The VC-Dimension of Axis-Parallel Boxes on the Torus

We show in this paper that the VC-dimension of the family of $d$-dimensional axis-parallel boxes and cubes on the $d$-dimensional torus are both asymptotically $d \log_2(d)$. This is especially surprising as the VC-dimension usually grows linearly with $d$ in similar settings.

Geometric Generalization Based Zero-Shot Learning Dataset Infinite World: Simple Yet Powerful

Raven's Progressive Matrices are one of the widely used tests in evaluating the human test taker's fluid intelligence. Analogously, this paper introduces geometric generalization based zero-shot learning tests to measure the rapid learning ability and the internal consistency of deep generative models. Our empirical research analysis on state-of-the-art generative models discern their ability to generalize concepts across classes. In the process, we introduce Infinite World, an evaluable, scalable, multi-modal, light-weight dataset and Zero-Shot Intelligence Metric ZSI. The proposed tests condenses human-level spatial and numerical reasoning tasks to its simplistic geometric forms. The dataset is scalable to a theoretical limit of infinity, in numerical features of the generated geometric figures, image size and in quantity. We systematically analyze state-of-the-art model's internal consistency, identify their bottlenecks and propose a pro-active optimization method for few-shot and zero-shot learning.