Self-Organization Towards $1/f$ Noise in Deep Neural Networks

Despite $1/f$ noise being ubiquitous in both natural and artificial systems, no general explanations for the phenomenon have received widespread acceptance. One well-known system where $1/f$ noise has been observed in is the human brain, with this 'noise' proposed by some to be important to the healthy function of the brain. As deep neural networks (DNNs) are loosely modelled after the human brain, and as they start to achieve human-level performance in specific tasks, it might be worth investigating if the same $1/f$ noise is present in these artificial networks as well. Indeed, we find the existence of $1/f$ noise in DNNs - specifically Long Short-Term Memory (LSTM) networks modelled on real world dataset - by measuring the Power Spectral Density (PSD) of different activations within the network in response to a sequential input of natural language. This was done in analogy to the measurement of $1/f$ noise in human brains with techniques such as electroencephalography (EEG) and functional Magnetic Resonance Imaging (fMRI). We further examine the exponent values in the $1/f$ noise in "inner" and "outer" activations in the LSTM cell, finding some resemblance in the variations of the exponents in the fMRI signal. In addition, comparing the values of the exponent at "rest" compared to when performing "tasks" of the LSTM network, we find a similar trend to that of the human brain where the exponent while performing tasks is less negative.