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Symbolic Music, akin to language, can be encoded in discrete symbols. Recent research has extended the application of large language models (LLMs) such as GPT-4 and Llama2 to the symbolic music domain including understanding and generation. Yet scant research explores the details of how these LLMs perform on advanced music understanding and conditioned generation, especially from the multi-step reasoning perspective, which is a critical aspect in the conditioned, editable, and interactive human-computer co-creation process. This study conducts a thorough investigation of LLMs' capability and limitations in symbolic music processing. We identify that current LLMs exhibit poor performance in song-level multi-step music reasoning, and typically fail to leverage learned music knowledge when addressing complex musical tasks. An analysis of LLMs' responses highlights distinctly their pros and cons. Our findings suggest achieving advanced musical capability is not intrinsically obtained by LLMs, and future research should focus more on bridging the gap between music knowledge and reasoning, to improve the co-creation experience for musicians., Comment: Accepted by ISMIR2024

This paper investigates the stability of the least squares approximation $P_m^n$ within the univariate polynomial space of degree $m$, denoted by ${\mathbb P}_m$. The approximation $P_m^n$ entails identifying a polynomial in ${\mathbb P}_m$ that approximates a function $f$ over a domain $X$ based on samples of $f$ taken at $n$ randomly selected points, according to a specified measure $\rho_X$. The primary goal is to determine the sampling rate necessary to ensure the stability of $P_m^n$. Assuming the sampling points are i.i.d. with respect to a Jacobi weight function, we present the sampling rates that guarantee the stability of $P_m^n$. Specifically, for uniform random sampling, we demonstrate that a sampling rate of $n \asymp m^2$ is required to maintain stability. By integrating these findings with those of Cohen-Davenport-Leviatan, we conclude that, for uniform random sampling, the optimal sampling rate for guaranteeing the stability of $P_m^n$ is $n \asymp m^2$, up to a $\log n$ factor. Motivated by this result, we extend the impossibility theorem, previously applicable to equally spaced samples, to the case of random samples, illustrating the balance between accuracy and stability in recovering analytic functions., Comment: 26 pages

We present InternLM-XComposer-2.5 (IXC-2.5), a versatile large-vision language model that supports long-contextual input and output. IXC-2.5 excels in various text-image comprehension and composition applications, achieving GPT-4V level capabilities with merely 7B LLM backend. Trained with 24K interleaved image-text contexts, it can seamlessly extend to 96K long contexts via RoPE extrapolation. This long-context capability allows IXC-2.5 to excel in tasks requiring extensive input and output contexts. Compared to its previous 2.0 version, InternLM-XComposer-2.5 features three major upgrades in vision-language comprehension: (1) Ultra-High Resolution Understanding, (2) Fine-Grained Video Understanding, and (3) Multi-Turn Multi-Image Dialogue. In addition to comprehension, IXC-2.5 extends to two compelling applications using extra LoRA parameters for text-image composition: (1) Crafting Webpages and (2) Composing High-Quality Text-Image Articles. IXC-2.5 has been evaluated on 28 benchmarks, outperforming existing open-source state-of-the-art models on 16 benchmarks. It also surpasses or competes closely with GPT-4V and Gemini Pro on 16 key tasks. The InternLM-XComposer-2.5 is publicly available at https://github.com/InternLM/InternLM-XComposer., Comment: Technical Report. https://github.com/InternLM/InternLM-XComposer

The Event Horizon Telescope has produced resolved images of the supermassive black holes Sgr A* and M87*, which present the largest shadows on the sky. In the next decade, technological improvements and extensions to the array will enable access to a greater number of sources, unlocking studies of a larger population of supermassive black holes through direct imaging. In this paper, we identify 12 of the most promising sources beyond Sgr A* and M87* based on their angular size and millimeter flux density. For each of these sources, we make theoretical predictions for their observable properties by ray tracing general relativistic magnetohydrodynamic models appropriately scaled to each target's mass, distance, and flux density. We predict that these sources would have somewhat higher Eddington ratios than M87*, which may result in larger optical and Faraday depths than previous EHT targets. Despite this, we find that visibility amplitude size constraints can plausibly recover masses within a factor of 2, although the unknown jet contribution remains a significant uncertainty. We find that the linearly polarized structure evolves substantially with Eddington ratio, with greater evolution at larger inclinations, complicating potential spin inferences for inclined sources. We discuss the importance of 345 GHz observations, milli-Jansky baseline sensitivity, and independent inclination constraints for future observations with upgrades to the Event Horizon Telescope (EHT) through ground updates with the next-generation EHT (ngEHT) program and extensions to space through the Black Hole Explorer (BHEX)., Comment: 21 pages, 9 figures, 1 table, submitted to ApJ

