The canonical theory of stochastic calculus under ambiguity, founded on sub-additivity, is insensitive to non-convex uncertainty structures, leading to an identifiability impasse. This paper develops a mathematical framework for an identifiable calculus sensitive to non-convex geometry. We introduce the $θ$-BSDE, a class of backward stochastic differential equations where the driver is determined by a pointwise maximization over a primitive, possibly non-convex, uncertainty set. The system's tractability is predicated not on convexity, but on a global analytic hypothesis: the existence of a unique and globally Lipschitz maximizer map for the driver function. Under this hypothesis, which carves out a tractable class of models, we establish well-posedness via a fixed-point argument. For a distinct, geometrically regular class of models, we prove a result of independent interest: under non-degeneracy conditions from Malliavin calculus, the maximizer is unique along any solution path, ensuring the model's internal consistency. We clarify the fundamental logical gap between this pathwise property and the global regularity required by our existence proof. The resulting valuation operator defines a dynamically consistent expectation, and we establish its connection to fully nonlinear PDEs via a Feynman-Kac formula.

This paper focuses on improving the mathematical interpretability of convolutional neural networks (CNNs) in the context of image classification. Specifically, we tackle the instability issue arising in their first layer, which tends to learn parameters that closely resemble oriented band-pass filters when trained on datasets like ImageNet. Subsampled convolutions with such Gabor-like filters are prone to aliasing, causing sensitivity to small input shifts. In this context, we establish conditions under which the max pooling operator approximates a complex modulus, which is nearly shift invariant. We then derive a measure of shift invariance for subsampled convolutions followed by max pooling. In particular, we highlight the crucial role played by the filter's frequency and orientation in achieving stability. We experimentally validate our theory by considering a deterministic feature extractor based on the dual-tree complex wavelet packet transform, a particular case of discrete Gabor-like decomposition.

The widely-used Extended Kalman Filter (EKF) provides a straightforward recipe to estimate the mean and covariance of the state given all past measurements in a causal and recursive fashion. For a wide variety of applications, the EKF is known to produce accurate estimates of the mean and typically inaccurate estimates of the covariance. For applications in visual inertial localization, we show that inaccuracies in the covariance estimates are \emph{systematic}, i.e. it is possible to learn a nonlinear map from the empirical ground truth to the estimated one. This is demonstrated on both a standard EKF in simulation and a Visual Inertial Odometry system on real-world data.