text
stringlengths 137
2.43k
| field
stringclasses 2
values | label
int64 0
1
|
|---|---|---|
Title: Domination structure in 3-connected graphs
Abstract: From a recent perspective, the structure of a 3-connected graph is studied in this paper. It stipulates the minimum dominating set of a 3-connected graph. Also, we count the number of structures, as a consequence, the upper bound is obtained. By it, the minimum dominating set of a 3-connected graph is determined in polynomial time. | math | 1 |
Title: Directional flow in perivascular networks: Mixed finite elements for reduced-dimensional models on graphs
Abstract: The flow of cerebrospinal fluid through the perivascular spaces of the brain is believed to play a crucial role in eliminating toxic waste proteins. While the driving forces of this flow have been enigmatic, experiments have shown that arterial wall motion is central. In this work, we present a network model for simulating pulsatile fluid flow in perivascular networks. We establish the well-posedness of this model in the primal and dual mixed variational settings, and show how it can be discretized using mixed finite elements. Further, we utilize this model to investigate fundamental questions concerning the physical mechanisms governing perivascular fluid flow. Notably, our findings reveal that arterial pulsations can induce directional flow in branching perivascular networks. | math | 0 |
Title: Radon-type transforms for holomorphic and Hermitian monogenic functions
Abstract: The standard Radon transform of holomorphic functions is not always well defined, as the integration of such functions over planes may not converge. In this paper, we introduce new Radon-type transforms of co-(real)dimension $2$ for harmonic and holomorphic functions on the unit ball. These transforms are abstractly defined as orthogonal projections onto spaces of complex harmonic and holomorphic plane waves, respectively. The inversion formulas are derived based on the dual transform, while the latter is defined as an integration on a complex Stiefel manifold. Our transforms are extended to the Fock space and give rise to a new transform defined on the entire $L^{2}(\mathbb{R}^{n})$ through the Segal-Bargmann transform. Furthermore, we develop these transforms for Hermitian monogenic functions on the unit ball, thereby refining the Szeg\"o-Radon transform for monogenic functions introduced by Colombo, Sabadini and Sommen. | math | 0 |
Title: Dispersive decay of small data solutions for the KdV equation
Abstract: We consider the Korteweg-de Vries (KdV) equation, and prove that small localized data yields solutions which have dispersive decay on a quartic time-scale. This result is optimal, in view of the emergence of solitons at quartic time, as predicted by inverse scattering theory. | math | 1 |
Title: Reuse-Aware Cache Partitioning Framework for Data-Sharing Multicore Systems
Abstract: Multi-core processors improve performance, but they can create unpredictability owing to shared resources such as caches interfering. Cache partitioning is used to alleviate the Worst-Case Execution Time (WCET) estimation by isolating the shared cache across each thread to reduce interference. It does, however, prohibit data from being transferred between parallel threads running on different cores. In this paper we present (SRCP) a cache replacement mechanism for partitioned caches that is aware of data being shared across threads, prevents shared data from being replicated across partitions and frequently used data from being evicted from caches. Our technique outperforms TA-DRRIP and EHC, which are existing state-of-the-art cache replacement algorithms, by 13.34% in cache hit-rate and 10.4% in performance over LRU (least recently used) cache replacement policy. | cs | 1 |
Title: Norm inflation for the Boussinesq system
Abstract: We prove the norm inflation phenomena for the Boussinesq system on $\mathbb T^3$. For arbitrarily small initial data $(u_0,\rho_0)$ in the negative-order Besov spaces $\dot{B}^{-1}_{\infty, \infty} \times \dot{B}^{-1}_{\infty, \infty}$, the solution can become arbitrarily large in a short time. Such largeness can be detected in $\rho$ in Besov spaces of any negative order: $\dot{B}^{-s}_{\infty, \infty}$ for any $s>0$. Notice that our initial data space is scaling critical for $u$ and is scaling subcritical for $\rho$. | math | 1 |
Title: Deep Learning based Multi-Label Image Classification of Protest Activities
Abstract: With the rise of internet technology amidst increasing rates of urbanization, sharing information has never been easier thanks to globally-adopted platforms for digital communication. The resulting output of massive amounts of user-generated data can be used to enhance our understanding of significant societal issues particularly for urbanizing areas. In order to better analyze protest behavior, we enhanced the GSR dataset and manually labeled all the images. We used deep learning techniques to analyze social media data to detect social unrest through image classification, which performed good in predict multi-attributes, then also used map visualization to display protest behaviors across the country. | cs | 1 |
Title: Distributed convergence detection based on global residual error under asynchronous iterations
Abstract: Convergence of classical parallel iterations is detected by performing a reduction operation at each iteration in order to compute a residual error relative to a potential solution vector. To efficiently run asynchronous iterations, blocking communication requests are avoided, which makes it hard to isolate and handle any global vector. While some termination protocols were proposed for asynchronous iterations, only very few of them are based on global residual computation and guarantee effective convergence. But the most effective and efficient existing solutions feature two reduction operations, which constitutes an important factor of termination delay. In this paper, we present new, non-intrusive, protocols to compute a residual error under asynchronous iterations, requiring only one reduction operation. Various communication models show that some heuristics can even be introduced and formally evaluated. Extensive experiments with up to 5600 processor cores confirm the practical effectiveness and efficiency of our approach. | cs | 0 |
Title: Approximating Single-Source Personalized PageRank with Absolute Error Guarantees
Abstract: Personalized PageRank (PPR) is an extensively studied and applied node proximity measure in graphs. For a pair of nodes $s$ and $t$ on a graph $G=(V,E)$, the PPR value $\pi(s,t)$ is defined as the probability that an $\alpha$-discounted random walk from $s$ terminates at $t$, where the walk terminates with probability $\alpha$ at each step. We study the classic Single-Source PPR query, which asks for PPR approximations from a given source node $s$ to all nodes in the graph. Specifically, we aim to provide approximations with absolute error guarantees, ensuring that the resultant PPR estimates $\hat{\pi}(s,t)$ satisfy $\max_{t\in V}\big|\hat{\pi}(s,t)-\pi(s,t)\big|\le\varepsilon$ for a given error bound $\varepsilon$. We propose an algorithm that achieves this with high probability, with an expected running time of - $\widetilde{O}\big(\sqrt{m}/\varepsilon\big)$ for directed graphs, where $m=|E|$; - $\widetilde{O}\big(\sqrt{d_{\mathrm{max}}}/\varepsilon\big)$ for undirected graphs, where $d_{\mathrm{max}}$ is the maximum node degree in the graph; - $\widetilde{O}\left(n^{\gamma-1/2}/\varepsilon\right)$ for power-law graphs, where $n=|V|$ and $\gamma\in\left(\frac{1}{2},1\right)$ is the extent of the power law. These sublinear bounds improve upon existing results. We also study the case when degree-normalized absolute error guarantees are desired, requiring $\max_{t\in V}\big|\hat{\pi}(s,t)/d(t)-\pi(s,t)/d(t)\big|\le\varepsilon_d$ for a given error bound $\varepsilon_d$, where the graph is undirected and $d(t)$ is the degree of node $t$. We give an algorithm that provides this error guarantee with high probability, achieving an expected complexity of $\widetilde{O}\left(\sqrt{\sum_{t\in V}\pi(s,t)/d(t)}\big/\varepsilon_d\right)$. This improves over the previously known $O(1/\varepsilon_d)$ complexity. | cs | 0 |
