Learning Mixed-Curvature Representations in Product Spaces
The quality of the representations achieved by embeddings is determined by how well the geometry of the embedding space matches the structure of the data. Euclidean space has been the workhorse for embeddings; recently hyperbolic and spherical spaces have gained popularity due to their ability to better embed new types of structured data---such as hierarchical data---but most data is not structured so uniformly. We address this problem by proposing learning embeddings in a product manifold combining multiple copies of these model spaces (spherical, hyperbolic, Euclidean), providing a space of heterogeneous curvature suitable for a wide variety of structures. We introduce a heuristic to estimate the sectional curvature of graph data and directly determine an appropriate signature---the number of component spaces and their dimensions---of the product manifold. Empirically, we jointly learn the curvature and the embedding in the product space via Riemannian optimization. We discuss how to define and compute intrinsic quantities such as means---a challenging notion for product manifolds---and provably learnable optimization functions. On a range of datasets and reconstruction tasks, our product space embeddings outperform single Euclidean or hyperbolic spaces used in previous works, reducing distortion by 32.55% on a Facebook social network dataset. We learn word embeddings and find that a product of hyperbolic spaces in 50 dimensions consistently improves on baseline Euclidean and hyperbolic embeddings, by 2.6 points in Spearman rank correlation on similarity tasks and 3.4 points on analogy accuracy.
 This paper proposes a new method to embed a graph onto a product of spherical/Euclidean/hyperbolic manifolds. The key is to use sectional curvature estimations to determine proper signature, i.e., all component manifolds, and then optimize over these manifolds. The results are validated on various synthetic and real graphs. The proposed idea is new, nontrivial, and is well supported by experimental evidence.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Learning Mixed-Curvature Representations in Product Spaces
The quality of the representations achieved by embeddings is determined by how well the geometry of the embedding space matches the structure of the data. Euclidean space has been the workhorse for embeddings; recently hyperbolic and spherical spaces have gained popularity due to their ability to better embed new types of structured data---such as hierarchical data---but most data is not structured so uniformly. We address this problem by proposing learning embeddings in a product manifold combining multiple copies of these model spaces (spherical, hyperbolic, Euclidean), providing a space of heterogeneous curvature suitable for a wide variety of structures. We introduce a heuristic to estimate the sectional curvature of graph data and directly determine an appropriate signature---the number of component spaces and their dimensions---of the product manifold. Empirically, we jointly learn the curvature and the embedding in the product space via Riemannian optimization. We discuss how to define and compute intrinsic quantities such as means---a challenging notion for product manifolds---and provably learnable optimization functions. On a range of datasets and reconstruction tasks, our product space embeddings outperform single Euclidean or hyperbolic spaces used in previous works, reducing distortion by 32.55% on a Facebook social network dataset. We learn word embeddings and find that a product of hyperbolic spaces in 50 dimensions consistently improves on baseline Euclidean and hyperbolic embeddings, by 2.6 points in Spearman rank correlation on similarity tasks and 3.4 points on analogy accuracy.
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>



 This paper proposes a new method to embed a graph onto a product of spherical/Euclidean/hyperbolic manifolds. The key is to use sectional curvature estimations to determine proper signature, i.e., all component manifolds, and then optimize over these manifolds. The results are validated on various synthetic and real graphs. The proposed idea is new, nontrivial, and is well supported by experimental evidence.
assistant
 This paper proposes a new method to embed a graph onto a product of spherical/Euclidean/hyperbolic manifolds. The key is to use sectional curvature estimations to determine proper signature, i.e., all component manifolds, and then optimize over these manifolds. The results are validated on various synthetic and real graphs. The proposed idea is new, nontrivial, and is well supported by experimental evidence.