In this section, we provide proofs for all of the theorems introduced in the main text. We begin with a simple extension of the results of [3] for the Bloom filter false positive and negative rates. Then, we prove our main claim, which is that the query time of our data structure is sublinear, given some relatively weak assumptions on the stability of the query. Theorem 1. Assuming the existence of an LSH family with collision probability s(x,y) = sim(x,y), the distance-sensitive Bloom filter solves the approximate membership query problem with p 1 exp 2m t/m+ SLH We begin with a brief explanation of the results from [3]. Recall that a distance-sensitive Bloom filter is a collection of mbit arrays. Array iis indexed using an independent LSH function li(x). To insert a point xinto the ith array, we set the bit at location li(x) to '1.' To query the filter, we calculate the mhash values of the query and return "true" when at least tof the corresponding bits are '1.' To bound p (the true positive rate) and q (the false positive rate), we bound the probability that a single array returns "true."

Let A be the map from RX to RX Y that sends a function f on X to the function f(x) p P(x,y)onX Y. Similarly,let B bethemapfrom RY toRX Y thatsendsafunction gto g(y) p P(x,y). Combining these, we get exactly the stated formula. Here we elaborate further on COPT's optimization routine. As the objective Equation 3.1 is not globally convex, gradient descent can fall into local minima. But this requires anontrivialnumber (e.g.

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We would appreciate any pointers to additional related work in the updated review.

We thank the reviewers for their detailed and thoughtful comments. All minor comments and corrections will be addressed in the final version. In the following, we address each reviewer's comments in detail one by one. Q1: Natural images may not have low-rank structures. A1: We did not model natural images by low-rank structures.

Dual discriminator generative adversarial networks (D2 GANs) were introduced to mitigate the problem of mode collapse in generative adversarial networks. In D2 GANs, two discriminators are employed alongside a generator: one discriminator rewards high scores for samples from the true data distribution, while the other favors samples from the generator. In this work, we first introduce dual discriminator $α$-GANs (D2 $α$-GANs), which combines the strengths of dual discriminators with the flexibility of a tunable loss function, $α$-loss. We further generalize this approach to arbitrary functions defined on positive reals, leading to a broader class of models we refer to as generalized dual discriminator generative adversarial networks. For each of these proposed models, we provide theoretical analysis and show that the associated min-max optimization reduces to the minimization of a linear combination of an $f$-divergence and a reverse $f$-divergence. This generalizes the known simplification for D2-GANs, where the objective reduces to a linear combination of the KL-divergence and the reverse KL-divergence. Finally, we perform experiments on 2D synthetic data and use multiple performance metrics to capture various advantages of our GANs.

Scientific text is complex as it contains technical terms by definition. Simplifying such text for non-domain experts enhances accessibility of innovation and information. Politicians could be enabled to understand new findings on topics on which they intend to pass a law, or family members of seriously ill patients could read about clinical trials. The SimpleText CLEF Lab focuses on exactly this problem of simplification of scientific text. Task 1.1 of the 2025 edition specifically handles the simplification of complex sentences, so very short texts with little context. To tackle this task we investigate the identification of complex terms in sentences which are rephrased using small Gemini and OpenAI large language models for non-expert readers.

Large language models (LLMs) can prove mathematical theorems formally by generating proof steps (\textit{a.k.a.} tactics) within a proof system. However, the space of possible tactics is vast and complex, while the available training data for formal proofs is limited, posing a significant challenge to LLM-based tactic generation. To address this, we introduce a neuro-symbolic tactic generator that synergizes the mathematical intuition learned by LLMs with domain-specific insights encoded by symbolic methods. The key aspect of this integration is identifying which parts of mathematical reasoning are best suited to LLMs and which to symbolic methods. While the high-level idea of neuro-symbolic integration is broadly applicable to various mathematical problems, in this paper, we focus specifically on Olympiad inequalities (Figure~1). We analyze how humans solve these problems and distill the techniques into two types of tactics: (1) scaling, handled by symbolic methods, and (2) rewriting, handled by LLMs. In addition, we combine symbolic tools with LLMs to prune and rank the proof goals for efficient proof search. We evaluate our framework on 161 challenging inequalities from multiple mathematics competitions, achieving state-of-the-art performance and significantly outperforming existing LLM and symbolic approaches without requiring additional training data.