NeurIPS 2024 Papers

Conference on Neural Information Processing Systems

Decision breakdown

4,562papers

Counted from the official paper list; may differ slightly from the number the conference announced.

  • Oral
    721.6%
  • Spotlight
    3828.4%
  • Poster
    4,06489.1%
  • Other / unlabeled
    441.0%

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PiSSA: Principal Singular Values and Singular Vectors Adaptation of Large Language Models

Fanxu Meng, Zhaohui Wang, Muhan Zhang

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To parameter-efficiently fine-tune (PEFT) large language models (LLMs), the low-rank adaptation (LoRA) method approximates the model changes $\Delta W \in \mathbb{R}^{m \times n}$ through the product of two matrices $A \in \mathbb{R}^{m \times r}$ and $B \in \mathbb{R}^{r \times n}$, where $r \ll \min(m, n)$, $A$ is initialized with Gaussian noise, and $B$ with zeros. LoRA **freezes the original model $W$** and **updates the "Noise \& Zero" adapter**, which may lead to slow convergence. To overcome this limitation, we introduce **P**r**i**ncipal **S**ingular values and **S**ingular vectors **A**daptation (PiSSA). PiSSA shares the same architecture as LoRA, but initializes the adaptor matrices $A$ and $B$ with the principal components of the original matrix $W$, and put the remaining components into a residual matrix $W^{res} \in \mathbb{R}^{m \times n}$ which is frozen during fine-tuning. Compared to LoRA, PiSSA **updates the principal components** while **freezing the "residual" parts**, allowing faster convergence and enhanced performance. Comparative experiments of PiSSA and LoRA across 11 different models, ranging from 184M to 70B, encompassing 5 NLG and 8 NLU tasks, reveal that PiSSA consistently outperforms LoRA under identical experimental setups. On the GSM8K benchmark, Gemma-7B fine-tuned with PiSSA achieves an accuracy of 77.7\%, surpassing LoRA's 74.53\% by 3.25\%. Due to the same architecture, PiSSA is also compatible with quantization to further reduce the memory requirement of fine-tuning. Compared to QLoRA, QPiSSA (PiSSA with 4-bit quantization) exhibits smaller quantization errors in the initial stages. Fine-tuning LLaMA-3-70B on GSM8K, QPiSSA attains an accuracy of 86.05\%, exceeding the performances of QLoRA at 81.73\%. Leveraging a fast SVD technique, PiSSA can be initialized in only a few seconds, presenting a negligible cost for transitioning from LoRA to PiSSA.

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LightGaussian: Unbounded 3D Gaussian Compression with 15x Reduction and 200+ FPS

Zhiwen Fan, Kevin Wang, Kairun Wen, Zehao Zhu, Dejia Xu, Zhangyang Wang

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Recent advances in real-time neural rendering using point-based techniques have enabled broader adoption of 3D representations. However, foundational approaches like 3D Gaussian Splatting impose substantial storage overhead, as Structure-from-Motion (SfM) points can grow to millions, often requiring gigabyte-level disk space for a single unbounded scene. This growth presents scalability challenges and hinders splatting efficiency. To address this, we introduce LightGaussian, a method for transforming 3D Gaussians into a more compact format. Inspired by Network Pruning, LightGaussian identifies Gaussians with minimal global significance on scene reconstruction, and applies a pruning and recovery process to reduce redundancy while preserving visual quality. Knowledge distillation and pseudo-view augmentation then transfer spherical harmonic coefficients to a lower degree, yielding compact representations. Gaussian Vector Quantization, based on each Gaussian’s global significance, further lowers bitwidth with minimal accuracy loss. LightGaussian achieves an average 15 times compression rate while boosting FPS from 144 to 237 within the 3D-GS framework, enabling efficient complex scene representation on the Mip-NeRF 360 and Tank & Temple datasets. The proposed Gaussian pruning approach is also adaptable to other 3D representations (e.g., Scaffold-GS), demonstrating strong generalization capabilities.