Title: Capacity of Lorentzian polynomials and distance to binomial distributions
Abstract: In this paper we study the capacity of Lorentzian polynomials. We give a new proof of a theorem of Br\"and\'en, Leake and Pak. Our approach is probabilistic in nature and uses a lemma about a certain distance of binomial distributions to distributions with fixed expected value. | math | 1 |
Title: Power-law bounds for increasing subsequences in Brownian separable permutons and homogeneous sets in Brownian cographons
Abstract: The Brownian separable permutons are a one-parameter family -- indexed by $p\in(0,1)$ -- of universal limits of random constrained permutations. We show that for each $p\in (0,1)$, there are explicit constants $1/2 < \alpha_*(p) \leq \beta^*(p) < 1$ such that the length of the longest increasing subsequence in a random permutation of size $n$ sampled from the Brownian separable permuton is between $n^{\alpha_*(p) - o(1)}$ and $n^{\beta^*(p) + o(1)}$ with probability tending to 1 as $n\to\infty$. In the symmetric case $p=1/2$, we have $\alpha_*(p) \approx 0.812$ and $\beta^*(p)\approx 0.975$. We present numerical simulations which suggest that the lower bound $\alpha_*(p)$ is close to optimal in the whole range $p\in(0,1)$. Our results work equally well for the closely related Brownian cographons. In this setting, we show that for each $p\in (0,1)$, the size of the largest clique (resp. independent set) in a random graph on $n$ vertices sampled from the Brownian cographon is between $n^{\alpha_*(p) - o(1)}$ and $n^{\beta^*(p) + o(1)}$ (resp. $n^{\alpha_*(1-p) - o(1)}$ and $n^{\beta^*(1-p) + o(1)}$) with probability tending to 1 as $n\to\infty$. Our proofs are based on the analysis of a fragmentation process embedded in a Brownian excursion introduced by Bertoin (2002). We expect that our techniques can be extended to prove similar bounds for uniform separable permutations and uniform cographs. | math | 0 |
Title: A convergence result for a local planning problem for mean field games and rigorous proof of a Freidlin-Ventchel-type Large Deviations Principle for the $1+1$ KPZ equation
Abstract: We prove the convergence of a viscous approximation to an one dimensional local mean field type planning problem with singular initial and terminal measures. Then we use this result to give a rigorous proof to a Freidlin-Ventchel-type Large Deviations Principle for the height of the $1+1$ KPZ equation. | math | 0 |
Title: On the Stability Functional for Conservation Laws
Abstract: This note is devoted to the explicit construction of a functional defined on all pairs of $\L1$ functions with small total variation, which is equivalent to the $\L1$ distance and non increasing along the trajectories of a given system of conservation laws. Two different constructions are provided, yielding an extension of the original stability functional by Bressan, Liu and Yang. | math | 1 |
Title: Large Language Model Capabilities in Perioperative Risk Prediction and Prognostication
Abstract: We investigate whether general-domain large language models such as GPT-4 Turbo can perform risk stratification and predict post-operative outcome measures using a description of the procedure and a patient's clinical notes derived from the electronic health record. We examine predictive performance on 8 different tasks: prediction of ASA Physical Status Classification, hospital admission, ICU admission, unplanned admission, hospital mortality, PACU Phase 1 duration, hospital duration, and ICU duration. Few-shot and chain-of-thought prompting improves predictive performance for several of the tasks. We achieve F1 scores of 0.50 for ASA Physical Status Classification, 0.81 for ICU admission, and 0.86 for hospital mortality. Performance on duration prediction tasks were universally poor across all prompt strategies. Current generation large language models can assist clinicians in perioperative risk stratification on classification tasks and produce high-quality natural language summaries and explanations. | cs | 0 |
Title: Singular SPDEs in domains with boundaries
Abstract: We study spaces of modelled distributions with singular behaviour near the boundary of a domain that, in the context of the theory of regularity structures, allow one to give robust solution theories for singular stochastic PDEs with boundary conditions. The calculus of modelled distributions established in Hairer (Invent. Math. 198, 2014) is extended to this setting. We formulate and solve fixed point problems in these spaces with a class of kernels that is sufficiently large to cover in particular the Dirichlet and Neumann heat kernels. These results are then used to provide solution theories for the KPZ equation with Dirichlet and Neumann boundary conditions and for the 2D generalised parabolic Anderson model with Dirichlet boundary conditions. In the case of the KPZ equation with Neumann boundary conditions, we show that, depending on the class of mollifiers one considers, a "boundary renormalisation" takes place. In other words, there are situations in which a certain boundary condition is applied to an approximation to the KPZ equation, but the limiting process is the Hopf-Cole solution to the KPZ equation with a different boundary condition. | math | 1 |
Title: Characters and chromatic symmetric functions
Abstract: Let $P$ be a poset, $inc(P)$ its incomparability graph, and $X_{inc(P)}$ the corresponding chromatic symmetric function, as defined by Stanley in {\em Adv. Math.}, {\bf 111} (1995) pp.~166--194. Certain conditions on $P$ imply that the expansions of $X_{inc(P)}$ in standard symmetric function bases yield coefficients which have simple combinatorial interpretations. By expressing these coefficients as character evaluations, we extend several of these interpretations to {\em all} posets $P$. Consequences include new combinatorial interpretations of the permanent and other immanants of totally nonnegative matrices, and of the sum of elementary coefficients in the Shareshian-Wachs chromatic quasisymmetric function $X_{inc(P),q}$ when $P$ is a unit interval order. | math | 1 |
Title: Strong Equivalence Relations for Iterated Models
Abstract: The Iterated Immediate Snapshot model (IIS), due to its elegant geometrical representation, has become standard for applying topological reasoning to distributed computing. Its modular structure makes it easier to analyze than the more realistic (non-iterated) read-write Atomic-Snapshot memory model (AS). It is known that AS and IIS are equivalent with respect to \emph{wait-free task} computability: a distributed task is solvable in AS if and only if it solvable in IIS. We observe, however, that this equivalence is not sufficient in order to explore solvability of tasks in \emph{sub-models} of AS (i.e. proper subsets of its runs) or computability of \emph{long-lived} objects, and a stronger equivalence relation is needed. In this paper, we consider \emph{adversarial} sub-models of AS and IIS specified by the sets of processes that can be \emph{correct} in a model run. We show that AS and IIS are equivalent in a strong way: a (possibly long-lived) object is implementable in AS under a given adversary if and only if it is implementable in IIS under the same adversary. %This holds whether the object is one-shot or long-lived. Therefore, the computability of any object in shared memory under an adversarial AS scheduler can be equivalently investigated in IIS. | cs | 1 |