We present the Black Hole Explorer (BHEX), a mission that will produce the sharpest images in the history of astronomy by extending submillimeter Very-Long-Baseline Interferometry (VLBI) to space. BHEX will discover and measure the bright and narrow "photon ring" that is predicted to exist in images of black holes, produced from light that has orbited the black hole before escaping. This discovery will expose universal features of a black hole's spacetime that are distinct from the complex astrophysics of the emitting plasma, allowing the first direct measurements of a supermassive black hole's spin. In addition to studying the properties of the nearby supermassive black holes M87* and Sgr A*, BHEX will measure the properties of dozens of additional supermassive black holes, providing crucial insights into the processes that drive their creation and growth. BHEX will also connect these supermassive black holes to their relativistic jets, elucidating the power source for the brightest and most efficient engines in the universe. BHEX will address fundamental open questions in the physics and astrophysics of black holes that cannot be answered without submillimeter space VLBI. The mission is enabled by recent technological breakthroughs, including the development of ultra-high-speed downlink using laser communications, and it leverages billions of dollars of existing ground infrastructure. We present the motivation for BHEX, its science goals and associated requirements, and the pathway to launch within the next decade., Comment: Proceedings for SPIE Astronomical Telescopes and Instrumentation

Large Language Models (LLMs) have made great strides in recent years to achieve unprecedented performance across different tasks. However, due to commercial interest, the most competitive models like GPT, Gemini, and Claude have been gated behind proprietary interfaces without disclosing the training details. Recently, many institutions have open-sourced several strong LLMs like LLaMA-3, comparable to existing closed-source LLMs. However, only the model's weights are provided with most details (e.g., intermediate checkpoints, pre-training corpus, and training code, etc.) being undisclosed. To improve the transparency of LLMs, the research community has formed to open-source truly open LLMs (e.g., Pythia, Amber, OLMo), where more details (e.g., pre-training corpus and training code) are being provided. These models have greatly advanced the scientific study of these large models including their strengths, weaknesses, biases and risks. However, we observe that the existing truly open LLMs on reasoning, knowledge, and coding tasks are still inferior to existing state-of-the-art LLMs with similar model sizes. To this end, we open-source MAP-Neo, a highly capable and transparent bilingual language model with 7B parameters trained from scratch on 4.5T high-quality tokens. Our MAP-Neo is the first fully open-sourced bilingual LLM with comparable performance compared to existing state-of-the-art LLMs. Moreover, we open-source all details to reproduce our MAP-Neo, where the cleaned pre-training corpus, data cleaning pipeline, checkpoints, and well-optimized training/evaluation framework are provided. Finally, we hope our MAP-Neo will enhance and strengthen the open research community and inspire more innovations and creativities to facilitate the further improvements of LLMs., Comment: https://map-neo.github.io/

We investigate a self-interacting dark matter (SIDM) model featuring a velocity-dependent cross section with an order-of-magnitude resonant enhancement of the cross section at $\sim 16\,{\rm km}\,{\rm s}^{-1}$. To understand the implications for the structure of dark matter halos, we perform N-body simulations of isolated dark matter halos of mass $\sim 10^8\,{\rm M}_\odot$, a halo mass selected to have a maximum response to the resonance. We track the core formation and the gravothermal collapse phases of the dark matter halo in this model and compare the halo evolving with the resonant cross section with halos evolving with velocity-independent cross sections. We show that dark matter halo evolution with the resonant cross section exhibits a deviation from universality that characterizes halo evolution with velocity-independent cross sections. The halo evolving under the influence of the resonance reaches a lower minimum central density during core formation. It subsequently takes about $20\%$ longer to reach its initial central density during the collapse phase. These results motivate a more detailed exploration of halo evolution in models with pronounced resonances., Comment: 10 pages, 6 figures, 1 table

