Message passing Graph Neural Networks (GNNs) are known to be limited in expressive power by the 1-WL color-refinement test for graph isomorphism. Other more expressive models either are computationally expensive or need preprocessing to extract structural features from the graph. In this work, we propose to make GNNs universal by guiding the learning process with exact isomorphism solver techniques which operate on the paradigm of Individualization and Refinement (IR), a method to artificially introduce asymmetry and further refine the coloring when 1-WL stops. Isomorphism solvers generate a search tree of colorings whose leaves uniquely identify the graph. However, the tree grows exponentially large and needs hand-crafted pruning techniques which are not desirable from a learning perspective. We take a probabilistic view and approximate the search tree of colorings (i.e. embeddings) by sampling multiple paths from root to leaves of the search tree. To learn more discriminative representations, we guide the sampling process with particle filter updates, a principled approach for sequential state estimation. Our algorithm is end-to-end differentiable, can be applied with any GNN as backbone and learns richer graph representations with only linear increase in runtime. Experimental evaluation shows that our approach consistently outperforms leading GNN models on both synthetic benchmarks for isomorphism detection as well as real-world datasets.
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We show that the known list-decoding algorithms for univariate multiplicity and folded Reed-Solomon codes can be made to run in $\tilde{O}(n)$ time. Univariate multiplicity codes and FRS codes are natural variants of Reed-Solomon codes that were discovered and studied for their applications to list decoding. It is known that for every $\epsilon>0$, and rate $r \in (0,1)$, there exist explicit families of these codes that have rate $r$ and can be list decoded from a $(1-r-\epsilon)$ fraction of errors with constant list size in polynomial time (Guruswami & Wang (IEEE Trans. Inform. Theory 2013) and Kopparty, Ron-Zewi, Saraf & Wootters (SIAM J. Comput. 2023)). In this work, we present randomized algorithms that perform the above list-decoding tasks in $\tilde{O}(n)$, where $n$ is the block-length of the code. Our algorithms have two main components. The first component builds upon the lattice-based approach of Alekhnovich (IEEE Trans. Inf. Theory 2005), who designed a $\tilde{O}(n)$ time list-decoding algorithm for Reed-Solomon codes approaching the Johnson radius. As part of the second component, we design $\tilde{O}(n)$ time algorithms for two natural algebraic problems: given a $(m+2)$-variate polynomial $Q(x,y_0,\dots,y_m) = \tilde{Q}(x) + \sum_{i=0}^m Q_i(x)\cdot y_i$ the first algorithm solves order-$m$ linear differential equations of the form $Q\left(x, f(x), \frac{df}{dx}, \dots,\frac{d^m f}{dx^m}\right) \equiv 0$ while the second solves functional equations of the form $Q\left(x, f(x), f(\gamma x), \dots,f(\gamma^m x)\right) \equiv 0$, where $m$ is an arbitrary constant and $\gamma$ is a field element of sufficiently high order. These algorithms can be viewed as generalizations of classical $\tilde{O}(n)$ time algorithms of Sieveking (Computing 1972) and Kung (Numer. Math. 1974) for computing the modular inverse of a power series, and might be of independent interest.
We present HiRA-Pro, a novel procedure to align, at high spatio-temporal resolutions, multimodal signals from real-world processes and systems that exhibit diverse transient, nonlinear stochastic dynamics, such as manufacturing machines. It is based on discerning and synchronizing the process signatures of salient kinematic and dynamic events in these disparate signals. HiRA-Pro addresses the challenge of aligning data with sub-millisecond phenomena, where traditional timestamp, external trigger, or clock-based alignment methods fall short. The effectiveness of HiRA-Pro is demonstrated in a smart manufacturing context, where it aligns data from 13+ channels acquired during 3D-printing and milling operations on an Optomec-LENS MTS 500 hybrid machine. The aligned data is then voxelized to generate 0.25 second aligned data chunks that correspond to physical voxels on the produced part. The superiority of HiRA-Pro is further showcased through case studies in additive manufacturing, demonstrating improved machine learning-based predictive performance due to precise multimodal data alignment. Specifically, testing classification accuracies improved by almost 35% with the application of HiRA-Pro, even with limited data, allowing for precise localization of artifacts. The paper also provides a comprehensive discussion on the proposed method, its applications, and comparative qualitative analysis with a few other alignment methods. HiRA-Pro achieves temporal-spatial resolutions of 10-1000 us and 100 um in order to generate datasets that register with physical voxels on the 3D-printed and milled part. These resolutions are at least an order of magnitude finer than the existing alignment methods that employ individual timestamps, statistical correlations, or common clocks, which achieve precision of hundreds of milliseconds.
In this paper, we propose a new modified likelihood ratio test (LRT) for simultaneously testing mean vectors and covariance matrices of two-sample populations in high-dimensional settings. By employing tools from Random Matrix Theory (RMT), we derive the limiting null distribution of the modified LRT for generally distributed populations. Furthermore, we compare the proposed test with existing tests using simulation results, demonstrating that the modified LRT exhibits favorable properties in terms of both size and power.
In this report, we present the latest model of the Gemini family, Gemini 1.5 Pro, a highly compute-efficient multimodal mixture-of-experts model capable of recalling and reasoning over fine-grained information from millions of tokens of context, including multiple long documents and hours of video and audio. Gemini 1.5 Pro achieves near-perfect recall on long-context retrieval tasks across modalities, improves the state-of-the-art in long-document QA, long-video QA and long-context ASR, and matches or surpasses Gemini 1.0 Ultra's state-of-the-art performance across a broad set of benchmarks. Studying the limits of Gemini 1.5 Pro's long-context ability, we find continued improvement in next-token prediction and near-perfect retrieval (>99%) up to at least 10M tokens, a generational leap over existing models such as Claude 2.1 (200k) and GPT-4 Turbo (128k). Finally, we highlight surprising new capabilities of large language models at the frontier; when given a grammar manual for Kalamang, a language with fewer than 200 speakers worldwide, the model learns to translate English to Kalamang at a similar level to a person who learned from the same content.
