TENSKEL: A Combinatorial Observable Tensor for Structured Measurement and Reconstruction

Abstract

Many imaging problems seek to reconstruct underlying configurations from partial observable measurements. While reconstruction algorithms operate on these measurements, the observable organization induced by the measurement process is rarely represented explicitly. We introduce TENSKEL, a combinatorial observable framework for structured measurement and reconstruction based on tensor representations defined over discrete domains. Starting from a binary latent ensemble, the framework constructs a hierarchy of tensors coupling measurement contexts to a latent Pascal organization through successive aggregation and folding operations. Each measurement context induces an observable partition of the same latent ensemble, and the resulting tensor formulation makes explicit the associated combinatorial multiplicities, shell organization, degeneracies, and induced reconstruction geometry. Rather than introducing a new reconstruction algorithm, this framework provides a mathematical representation of how latent configurations become organized under observation. The induced tensor kernel characterizes similarities between latent coordinates through their measurement-context responses, while regularized inversion provides a structured reconstruction of the latent representation from observable measurements. The binary construction further admits a natural multinomial extension to discrete simplex-supported latent representations. Connections to Pascal cellular automata and structured discrete color mappings illustrate respectively compressed and multinomial realizations of the framework. More generally, TENSKEL provides a combinatorial basis for reasoning about the organization induced by observation, with potential relevance to computational imaging, inverse problems, structured sensing, and quantum-inspired measurement formulations.