Interpretable Measurement of CNN Deep Feature Density using Copula and the Generalized Characteristic Function

We present a novel empirical approach toward measuring the Probability Density Function (PDF) of the deep features of Convolutional Neural Networks (CNNs). Measurement of the deep feature PDF is a valuable problem for several reasons. Notably, a. Understanding the deep feature PDF yields new insight into deep representations. b. Feature density methods are important for tasks such as anomaly detection which can improve the robustness of deep learning models in the wild. Interpretable measurement of the deep feature PDF is challenging due to the Curse of Dimensionality (CoD), and the Spatial intuition Limitation. Our novel measurement technique combines copula analysis with the Method of Orthogonal Moments (MOM), in order to directly measure the Generalized Characteristic Function (GCF) of the multivariate deep feature PDF. We find that, surprisingly, the one-dimensional marginals of non-negative deep CNN features after major blocks are not well approximated by a Gaussian distribution, and that these features increasingly approximate an exponential distribution with increasing network depth. Furthermore, we observe that deep features become increasingly independent with increasing network depth within their typical ranges. However, we surprisingly also 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 propose a new hypothesis that exponentially infrequent large valued features correspond to strong computer vision detections of semantic targets, which would imply that these large-valued features are not outliers but rather an important detection signal.

A Method of Moments Embedding Constraint and its Application to Semi-Supervised Learning

Discriminative deep learning models with a linear+softmax final layer have a problem: the latent space only predicts the conditional probabilities $p(Y|X)$ but not the full joint distribution $p(Y,X)$, which necessitates a generative approach. The conditional probability cannot detect outliers, causing outlier sensitivity in softmax networks. This exacerbates model over-confidence impacting many problems, such as hallucinations, confounding biases, and dependence on large datasets. To address this we introduce a novel embedding constraint based on the Method of Moments (MoM). We investigate the use of polynomial moments ranging from 1st through 4th order hyper-covariance matrices. Furthermore, we use this embedding constraint to train an Axis-Aligned Gaussian Mixture Model (AAGMM) final layer, which learns not only the conditional, but also the joint distribution of the latent space. We apply this method to the domain of semi-supervised image classification by extending FlexMatch with our technique. We find our MoM constraint with the AAGMM layer is able to match the reported FlexMatch accuracy, while also modeling the joint distribution, thereby reducing outlier sensitivity. We also present a preliminary outlier detection strategy based on Mahalanobis distance and discuss future improvements to this strategy. Code is available at: \url{https://github.com/mmajurski/ssl-gmm}