While end-to-end self-supervised learning with backpropagation (global BP-SSL) has become central for training modern AI systems, theories of local self-supervised learning (local-SSL) have struggled to build functional representations in deep neural networks. To establish a link between global and local rules, we first develop a theory for deep linear networks: we identify conditions for local-SSL algorithms (like Forward-forward or CLAPP) to implement exactly the same weight update as a global BP-SSL. Starting from the theoretical insights, we then develop novel variants of local-SSL algorithms to approximate global BP-SSL in deep non-linear convolutional neural networks. Variants that improve the similarity between gradient updates of local-SSL with those of global BP-SSL also show better performance on image datasets (CIFAR-10, STL-10, and Tiny ImageNet). The best local-SSL rule with the CLAPP loss function matches the performance of a comparable global BP-SSL with InfoNCE or CPC-like loss functions, and improves upon state-of-the-art for local SSL on these benchmarks.
In end-to-end autonomous driving,the motion prediction plays a pivotal role in ego-vehicle planning. However, existing methods often rely on globally aggregated motion features, ignoring the fact that planning decisions are primarily influenced by a small number of locally interacting agents. Failing to attend to these critical local interactions can obscure potential risks and undermine planning reliability. In this work, we propose FocalAD, a novel end-to-end autonomous driving framework that focuses on critical local neighbors and refines planning by enhancing local motion representations. Specifically, FocalAD comprises two core modules: the Ego-Local-Agents Interactor (ELAI) and the Focal-Local-Agents Loss (FLA Loss). ELAI conducts a graph-based ego-centric interaction representation that captures motion dynamics with local neighbors to enhance both ego planning and agent motion queries. FLA Loss increases the weights of decision-critical neighboring agents, guiding the model to prioritize those more relevant to planning. Extensive experiments show that FocalAD outperforms existing state-of-the-art methods on the open-loop nuScenes datasets and closed-loop Bench2Drive benchmar
Quantum nonlocality has different manifestations that, in general, are revealed by local measurements of the parts of a composite system. In this paper, we study nonlocality arising from a set of orthogonal states that cannot be perfectly distinguished by local operations and classical communication (LOCC). Such a set is deemed nonlocal, for a joint measurement on the whole system is necessary for perfect discrimination of the states with certainty. On the other hand, a set of orthogonal states that can be perfectly distinguished by LOCC is believed to be devoid of nonlocal properties. Here, we show that there exist orthogonal sets that are locally distinguishable but without local redundancy (i.e., they become nonorthogonal on discarding one or more subsystems) whose nonlocality can be activated by local measurements. In particular, a state chosen from such a set can be locally converted, with certainty, into another state, the identity of which can now only be ascertained by global measurement and no longer by LOCC. In other words, a locally distinguishable set without local redundancy may be locally converted into a locally indistinguishable set with certainty. We also suggest a
The locality of thermal quantum states has emerged as a key input for applications to thermalization, response theory, and efficient simulability. Locality is either captured by the decay of correlations or by local indistinguishability, which allows to approximate local expectation values by those of local thermal states. Most techniques for deriving locality bounds deteriorate at small temperature, a physically highly relevant regime and so it is of interest to identify conditions for uniform-in-temperature bounds. Here we prove that a class of weakly interacting quantum Hamiltonians satisfies exponential decay of correlations and local indistinguishability uniformly in the temperature. The proof uses a low-temperature cluster expansion and a quantum version of a probabilistic swapping trick developed by the first author and Cao (Ann. Probab. 53, 2025) in the context of lattice gauge theories.
Out-of-Distribution (OOD) detection, aiming to distinguish outliers from known categories, has gained prominence in practical scenarios. Recently, the advent of vision-language models (VLM) has heightened interest in enhancing OOD detection for VLM through few-shot tuning. However, existing methods mainly focus on optimizing global prompts, ignoring refined utilization of local information with regard to outliers. Motivated by this, we freeze global prompts and introduce Local-Prompt, a novel coarse-to-fine tuning paradigm to emphasize regional enhancement with local prompts. Our method comprises two integral components: global prompt guided negative augmentation and local prompt enhanced regional regularization. The former utilizes frozen, coarse global prompts as guiding cues to incorporate negative augmentation, thereby leveraging local outlier knowledge. The latter employs trainable local prompts and a regional regularization to capture local information effectively, aiding in outlier identification. We also propose regional-related metric to empower the enrichment of OOD detection. Moreover, since our approach explores enhancing local prompts only, it can be seamlessly integra
Cumulative observables often exhibit saturation in systems involving propagation or spreading with local dissipation. This work shows that bounded cumulative response follows directly from local linear relaxation. Linear cumulative observables accumulated over the lifetime of a relaxing signal are limited by a scale set by the relaxation time, independent of geometry, dimensionality, or microscopic transport dynamics. When relaxation is mapped to space through transport or spreading, this temporal bound yields a corresponding spatial saturation scale determined by the transport law. The result shows that cumulative saturation follows directly from exponential local relaxation and does not depend on the specific transport mechanism.
