Abstract
We present a novel empirical approach toward estimating the Probability Density Function (PDF) of the deep features of Convolutional Neural Networks (CNNs). Estimating the PDF of deep CNN features is an important task, because it will yield new insight into deep representations. Moreover, characterizing the statistical behavior has implications for the feasibility of promising downstream tasks such as density based anomaly detection. Expressive, yet interpretable estimation of the deep feature PDF is challenging due to the Curse of Dimensionality (CoD) as well as our limited ability to comprehend high-dimensional inter-dependencies. Our novel estimation technique combines copula analysis with the Method of Orthogonal Moments (MOM), in order to directly estimate the Generalized Characteristic Function (GCF) of the multivariate deep feature PDF. We find that the one-dimensional marginals of non-negative deep CNN features after major blocks are not well approximated by a Gaussian distribution, and that the features of deep layers are much better approximated by the Exponential, Gamma, and/or Weibull distributions. Furthermore, we observe that deep features become increasingly long-tailed with network depth, although surprisingly the rate of this increase is much slower than theoretical estimates. Finally, we observe that many deep features exhibit strong dependence (either correlation or anti-correlation) with other extremely strong detections, even if these features are independent within typical ranges. We elaborate on these findings in our discussion, where we hypothesize that the long-tail of large valued features corresponds to the strongest computer vision detections of semantic targets, which would imply that these large-valued features are not outliers but rather an important detection signal.