By making the appropriate changes to the proofs of Lemma B.3 and Lemma B.4, we get These two lemmas immediately imply the following generalization of Lemma B.5. The upper bound on cost(σ) given in Lemma B.6 can be generalized by noticing that cost(σ, C The lower bound on opt(U) given in Lemma B.10 holds for ρ-metric spaces with no modifications. By making the appropriate modifications to the proof of Theorem C.1, we can extend this theorem to In particular, we can obtain a proof of Theorem A.5 by taking the proof of Theorem C.1 and adding extra ρ factors whenever the triangle inequality is applied. We first prove Lemma B.1, which shows that the sizes of the sets U By Lemma B.2, we get that Henceforth, we fix some positive ξ and sufficiently large α such that Lemma B.3 holds. By now applying Lemma B.4 it follows that µ Lower Bounding opt(U) (Lemma B.10): Let r denote log For the rest of this subsection we fix an arbitrary S U of size k.

One of the fundamental problems in Artificial Intelligence is to perform complex multi-hop logical reasoning over the facts captured by a knowledge graph (KG). This problem is challenging, because KGs can be massive and incomplete. Recent approaches embed KG entities in a low dimensional space and then use these embeddings to find the answer entities. However, it has been an outstanding challenge of how to handle arbitrary first-order logic (FOL) queries as present methods are limited to only a subset of FOL operators. In particular, the negation operator is not supported. An additional limitation of present methods is also that they cannot naturally model uncertainty.

We thank the reviewers for their time and valuable feedback. Below, we clarify a number of important points raised by the reviewers. Reviewers raise concern on multi-modal embeddings. We will highlight this limitation in Sec. R3 suggests that "the authors can adapt the FOL queries to other We argue the differences in tasks and setups below.

The Mixture of Experts (MoE) paradigm provides a powerful way to decompose dense layers into smaller, modular computations often more amenable to human interpretation, debugging, and editability. However, a major challenge lies in the computational cost of scaling the number of experts high enough to achieve finegrained specialization. In this paper, we propose the Multilinear Mixture of Experts (µMoE) layer to address this, focusing on vision models.