Smoothing the Geometry of Probabilistic Box Embeddings
There is growing interest in geometrically-inspired embeddings for learning hierarchies, partial orders, and lattice structures, with natural applications to transitive relational data such as entailment graphs. Recent work has extended these ideas beyond deterministic hierarchies to probabilistically calibrated models, which enable learning from uncertain supervision and inferring soft-inclusions among concepts, while maintaining the geometric inductive bias of hierarchical embedding models. We build on the Box Lattice model of Vilnis et al. (2018), which showed promising results in modeling soft-inclusions through an overlapping hierarchy of sets, parameterized as high-dimensional hyperrectangles (boxes). However, the hard edges of the boxes present difficulties for standard gradient based optimization; that work employed a special surrogate function for the disjoint case, but we find this method to be fragile. In this work, we present a novel hierarchical embedding model, inspired by a relaxation of box embeddings into parameterized density functions using Gaussian convolutions over the boxes. Our approach provides an alternative surrogate to the original lattice measure that improves the robustness of optimization in the disjoint case, while also preserving the desirable properties with respect to the original lattice. We demonstrate increased or matching performance on WordNet hypernymy prediction, Flickr caption entailment, and a MovieLens-based market basket dataset. We show especially marked improvements in the case of sparse data, where many conditional probabilities should be low, and thus boxes should be nearly disjoint.
 Post-rebuttal revision: All my concerns were adressed by the authors. This is a great paper and should be accepted. ------ The paper presents smoothing probabilistic box embeddings with softplus functions, which make the optimization landscape continuous, while also presenting the theoretical background of the proposed method well. The paper presents the overall idea beautifully and is very easy to follow. The overall idea of smoothed sotfplus boxes is well-founded, elegant and practical. The results on standard WordNet do not improve upon state-of-the-art, however imbalanced WordNet with abundance of negative examples gain remarkable improvements. Similarly in Flickr and MovieLens the method performs well. This paper presents a novel, theoretically well-justified idea with excellent results, and is likely going to be a high-impact paper. An illustrating figure would still be nice to include, also for the convolutions of eq 2. The paper does not comment on running times, some kind of scalability comparison should be included since the paper claims that the model is easier to train. The paper should clarify that the .prod in 3.3. meet and join definitions seems to refer to a set product, while the p(a) equation has a standard product (or does it?). What is the “a” in the p(a), should it be "p(x)” ? I have trouble understanding eq 1: the difference inside the function is always negative, while the hinge function seems to clip negative values away. The definition of the m(x) is too clever, please clarify the function in more conventional notation.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Smoothing the Geometry of Probabilistic Box Embeddings
There is growing interest in geometrically-inspired embeddings for learning hierarchies, partial orders, and lattice structures, with natural applications to transitive relational data such as entailment graphs. Recent work has extended these ideas beyond deterministic hierarchies to probabilistically calibrated models, which enable learning from uncertain supervision and inferring soft-inclusions among concepts, while maintaining the geometric inductive bias of hierarchical embedding models. We build on the Box Lattice model of Vilnis et al. (2018), which showed promising results in modeling soft-inclusions through an overlapping hierarchy of sets, parameterized as high-dimensional hyperrectangles (boxes). However, the hard edges of the boxes present difficulties for standard gradient based optimization; that work employed a special surrogate function for the disjoint case, but we find this method to be fragile. In this work, we present a novel hierarchical embedding model, inspired by a relaxation of box embeddings into parameterized density functions using Gaussian convolutions over the boxes. Our approach provides an alternative surrogate to the original lattice measure that improves the robustness of optimization in the disjoint case, while also preserving the desirable properties with respect to the original lattice. We demonstrate increased or matching performance on WordNet hypernymy prediction, Flickr caption entailment, and a MovieLens-based market basket dataset. We show especially marked improvements in the case of sparse data, where many conditional probabilities should be low, and thus boxes should be nearly disjoint.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



 Post-rebuttal revision: All my concerns were adressed by the authors. This is a great paper and should be accepted. ------ The paper presents smoothing probabilistic box embeddings with softplus functions, which make the optimization landscape continuous, while also presenting the theoretical background of the proposed method well. The paper presents the overall idea beautifully and is very easy to follow. The overall idea of smoothed sotfplus boxes is well-founded, elegant and practical. The results on standard WordNet do not improve upon state-of-the-art, however imbalanced WordNet with abundance of negative examples gain remarkable improvements. Similarly in Flickr and MovieLens the method performs well. This paper presents a novel, theoretically well-justified idea with excellent results, and is likely going to be a high-impact paper. An illustrating figure would still be nice to include, also for the convolutions of eq 2. The paper does not comment on running times, some kind of scalability comparison should be included since the paper claims that the model is easier to train. The paper should clarify that the .prod in 3.3. meet and join definitions seems to refer to a set product, while the p(a) equation has a standard product (or does it?). What is the “a” in the p(a), should it be "p(x)” ? I have trouble understanding eq 1: the difference inside the function is always negative, while the hinge function seems to clip negative values away. The definition of the m(x) is too clever, please clarify the function in more conventional notation.
assistant
 Post-rebuttal revision: All my concerns were adressed by the authors. This is a great paper and should be accepted. ------ The paper presents smoothing probabilistic box embeddings with softplus functions, which make the optimization landscape continuous, while also presenting the theoretical background of the proposed method well. The paper presents the overall idea beautifully and is very easy to follow. The overall idea of smoothed sotfplus boxes is well-founded, elegant and practical. The results on standard WordNet do not improve upon state-of-the-art, however imbalanced WordNet with abundance of negative examples gain remarkable improvements. Similarly in Flickr and MovieLens the method performs well. This paper presents a novel, theoretically well-justified idea with excellent results, and is likely going to be a high-impact paper. An illustrating figure would still be nice to include, also for the convolutions of eq 2. The paper does not comment on running times, some kind of scalability comparison should be included since the paper claims that the model is easier to train. The paper should clarify that the .prod in 3.3. meet and join definitions seems to refer to a set product, while the p(a) equation has a standard product (or does it?). What is the “a” in the p(a), should it be "p(x)” ? I have trouble understanding eq 1: the difference inside the function is always negative, while the hinge function seems to clip negative values away. The definition of the m(x) is too clever, please clarify the function in more conventional notation.