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ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networks

Yuezhu Xu, S Sivaranjani

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The Lipschitz constant plays a crucial role in certifying the robustness of neural networks to input perturbations. Since calculating the exact Lipschitz constant is NP-hard, efforts have been made to obtain tight upper bounds on the Lipschitz constant. Typically, this involves solving a large matrix verification problem, the computational cost of which grows significantly for both deeper and wider networks. In this paper, we provide a compositional approach to estimate Lipschitz constants for deep feed-forward neural networks. We first obtain an exact decomposition of the large matrix verification problem into smaller sub-problems. Then, leveraging the underlying cascade structure of the network, we develop two algorithms. The first algorithm explores the geometric features of the problem and enables us to provide Lipschitz estimates that are comparable to existing methods by solving small semidefinite programs (SDPs) that are only as large as the size of each layer. The second algorithm relaxes these sub-problems and provides a closed-form solution to each sub-problem for extremely fast estimation, altogether eliminating the need to solve SDPs. The two algorithms represent different levels of trade-offs between efficiency and accuracy. Finally, we demonstrate that our approach provides a steep reduction in computation time (as much as several thousand times faster, depending on the algorithm for deeper networks) while yielding Lipschitz bounds that are very close to or even better than those achieved by state-of-the-art approaches in a broad range of experiments. In summary, our approach considerably advances the scalability and efficiency of certifying neural network robustness, making it particularly attractive for online learning tasks.

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Scalable and Effective Arithmetic Tree Generation for Adder and Multiplier Designs

Yao Lai, Jinxin Liu, David Z. Pan, Ping Luo

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Across a wide range of hardware scenarios, the computational efficiency and physical size of the arithmetic units significantly influence the speed and footprint of the overall hardware system. Nevertheless, the effectiveness of prior arithmetic design techniques proves inadequate, as they do not sufficiently optimize speed and area, resulting in increased latency and larger module size. To boost computing performance, this work focuses on the two most common and fundamental arithmetic modules, adders and multipliers. We cast the design tasks as single-player tree generation games, leveraging reinforcement learning techniques to optimize their arithmetic tree structures. This tree generation formulation allows us to efficiently navigate the vast search space and discover superior arithmetic designs that improve computational efficiency and hardware size within just a few hours. Our proposed method, **ArithTreeRL**, achieves significant improvements for both adders and multipliers. For adders, our approach discovers designs of 128-bit adders that achieve Pareto optimality in theoretical metrics. Compared with PrefixRL, it reduces delay and size by up to 26% and 30%, respectively. For multipliers, compared to RL-MUL, our method enhances speed and reduces size by as much as 49% and 45%. Additionally, ArithTreeRL's flexibility and scalability enable seamless integration into 7nm technology. We believe our work will offer valuable insights into hardware design, further accelerating speed and reducing size through the refined search space and our tree generation methodologies.

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Reinforcement Learning Gradients as Vitamin for Online Finetuning Decision Transformers

Kai Yan, Alex Schwing, Yu-Xiong Wang

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Decision Transformers have recently emerged as a new and compelling paradigm for offline Reinforcement Learning (RL), completing a trajectory in an autoregressive way. While improvements have been made to overcome initial shortcomings, online finetuning of decision transformers has been surprisingly under-explored. The widely adopted state-of-the-art Online Decision Transformer (ODT) still struggles when pretrained with low-reward offline data. In this paper, we theoretically analyze the online-finetuning of the decision transformer, showing that the commonly used Return-To-Go (RTG) that's far from the expected return hampers the online fine-tuning process. This problem, however, is well-addressed by the value function and advantage of standard RL algorithms. As suggested by our analysis, in our experiments, we hence find that simply adding TD3 gradients to the finetuning process of ODT effectively improves the online finetuning performance of ODT, especially if ODT is pretrained with low-reward offline data. These findings provide new directions to further improve decision transformers.

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Text2CAD: Generating Sequential CAD Designs from Beginner-to-Expert Level Text Prompts

Mohammad Sadil Khan, Sankalp Sinha, Sheikh Talha Uddin, Didier Stricker, Sk Aziz Ali, Muhammad Zeshan Afzal

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Prototyping complex computer-aided design (CAD) models in modern softwares can be very time-consuming. This is due to the lack of intelligent systems that can quickly generate simpler intermediate parts. We propose Text2CAD, the first AI framework for generating text-to-parametric CAD models using designer-friendly instructions for all skill levels. Furthermore, we introduce a data annotation pipeline for generating text prompts based on natural language instructions for the DeepCAD dataset using Mistral and LLaVA-NeXT. The dataset contains $\sim170$K models and $\sim660$K text annotations, from abstract CAD descriptions (e.g., _generate two concentric cylinders_) to detailed specifications (e.g., _draw two circles with center_ $(x,y)$ and _radius_ $r_{1}$, $r_{2}$, \textit{and extrude along the normal by} $d$...). Within the Text2CAD framework, we propose an end-to-end transformer-based auto-regressive network to generate parametric CAD models from input texts. We evaluate the performance of our model through a mixture of metrics, including visual quality, parametric precision, and geometrical accuracy. Our proposed framework shows great potential in AI-aided design applications. Project page is available at https://sadilkhan.github.io/text2cad-project/.