The Large Vision-Language Model (LVLM) field has seen significant advancements, yet its progression has been hindered by challenges in comprehending fine-grained visual content due to limited resolution. Recent efforts have aimed to enhance the high-resolution understanding capabilities of LVLMs, yet they remain capped at approximately 1500 x 1500 pixels and constrained to a relatively narrow resolution range. This paper represents InternLM-XComposer2-4KHD, a groundbreaking exploration into elevating LVLM resolution capabilities up to 4K HD (3840 x 1600) and beyond. Concurrently, considering the ultra-high resolution may not be necessary in all scenarios, it supports a wide range of diverse resolutions from 336 pixels to 4K standard, significantly broadening its scope of applicability. Specifically, this research advances the patch division paradigm by introducing a novel extension: dynamic resolution with automatic patch configuration. It maintains the training image aspect ratios while automatically varying patch counts and configuring layouts based on a pre-trained Vision Transformer (ViT) (336 x 336), leading to dynamic training resolution from 336 pixels to 4K standard. Our research demonstrates that scaling training resolution up to 4K HD leads to consistent performance enhancements without hitting the ceiling of potential improvements. InternLM-XComposer2-4KHD shows superb capability that matches or even surpasses GPT-4V and Gemini Pro in 10 of the 16 benchmarks. The InternLM-XComposer2-4KHD model series with 7B parameters are publicly available at https://github.com/InternLM/InternLM-XComposer., Comment: Code and models are publicly available at https://github.com/InternLM/InternLM-XComposer

The evolution of Large Language Models (LLMs) like ChatGPT and GPT-4 has sparked discussions on the advent of Artificial General Intelligence (AGI). However, replicating such advancements in open-source models has been challenging. This paper introduces InternLM2, an open-source LLM that outperforms its predecessors in comprehensive evaluations across 6 dimensions and 30 benchmarks, long-context modeling, and open-ended subjective evaluations through innovative pre-training and optimization techniques. The pre-training process of InternLM2 is meticulously detailed, highlighting the preparation of diverse data types including text, code, and long-context data. InternLM2 efficiently captures long-term dependencies, initially trained on 4k tokens before advancing to 32k tokens in pre-training and fine-tuning stages, exhibiting remarkable performance on the 200k ``Needle-in-a-Haystack" test. InternLM2 is further aligned using Supervised Fine-Tuning (SFT) and a novel Conditional Online Reinforcement Learning from Human Feedback (COOL RLHF) strategy that addresses conflicting human preferences and reward hacking. By releasing InternLM2 models in different training stages and model sizes, we provide the community with insights into the model's evolution.

Recently, LLM-powered driver agents have demonstrated considerable potential in the field of autonomous driving, showcasing human-like reasoning and decision-making abilities.However, current research on aligning driver agent behaviors with human driving styles remains limited, partly due to the scarcity of high-quality natural language data from human driving behaviors.To address this research gap, we propose a multi-alignment framework designed to align driver agents with human driving styles through demonstrations and feedback. Notably, we construct a natural language dataset of human driver behaviors through naturalistic driving experiments and post-driving interviews, offering high-quality human demonstrations for LLM alignment. The framework's effectiveness is validated through simulation experiments in the CARLA urban traffic simulator and further corroborated by human evaluations. Our research offers valuable insights into designing driving agents with diverse driving styles.The implementation of the framework and details of the dataset can be found at the link.

While quantum computing has strong potential in data-driven fields, the privacy issue of sensitive or valuable information involved in the quantum algorithm should be considered. Differential privacy (DP), which is a fundamental privacy tool widely used in the classical scenario, has been extended to the quantum domain, i.e., quantum differential privacy (QDP). QDP may become one of the most promising approaches toward privacy-preserving quantum computing since it is not only compatible with classical DP mechanisms but also achieves privacy protection by exploiting unavoidable quantum noise in noisy intermediate-scale quantum (NISQ) devices. This paper provides an overview of the various implementations of QDP and their performance in terms of privacy parameters under the DP setting. Specifically, we propose a taxonomy of QDP techniques, categorizing the literature on whether internal or external randomization is used as a source to achieve QDP and how these implementations are applied to each phase of the quantum algorithm. We also discuss challenges and future directions for QDP. By summarizing recent advancements, we hope to provide a comprehensive, up-to-date review for researchers venturing into this field., Comment: To be published in QCE24 (IEEE Quantum Week 2024)

