Most mathematical distortions used in ML are fundamentally integral in
nature: $f$-divergences, Bregman divergences, (regularized) optimal transport
distances, integral probability metrics, geodesic distances, etc. In this
paper, we unveil a grounded theory and tools which can help improve these
distortions to better cope with ML requirements. We start with a generalization
of Riemann integration that also encapsulates functions that are not strictly
additive but are, more generally, $t$-additive, as in nonextensive statistical
mechanics. Notably, this recovers Volterra's product integral as a special
case. We then generalize the Fundamental Theorem of calculus using an extension
of the (Euclidean) derivative. This, along with a series of more specific
Theorems, serves as a basis for results showing how one can specifically
design, alter, or change fundamental properties of distortion measures in a
simple way, with a special emphasis on geometric- and ML-related properties
that are the metricity, hyperbolicity, and encoding. We show how to apply it to
a problem that has recently gained traction in ML: hyperbolic embeddings with a
"cheap" and accurate encoding along the hyperbolic vs Euclidean scale. We
unveil a new application for which the Poincar\'e disk model has very appealing
features, and our theory comes in handy: \textit{model} embeddings for boosted
combinations of decision trees, trained using the log-loss (trees) and logistic
loss (combinations).

改进数学扭曲来适应机器学习需求，包括推广黎曼积分、使用欧几里得导数扩展基本微积分定理、设计、改变扭曲度量的基本性质以及在超几何和机器学习中应用超几何嵌入。