Keskeiset käsitteet
Despite the complex and varied nature of Gaussian mixture distributions, neural networks exhibit asymptotic behaviors in line with predictions under simple Gaussian frameworks, particularly when inputs are standardized.
Tiivistelmä
This study investigates the dynamics of neural networks when the input data follows a Gaussian mixture distribution, rather than a simple Gaussian distribution. The key findings are:
Applying standardization to Gaussian mixture inputs reveals convergence of the neural network dynamics towards the predictions of conventional theories based on simple Gaussian structures.
This observed non-divergence is attributed to the distinctive characteristics of the nonlinear functions utilized in deep learning, which make the network dynamics predominantly influenced by the distribution's lower-order cumulants.
The authors first analyze how the dynamics of neural networks under Gaussian mixture-structured inputs diverge from the predictions of conventional theories based on simple Gaussian structures. However, they then demonstrate that despite the complex and varied nature of Gaussian mixture distributions, neural networks exhibit asymptotic behaviors in line with predictions under simple Gaussian frameworks, particularly when the inputs are standardized.
The mathematical analysis shows that for specific nonlinear functions like ReLU, the expectation values of function correlations involving the preactivations are predominantly determined by the first and second moments of the input distribution. This allows the dynamics under Gaussian mixture inputs to align with the predictions of conventional theories developed for simple Gaussian inputs.
The findings suggest a newfound universality, where the insights drawn from function correlation and the Gaussian Equivalence Property retain their validity even when the input distribution deviates from a simple Gaussian form, as long as the distribution's lower-order cumulants are preserved through standardization.
Tilastot
The study does not provide any specific numerical data or statistics to support the key findings. The analysis is primarily based on theoretical derivations and mathematical proofs.
Lainaukset
"Applying 'standardization' to input datasets with inherent Gaussian mixture properties reveals convergence to the predicted dynamics outcomes of existing theories."
"This observed non-divergence is attributed to the distinctive characteristics of nonlinear functions utilized in deep learning network and dataset modeling process, makes deep learning dynamics are predominantly influenced by the distribution's lower-order cumulants."