We introduce InternLM-XComposer2, a cutting-edge vision-language model excelling in free-form text-image composition and comprehension. This model goes beyond conventional vision-language understanding, adeptly crafting interleaved text-image content from diverse inputs like outlines, detailed textual specifications, and reference images, enabling highly customizable content creation. InternLM-XComposer2 proposes a Partial LoRA (PLoRA) approach that applies additional LoRA parameters exclusively to image tokens to preserve the integrity of pre-trained language knowledge, striking a balance between precise vision understanding and text composition with literary talent. Experimental results demonstrate the superiority of InternLM-XComposer2 based on InternLM2-7B in producing high-quality long-text multi-modal content and its exceptional vision-language understanding performance across various benchmarks, where it not only significantly outperforms existing multimodal models but also matches or even surpasses GPT-4V and Gemini Pro in certain assessments. This highlights its remarkable proficiency in the realm of multimodal understanding. The InternLM-XComposer2 model series with 7B parameters are publicly available at https://github.com/InternLM/InternLM-XComposer., Comment: Code and models are available at https://github.com/InternLM/InternLM-XComposer

In this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem \begin{equation}\label{1.1ab} \left\{ \begin{aligned} -\Delta {u}-\frac{1}{2}(x \cdot{\nabla u})&= \lambda{|u|^{{2}^{*}-2}u}+{\mu {|u|^{p-2}u}}& \ \ \mbox{in} \ \ \ {{\mathbb{R}}^{N}_{+}}, \frac{{\partial u}}{{\partial n}}&=\sqrt{\lambda}|u|^{{2}_{*}-2}u \ & \mbox{on}\ {{\partial {{\mathbb{R}}^{N}_{+}}}}, \end{aligned} \right. \end{equation} where $ \mathbb{R}^{N}_{+}=\{(x{'}, x_{N}): x{'}\in {\mathbb{R}}^{N-1}, x_{N}>0\}$, $N\geq3$, $\lambda>0$, $\mu\in \mathbb{R}$, $2< p <{2}^{*}$, $n$ is the outward normal vector at the boundary ${{\partial {{\mathbb{R}}^{N}_{+}}}}$, $2^{*}=\frac{2N}{N-2}$ is the usual critical exponent for the Sobolev embedding $D^{1,2}({\mathbb{R}}^{N}_{+})\hookrightarrow {L^{{2}^{*}}}({\mathbb{R}}^{N}_{+})$ and ${2}_{*}=\frac{2(N-1)}{N-2}$ is the critical exponent for the Sobolev trace embedding $D^{1,2}({\mathbb{R}}^{N}_{+})\hookrightarrow {L^{{2}_{*}}}(\partial \mathbb{R}^{N}_{+})$. By establishing an improved Pohozaev identity, we show that the problem has no nontrivial solution if $\mu \le 0$; By applying the Mountain Pass Theorem without $(PS)$ condition and the delicate estimates for Mountain Pass level, we obtain the existence of a positive solution for all $\lambda>0$ and the different values of the parameters $p$ and ${\mu}>0$. Particularly, for $\lambda >0$, $N\ge 4$, $20$. Moreover, the existence of multiple solutions for the problem is also obtained by dual variational principle for all $\mu>0$ and suitable $\lambda$.

Audio is an essential part of our life, but creating it often requires expertise and is time-consuming. Research communities have made great progress over the past year advancing the performance of large scale audio generative models for a single modality (speech, sound, or music) through adopting more powerful generative models and scaling data. However, these models lack controllability in several aspects: speech generation models cannot synthesize novel styles based on text description and are limited on domain coverage such as outdoor environments; sound generation models only provide coarse-grained control based on descriptions like "a person speaking" and would only generate mumbling human voices. This paper presents Audiobox, a unified model based on flow-matching that is capable of generating various audio modalities. We design description-based and example-based prompting to enhance controllability and unify speech and sound generation paradigms. We allow transcript, vocal, and other audio styles to be controlled independently when generating speech. To improve model generalization with limited labels, we adapt a self-supervised infilling objective to pre-train on large quantities of unlabeled audio. Audiobox sets new benchmarks on speech and sound generation (0.745 similarity on Librispeech for zero-shot TTS; 0.77 FAD on AudioCaps for text-to-sound) and unlocks new methods for generating audio with novel vocal and acoustic styles. We further integrate Bespoke Solvers, which speeds up generation by over 25 times compared to the default ODE solver for flow-matching, without loss of performance on several tasks. Our demo is available at https://audiobox.metademolab.com/

