PCA doesn't give you the vectors in your dataset. From Wikipedia : Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. The number of principal components is less than or equal to the number of original variables. This transformation is defined in such a way that the first principal component has the largest possible variance (that is, accounts for as much of the variability in the data as possible), and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components.
Say you have a column vector V containing ONE flattened spectrogram. PCA will find a matrix M whose columns are orthogonal vectors (think of them as being at right angles to every other column in M).
Multiplying M and T will give you a vector of "scores", which can be used to determine how much variance each column of M captures from the original data and each column of M captures progressively less variance in the data.
Multiplying matrix M' (the first 2 columns of M) by V will produce a 2x1 vector T' representing the "dimension-reduced spectrogram". You could reconstruct an approximation of V by multiplying T' by the inverse of M'. This would work if you had a matrix of spectrograms, too. Keeping only two principal components would produce an extremely lossy compression of your data.
But what if you want to add a new song to your dataset? Unless it is very much like the original song (meaning it introduces little variance to the original data set), there's no reason to think that the vectors of M will describe the new song well. For that matter, even multiplying all the elements of V by a constant would render M useless. PCA is quite data specific. Which is why it's not used in image/audio compression.
The good news? You can use a Discrete Cosine transform to compress your training data. Instead of lines, it finds cosines that form a descriptive basis, and doesn't suffer from the data specific limitation. DCT is used in jpeg, mp3 and other compression schemes.
The Cocktail Party Problem is a Blind Source Separation (BSS) problem.
Given a linear mixture of signals:
$$ \boldsymbol{y} \left[ n \right] = A \boldsymbol{x} \left[ n \right] $$
We're trying to estimate the signal $ \boldsymbol{x} \left[ n \right] $.
The model can get even more complex with $ A $ being time varying:
$$ \boldsymbol{y} \left[ n \right] = A \left[ n \right] \boldsymbol{x} \left[ n \right] $$
We have 3 main approaches to this problem:
- Probabilistic Approach
Looking at the signals as an ensemble of points of a distribution and find the linear coordinate transform to guarantee some property. The PCA approach tries to remove correlation (2nd moment information) while the ICA tries to remove correlation in higher moments (Basically statistical independence). - Time Signal Processing Approach
In case of 2 signals one of them being a reference we can use the adaptive decorrelation filter. Basically we're after removing any time correlation from the signals. - Spatial Signal Processing Approach
We can utilize the known location of the microphones in the room to create adaptive beamforming. The idea is that delayed adaptive summation of the data can change the spatial curve of the array and making matching a certain direction.
Of course in late years we can find work on the subject utilizing Deep Learning approaches. Their main advantage is being able to incorporate additional information (Like using the properties of the signal, be it in a certain language, incorporating visual data on the scene like images and videos [Who is moving the lips when?]).
This is a vast subject and the main idea is to tailor the solution to the specific case of yours.
Modern Robust ICA and IVA (Independent Vector Analysis) can be very effective.
I'd try them first unless you have the case which matches the Adaptive Filter (Which can be proven to be matching the Beam Forming solution under some conditions).
Speech Source Separation (SSS) or Audio Source Separation (ASS) can be seen as a specialized version of source separation. I mention these expressions under which one can find additional works. One acceptation of the "Cocktail Party Problem" is the task of hearing/recovering one specific sound of interest in a complex environment (one-source versus all), while your objective seems more ambitious: unmixing all sources (another common term). The 2015 paper The cocktail-party problem revisited: early processing and selection of multi-talker speech reviews speech-related and perception issues.
The possibility of their identification depends a lot on the quality and quantity of observations, and the formation model of the observed signals. If the model is linear and instantaneous, this is already complicated. Single channel source separation is a specific topic of interest. In non-linear environments, when the number of sources is greater than the observations, when convolutive effects or echoes happen, when noise is difficult to tackle, blind source separation techniques are unlikely to succeed without additional priors and ancillary information/models.
Humans use their binaural features a lot in this context. Therefore, using spatial information from similar sensors is useful, but this can be insufficient. Indeed, there is a whole domain of Audio-Visual Speech Source Separation because:
The separation of speech signals measured at multiple microphones in noisy and reverberant environments using only the audio modality has limitations because there is generally insufficient information to discriminate fully the different sound sources. Humans mitigate this problem by exploiting the visual modality which is insensitive to background noise and can provide contextual information about the audio scene.
Combining audio and visual sensors is an instance of increasing the diversity in the sources: the more the sources can avoid to overlap in the recorded domain, the highest the chance of separation. PCA is very limited to that respect because it is too-strongly-tied to correlation and orthogonality. It is linear, nonparametric and cannot (easily) incorporate prior knowledge. It can estimate decorrelated components, up to a rotation. In other words, suppose that we two are talking. A PCA could detect the following two sources: 1) your voice minus mine 2) your voice added to mine, not what you'd expect. However, as a whitening or compression method, PCA can be used as a preprocessing to other methods like Independent Component Analysis (ICA), see for instance A. Hyvarinen, J. arhunen, E. Oja, Independent Component Analysis, 2001.
Additional speech features, like sparsity in a time-frequency domain (see Sparse Representations for the Cocktail Party Problem, 2006), more informed (less-blind) source separation (see From blind to guided audio source separation: How models and side information can improve the separation of sound) can help.
Finally, the link Deep Learning Machine Solves the Cocktail Party Problem may provide some pointers to the use of machine learning and artificial intelligence.