Converting mel spectrogram to spectrogram I have a set of songs for which I extracted the STFT (Short-Time Fourier Transform) and used the magnitude spectrum $|S|$ to calculate the mel spectrogram by using a mel filterbank matrix $M$, so $X=\log(M\times |S|)$. I want to know is there any method to reverse this process, i.e. convert from the mel spectrogram back to the spectrogram. I performed some dimensionality reduction on the mel spectrogram, and reconstructed the mel spectrogram from lower dimensions. Now I want to regenerate the audio signal from the reconstructed mel spectrogram, so I guess first reconstruct the spectrogram and then the audio signal.

The problem is that the mel filter bank matrix is not a square matrix, since we the reduce the no of frequency bins, so inverse of $M$ cant be used like this : $ \hat{S}=M^{-1}\exp(X)$. So is there any way to generate the inverse mapping, like some inverse transfer function that can convert from $X$ to $S$?
Both taking a magnitude spectrogram and a Mel filter bank are lossy processes.  Important information needed to reconstruct the original will have been lost.  Thus you need to go back and use the original audio samples to do the reconstruction by determining a time or frequency domain filter equivalent to your dimensionality reduction.

You can make assumptions about the lost information, but those assumptions themselves usually sound inaccurate, artificial and/or robotic.  Or you can use only specially synthesized input, where the assumptions will be correct by design of that input.