Title: Sharp weighted inequalities for iterated commutators of a class of multilinear operators
Abstract: In this paper, the sharp quantitative weighted bounds for the iterated commutators of a class of multilinear operators were systematically studied. This class of operators contains multilinear Calder\'{o}n-Zygmund operators, multilinear Fourier integral operators, and multilinear Littlewood-Paley square operators as its typical examples. These were done only under two pretty much general assumptions of pointwise sparse domination estimates. We first established local decay estimates and quantitative weak $A_\infty$ decay estimates for iterated commutators of this class of operators. Then, we considered the corresponding Coifman-Fefferman inequalities and the mixed weak type estimates associated with Sawyer's conjecture. Beyond that, the Fefferman-Stein inequalities with respect to arbitrary weights and weighted modular inequalities were also given. As applications, it was shown that all the conclusions aforementioned can be applied to multilinear $\omega$-Calder\'{o}n-Zygmund operators, multilinear maximal singular integral operators, multilinear pseudo-differential operators, Stein's square functions, and higher order Calder\'{o}n commutators. | math | 0 |
Title: Morita Equivalence and Spectral Triples on Noncommutative Orbifolds
Abstract: Let $G$ be a finite group. Noncommutative geometry of unital $G$-algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical orbifold groupoids, a Morita equivalence for the crossed product spectral triples is developed. Noncommutative orbifolds are Morita equivalence classes of the crossed product spectral triples. As a special case of this Morita theory one can study freeness of the $G$-action on the noncommutative level. In the case of a free action, the crossed product formalism reduced to the usual spectral triple formalism on the algebra of $G$-invariant functions. | math | 1 |
Title: Quantum Team Logic and Bell's Inequalities
Abstract: A logical approach to Bell's Inequalities of quantum mechanics has been introduced by Abramsky and Hardy [2]. We point out that the logical Bell's Inequalities of [2] are provable in the probability logic of Fagin, Halpern and Megiddo [4]. Since it is now considered empirically established that quantum mechanics violates Bell's Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bell's Inequalities are not provable, and prove a Completeness Theorem for this logic. For this end we generalise the team semantics of dependence logic [7] first to probabilistic team semantics, and then to what we call quantum team semantics. | math | 1 |
Title: Balancing Specialization and Adaptation in a Transforming Scientific Landscape
Abstract: How scientists navigate between the need to capitalize on their prior knowledge by specializing, and the urge to adapt to evolving research opportunities? Drawing from diverse perspectives on adaptation, in particular from institutional change and cultural evolution, this paper proposes a Bayesian model of the evolution of scientists' research portfolios in response to transformations in their field. The model relies on scientific abstracts and authorship data to evaluate the influence of intellectual, social, and institutional resources on scientists' trajectories within a cohort of $2\,195$ high-energy physicists between 2000 and 2019. The reallocation of research efforts is shown to be structured by learning costs, thus enhancing the utility of the scientific capital disseminated among scientists. Two dimensions of social capital, namely ``diversity'' and ``power'', have opposite effects on the magnitude of change in scientists' research interests: while ``diversity'' disrupts and expands research interests, ``power'' stabilizes physicists' research agendas -- as does institutional stability. Social capital plays a more crucial role in shifts between cognitively distant research areas. | cs | 0 |
Title: A Novel Dual-Stage Evolutionary Algorithm for Finding Robust Solutions
Abstract: In robust optimization problems, the magnitude of perturbations is relatively small. Consequently, solutions within certain regions are less likely to represent the robust optima when perturbations are introduced. Hence, a more efficient search process would benefit from increased opportunities to explore promising regions where global optima or good local optima are situated. In this paper, we introduce a novel robust evolutionary algorithm named the dual-stage robust evolutionary algorithm (DREA) aimed at discovering robust solutions. DREA operates in two stages: the peak-detection stage and the robust solution-searching stage. The primary objective of the peak-detection stage is to identify peaks in the fitness landscape of the original optimization problem. Conversely, the robust solution-searching stage focuses on swiftly identifying the robust optimal solution using information obtained from the peaks discovered in the initial stage. These two stages collectively enable the proposed DREA to efficiently obtain the robust optimal solution for the optimization problem. This approach achieves a balance between solution optimality and robustness by separating the search processes for optimal and robust optimal solutions. Experimental results demonstrate that DREA significantly outperforms five state-of-the-art algorithms across 18 test problems characterized by diverse complexities. Moreover, when evaluated on higher-dimensional robust optimization problems (100-$D$ and 200-$D$), DREA also demonstrates superior performance compared to all five counterpart algorithms. | cs | 0 |
Title: Fast and Continual Learning for Hybrid Control Policies using Generalized Benders Decomposition
Abstract: Hybrid model predictive control with both continuous and discrete variables is widely applicable to robotic control tasks, especially those involving contact with the environment. Due to the combinatorial complexity, the solving speed of hybrid MPC can be insufficient for real-time applications. In this paper, we proposed a hybrid MPC solver based on Generalized Benders Decomposition (GBD). The algorithm enumerates and stores cutting planes online inside a finite buffer. After a short cold-start phase, the stored cuts provide warm-starts for the new problem instances to enhance the solving speed. Despite the disturbance and randomly changing environment, the solving speed maintains. Leveraging on the sparsity of feasibility cuts, we also propose a fast algorithm for Benders master problems. Our solver is validated through controlling a cart-pole system with randomly moving soft contact walls, and a free-flying robot navigating around obstacles. The results show that with significantly less data than previous works, the solver reaches competitive speeds to the off-the-shelf solver Gurobi despite the Python overhead. | cs | 0 |
Title: An Experimental Study of ILP Formulations for the Longest Induced Path Problem
Abstract: Given a graph $G=(V,E)$, the longest induced path problem asks for a maximum cardinality node subset $W\subseteq V$ such that the graph induced by $W$ is a path. It is a long established problem with applications, e.g., in network analysis. We propose novel integer linear programming (ILP) formulations for the problem and discuss efficient implementations thereof. Comparing them with known formulations from literature, we prove that they are beneficial in theory, yielding stronger relaxations. Moreover, our experiments show their practical superiority. | cs | 1 |
Title: On a class of stochastic hyperbolic equations with double characteristics