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Unlocking Tokens as Data Points for Generalization Bounds on Larger Language Models

Sanae Lotfi, Yilun Kuang, Marc Anton Finzi, Brandon Amos, Micah Goldblum, Andrew Gordon Wilson

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Large language models (LLMs) with billions of parameters excel at predicting the next token in a sequence. Recent work computes non-vacuous compression-based generalization bounds for LLMs, but these bounds are vacuous for large models at the billion-parameter scale. Moreover, these bounds are obtained through restrictive compression techniques, bounding compressed models that generate low-quality text. Additionally, the tightness of these existing bounds depends on the number of IID documents in a training set rather than the much larger number of non-IID constituent tokens, leaving untapped potential for tighter bounds. In this work, we instead use properties of martingales to derive generalization bounds that benefit from the vast number of tokens in LLM training sets. Since a dataset contains far more tokens than documents, our generalization bounds not only tolerate but actually benefit from far less restrictive compression schemes. With Monarch matrices, Kronecker factorizations, and post-training quantization, we achieve non-vacuous generalization bounds for LLMs as large as LLaMA2-70B. Unlike previous approaches, our work achieves the first non-vacuous bounds for models that are deployed in practice and generate high-quality text.

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Peri-midFormer: Periodic Pyramid Transformer for Time Series Analysis

Qiang Wu, Gechang Yao, Zhixi Feng, Shuyuan Yang

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Time series analysis finds wide applications in fields such as weather forecasting, anomaly detection, and behavior recognition. Previous methods attempted to model temporal variations directly using 1D time series. However, this has been quite challenging due to the discrete nature of data points in time series and the complexity of periodic variation. In terms of periodicity, taking weather and traffic data as an example, there are multi-periodic variations such as yearly, monthly, weekly, and daily, etc. In order to break through the limitations of the previous methods, we decouple the implied complex periodic variations into inclusion and overlap relationships among different level periodic components based on the observation of the multi-periodicity therein and its inclusion relationships. This explicitly represents the naturally occurring pyramid-like properties in time series, where the top level is the original time series and lower levels consist of periodic components with gradually shorter periods, which we call the periodic pyramid. To further extract complex temporal variations, we introduce self-attention mechanism into the periodic pyramid, capturing complex periodic relationships by computing attention between periodic components based on their inclusion, overlap, and adjacency relationships. Our proposed Peri-midFormer demonstrates outstanding performance in five mainstream time series analysis tasks, including short- and long-term forecasting, imputation, classification, and anomaly detection.

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Toxicity Detection for Free

Zhanhao Hu, Julien Piet, Geng Zhao, Jiantao Jiao, David Wagner

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Current LLMs are generally aligned to follow safety requirements and tend to refuse toxic prompts. However, LLMs can fail to refuse toxic prompts or be overcautious and refuse benign examples. In addition, state-of-the-art toxicity detectors have low TPRs at low FPR, incurring high costs in real-world applications where toxic examples are rare. In this paper, we introduce Moderation Using LLM Introspection (MULI), which detects toxic prompts using the information extracted directly from LLMs themselves. We found we can distinguish between benign and toxic prompts from the distribution of the first response token's logits. Using this idea, we build a robust detector of toxic prompts using a sparse logistic regression model on the first response token logits. Our scheme outperforms SOTA detectors under multiple metrics.

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EHRCon: Dataset for Checking Consistency between Unstructured Notes and Structured Tables in Electronic Health Records

Yeonsu Kwon, Jiho Kim, Gyubok Lee, Seongsu Bae, Daeun Kyung, Wonchul Cha, Tom Pollard, ALISTAIR JOHNSON, Edward Choi

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Electronic Health Records (EHRs) are integral for storing comprehensive patient medical records, combining structured data (e.g., medications) with detailed clinical notes (e.g., physician notes). These elements are essential for straightforward data retrieval and provide deep, contextual insights into patient care. However, they often suffer from discrepancies due to unintuitive EHR system designs and human errors, posing serious risks to patient safety. To address this, we developed EHRCon, a new dataset and task specifically designed to ensure data consistency between structured tables and unstructured notes in EHRs. EHRCon was crafted in collaboration with healthcare professionals using the MIMIC-III EHR dataset, and includes manual annotations of 3,943 entities across 105 clinical notes checked against database entries for consistency. EHRCon has two versions, one using the original MIMIC-III schema, and another using the OMOP CDM schema, in order to increase its applicability and generalizability. Furthermore, leveraging the capabilities of large language models, we introduce CheckEHR, a novel framework for verifying the consistency between clinical notes and database tables. CheckEHR utilizes an eight-stage process and shows promising results in both few-shot and zero-shot settings. The code is available at \url{https://github.com/dustn1259/EHRCon}.