We utilize the theory of non-Kahler loci by S. Boucksom to construct an integral 2-cohomology class whose restriction to a general fiber is big, and then construct a relatively big line bundle via the exponential sequence. This leads to a local Moishezonness criterion for fibrations whose total spaces are in Fujiki class C, generalizing the bimeromorphic version of F. Campana's local projectivity theorem. We further combine a similar idea with the singular Demailly-Paun theorem by T. Collins-V. Tosatti to obtain a local projectivity criterion for fibrations from compact Kahler manifolds, yielding a new proof and a generalization of F. Campana's local projectivity theorem.
A locally testable semigroup S is a semigroup with the property that for some nonnegative integer k, called the order or level of local testability, two words u and v in some set of generators for semigroup S are equal in the semigroup if (1) the prefix and suffix of the words of length k coincide, and (2) the set of intermediate substrings of length k of the words coincide. The local testability problem for semigroups is, given a finite semigroup, to decide, if the semigroup is locally testable or not. Recently, we introduced a polynomial time algorithm for the local testability problem and to find the level of local testability for semigroups based on our previous description of identities of $k$-testable semigroups and the structure of locally testable semigroups. The first part of the algorithm we introduce solves the local testability problem. The second part of the algorithm finds the order of local testability of a semigroup. The algorithm is of quadratic order where n is the order of the semigroup.
Local feature matching enjoys wide-ranging applications in the realm of computer vision, encompassing domains such as image retrieval, 3D reconstruction, and object recognition. However, challenges persist in improving the accuracy and robustness of matching due to factors like viewpoint and lighting variations. In recent years, the introduction of deep learning models has sparked widespread exploration into local feature matching techniques. The objective of this endeavor is to furnish a comprehensive overview of local feature matching methods. These methods are categorized into two key segments based on the presence of detectors. The Detector-based category encompasses models inclusive of Detect-then-Describe, Joint Detection and Description, Describe-then-Detect, as well as Graph Based techniques. In contrast, the Detector-free category comprises CNN Based, Transformer Based, and Patch Based methods. Our study extends beyond methodological analysis, incorporating evaluations of prevalent datasets and metrics to facilitate a quantitative comparison of state-of-the-art techniques. The paper also explores the practical application of local feature matching in diverse domains such a
We investigate the asymptotic properties of permutations drawn from the Luce model, a natural probabilistic framework in which permutations are generated sequentially by sampling without replacement, with selection probabilities proportional to prescribed positive weights. These permutations arise in applications such as ranking models, the Tsetlin library, and related Markov processes. Under minimal assumptions on the weights, we establish a permuton limit theorem describing the global behavior of Luce-distributed permutations and derive an explicit density of the limiting permuton. We further compute limiting pattern densities and analyze the differences between exact Luce permutations and their permuton approximations. We also study the local convergence of these permutations, proving a quenched Benjamini--Schramm limit and a central limit theorem for consecutive pattern occurrences. Finally, we prove a central limit theorem for the number of inversions.
Much of the theory of entanglement concerns the transformations that are possible to a state under local operations with classical communication (LOCC); however, this set of operations is complicated and difficult to describe mathematically. An idea which has proven very useful is that of the {\it entanglement monotone}: a function of the state which is invariant under local unitary transformations and always decreases (or increases) on average after any local operation. In this paper we look on LOCC as the set of operations generated by {\it infinitesimal local operations}, operations which can be performed locally and which leave the state little changed. We show that a necessary and sufficient condition for a function of the state to be an entanglement monotone under local operations that do not involve information loss is that the function be a monotone under infinitesimal local operations. We then derive necessary and sufficient differential conditions for a function of the state to be an entanglement monotone. We first derive two conditions for local operations without information loss, and then show that they can be extended to more general operations by adding the requireme
I discuss Local Group galaxies from the perspective of external galaxies that define benchmark scaling relations. Making use of this information leads to a model for the Milky Way that includes bumps and wiggles due to spiral arms. This model reconciles the terminal velocities observed in the interstellar medium with the rotation curve derived from stars, correctly predicts the gradual decline of the outer rotation curve ($dV/dR = -1.7\;\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{kpc}^{-1}$), and extrapolates well out to 50 kpc. Rotationally supported Local Group galaxies are in excellent agreement with the baryonic Tully-Fisher relation. Pressure supported dwarfs that are the most likely to be in dynamical equilibrium also align with this relation. Local Group galaxies thus appear to be normal members of the low redshift galaxy population. There is, however, a serious tension between the dynamical masses of the Milky Way and M31 ($M_{200} \approx 1.4$ and $1.6 \times 10^{12}\;\mathrm{M}_{\odot}$, respectively) and those expected from the stellar mass-halo mass relation of abundance matching ($M_{200} \approx 3$ and $20 \times 10^{12}\;\mathrm{M}_{\odot}$, respectively).
