Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dots
Théo Bourdais, Pau Batlle, Xianjin Yang, Ricardo Baptista, Nicolas Rouquette, Houman Owhadi
TL;DR
The paper introduces a Gaussian Process framework for Type 3 problems that discovers unknown hypergraph connectivity and nonlinear functional relationships among $d$ variables from data without requiring randomized sampling. It leverages additive kernels, kernel mode decomposition, and uncertainty quantification to identify ancestor sets and recover interpretable, sparse dependency graphs, with polynomial complexity in $d$ under practical assumptions. The method uses signal-to-noise ratio and $Z$-test bounds to decide whether a node has ancestors and to prune the graph, while kernel-PCA and KMD generalizations enable handling of nonlinear and implicit dependencies. Demonstrations across dynamical systems, chemical reaction networks, and real-world data (e.g., Google Covid-19, Sachs cell signaling, BCR networks) illustrate accurate structure recovery and scalable computation, though limitations include nonidentifiability and the absence of causal guarantees. The approach complements causal inference by focusing on functional rather than causal structure and offers a scalable pathway to discovering complex networks in genomics and systems biology.
Abstract
Most problems within and beyond the scientific domain can be framed into one of the following three levels of complexity of function approximation. Type 1: Approximate an unknown function given input/output data. Type 2: Consider a collection of variables and functions, some of which are unknown, indexed by the nodes and hyperedges of a hypergraph (a generalized graph where edges can connect more than two vertices). Given partial observations of the variables of the hypergraph (satisfying the functional dependencies imposed by its structure), approximate all the unobserved variables and unknown functions. Type 3: Expanding on Type 2, if the hypergraph structure itself is unknown, use partial observations of the variables of the hypergraph to discover its structure and approximate its unknown functions. These hypergraphs offer a natural platform for organizing, communicating, and processing computational knowledge. While most scientific problems can be framed as the data-driven discovery of unknown functions in a computational hypergraph whose structure is known (Type 2), many require the data-driven discovery of the structure (connectivity) of the hypergraph itself (Type 3). We introduce an interpretable Gaussian Process (GP) framework for such (Type 3) problems that does not require randomization of the data, access to or control over its sampling, or sparsity of the unknown functions in a known or learned basis. Its polynomial complexity, which contrasts sharply with the super-exponential complexity of causal inference methods, is enabled by the nonlinear ANOVA capabilities of GPs used as a sensing mechanism.