized in Fig. 2
using Principal Component Analysis. We observe that there exist “bridges” between some of the sources. For example, other and vocals share many timefrequency points, possibly due to their source similarity. Both sources contain harmonic sounds and sometimes leading melodic instruments. However, unlike other embedding spaces (e.g., word2vec) where things that are similar are near each other in the embedding space, we instead have learned a separation space, where sources that are similar (but different) seem to be placed far from each other in the embedding space. We hypothesize that this is to optimize the separation objective. In
[8], it is observed that attractors for speaker separation come in two pairs, across from each other. Our work suggests that the two pairs may correspond to similar sources (e.g., separating female speakers from one another and separating male speakers from one another). Verifying this and understanding embedding spaces learned by embedding networks will be the subject of future work.We hypothesize that the reason the simplest covariance model (tied spherical) performs best in Table 1 is that for the diagonal case, the variances collapse in all but a few embedding dimensions. Embedding dimensions with the lowest variance contribute most to the overall mask. As a result, they essentially become the mask by themselves, reducing the network more to mask inference rather than an embedding space. An example of this can be seen in Fig. 3, where the embedding dimension has essentially reduced to a mask for the vocals source. With a spherical covariance model, each embedding dimension must be treated equally, and the embedding space cannot collapse to mask inference. A possible reason tied spherical performs better than untied spherical, may be that the network becomes overly confident (low variance) for certain classes. With a tied spherical covariance structure, all embedding dimensions and instrument classes are equally weighted, forcing the network to use more of the embedding space, perhaps leading to better performance.



Approach  Vocals  Drums  Bass  Other 


BLSTM  
DC/GMM  diag. (untied)  
DC/GMM  diag. (tied)  
DC/GMM  sphr. (untied)  
DC/GMM  sphr. (tied)  

4.3 Querying with isolated sources
To query a mixture with isolated sources, we propose a simple approach that leverages an already trained classconditional embedding network. We take the query audio and pass it through the network to produce an embedding space. Then, we fit a Gaussian with a single component to the resultant embedding space. Next, we take a mixture that may contain audio of the same type as the query, but not the exact instance of the query, and pass that through the same network. This produces an embedding for the mixture. To extract similar content to the query from the mixture, we take the Gaussian that was fit to the query embeddings and run the expectation step of EM by calculating the likelihood of the mixture’s embeddings under the query’s Gaussian. Because there is only one component in this mixture model, calculating posteriors gives a mask of all ones. To alleviate this, we use the likelihood under the query Gaussian as the mask on the mixture and normalize it to by dividing each likelihood by the maximum observed likelihood value in the mixture.
An example of query by isolated source can be seen in Fig. 4. We use a recording of solo snare drum as our query. The snare drum in the query is from an unrelated recording found on YouTube. The mixture recording is of a song with simultaneous vocals, drums, bass, and guitar (Heart of Gold  Neil Young). The Gaussian is fit to the snare drum embeddings and transferred to the mixture embeddings. The mask produced is similar to the query as is the extracted part of the mixture. This invariance of embedding location was a result of conditioning the embeddings on the class.
5 Conclusion
We have presented a method for conditioning on class an embedding space for source separation. We have extended the formulation of deep attractor networks and other embedding networks to accommodate Gaussian mixture models with different covariances. We test our method on musical mixtures and found that it outperforms a mask inference baseline. We find that the embeddings found by the network are interpretable to an extent and hypothesize that embeddings are learned such that source classes that have similar characteristics are kept far from each other in order to optimize the separation objective. Our model can be easily adapted to a querying task using an isolated source. In future work, we hope to investigate the dynamics of embedding spaces for source separation, apply our approach to more general audio classes, and explore the querying task further.
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