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In this note we show examples of total Boolean functions that depend on $n$ variables and have spectral sensitivity $\Theta(\sqrt{\log n})$, which is asymptotically minimal.

In the field of autonomous driving, a variety of sensor data types exist, each representing different modalities of the same scene. Therefore, it is feasible to utilize data from other sensors to facilitate image compression. However, few techniques have explored the potential benefits of utilizing inter-modality correlations to enhance the image compression performance. In this paper, motivated by the recent success of learned image compression, we propose a new framework that uses sparse point clouds to assist in learned image compression in the autonomous driving scenario. We first project the 3D sparse point cloud onto a 2D plane, resulting in a sparse depth map. Utilizing this depth map, we proceed to predict camera images. Subsequently, we use these predicted images to extract multi-scale structural features. These features are then incorporated into learned image compression pipeline as additional information to improve the compression performance. Our proposed framework is compatible with various mainstream learned image compression models, and we validate our approach using different existing image compression methods. The experimental results show that incorporating point cloud assistance into the compression pipeline consistently enhances the performance.

The famous Ryser--Brualdi--Stein conjecture asserts that every $n \times n$ Latin square contains a partial transversal of size $n-1$. Since its appearance, the conjecture has attracted significant interest, leading to several generalizations. One of the most notable extensions is to matroid intersection given by Aharoni, Kotlar, and Ziv, focusing on the existence of a common independent transversal of the common independent sets of two matroids. In this paper, we study a special case of this setting, the Rainbow Arborescence Conjecture, which states that any graph on $n$ vertices formed by the union of $n-1$ spanning arborescences contains an arborescence using exactly one arc from each. We prove that the computational problem of testing the existence of such an arborescence with a fixed root is NP-complete, verify the conjecture in several cases, and explore relaxed versions of the problem.

We consider the conditional generation of 3D drug-like molecules with \textit{explicit control} over molecular properties such as drug-like properties (e.g., Quantitative Estimate of Druglikeness or Synthetic Accessibility score) and effectively binding to specific protein sites. To tackle this problem, we propose an E(3)-equivariant Wasserstein autoencoder and factorize the latent space of our generative model into two disentangled aspects: molecular properties and the remaining structural context of 3D molecules. Our model ensures explicit control over these molecular attributes while maintaining equivariance of coordinate representation and invariance of data likelihood. Furthermore, we introduce a novel alignment-based coordinate loss to adapt equivariant networks for auto-regressive de-novo 3D molecule generation from scratch. Extensive experiments validate our model's effectiveness on property-guided and context-guided molecule generation, both for de-novo 3D molecule design and structure-based drug discovery against protein targets.

We consider a nonparametric model $\mathcal{E}^{n},$ generated by independent observations $X_{i},$ $i=1,...,n,$ with densities $p(x,\theta_{i}),$ $i=1,...,n,$ the parameters of which $\theta _{i}=f(i/n)\in \Theta $ are driven by the values of an unknown function $f:[0,1]\rightarrow \Theta $ in a smoothness class. The main result of the paper is that, under regularity assumptions, this model can be approximated, in the sense of the Le Cam deficiency pseudodistance, by a nonparametric Gaussian shift model $Y_{i}=\Gamma (f(i/n))+\varepsilon _{i},$ where $\varepsilon_{1},...,\varepsilon _{n}$ are i.i.d. standard normal r.v.'s, the function $\Gamma (\theta ):\Theta \rightarrow \mathrm{R}$ satisfies $\Gamma ^{\prime}(\theta )=\sqrt{I(\theta )}$ and $I(\theta )$ is the Fisher information corresponding to the density $p(x,\theta ).$

We propose a method for estimating a log-concave density on $\mathbb R^d$ from samples, under the assumption that there exists an orthogonal transformation that makes the components of the random vector independent. While log-concave density estimation is hard both computationally and statistically, the independent components assumption alleviates both issues, while still maintaining a large non-parametric class. We prove that under mild conditions, at most $\tilde{\mathcal{O}}(\epsilon^{-4})$ samples (suppressing constants and log factors) suffice for our proposed estimator to be within $\epsilon$ of the original density in squared Hellinger distance. On the computational front, while the usual log-concave maximum likelihood estimate can be obtained via a finite-dimensional convex program, it is slow to compute -- especially in higher dimensions. We demonstrate through numerical experiments that our estimator can be computed efficiently, making it more practical to use.

