Many popular network models rely on the assumption of (vertex) exchangeability, in which the distribution of the graph is invariant to relabelings of the vertices. However, the Aldous-Hoover theorem guarantees that these graphs are dense or empty with probability one, whereas many real-world graphs are sparse. We present an alternative notion of exchangeability for random graphs, which we call edge exchangeability, in which the distribution of a graph sequence is invariant to the order of the edges. We demonstrate that edge-exchangeable models, unlike models that are traditionally vertex exchangeable, can exhibit sparsity. To do so, we outline a general framework for graph generative models; by contrast to the pioneering work of Caron and Fox (2015), models within our framework are stationary across steps of the graph sequence. In particular, our model grows the graph by instantiating more latent atoms of a single random measure as the dataset size increases, rather than adding new atoms to the measure.

We study the fundamental problem of clustering $n$ points into $K$ groups drawn from a mixture of isotropic Gaussians in $\mathbb{R}^d$. Specifically, we investigate the requisite minimal distance $Δ$ between mean vectors to partially recover the underlying partition. While the minimax-optimal threshold for $Δ$ is well-established, a significant gap exists between this information-theoretic limit and the performance of known polynomial-time procedures. Although this gap was recently characterized in the high-dimensional regime ($n \leq dK$), it remains largely unexplored in the moderate-dimensional regime ($n \geq dK$). In this manuscript, we address this regime by establishing a new low-degree polynomial lower bound for the moderate-dimensional case when $d \geq K$. We show that while the difficulty of clustering for $n \leq dK$ is primarily driven by dimension reduction and spectral methods, the moderate-dimensional regime involves more delicate phenomena leading to a "non-parametric rate". We provide a novel non-spectral algorithm matching this rate, shedding new light on the computational limits of the clustering problem in moderate dimension.

We collect 312 programs from various sources, including daily programs from college homework, the international competition (SV -COMP), benchmarks from previous papers (SLING), and programs from real-world software systems (Linux Kernel, GlibC, LiteOS, and Zephyr).

We consider the problem of crystal materials generation using language models (LMs).

Molecular representation learning lays the foundation for drug discovery. However, existing methods suffer from poor out-of-distribution (OOD) generalization, particularly when data for training and testing originate from different environments.

Further taking the usual assumption that X is compact. Let us start with Proposition 3, a central observation needed in Theorem 2. Put into words Now, we can proceed to prove the universality part of Theorem 2. Since the task admits a smooth separator, By Fubini's theorem and Proposition 3, we have F The reader can think of λ as a uniform distribution over G. (as in Theorem 2). The result follows directly from the combination of de Finetti's theorem [ Combining this with Kallenberg's noise transfer theorem we have that the weights and Assumption 1 or ii) is an inner-product decision graph problem as in Definition 3. Further, the task has infinitely (as in Theorem 2). Finally, we follow Proposition 2's proof by simply replacing de Finetti's with Aldous-Hoover's theorem. Define an RLC that samples the linear coefficients as follows.

Combined, these features provide both theoretical and practical simplicity.

Any explanation that faithfully explains this type of model needs to be in agreement with this invariance property.