In this work we produce a framework for constructing universal function approximators on graph isomorphism classes. We prove how this framework comes with a collection of theoretically desirable properties and enables novel analysis. We show how this allows us to achieve state-of-the-art performance on four different well-known datasets in graph classification and separate classes of graphs that other graph-learning methods cannot. Our approach is inspired by persistent homology, dependency parsing for NLP, and multivalued functions.

In this work we produce a framework for constructing universal function approxi-mators on graph isomorphism classes.

Additional Feedback: page 1, last para: this is confusing to read. The reference cited here is Babai's paper. Why is the slowness of current graph isomorphism algorithms relevant to the problem of producing isomorphism-injective graph representations? Definition 3: a minor point, but it is useful to say what is meant by size (e.g., #edges, or size of description of the graph) After definition 7, it is useful to formally define the notion of "universal function approximator", as this could be interpreted in different ways The notation of multi-function in Definition 6 uses a double arrow, but it doesn't seem to get used consistently like that. It is used with a single arrow in Definition 7. Also the notion is confusing, since it is not a function into the range but into its power set.

Each reviewer believes that the paper is poorly written. The reviewers, though, have agreed that (i) the problem is interesting, that (ii) the theoretical results seem to hold, and that (iii) they are interesting. On the other hand, it is not clear whether a quick revision would solve all or many of the readability issues.

In this work we produce a framework for constructing universal function approximators on graph isomorphism classes. We prove how this framework comes with a collection of theoretically desirable properties and enables novel analysis. We show how this allows us to achieve state-of-the-art performance on four different well-known datasets in graph classification and separate classes of graphs that other graph-learning methods cannot. Our approach is inspired by persistent homology, dependency parsing for NLP, and multivalued functions.

In this work we produce a framework for constructing universal function approximators on graph isomorphism classes. Additionally, we prove how this framework comes with a collection of theoretically desirable properties and enables novel analysis. We show how this allows us to outperform state of the art on four different well known datasets in graph classification and how our method can separate classes of graphs that other graph-learning methods cannot. Our approach is inspired by persistence homology, dependency parsing for Natural Language Processing, and multivalued functions. The complexity of the underlying algorithm is O(mn) and code is publicly available.

This article concerns the expressive power of depth in neural nets with ReLU activations and bounded width. We are particularly interested in the following questions: what is the minimal width $w_{\text{min}}(d)$ so that ReLU nets of width $w_{\text{min}}(d)$ (and arbitrary depth) can approximate any continuous function on the unit cube $[0,1]^d$ aribitrarily well? For ReLU nets near this minimal width, what can one say about the depth necessary to approximate a given function? Our approach to this paper is based on the observation that, due to the convexity of the ReLU activation, ReLU nets are particularly well-suited for representing convex functions. In particular, we prove that ReLU nets with width $d+1$ can approximate any continuous convex function of $d$ variables arbitrarily well. These results then give quantitative depth estimates for the rate of approximation of any continuous scalar function on the $d$-dimensional cube $[0,1]^d$ by ReLU nets with width $d+3.$

Arthur C. Clarke famously stated that "any sufficiently advanced technology is indistinguishable from magic." No current technology embodies this statement more than neural networks and deep learning. And like any good magic it not only dazzles and inspires but also puts fear into people's hearts. One known property of artificial neural networks (ANNs) is that they are universal function approximators. This means that any mathematical function can be represented by a neural network.

Arthur C. Clarke famously stated that "any sufficiently advanced technology is indistinguishable from magic." No current technology embodies this statement more than neural networks and deep learning. And like any good magic it not only dazzles and inspires but also puts fear into people's hearts. One known property of artificial neural networks (ANNs) is that they are universal function approximators. This means that any mathematical function can be represented by a neural network.

Arthur C. Clarke famously stated that "any sufficiently advanced technology is indistinguishable from magic." No current technology embodies this statement more than neural networks and deep learning. And like any good magic it not only dazzles and inspires but also puts fear into people's hearts. One known property of artificial neural networks (ANNs) is that they are universal function approximators. This means that any mathematical function can be represented by a neural network.