The Church Problem asks for the construction of a procedure which, given a logical specification A(I,O) between input omega-strings I and output omega-strings O, determines whether there exists an operator F that implements the specification in the sense that A(I, F(I)) holds for all inputs I. Buchi and Landweber provided a procedure to solve the Church problem for MSO specifications and operators computable by finite-state automata. We investigate a generalization of the Church synthesis problem to the continuous time domain of the non-negative reals. We show that in the continuous time domain there are phenomena which are very different from the canonical discrete time domain of the natural numbers.

Given a finite set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can we find? We make progress on three instances of subset selection problems in planar point sets. Let $n,s\in\mathbb{N}$ with $n\geq s$, and let $P\subseteq\mathbb{R}^2$ be a set of $n$ points, where at most $s$ points lie on the same line. Firstly, we select a general position subset of $P$, i.e., a subset containing no $3$ points on the same line. This problem was proposed by Erd\H{o}s under the regime when $s$ is a constant. For $s$ being non-constant, we give new lower and upper bounds on the maximum size of such a subset. In particular, we show that in the worst case such a set can have size at most $O(n/s)$ when $n^{1/3}\leq s\leq n$ and $O(n^{5/6+o(1)}/\sqrt{s})$ when $3\leq s\leq n^{1/3}$. Secondly, we select a monotone general position subset of $P$, that is, a subset in general position where the points are ordered from left to right and their $y$-coordinates are either non-decreasing or non-increasing. We present bounds on the maximum size of such a subset. In particular, when $s=\Theta(\sqrt{n})$, our upper and lower bounds differ only by a logarithmic factor. Lastly, we select a subset of $P$ with pairwise distinct slopes. This problem was initially studied by Erd\H{o}s, Graham, Ruzsa, and Taylor on the grid. We show that for $s=O(\sqrt{n})$ such a subset of size $\Omega((n/\log{s})^{1/3})$ can always be found in $P$. When $s=\Theta(\sqrt{n})$, this matches a lower bound given by Zhang on the grid. As for the upper bound, we show that in the worst case such a subset has size at most $O(\sqrt{n})$ for $2\leq s\leq n^{3/8}$ and $O((n/s)^{4/5})$ for $n^{3/8}\leq s=O(\sqrt{n})$. The proofs use a wide range of tools such as incidence geometry, probabilistic methods, the hypergraph container method, and additive combinatorics.

We consider temporal numeric planning problems $\Pi$ expressed in PDDL2.1 level 3, and show how to produce SMT formulas $(i)$ whose models correspond to valid plans of $\Pi$, and $(ii)$ that extend the recently proposed planning with patterns approach from the numeric to the temporal case. We prove the correctness and completeness of the approach and show that it performs very well on 10 domains with required concurrency.

We present {\em generative clustering} (GC) for clustering a set of documents, $\mathrm{X}$, by using texts $\mathrm{Y}$ generated by large language models (LLMs) instead of by clustering the original documents $\mathrm{X}$. Because LLMs provide probability distributions, the similarity between two documents can be rigorously defined in an information-theoretic manner by the KL divergence. We also propose a natural, novel clustering algorithm by using importance sampling. We show that GC achieves the state-of-the-art performance, outperforming any previous clustering method often by a large margin. Furthermore, we show an application to generative document retrieval in which documents are indexed via hierarchical clustering and our method improves the retrieval accuracy.