Variable binding is not a property of representations

Variable binding is the capacity of a network to hold each attribute of an object as a separable part of the object, so that attributes and objects can be recombined into novel arrangements. The field has long believed that binding is an intrinsic property of a representation and thus, sought to measure and certify it. However, we demonstrate it is not such a property. We first prove that how cleanly the attributes of an object can be separated is fixed by the observation itself. This is done before any encoder is trained, so no optimizer or architecture can improve the separability. If the objects in an observation are separable, a representation can bind them and where they are not, it can’t. We then show that the leakage metric the field currently uses to certify that a network binds changes from zero to positive under a lossless relabeling, and thus, no architecture-agnostic certificate can exist. Across seven architectures, including the tensor-product representation, an information-preserving relabel is able to change each model's binding verdict showing that variable binding is truly not an intrinsic property and the object entanglement predicts the out-of-distribution gap on real photographs, and reducing this entanglement closes the binding gap.