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TOPA: Extending Large Language Models for Video Understanding via Text-Only Pre-Alignment

Wei Li, Hehe Fan, Yongkang Wong, Mohan Kankanhalli, Yi Yang

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Recent advancements in image understanding have benefited from the extensive use of web image-text pairs. However, video understanding remains a challenge despite the availability of substantial web video-text data. This difficulty primarily arises from the inherent complexity of videos and the inefficient language supervision in recent web-collected video-text datasets. In this paper, we introduce Text-Only Pre-Alignment (TOPA), a novel approach to extend large language models (LLMs) for video understanding, without the need for pre-training on real video data. Specifically, we first employ an advanced LLM to automatically generate Textual Videos comprising continuous textual frames, along with corresponding annotations to simulate real video-text data. Then, these annotated textual videos are used to pre-align a language-only LLM with the video modality. To bridge the gap between textual and real videos, we employ the CLIP model as the feature extractor to align image and text modalities. During text-only pre-alignment, the continuous textual frames, encoded as a sequence of CLIP text features, are analogous to continuous CLIP image features, thus aligning the LLM with real video representation. Extensive experiments, including zero-shot evaluation and finetuning on various video understanding tasks, demonstrate that TOPA is an effective and efficient framework for aligning video content with LLMs. In particular, without training on any video data, the TOPA-Llama2-13B model achieves a Top-1 accuracy of 51.0% on the challenging long-form video understanding benchmark, Egoschema. This performance surpasses previous video-text pre-training approaches and proves competitive with recent GPT-3.5 based video agents.

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VLM Agents Generate Their Own Memories: Distilling Experience into Embodied Programs of Thought

Gabriel Herbert Sarch, Lawrence Jang, Michael J. Tarr, William W. Cohen, Kenneth Marino, Katerina Fragkiadaki

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Large-scale generative language and vision-language models (LLMs and VLMs) excel in few-shot in-context learning for decision making and instruction following. However, they require high-quality exemplar demonstrations to be included in their context window. In this work, we ask: Can LLMs and VLMs generate their own examples from generic, sub-optimal demonstrations? We propose In-Context Abstraction Learning (ICAL), a method that builds a memory of multimodal experience from sub-optimal demonstrations and human feedback. Given a task demonstration that may contain inefficiencies or mistakes, a VLM abstracts the trajectory into a generalized program by correcting inefficient actions and annotating cognitive abstractions: causal relationships, object state changes, temporal subgoals, and task-relevant visual elements. These abstractions are iteratively improved and adapted through human feedback while the agent attempts to execute the trajectory in a similar environment. The resulting examples, when used as exemplars in the prompt, significantly improve decision-making in retrieval-augmented LLM and VLM agents. Moreover, as the agent's library of examples grows, it becomes more efficient, relying less on human feedback and requiring fewer environment interactions per demonstration. Our ICAL agent surpasses the state-of-the-art in dialogue-based instruction following in TEACh, multimodal web agents in VisualWebArena, and action anticipation in Ego4D. In TEACh, we achieve a 12.6% improvement in goal-condition success. In VisualWebArena, our task success rate improves over the SOTA from 14.3% to 22.7% using GPT4V. In Ego4D action forecasting, we improve over few-shot GPT-4V and remain competitive with supervised models. We show finetuning our retrieval-augmented in-context agent yields additional improvements. Our approach significantly reduces reliance on manual prompt engineering and consistently outperforms in-context learning from action plans that lack such abstractions.