Abstract: We study the effect of Gaussian perturbations on a hyperbolic partial differential equation with double characteristics in two spatial dimensions. The coefficients of our partial differential operator depend polynomially on the space variables, while the noise is additive, white in time and coloured in space. We provide a sufficient condition on the spectral measure of the covariance functional describing the noise that allows for the existence of a random field solution for the resulting stochastic partial differential equation. Our approach is based on explicit computations for the fundamental solution of the partial differential operator and its Fourier transform. | math | 1 |
Title: Representations of Lie groups
Abstract: These are notes of a graduate course on representations of non-compact semisimple Lie groups given by the author at MIT. | math | 0 |
Title: Where You Are Is Who You Are: User Identification by Matching Statistics
Abstract: Most users of online services have unique behavioral or usage patterns. These behavioral patterns can be exploited to identify and track users by using only the observed patterns in the behavior. We study the task of identifying users from statistics of their behavioral patterns. Specifically, we focus on the setting in which we are given histograms of users' data collected during two different experiments. We assume that, in the first dataset, the users' identities are anonymized or hidden and that, in the second dataset, their identities are known. We study the task of identifying the users by matching the histograms of their data in the first dataset with the histograms from the second dataset. In recent works, the optimal algorithm for this user identification task is introduced. In this paper, we evaluate the effectiveness of this method on three different types of datasets and in multiple scenarios. Using datasets such as call data records, web browsing histories, and GPS trajectories, we show that a large fraction of users can be easily identified given only histograms of their data; hence these histograms can act as users' fingerprints. We also verify that simultaneous identification of users achieves better performance compared to one-by-one user identification. We show that using the optimal method for identification gives higher identification accuracy than heuristics-based approaches in practical scenarios. The accuracy obtained under this optimal method can thus be used to quantify the maximum level of user identification that is possible in such settings. We show that the key factors affecting the accuracy of the optimal identification algorithm are the duration of the data collection, the number of users in the anonymized dataset, and the resolution of the dataset. We analyze the effectiveness of k-anonymization in resisting user identification attacks on these datasets. | cs | 1 |
Title: Efficient Checking of Timed Order Compliance Rules over Graph-encoded Event Logs
Abstract: Validation of compliance rules against process data is a fundamental functionality for business process management. Over the years, the problem has been addressed for different types of process data, i.e., process models, process event data at runtime, and event logs representing historical execution. Several approaches have been proposed to tackle compliance checking over process logs. These approaches have been based on different data models and storage technologies including relational databases, graph databases, and proprietary formats. Graph-based encoding of event logs is a promising direction that turns several process analytics tasks into queries on the underlying graph. Compliance checking is one class of such analysis tasks. In this paper, we argue that encoding log data as graphs alone is not enough to guarantee efficient processing of queries on this data. Efficiency is important due to the interactive nature of compliance checking. Thus, compliance checking would benefit from sub-linear scanning of the data. Moreover, as more data are added, e.g., new batches of logs arrive, the data size should grow sub-linearly to optimize both the space of storage and time for querying. We propose two encoding methods using graph representation, realized in Neo4J, and show the benefits of these encoding on a special class of queries, namely timed order compliance rules. Compared to a baseline encoding, our experiments show up to 5x speed up in the querying time as well as a 3x reduction in the graph size. | cs | 1 |
Title: Optimal Decay Estimates for the Radially Symmetric Compressible Navier-Stokes Equations
Abstract: We examine the large-time behaviour of solutions to the compressible Navier-Stokes equations under the assumption of radial symmetry. In particular, we calculate a fast time-decay estimate of the norm of the nonlinear part of the solution. This allows us to obtain a bound from below for the time-decay of the solution in $L^\infty$, proving that our decay estimate in that space is sharp. The decay rate is the same as that of the linear problem for curl-free flow. We also obtain an estimate for a scalar system related to curl-free solutions to the compressible Navier-Stokes equations in a weighted Lebesgue space. | math | 0 |
Title: Adams spectral sequences and Franke's algebraicity conjecture
Abstract: To any well-behaved homology theory we associate a derived $\infty$-category which encodes its Adams spectral sequence. As applications, we prove a conjecture of Franke on algebraicity of certain homotopy categories and establish homotopy-coherent monoidality of the Adams filtration. | math | 1 |
Title: Non-contractible closed geodesics on compact Finsler space forms without self-intersections
Abstract: Let $M=S^n/ \Gamma$ and $h \in \pi_1(M)$ be a non-trivial element of finite order $p$, where the integers $n, p\geq2$ and $\Gamma$ is a finite abelian group which acts on the sphere freely and isometrically. Therefore $M$ is diffeomorphic to a compact space form which is a typical non-simply connected manifold. For $\Gamma =\mathbb{Z}_2$, we obtain there are at least two non-contractible closed geodesics on $\mathbb{R}P^2$ whose lengths are bounded by geometry of the manifold from above. Moreover, suppose $g_0$ is standard Riemannian metric. We prove that there exists at least $n$ prime non-contractible closed geodesics on $(M,F)$ of prescribed class $[h]$ without self-intersections, provided $F^2 <(\frac{\lambda+1}{\lambda})^2 g_0$ and \[(\frac{\lambda}{\lambda+1})^2 < K \leq 1 \text{ for $n$ is odd or }\; 0<K \leq 1 \text{ for $n$ is even}, \] where $\lambda$ is the reversibility and $K$ is the flag curvature. Some results on stability are obtained. | math | 0 |
Title: Lookahead: An Inference Acceleration Framework for Large Language Model with Lossless Generation Accuracy
Abstract: As Large Language Models (LLMs) have made significant advancements across various tasks, such as question answering, translation, text summarization, and dialogue systems, the need for accuracy in information becomes crucial, especially for serious financial products serving billions of users like Alipay. To address this, Alipay has developed a Retrieval-Augmented Generation (RAG) system that grounds LLMs on the most accurate and up-to-date information. However, for a real-world product serving millions of users, the inference speed of LLMs becomes a critical factor compared to a mere experimental model. Hence, this paper presents a generic framework for accelerating the inference process, resulting in a substantial increase in speed and cost reduction for our RAG system, with lossless generation accuracy. In the traditional inference process, each token is generated sequentially by the LLM, leading to a time consumption proportional to the number of generated tokens. To enhance this process, our framework, named \textit{lookahead}, introduces a \textit{multi-branch} strategy. Instead of generating a single token at a time, we propose a \textit{Trie-based Retrieval} (TR) process that enables the generation of multiple branches simultaneously, each of which is a sequence of tokens. Subsequently, for each branch, a \textit{Verification and Accept} (VA) process is performed to identify the longest correct sub-sequence as the final output. Our strategy offers two distinct advantages: (1) it guarantees absolute correctness of the output, avoiding any approximation algorithms, and (2) the worst-case performance of our approach is equivalent to the conventional process. We conduct extensive experiments to demonstrate the significant improvements achieved by applying our inference acceleration framework. Code is avaliable: https://github.com/alipay/PainlessInferenceAcceleration. | cs | 0 |