Quantum computing is a promising paradigm for efficiently solving large and high-complexity problems. To protect quantum computing privacy, pioneering research efforts proposed to redefine differential privacy (DP) in quantum computing, i.e., quantum differential privacy (QDP), and harvest inherent noises generated by quantum computing to implement QDP. However, such an implementation approach is limited by the amount of inherent noises, which makes the privacy budget of the QDP mechanism fixed and uncontrollable. To address this issue, in this paper, we propose to leverage quantum error correction (QEC) techniques to reduce quantum computing errors, while tuning the privacy protection levels in QDP. In short, we gradually decrease the quantum noise error rate by deciding whether to apply QEC operations on the gate in a multiple single qubit gates circuit. We have derived a new calculation formula for the general error rate and corresponding privacy budgets after QEC operation. Then, we expand to achieve further noise reduction using multi-level concatenated QEC operation. Through extensive numerical simulations, we demonstrate that QEC is a feasible way to regulate the degree of privacy protection in quantum computing.

Materialisation facilitates Datalog reasoning by precomputing all consequences of the facts and the rules so that queries can be directly answered over the materialised facts. However, storing all materialised facts may be infeasible in practice, especially when the rules are complex and the given set of facts is large. We observe that for certain combinations of rules, there exist data structures that compactly represent the reasoning result and can be efficiently queried when necessary. In this paper, we present a general framework that allows for the integration of such optimised storage schemes with standard materialisation algorithms. Moreover, we devise optimised storage schemes targeting at transitive rules and union rules, two types of (combination of) rules that commonly occur in practice. Our experimental evaluation shows that our approach significantly improves memory consumption, sometimes by orders of magnitude, while remaining competitive in terms of query answering time., Comment: 19 pages

Quantum computing revolutionizes the way of solving complex problems and handling vast datasets, which shows great potential to accelerate the machine learning process. However, data leakage in quantum machine learning (QML) may present privacy risks. Although differential privacy (DP), which protects privacy through the injection of artificial noise, is a well-established approach, its application in the QML domain remains under-explored. In this paper, we propose to harness inherent quantum noises to protect data privacy in QML. Especially, considering the Noisy Intermediate-Scale Quantum (NISQ) devices, we leverage the unavoidable shot noise and incoherent noise in quantum computing to preserve the privacy of QML models for binary classification. We mathematically analyze that the gradient of quantum circuit parameters in QML satisfies a Gaussian distribution, and derive the upper and lower bounds on its variance, which can potentially provide the DP guarantee. Through simulations, we show that a target privacy protection level can be achieved by running the quantum circuit a different number of times., Comment: 6 pages, 4 figures

Quantum computing has been widely applied in various fields, such as quantum physics simulations, quantum machine learning, and big data analysis. However, in the domains of data-driven paradigm, how to ensure the privacy of the database is becoming a vital problem. For classical computing, we can incorporate the concept of differential privacy (DP) to meet the standard of privacy preservation by manually adding the noise. In the quantum computing scenario, researchers have extended classic DP to quantum differential privacy (QDP) by considering the quantum noise. In this paper, we propose a novel approach to satisfy the QDP definition by considering the errors generated by the projection operator measurement, which is denoted as shot noises. Then, we discuss the amount of privacy budget that can be achieved with shot noises, which serves as a metric for the level of privacy protection. Furthermore, we provide the QDP of shot noise in quantum circuits with depolarizing noise. Through numerical simulations, we show that shot noise can effectively provide privacy protection in quantum computing.

The Metaverse is a virtual world, an immersive experience, a new human-computer interaction, built upon various advanced technologies. How to protect Metaverse personal information and virtual properties is also facing new challenges, such as new attacks and new expectations of user experiences. While traditional methods (e.g., those employed in smartphone authentication) generally pass the basic design considerations, they are repeatedly reported to be either unsafe or inconvenient in the Metaverse. In this paper, we address this discrepancy by introducing CAN: a new design consideration especially for the Metaverse. Specifically, we focus on the legacy and novel biometric authentication systems and evaluate them thoroughly with basic and CAN considerations. We also propose an ear-based method as one example of CAN systems. To conclude, a continuous, active and non-intrusive biometric system is suggested for Metaverse authentication for its capability in continuous sessions, against imposters, and immersive experience., Comment: 6 pages, 3 figures