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Motion Forecasting in Continuous Driving

Nan Song, Bozhou Zhang, Xiatian Zhu, Li Zhang

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Motion forecasting for agents in autonomous driving is highly challenging due to the numerous possibilities for each agent's next action and their complex interactions in space and time. In real applications, motion forecasting takes place repeatedly and continuously as the self-driving car moves. However, existing forecasting methods typically process each driving scene within a certain range independently, totally ignoring the situational and contextual relationships between successive driving scenes. This significantly simplifies the forecasting task, making the solutions suboptimal and inefficient to use in practice. To address this fundamental limitation, we propose a novel motion forecasting framework for continuous driving, named RealMotion. It comprises two integral streams both at the scene level: (1) The scene context stream progressively accumulates historical scene information until the present moment, capturing temporal interactive relationships among scene elements. (2) The agent trajectory stream optimizes current forecasting by sequentially relaying past predictions. Besides, a data reorganization strategy is introduced to narrow the gap between existing benchmarks and real-world applications, consistent with our network. These approaches enable exploiting more broadly the situational and progressive insights of dynamic motion across space and time. Extensive experiments on Argoverse series with different settings demonstrate that our RealMotion achieves state-of-the-art performance, along with the advantage of efficient real-world inference.

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Resolving Discrepancies in Compute-Optimal Scaling of Language Models

Tomer Porian, Mitchell Wortsman, Jenia Jitsev, Ludwig Schmidt, Yair Carmon

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Kaplan et al. and Hoffmann et al. developed influential scaling laws for the optimal model size as a function of the compute budget, but these laws yield substantially different predictions. We explain the discrepancy by reproducing the Kaplan scaling law on two datasets (OpenWebText2 and RefinedWeb) and identifying three factors causing the difference: last layer computational cost, warmup duration, and scale-dependent optimizer tuning. With these factors corrected, we obtain excellent agreement with the Hoffmann et al. (i.e., "Chinchilla") scaling law. Counter to a hypothesis of Hoffmann et al., we find that careful learning rate decay is not essential for the validity of their scaling law. As a secondary result, we derive scaling laws for the optimal learning rate and batch size, finding that tuning the AdamW $\beta_2$ parameter is essential at lower batch sizes.

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A Near-optimal Algorithm for Learning Margin Halfspaces with Massart Noise

Ilias Diakonikolas, Nikos Zarifis

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We study the problem of PAC learning $\gamma$-margin halfspaces in the presence of Massart noise. Without computational considerations, the sample complexity of this learning problem is known to be $\widetilde{\Theta}(1/(\gamma^2 \epsilon))$. Prior computationally efficient algorithms for the problem incur sample complexity $\tilde{O}(1/(\gamma^4 \epsilon^3))$ and achieve 0-1 error of $\eta+\epsilon$, where $\eta<1/2$ is the upper bound on the noise rate. Recent work gave evidence of an information-computation tradeoff, suggesting that a quadratic dependence on $1/\epsilon$ is required for computationally efficient algorithms. Our main result is a computationally efficient learner with sample complexity $\widetilde{\Theta}(1/(\gamma^2 \epsilon^2))$, nearly matching this lower bound. In addition, our algorithm is simple and practical, relying on online SGD on a carefully selected sequence of convex losses.

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Private Edge Density Estimation for Random Graphs: Optimal, Efficient and Robust

Hongjie Chen, Jingqiu Ding, Yiding Hua, David Steurer

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We give the first polynomial-time, differentially node-private, and robust algorithm for estimating the edge density of Erdős-Rényi random graphs and their generalization, inhomogeneous random graphs. We further prove information-theoretical lower bounds, showing that the error rate of our algorithm is optimal up to logarithmic factors. Previous algorithms incur either exponential running time or suboptimal error rates. Two key ingredients of our algorithm are (1) a new sum-of-squares algorithm for robust edge density estimation, and (2) the reduction from privacy to robustness based on sum-of-squares exponential mechanisms due to Hopkins et al. (STOC 2023).

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Linear Regression using Heterogeneous Data Batches

Ayush Jain, Rajat Sen, Weihao Kong, Abhimanyu Das, Alon Orlitsky

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In many learning applications, data are collected from multiple sources, each providing a \emph{batch} of samples that by itself is insufficient to learn its input-output relationship. A common approach assumes that the sources fall in one of several unknown subgroups, each with an unknown input distribution and input-output relationship. We consider one of this setup's most fundamental and important manifestations where the output is a noisy linear combination of the inputs, and there are $k$ subgroups, each with its own regression vector. Prior work [KSS$^+$20] showed that with abundant small-batches, the regression vectors can be learned with only few, $\tilde\Omega( k^{3/2})$, batches of medium-size with $\tilde\Omega(\sqrt k)$ samples each. However, the paper requires that the input distribution for all $k$ subgroups be isotropic Gaussian, and states that removing this assumption is an ``interesting and challenging problem". We propose a novel gradient-based algorithm that improves on the existing results in several ways. It extends the applicability of the algorithm by: (1) allowing the subgroups' underlying input distributions to be different, unknown, and heavy-tailed; (2) recovering all subgroups followed by a significant proportion of batches even for infinite $k$; (3) removing the separation requirement between the regression vectors; (4) reducing the number of batches and allowing smaller batch sizes.