Title: Closed categories vs. closed multicategories
Abstract: We prove that the 2-category of closed categories of Eilenberg and Kelly is equivalent to a suitable full 2-subcategory of the 2-category of closed multicategories. | math | 1 |
Title: Unsupervised Learning of the Total Variation Flow
Abstract: The total variation (TV) flow generates a scale-space representation of an image based on the TV functional. This gradient flow observes desirable features for images such as sharp edges and enables spectral, scale, and texture analysis. The standard numerical approach for TV flow requires solving multiple non-smooth optimisation problems. Even with state-of-the-art convex optimisation techniques, this is often prohibitively expensive and strongly motivates the use of alternative, faster approaches. Inspired by and extending the framework of physics-informed neural networks (PINNs), we propose the TVflowNET, a neural network approach to compute the solution of the TV flow given an initial image and a time instance. We significantly speed up the computation time by more than one order of magnitude and show that the TVflowNET approximates the TV flow solution with high fidelity. This is a preliminary report, more details are to follow. | cs | 1 |
Title: Conformal invariants of $3$-Braids and Counting Functions
Abstract: We consider a conformal invariant of braids, the extremal length with totally real horizontal boundary values $\lambda_{tr}$. The invariant descends to an invariant of elements of $\mathcal{B}_n\diagup\mathcal{Z}_n$, the braid group modulo its center. We prove that the number of elements of $\mathcal{B}_3\diagup\mathcal{Z}_3$ of positive $\lambda_{tr}$ grows exponentially. The estimate applies to obtain effective finiteness theorems in the spirit of the geometric Shafarevich conjecture over Riemann surfaces of second kind. As a corollary we obtain another proof of the exponential growth of the number of conjugacy classes of $\mathcal{B}_3\diagup\mathcal{Z}_3$ with positive entropy not exceeding $Y$. | math | 1 |
Title: Almost dominant generalized slices and convolution diagrams over them
Abstract: Let $G$ be a connected reductive complex algebraic group with a maximal torus $T$. We denote by $\Lambda$ the cocharacter lattice of $(T,G)$. Let $\Lambda^+ \subset \Lambda$ be the submonoid of dominant coweights. For $\lambda \in \Lambda^+,\,\mu \in \Lambda,\,\mu \leqslant \lambda$, in arXiv:1604.03625, authors defined a generalized transversal slice $\overline{\mathcal{W}}^\lambda_\mu$. This is an algebraic variety of the dimension $\langle 2\rho^{\vee}, \lambda-\mu \rangle$, where $2\rho^{\vee}$ is the sum of positive roots of $G$. In this paper, we construct an isomorphism $\overline{\mathcal{W}}^\lambda_\mu \simeq \overline{\mathcal{W}}^\lambda_{\mu^+} \times {\mathbb{A}}^{\langle 2\rho^{\vee},\, \mu^+-\mu\rangle}$ for $\mu \in \Lambda$ such that $\langle \alpha^{\vee},\mu\rangle \geqslant -1$ for any positive root $\alpha^{\vee}$, here $\mu^+ \in W\mu$ is the dominant representative in the Weyl group orbit of $\mu$. We consider the example when $\lambda$ is minuscule, $\mu \in W\lambda$ and describe natural coordinates, Poisson structure on $\overline{\mathcal{W}}^\lambda_\mu \simeq {\mathbb{A}}^{\langle 2\rho^\vee,\,\lambda-\mu \rangle}$ and its $T\times {\mathbb{C}}^\times$-character. We apply these results to compute $T \times {\mathbb{C}}^\times$-characters of tangent spaces at fixed points of convolution diagrams $\widetilde{\mathcal{W}}^{\underline{\lambda}}_\mu$ with minuscule $\lambda_i$. We also apply our results to construct open coverings by affine spaces of convolution diagrams $\widetilde{\mathcal{W}}^{\underline{\lambda}}_\mu$ over slices with $\mu$ such that $\langle \alpha^{\vee},\mu\rangle \geqslant -1$ for any positive root $\alpha^{\vee}$ and minuscule $\lambda_i$ and to compute Poincar\'e polynomials of such convolution diagrams $\widetilde{\mathcal{W}}^{\underline{\lambda}}_{\mu}$. | math | 1 |
Title: Hurewicz sets of reals without perfect subsets
Abstract: We show that even for subsets X of the real line which do not contain perfect sets, the Hurewicz property does not imply the property S1(Gamma,Gamma), asserting that for each countable family of open gamma-covers of X, there is a choice function whose image is a gamma-cover of X. This settles a problem of Just, Miller, Scheepers, and Szeptycki. Our main result also answers a question of Bartoszynski and Tsaban, and implies that for C_p(X), the conjunction of Sakai's strong countable fan tightness and the Reznichenko property does not imply Arhangelskii's property alpha_2. | math | 1 |
Title: Parametric robust structured control design
Abstract: We present a new approach to parametric robust controller design, where we compute controllers of arbitrary order and structure which minimize the worst-case $H_\infty$ norm over a pre-specified set of uncertain parameters. At the core of our method is a nonsmooth minimization method tailored to functions which are semi-infinite minima of smooth functions. A rich test bench and a more detailed example illustrate the potential of the technique, which can deal with complex problems involving multiple possibly repeated uncertain parameters. | math | 1 |
Title: Towards Multi-Objective High-Dimensional Feature Selection via Evolutionary Multitasking
Abstract: Evolutionary Multitasking (EMT) paradigm, an emerging research topic in evolutionary computation, has been successfully applied in solving high-dimensional feature selection (FS) problems recently. However, existing EMT-based FS methods suffer from several limitations, such as a single mode of multitask generation, conducting the same generic evolutionary search for all tasks, relying on implicit transfer mechanisms through sole solution encodings, and employing single-objective transformation, which result in inadequate knowledge acquisition, exploitation, and transfer. To this end, this paper develops a novel EMT framework for multiobjective high-dimensional feature selection problems, namely MO-FSEMT. In particular, multiple auxiliary tasks are constructed by distinct formulation methods to provide diverse search spaces and information representations and then simultaneously addressed with the original task through a multi-slover-based multitask optimization scheme. Each task has an independent population with task-specific representations and is solved using separate evolutionary solvers with different biases and search preferences. A task-specific knowledge transfer mechanism is designed to leverage the advantage information of each task, enabling the discovery and effective transmission of high-quality solutions during the search process. Comprehensive experimental results demonstrate that our MO-FSEMT framework can achieve overall superior performance compared to the state-of-the-art FS methods on 26 datasets. Moreover, the ablation studies verify the contributions of different components of the proposed MO-FSEMT. | cs | 0 |
Title: A Hybrid Neural Coding Approach for Pattern Recognition with Spiking Neural Networks
