Any-Dimensional Learning by Sampling

Imagine you're building a machine learning model that needs to handle inputs of varying sizes, like point clouds with different numbers of points or sequences of text of different lengths. The challenge is that these models are often trained on a limited number of examples, which makes it hard to know how well they will perform on larger inputs that they haven't seen before. Evaluating these models on larger inputs can also be expensive, which adds another layer of difficulty. This situation leads to a problem known as generalization failure, where the model struggles to adapt to new input sizes effectively.

Currently, when faced with this issue, people might try to evaluate their models on larger inputs directly, but this can be costly and inefficient. Alternatively, they might use techniques to downsample or simplify the inputs, but these methods often fail to capture the essential characteristics of the larger inputs. This is what's called input approximation failure, where the model's performance on smaller inputs doesn't reflect its performance on larger, more complex inputs.

The approach presented in this paper offers a fresh perspective on these challenges. It introduces random sampling maps as a way to compare inputs of different sizes and to create smaller, representative sketches of larger inputs. This method allows for a more nuanced understanding of how models can generalize from small to large inputs, and it provides explicit rates for generalization and sketching that can be applied across various domains, including sequences, graphs, and tensors.

What sets this work apart from previous methods is its unified framework that characterizes different types of sampling based on the relationships between problem instances of varying sizes. This means that builders can now have a clearer guideline on how to approach the evaluation of their models, potentially leading to more efficient and effective use of resources when dealing with large datasets.

The paper introduces a unified approach to generalization and sketching for models with variable input sizes, extending existing methods.

The claims are supported by a solid framework and examples, though specific experimental validation is less detailed.