We prove the absolute convergence, functional equations and meromorphic continuation of local intertwining periods on parabolically induced representations of finite length for certain symmetric spaces over local fields of characteristic zero, including Galois pairs as well as pairs of Prasad and Takloo-Bighash type. Furthermore, for a general symmetric space we prove a sufficient condition for distinction of an induced representation in terms of distinction of its inducing data. Both results generalize previous results of the first two named authors. In particular, for both we remove a boundedness assumption on the inducing data and for the second we further remove any assumption on the symmetric space. Moreover, when the inducing representation is uniformly bounded, we extend the field of cofficients from p-adic to any local field of characteristic zero. In fact this extension holds for all finite length representations under a natural generic irreducibility assumption for parabolic induction. In the case of p-adic symmetric spaces, combined with the necessary conditions for distinction that follow from the geometric lemma, this provides a necessary and sufficient condition for d
In this article, we establish that any uniformly local Mizoguchi-Takahashi contraction is actually a set-valued contraction due to Feng and Liu on a metrically convex complete metric space. Through an example, we demonstrate that this result need not hold on any arbitrary metric space. Furthermore, when the metric space is compact, we derive that any Mizoguchi-Takahashi local contraction and Nadler local contraction are equivalent. Moreover, a result related to invariant best approximation is established.
Local feature matching is challenging due to textureless and repetitive patterns. Existing methods focus on using appearance features and global interaction and matching, while the importance of geometry priors in local feature matching has not been fully exploited. Different from these methods, in this paper, we delve into the importance of geometry prior and propose Structured Epipolar Matcher (SEM) for local feature matching, which can leverage the geometric information in an iterative matching way. The proposed model enjoys several merits. First, our proposed Structured Feature Extractor can model the relative positional relationship between pixels and high-confidence anchor points. Second, our proposed Epipolar Attention and Matching can filter out irrelevant areas by utilizing the epipolar constraint. Extensive experimental results on five standard benchmarks demonstrate the superior performance of our SEM compared to state-of-the-art methods. Project page: https://sem2023.github.io.
In this paper, we present an overview of the development of one of the most dynamic areas of mathematics today: local differential operators of non-integer order. The underlying question is whether we are witnessing a period of consolidation and are on the way to a general formulation of this fascinating topic.
After extending the theory of Rankin-Selberg local factors to pairs of $\ell$-modular representations of Whittaker type, of general linear groups over a non-archimedean local field, we study the reduction modulo $\ell$ of $\ell$-adic local factors and their relation to these $\ell$-modular local factors. While the $\ell$-modular local $γ$-factor we associate to such a pair turns out to always coincide with the reduction modulo $\ell$ of the $\ell$-adic $γ$-factor of any Whittaker lifts of this pair, the local $L$-factor exhibits a more interesting behaviour; always dividing the reduction modulo-$\ell$ of the $\ell$-adic $L$-factor of any Whittaker lifts, but with the possibility of a strict division occurring. In our main results, we completely describe $\ell$-modular $L$-factors in the generic case. We obtain two simple to state nice formulae: Let $π,π'$ be generic $\ell$-modular representations; then, writing $π_b,π'_b$ for their banal parts, we have \[L(X,π,π')=L(X,π_b,π_b').\] Using this formula, we obtain the inductivity relations for local factors of generic representations. Secondly, we show that \[L(X,π,π')=\mathbf{GCD}(r_{\ell}(L(X,τ,τ'))),\] where the divisor is over all
We introduce a novel regularization for localizing an elastic-energy-driven deformation to only those regions being manipulated by the user. Our local deformation features a natural region of influence, which is automatically adaptive to the geometry of the shape, the size of the deformation and the elastic energy in use. We further propose a three-block ADMM-based optimization to efficiently minimize the energy and achieve interactive frame rates. Our approach avoids the artifacts of other alternative methods, is simple and easy to implement, does not require tedious control primitive setup and generalizes across different dimensions and elastic energies. We demonstrates the effectiveness and efficiency of our localized deformation tool through a variety of local editing scenarios, including 1D, 2D, 3D elasticity and cloth deformation.
In this paper we study the novel notion of thin polytopes: lattice polytopes whose local $h^*$-polynomials vanish. The local $h^*$-polynomial is an important invariant in modern Ehrhart theory. Its definition goes back to Stanley with fundamental results achieved by Karu, Borisov & Mavlyutov, Schepers, and Katz & Stapledon. The study of thin simplices was originally proposed by Gelfand, Kapranov and Zelevinsky, where in this case the local $h^*$-polynomial simply equals its so-called box polynomial. Our main results are the complete classification of thin polytopes up to dimension 3 and the characterization of thinness for Gorenstein polytopes. The paper also includes an introduction to the local $h^*$-polynomial with a survey of previous results.
We investigate the local geometry of a pair of independent contact structures on 3-manifolds under maps that independently preserve each contact structure. We discover that such maps are homotheties on the contact 1-forms and we discover differential invariants associated to such structures under these equivalences. This allows us to generalize the notion of contact circles and (equilateral) hyperbolas to contact ellipses and hyperbolas. Moreover, these invariants may sometimes be used to define a complete local normal form and in at least one case are related to symplectic structures through a natural $e$-structure on a bundle arising from the Cartan equivalence method. Finally, there is a type of natural Riemannian metric (but not necessarily the well-known associated contact metric) and we discover certain curvatures may be written in terms of the bi-contact differential invariants.