Path eccentricity of $k$-AT-free graphs and application on graphs with the consecutive ones property
The central path problem is a variation on the single facility location problem. The aim is to find, in a given connected graph $G$, a path $P$ minimizing its eccentricity, which is the maximal distance from $P$ to any vertex of the graph $G$. The path eccentricity of $G$ is the minimal eccentricity achievable over all paths in $G$. In this article we consider the path eccentricity of the class of the $k$-AT-free graphs. They are graphs in which any set of three vertices contains a pair for which every path between them uses at least one vertex of the closed neighborhood at distance $k$ of the third. We prove that they have path eccentricity bounded by $k$. Moreover, we answer a question of G\'omez and Guti\'errez asking if there is a relation between path eccentricity and the consecutive ones property. The latter is the property for a binary matrix to admit a permutation of the rows placing the 1's consecutively on the columns. It was already known that graphs whose adjacency matrices have the consecutive ones property have path eccentricity at most 1, and that the same remains true when the augmented adjacency matrices (with ones on the diagonal) has the consecutive ones property. We generalize these results as follow. We study graphs whose adjacency matrices can be made to satisfy the consecutive ones property after changing some values on the diagonal, and show that those graphs have path eccentricity at most 2, by showing that they are 2-AT-free.
This paper concerns an expansion of first-order Belnap-Dunn logic whose connectives and quantifiers are all familiar from classical logic. The language and logical consequence relation of the logic are defined, a proof system for the defined logic is presented, and the soundness and completeness of the presented proof system is established. The close relationship between the logical consequence relations of the defined logic and the version of classical logic with the same language is illustrated by the minor differences between the presented proof system and a sound and complete proof system for the version of classical logic with the same language. Moreover, fifteen classical laws of logical equivalence are given by which the logical equivalence relation of the defined logic distinguishes itself from the logical equivalence relation of many logics that are closely related at first glance.
Advanced techniques using Neural Radiance Fields (NeRF), Signed Distance Fields (SDF), and Occupancy Fields have recently emerged as solutions for 3D indoor scene reconstruction. We introduce a novel two-phase learning approach, H2O-SDF, that discriminates between object and non-object regions within indoor environments. This method achieves a nuanced balance, carefully preserving the geometric integrity of room layouts while also capturing intricate surface details of specific objects. A cornerstone of our two-phase learning framework is the introduction of the Object Surface Field (OSF), a novel concept designed to mitigate the persistent vanishing gradient problem that has previously hindered the capture of high-frequency details in other methods. Our proposed approach is validated through several experiments that include ablation studies.
I show that a one-dimensional (1D) conditional generative adversarial network (cGAN) with an adversarial training architecture is capable of unpaired signal-to-signal ("sig2sig") translation. Using a simplified CycleGAN model with 1D layers and wider convolutional kernels, mirroring WaveGAN to reframe two-dimensional (2D) image generation as 1D audio generation, I show that recasting the 2D image-to-image translation task to a 1D signal-to-signal translation task with deep convolutional GANs is possible without substantial modification to the conventional U-Net model and adversarial architecture developed as CycleGAN. With this I show for a small tunable dataset that noisy test signals unseen by the 1D CycleGAN model and without paired training transform from the source domain to signals similar to paired test signals in the translated domain, especially in terms of frequency, and I quantify these differences in terms of correlation and error.
Recently pre-trained language representation models such as BERT have shown great success when fine-tuned on downstream tasks including information retrieval (IR). However, pre-training objectives tailored for ad-hoc retrieval have not been well explored. In this paper, we propose Pre-training with Representative wOrds Prediction (PROP) for ad-hoc retrieval. PROP is inspired by the classical statistical language model for IR, specifically the query likelihood model, which assumes that the query is generated as the piece of text representative of the "ideal" document. Based on this idea, we construct the representative words prediction (ROP) task for pre-training. Given an input document, we sample a pair of word sets according to the document language model, where the set with higher likelihood is deemed as more representative of the document. We then pre-train the Transformer model to predict the pairwise preference between the two word sets, jointly with the Masked Language Model (MLM) objective. By further fine-tuning on a variety of representative downstream ad-hoc retrieval tasks, PROP achieves significant improvements over baselines without pre-training or with other pre-training methods. We also show that PROP can achieve exciting performance under both the zero- and low-resource IR settings. The code and pre-trained models are available at //github.com/Albert-Ma/PROP.
Recent work pre-training Transformers with self-supervised objectives on large text corpora has shown great success when fine-tuned on downstream NLP tasks including text summarization. However, pre-training objectives tailored for abstractive text summarization have not been explored. Furthermore there is a lack of systematic evaluation across diverse domains. In this work, we propose pre-training large Transformer-based encoder-decoder models on massive text corpora with a new self-supervised objective. In PEGASUS, important sentences are removed/masked from an input document and are generated together as one output sequence from the remaining sentences, similar to an extractive summary. We evaluated our best PEGASUS model on 12 downstream summarization tasks spanning news, science, stories, instructions, emails, patents, and legislative bills. Experiments demonstrate it achieves state-of-the-art performance on all 12 downstream datasets measured by ROUGE scores. Our model also shows surprising performance on low-resource summarization, surpassing previous state-of-the-art results on 6 datasets with only 1000 examples. Finally we validated our results using human evaluation and show that our model summaries achieve human performance on multiple datasets.
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