An infinite structure has the finite length property (over a given field) if, for each of its finite powers, chains of equivariant subspaces in the corresponding free vector space are bounded in length. Prior work showed that the countable pure set and the countable dense linear order without endpoints have this property. We generalise these results to (a) any structure approximated by finite substructures with few orbits, provided the field is of characteristic zero, and (b) any Fraïssé limit with free amalgamation in a finite vocabulary consisting of unary and binary relations, possibly expanded with a generic total order. As a special case, we deduce the finite length property of the Rado graph using both methods. We also describe some connections with function spaces, weighted register automata, and orbit-finite systems of linear equations.

This paper presents a weakly supervised regression model for the visual house appraisal problem, which aims to predict the value of a house from its photos and textual descriptions (e.g., number of bedrooms). The key idea of our approach is a multi-layer neural network, called multi-instance Deep Ranking and Regression (MiDRR) net, which jointly solves two coupled tasks: ranking and regression, in the multiple instance setting. The network is trained using weakly supervised data, which do not require intensive human annotations. We also design a set of human heuristics to promote deep features through imposing constraints over the solution space, e.g., a house with three bedrooms often has a higher value than that with only two bedrooms. While these constraints are specific to the studied problem, the developed formula can be easily generalized to the other regression applications. For test and evaluation purposes, we collect a comprehensive house image benchmark that includes 900,000 photos from 30,000 houses recently traded in the USA, and apply the proposed MiDRR net to predict house values. Extensive evaluations with comparisons demonstrate that additional usage of imagery data as well as human heuristics can significantly boost system performance and that the proposed MiDRR net clearly outperforms the alternative methods.