  Some extremely low-dimensional yet crucial geometric eigen-lengths often
determine the success of some geometric tasks. For example, the height of an
object is important to measure to check if it can fit between the shelves of a
cabinet, while the width of a couch is crucial when trying to move it through a
doorway. Humans have materialized such crucial geometric eigen-lengths in
common sense since they are very useful in serving as succinct yet effective,
highly interpretable, and universal object representations. However, it remains
obscure and underexplored if learning systems can be equipped with similar
capabilities of automatically discovering such key geometric quantities from
doing tasks. In this work, we therefore for the first time formulate and
propose a novel learning problem on this question and set up a benchmark suite
including tasks, data, and evaluation metrics for studying the problem. We
focus on a family of common fitting tasks as the testbed for the proposed
learning problem. We explore potential solutions and demonstrate the
feasibility of learning eigen-lengths from simply observing successful and
failed fitting trials. We also attempt geometric grounding for more accurate
eigen-length measurement and study the reusability of the learned eigen-lengths
across multiple tasks. Our work marks the first exploratory step toward
learning crucial geometric eigen-lengths and we hope it can inspire future
research in tackling this important yet underexplored problem.
