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Convex sets arising in a variety of applications are well-defined for everyrelevant dimension. Examples include the simplex and the spectraplex thatcorrespond, respectively, to probability distributions and to quantum states;combinatorial polytopes and their associated relaxations such as the cutpolytope and the elliptope in integer programming; and unit balls ofcommonly-employed regularizers such as the $\ell_p$ and Schatten norms ininverse problems. Moreover, these sets are often specified using conicdescriptions that can be obviously instantiated in any dimension. We develop asystematic framework to study such free descriptions of convex sets. We showthat free descriptions arise from a recently-identified phenomenon in algebraictopology called representation stability, which relates invariants acrossdimensions in a sequence of group representations. Our framework yields aprocedure to obtain parametric families of freely-described convex sets whosestructure is adapted to a given application; illustrations are provided viaexamples that arise in the literature as well as new families that are derivedusing our procedure. We demonstrate the utility of our framework in twocontexts. First, we develop an algorithm for a free analog of the convexregression problem, where a convex function is fit to input-output trainingdata; in our setting, the inputs may be of different dimensions and we seek aconvex function that is well-defined for inputs of any dimension (includingthose that are not in the training set). Second, we prove that many sequencesof symmetric conic programs can be solved in constant time, which unifies andstrengthens several disparate results in the literature. Our work extensivelyuses ideas and results from representation stability, and it can be seen as anew point of contact between representation stability and convex geometry viaconic descriptions.

Free Descriptions of Convex Sets