We propose InternLM-XComposer, a vision-language large model that enables advanced image-text comprehension and composition. The innovative nature of our model is highlighted by three appealing properties: 1) Interleaved Text-Image Composition: InternLM-XComposer can effortlessly generate coherent and contextual articles that seamlessly integrate images, providing a more engaging and immersive reading experience. Simply provide a writing instruction, and our system will generate the corresponding manuscript. It can intelligently identify the areas in the text where images would enhance the content and automatically insert the most appropriate visual candidates. 2) Comprehension with Rich Multilingual Knowledge: The text-image comprehension is empowered by training on an extensive multi-modal multilingual database with carefully crafted strategies, resulting in a deep understanding of visual content. 3) State-of-the-art Performance: Our model consistently achieves state-of-the-art results across various mainstream benchmarks for vision-language foundational models, including MME Benchmark, MMBench, MMBench-CN, Seed-Bench, CCBench (Chinese Cultural Benchmark), QBench and Tiny LVLM. Owing to the absence of established metrics for quantitatively assessing text-image composition, we have devised a robust evaluation procedure that comprises both human and GPT4-Vision (GPT4-V) to ensure reliability. Notably, our InternLM-XComposer achieves competitive text-image composition scores compared to public solutions, including GPT4-V and GPT3.5. Collectively, InternLM-XComposer seamlessly blends advanced text-image comprehension and composition, revolutionizing vision-language interaction and offering new insights and opportunities. The InternLM-XComposer model series are publicly available at https://github.com/InternLM/InternLM-XComposer., Comment: Code and models are available at https://github.com/InternLM/InternLM-XComposer

Due to the absence of more efficient diagnostic tools, the spread of mpox continues to be unchecked. Although related studies have demonstrated the high efficiency of deep learning models in diagnosing mpox, key aspects such as model inference speed and parameter size have always been overlooked. Herein, an ultrafast and ultralight network named Fast-MpoxNet is proposed. Fast-MpoxNet, with only 0.27M parameters, can process input images at 68 frames per second (FPS) on the CPU. To detect subtle image differences and optimize model parameters better, Fast-MpoxNet incorporates an attention-based feature fusion module and a multiple auxiliary losses enhancement strategy. Experimental results indicate that Fast-MpoxNet, utilizing transfer learning and data augmentation, produces 98.40% classification accuracy for four classes on the mpox dataset. Furthermore, its Recall for early-stage mpox is 93.65%. Most importantly, an application system named Mpox-AISM V2 is developed, suitable for both personal computers and smartphones. Mpox-AISM V2 can rapidly and accurately diagnose mpox and can be easily deployed in various scenarios to offer the public real-time mpox diagnosis services. This work has the potential to mitigate future mpox outbreaks and pave the way for developing real-time diagnostic tools in the healthcare field., Comment: Accepted by Advanced Intelligent Systems

Datalog reasoning based on the semina\"ive evaluation strategy evaluates rules using traditional join plans, which often leads to redundancy and inefficiency in practice, especially when the rules are complex. Hypertree decompositions help identify efficient query plans and reduce similar redundancy in query answering. However, it is unclear how this can be applied to materialisation and incremental reasoning with recursive Datalog programs. Moreover, hypertree decompositions require additional data structures and thus introduce nonnegligible overhead in both runtime and memory consumption. In this paper, we provide algorithms that exploit hypertree decompositions for the materialisation and incremental evaluation of Datalog programs. Furthermore, we combine this approach with standard Datalog reasoning algorithms in a modular fashion so that the overhead caused by the decompositions is reduced. Our empirical evaluation shows that, when the program contains complex rules, the combined approach is usually significantly faster than the baseline approach, sometimes by orders of magnitude.

Branch-and-bound-based consensus maximization stands out due to its important ability of retrieving the globally optimal solution to outlier-affected geometric problems. However, while the discovery of such solutions caries high scientific value, its application in practical scenarios is often prohibited by its computational complexity growing exponentially as a function of the dimensionality of the problem at hand. In this work, we convey a novel, general technique that allows us to branch over an n-1 dimensional space for an n-dimensional problem. The remaining degree of freedom can be solved globally optimally within each bound calculation by applying the efficient interval stabbing technique. While each individual bound derivation is harder to compute owing to the additional need for solving a sorting problem, the reduced number of intervals and tighter bounds in practice lead to a significant reduction in the overall number of required iterations. Besides an abstract introduction of the approach, we present applications to four fundamental geometric computer vision problems: camera resectioning, relative camera pose estimation, point set registration, and rotation and focal length estimation. Through our exhaustive tests, we demonstrate significant speed-up factors at times exceeding two orders of magnitude, thereby increasing the viability of globally optimal consensus maximizers in online application scenarios., Comment: Accepted by IEEE Transactions on Pattern Analysis and Machine Intelligence, 2024