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A range of data insight analytical tasks involves analyzing a large set oftables of different schemas, possibly induced by various groupings, to findsalient patterns. This paper presents Multi-Relational Algebra, an extension ofthe classic Relational Algebra, to facilitate such transformations and theircompositions. Multi-Relational Algebra has two main characteristics: (1)Information Unit. The information unit is a slice $(r, X)$, where $r$ is a(region) tuple, and $X$ is a (feature) table. Specifically, a slice canencompass multiple columns, which surpasses the information unit of "a singletuple" or "a group of tuples of one column" in the classic relational algebra,(2) Schema Flexibility. Slices can have varying schemas, not constrained to asingle schema. This flexibility further expands the expressive power of thealgebra. Through various examples, we show that multi-relational algebra caneffortlessly express many complex analytic problems, some of which are beyondthe scope of traditional relational analytics. We have implemented and deployeda service for multi-relational analytics. Due to a unified logical design, weare able to conduct systematic optimization for a variety of seeminglydifferent tasks. Our service has garnered interest from numerous internal teamswho have developed data-insight applications using it, and serves millions ofoperators daily.

Multi-Relational Algebra and Its Applications to Data Insights