Abstract: Recently, brain-inspired spiking neural networks (SNNs) have demonstrated promising capabilities in solving pattern recognition tasks. However, these SNNs are grounded on homogeneous neurons that utilize a uniform neural coding for information representation. Given that each neural coding scheme possesses its own merits and drawbacks, these SNNs encounter challenges in achieving optimal performance such as accuracy, response time, efficiency, and robustness, all of which are crucial for practical applications. In this study, we argue that SNN architectures should be holistically designed to incorporate heterogeneous coding schemes. As an initial exploration in this direction, we propose a hybrid neural coding and learning framework, which encompasses a neural coding zoo with diverse neural coding schemes discovered in neuroscience. Additionally, it incorporates a flexible neural coding assignment strategy to accommodate task-specific requirements, along with novel layer-wise learning methods to effectively implement hybrid coding SNNs. We demonstrate the superiority of the proposed framework on image classification and sound localization tasks. Specifically, the proposed hybrid coding SNNs achieve comparable accuracy to state-of-the-art SNNs, while exhibiting significantly reduced inference latency and energy consumption, as well as high noise robustness. This study yields valuable insights into hybrid neural coding designs, paving the way for developing high-performance neuromorphic systems. | cs | 0 |
Title: On the 2D Yang-Mills/Hurwitz Correspondence
Abstract: In this paper, we show that in the large $N$ limit two-dimensional Yang-Mills theory with $U(N)$ gauge group becomes mixed Hurwitz theory, in the sense that the $1/N$ expansion of the chiral partition function receives contributions from both classical and monotone Hurwitz theory for all but finitely many compact orientable spacetimes. | math | 0 |
Title: Strong Products of Hypergraphs: Unique Prime Factorization Theorems and Algorithms
Abstract: It is well-known that all finite connected graphs have a unique prime factor decomposition (PFD) with respect to the strong graph product which can be computed in polynomial time. Essential for the PFD computation is the construction of the so-called Cartesian skeleton of the graphs under investigation. In this contribution, we show that every connected thin hypergraph H has a unique prime factorization with respect to the normal and strong (hypergraph) product. Both products coincide with the usual strong graph product whenever H is a graph. We introduce the notion of the Cartesian skeleton of hypergraphs as a natural generalization of the Cartesian skeleton of graphs and prove that it is uniquely defined for thin hypergraphs. Moreover, we show that the Cartesian skeleton of hypergraphs can be determined in O(|E|^2) time and that the PFD can be computed in O(|V|^2|E|) time, for hypergraphs H = (V,E) with bounded degree and bounded rank. | cs | 1 |
Title: Significance of Anatomical Constraints in Virtual Try-On
Abstract: The system of Virtual Try-ON (VTON) allows a user to try a product virtually. In general, a VTON system takes a clothing source and a person's image to predict the try-on output of the person in the given clothing. Although existing methods perform well for simple poses, in case of bent or crossed arms posture or when there is a significant difference between the alignment of the source clothing and the pose of the target person, these methods fail by generating inaccurate clothing deformations. In the VTON methods that employ Thin Plate Spline (TPS) based clothing transformations, this mainly occurs for two reasons - (1)~the second-order smoothness constraint of TPS that restricts the bending of the object plane. (2)~Overlaps among different clothing parts (e.g., sleeves and torso) can not be modeled by a single TPS transformation, as it assumes the clothing as a single planar object; therefore, disregards the independence of movement of different clothing parts. To this end, we make two major contributions. Concerning the bending limitations of TPS, we propose a human AnaTomy-Aware Geometric (ATAG) transformation. Regarding the overlap issue, we propose a part-based warping approach that divides the clothing into independently warpable parts to warp them separately and later combine them. Extensive analysis shows the efficacy of this approach. | cs | 0 |
Title: Good Things Come to Those Who Swap Objects on Paths
Abstract: We study a simple exchange market, introduced by Gourv\'{e}s, Lesca and Wilczynski (IJCAI-17), where every agent initially holds a single object. The agents have preferences over the objects, and two agents may swap their objects if they both prefer the object of the other agent. The agents live in an underlying social network that governs the structure of the swaps: Two agents can only swap their objects if they are adjacent. We investigate the REACHABLE OBJECT problem, which asks whether a given starting situation can ever lead, by means of a sequence of swaps, to a situation where a given agent obtains a given object. Our results answer several central open questions on the complexity of REACHABLE OBJECT. First, the problem is polynomial-time solvable if the social network is a path. Second, the problem is NP-hard on cliques and generalized caterpillars. Finally, we establish a three-versus-four dichotomy result for preference lists of bounded length: The problem is easy if all preference lists have length at most three, and the problem becomes NP-hard even if all agents have preference lists of length at most four. | cs | 1 |
Title: Enabling Digitalization in Modular Robotic Systems Integration
Abstract: Integrating robot systems into manufacturing lines is a time-consuming process. In the era of digitalization, the research and development of new technologies is crucial for improving integration processes. Numerous challenges, including the lack of standardization, as well as intricate stakeholder relationships, complicate the process of robotic systems integration. This process typically consists of acquisition, integration, and deployment of the robot systems. This thesis focuses on three areas that help automate and simplify robotic systems integration. In the first area, related to acquisition, a constraint-based configurator is demonstrated that resolves compatibility challenges between robot devices, and automates the configuration process. This reduces the risk of integrating incompatible devices and decreases the need for experts during the configuration phase. In the second area, related to integration, the interoperable modeling format, Unified Robot Description Format (URDF), is investigated, where a detailed analysis is performed, revealing significant inconsistencies and critical improvements. This format is widely used for kinematic modeling and 3D visualization of robots, and its models can be reused across simulation tools. Improving this format benefits a wide range of users, including robotics engineers, researchers, and students. In the third area, related to deployment, Digital Twins (DTs) for robot systems are explored, as these improve efficiency and reduce downtime. A comprehensive literature review of DTs is conducted, and a case study of modular robot systems is developed. This research can accelerate the adoption of DTs in the robotics industry. These insights and approaches improve the process of robotic systems integration, offering valuable contributions that future research can build upon, ultimately driving efficiency, and reducing costs. | cs | 0 |
Title: A survey on set-theoretical solutions to the pentagon equation
Abstract: This survey aims to collect the main results of the theory of the set-theoretical solutions to the pentagon equation obtained up to now in the literature. In particular, we present some classes of these solutions and raise some questions. | math | 0 |
Title: MULTI-CASE: A Transformer-based Ethics-aware Multimodal Investigative Intelligence Framework