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Expectile Regularization for Fast and Accurate Training of Neural Optimal Transport

Nazar Buzun, Maksim Bobrin, Dmitry V. Dylov

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We present a new approach for Neural Optimal Transport (NOT) training procedure, capable of accurately and efficiently estimating optimal transportation plan via specific regularization on dual Kantorovich potentials. The main bottleneck of existing NOT solvers is associated with the procedure of finding a near-exact approximation of the conjugate operator (i.e., the c-transform), which is done either by optimizing over non-convex max-min objectives or by the computationally intensive fine-tuning of the initial approximated prediction. We resolve both issues by proposing a new theoretically justified loss in the form of expectile regularization which enforces binding conditions on the learning process of the dual potentials. Such a regularization provides the upper bound estimation over the distribution of possible conjugate potentials and makes the learning stable, completely eliminating the need for additional extensive fine-tuning. Proposed method, called Expectile-Regularized Neural Optimal Transport (ENOT), outperforms previous state-of-the-art approaches in the established Wasserstein-2 benchmark tasks by a large margin (up to a 3-fold improvement in quality and up to a 10-fold improvement in runtime). Moreover, we showcase performance of ENOT for various cost functions in different tasks, such as image generation, demonstrating generalizability and robustness of the proposed algorithm.

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SA3DIP: Segment Any 3D Instance with Potential 3D Priors

Xi Yang, Xu Gu, Xingyilang Yin, Xinbo Gao

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The proliferation of 2D foundation models has sparked research into adapting them for open-world 3D instance segmentation. Recent methods introduce a paradigm that leverages superpoints as geometric primitives and incorporates 2D multi-view masks from Segment Anything model (SAM) as merging guidance, achieving outstanding zero-shot instance segmentation results. However, the limited use of 3D priors restricts the segmentation performance. Previous methods calculate the 3D superpoints solely based on estimated normal from spatial coordinates, resulting in under-segmentation for instances with similar geometry. Besides, the heavy reliance on SAM and hand-crafted algorithms in 2D space suffers from over-segmentation due to SAM's inherent part-level segmentation tendency. To address these issues, we propose SA3DIP, a novel method for Segmenting Any 3D Instances via exploiting potential 3D Priors. Specifically, on one hand, we generate complementary 3D primitives based on both geometric and textural priors, which reduces the initial errors that accumulate in subsequent procedures. On the other hand, we introduce supplemental constraints from the 3D space by using a 3D detector to guide a further merging process. Furthermore, we notice a considerable portion of low-quality ground truth annotations in ScanNetV2 benchmark, which affect the fair evaluations. Thus, we present ScanNetV2-INS with complete ground truth labels and supplement additional instances for 3D class-agnostic instance segmentation. Experimental evaluations on various 2D-3D datasets demonstrate the effectiveness and robustness of our approach. Our code and proposed ScanNetV2-INS dataset are available HERE.

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Nearly Optimal Approximation of Matrix Functions by the Lanczos Method

Noah Amsel, Tyler Chen, Anne Greenbaum, Cameron N Musco, Christopher Musco

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Approximating the action of a matrix function $f(\vec{A})$ on a vector $\vec{b}$ is an increasingly important primitive in machine learning, data science, and statistics, with applications such as sampling high dimensional Gaussians, Gaussian process regression and Bayesian inference, principle component analysis, and approximating Hessian spectral densities. Over the past decade, a number of algorithms enjoying strong theoretical guarantees have been proposed for this task. Many of the most successful belong to a family of algorithms called Krylov subspace methods. Remarkably, a classic Krylov subspace method, called the Lanczos method for matrix functions (Lanczos-FA), frequently outperforms newer methods in practice. Our main result is a theoretical justification for this finding: we show that, for a natural class of rational functions, Lanczos-FA matches the error of the best possible Krylov subspace method up to a multiplicative approximation factor. The approximation factor depends on the degree of $f(x)$'s denominator and the condition number of $\vec{A}$, but not on the number of iterations $k$. Our result provides a strong justification for the excellent performance of Lanczos-FA, especially on functions that are well approximated by rationals, such as the matrix square root.

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