Abstract: AI-driven models are increasingly deployed in operational analytics solutions, for instance, in investigative journalism or the intelligence community. Current approaches face two primary challenges: ethical and privacy concerns, as well as difficulties in efficiently combining heterogeneous data sources for multimodal analytics. To tackle the challenge of multimodal analytics, we present MULTI-CASE, a holistic visual analytics framework tailored towards ethics-aware and multimodal intelligence exploration, designed in collaboration with domain experts. It leverages an equal joint agency between human and AI to explore and assess heterogeneous information spaces, checking and balancing automation through Visual Analytics. MULTI-CASE operates on a fully-integrated data model and features type-specific analysis with multiple linked components, including a combined search, annotated text view, and graph-based analysis. Parts of the underlying entity detection are based on a RoBERTa-based language model, which we tailored towards user requirements through fine-tuning. An overarching knowledge exploration graph combines all information streams, provides in-situ explanations, transparent source attribution, and facilitates effective exploration. To assess our approach, we conducted a comprehensive set of evaluations: We benchmarked the underlying language model on relevant NER tasks, achieving state-of-the-art performance. The demonstrator was assessed according to intelligence capability assessments, while the methodology was evaluated according to ethics design guidelines. As a case study, we present our framework in an investigative journalism setting, supporting war crime investigations. Finally, we conduct a formative user evaluation with domain experts in law enforcement. Our evaluations confirm that our framework facilitates human agency and steering in security-sensitive applications. | cs | 0 |
Title: Co-compact Gabor systems on locally compact abelian groups
Abstract: In this work we extend classical structure and duality results in Gabor analysis on the euclidean space to the setting of second countable locally compact abelian (LCA) groups. We formulate the concept of rationally oversampling of Gabor systems in an LCA group and prove corresponding characterization results via the Zak transform. From these results we derive non-existence results for critically sampled continuous Gabor frames. We obtain general characterizations in time and in frequency domain of when two Gabor generators yield dual frames. Moreover, we prove the Walnut and Janssen representation of the Gabor frame operator and consider the Wexler-Raz biorthogonality relations for dual generators. Finally, we prove the duality principle for Gabor frames. Unlike most duality results on Gabor systems, we do not rely on the fact that the translation and modulation groups are discrete and co-compact subgroups. Our results only rely on the assumption that either one of the translation and modulation group (in some cases both) are co-compact subgroups of the time and frequency domain. This presentation offers a unified approach to the study of continuous and the discrete Gabor frames. | math | 1 |
Title: Classification of algebras of level two in the variety of nilpotent algebras and Leibniz algebras
Abstract: This paper is devoted to the description of complex finite-dimensional algebras of level two. We obtain the classification of algebras of level two in the variety of Leibniz algebras. It is shown that, up to isomorphism, there exist three Leibniz algebras of level two, one of which is solvable, and two of which are nilpotent. Moreover we describe all algebras of level two in the variety of nilpotent algebras. | math | 1 |
Title: Leader Election Problem Versus Pattern Formation Problem
Abstract: Leader election and arbitrary pattern formation are funda- mental tasks for a set of autonomous mobile robots. The former consists in distinguishing a unique robot, called the leader. The latter aims in arranging the robots in the plane to form any given pattern. The solv- ability of both these tasks turns out to be necessary in order to achieve more complex tasks. In this paper, we study the relationship between these two tasks in a model, called CORDA, wherein the robots are weak in several aspects. In particular, they are fully asynchronous and they have no direct means of communication. They cannot remember any previous observation nor computation performed in any previous step. Such robots are said to be oblivious. The robots are also uniform and anonymous, i.e, they all have the same program using no global parameter (such as an identity) allowing to differentiate any of them. Moreover, we assume that none of them share any kind of common coordinate mechanism or common sense of direction and we discuss the influence of a common handedness (i.e., chirality). In such a system, Flochini et al. proved in [11] that it is possible to elect a leader for n \geq 3 robots if it is possible to form any pattern for n \geq 3. In this paper, we show that the converse is true for n \geq 4 when the robots share a common handedness and for n \geq 5 when they do not. Thus, we deduce that with chirality (resp. without chirality) both problems are equivalent for n \geq 4 (resp. n \geq 5) in CORDA. | cs | 1 |
Title: Supersonic flow past an airfoil
Abstract: We consider a supersonic flow past an airfoil in the context of the steady, isentropic and irrotational compressible Euler equations. We show that for appropriate data, either a shock forms, in which case certain derivatives of the solution blow up, or a sonic line forms, which translates to the fact that the hyperbolic character of the equations degenerates. | math | 0 |
Title: Emerging Language Spaces Learned From Massively Multilingual Corpora
Abstract: Translations capture important information about languages that can be used as implicit supervision in learning linguistic properties and semantic representations. In an information-centric view, translated texts may be considered as semantic mirrors of the original text and the significant variations that we can observe across various languages can be used to disambiguate a given expression using the linguistic signal that is grounded in translation. Parallel corpora consisting of massive amounts of human translations with a large linguistic variation can be applied to increase abstractions and we propose the use of highly multilingual machine translation models to find language-independent meaning representations. Our initial experiments show that neural machine translation models can indeed learn in such a setup and we can show that the learning algorithm picks up information about the relation between languages in order to optimize transfer leaning with shared parameters. The model creates a continuous language space that represents relationships in terms of geometric distances, which we can visualize to illustrate how languages cluster according to language families and groups. Does this open the door for new ideas of data-driven language typology with promising models and techniques in empirical cross-linguistic research? | cs | 1 |
Title: Signal Detection for Ultra-Massive MIMO: An Information Geometry Approach
Abstract: In this paper, we propose an information geometry approach (IGA) for signal detection (SD) in ultra-massive multiple-input multiple-output (MIMO) systems. We formulate the signal detection as obtaining the marginals of the a posteriori probability distribution of the transmitted symbol vector. Then, a maximization of the a posteriori marginals (MPM) for signal detection can be performed. With the information geometry theory, we calculate the approximations of the a posteriori marginals. It is formulated as an iterative m-projection process between submanifolds with different constraints. We then apply the central-limit-theorem (CLT) to simplify the calculation of the m-projection since the direct calculation of the m-projection is of exponential-complexity. With the CLT, we obtain an approximate solution of the m-projection, which is asymptotically accurate. Simulation results demonstrate that the proposed IGA-SD emerges as a promising and efficient method to implement the signal detector in ultra-massive MIMO systems. | cs | 0 |
Title: QoE-oriented Dependent Task Scheduling under Multi-dimensional QoS Constraints over Distributed Networks
Abstract: Task scheduling as an effective strategy can improve application performance on computing resource-limited devices over distributed networks. However, existing evaluation mechanisms fail to depict the complexity of diverse applications, which involve dependencies among tasks, computing resource requirements, and multi-dimensional quality of service (QoS) constraints. Furthermore, traditional QoS-oriented task scheduling strategies struggle to meet the performance requirements without considering differences in satisfaction and acceptance of application, leading application failures and resource wastage. To tackle these issues, a quality of experience (QoE) cost model is designed to evaluate application completion, depicting the relationship among application satisfaction, communications, and computing resources in the distributed networks. Specifically, considering the sensitivity and preference of QoS, we model the different dimensional QoS degradation cost functions for dependent tasks, which are then integrated into the QoE cost model. Based on the QoE model, the dependent task scheduling problem is formulated as the minimization of overall QoE cost, aiming to improve the application performance in the distributed networks, which is proven Np-hard. Moreover, a heuristic Hierarchical Multi-queue Task Scheduling Algorithm (HMTSA) is proposed to address the QoE-oriented task scheduling problem among multiple dependent tasks, which utilizes hierarchical multiple queues to determine the optimal task execution order and location according to different dimensional QoS priorities. Finally, extensive experiments demonstrate that the proposed algorithm can significantly improve the satisfaction of applications. | cs | 0 |
Title: Identification of the Heat Transfer Coefficient Using an Inverse Heat Conduction Model
Abstract: Inverse problems of recovering heat transfer coefficient from integral measurements are considered. The heat transfer coefficient occurs in the transmission conditions of imperfect contact type or the Robin type boundary conditions. It is representable as a finite part of the Fourier series with time dependent coefficients. The additional measurements are integrals of a solution multiplied by some weights. Existence and uniqueness of solutions in Sobolev classes are proven and the conditions on the data are sharp. These conditions include smoothness and consistency conditions on the data and additional conditions on the kernels of the integral operators used in additional measurements. The proof relies on a priori bounds and the contraction mapping principle. The existence and uniqueness theorems are local in time. | math | 0 |
Title: A Non-Holonomic Systems Approach to Special Function Identities
Abstract: We extend Zeilberger's approach to special function identities to cases that are not holonomic. The method of creative telescoping is thus applied to definite sums or integrals involving Stirling or Bernoulli numbers, incomplete Gamma function or polylogarithms, which are not covered by the holonomic framework. The basic idea is to take into account the dimension of appropriate ideals in Ore algebras. This unifies several earlier extensions and provides algorithms for summation and integration in classes that had not been accessible to computer algebra before. | cs | 1 |
Title: Rigid Components of Random Graphs
Abstract: The planar rigidity problem asks, given a set of m pairwise distances among a set P of n unknown points, whether it is possible to reconstruct P, up to a finite set of possibilities (modulo rigid motions of the plane). The celebrated Maxwell-Laman Theorem from Rigidity Theory says that, generically, the rigidity problem has a combinatorial answer: the underlying combinatorial structure must contain a spanning minimally-rigid graph (Laman graph). In the case where the system is not rigid, its inclusion-wise maximal rigid substructures (rigid components) are also combinatorially characterized via the Maxwell-Laman theorem, and may be found efficiently. Physicists have used planar combinatorial rigidity has been used to study the phase transition between liquid and solid in network glasses. The approach has been to generate a graph via a stochastic process and then experimentally analyze its rigidity properties. Of particular interest is the size of the largest rigid components. In this paper, we study the emergence of rigid components in an Erdos-Renyi random graph G(n,p), using the parameterization p=c/n for a fixed constant c>0. Our first result is that for all c>0, almost surely all rigid components have size 2, 3 or Omega(n). We also show that for c>4, almost surely the largest rigid components have size at least n/10. While the G(n,p) model is simpler than those appearing in the physics literature, these results are the first of this type where the distribution is over all graphs on n vertices and the expected number of edges is O(n). | math | 1 |
Title: Open Gromov-Witten invariants from the Fukaya category
Abstract: This paper proposes a framework to show that the Fukaya category of a symplectic manifold $X$ determines the open Gromov-Witten invariants of Lagrangians $L \subset X$. We associate to an object in an $A_\infty$-category an extension of the negative cyclic homology, called \emph{relative cyclic homology}. We extend the Getzler-Gauss-Manin connection to relative cyclic homology. Then, we construct (under simplifying technical assumptions) a relative cyclic open-closed map, which maps the relative cyclic homology of a Lagrangian $L$ in the Fukaya category of a symplectic manifold $X$ to the $S^1$-equivariant relative quantum homology of $(X,L)$. Relative quantum homology is the dual to the relative quantum cohomology constructed by Solomon-Tukachinsky. This is an extension of quantum cohomology, and comes equipped with a connection extending the quantum connection. We prove that the relative open-closed map respects connections. As an application of this framework, we show, assuming a construction of the relative cyclic open-closed map in a broader technical setup, that the Fukaya category of a Calabi-Yau variety determines the open Gromov-Witten invariants with one interior marked point for any null-homologous Lagrangian brane. | math | 1 |
Title: Advanced Determinant Calculus: A Complement
Abstract: This is a complement to my previous article "Advanced Determinant Calculus" (S\'eminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12 (2003), 441-456). Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several conjectures and open problems. | math | 1 |
Title: Cellular Automata to More Efficiently Compute the Collatz Map
Abstract: The Collatz, or 3x+1, Conjecture claims that for every positive integer n, there exists some k such that T^k(n)=1, where T is the Collatz map. We present three cellular automata (CA) that transform the global problem of mimicking the Collatz map in bases 2, 3, and 4 into a local one of transforming the digits of iterates. The CAs streamline computation first by bypassing calculation of certain parts of trajectories: the binary CA bypasses division by two altogether. In addition, they allow for multiple trajectories to be calculated simultaneously, representing both a significant improvement upon existing sequential methods of computing the Collatz map and a demonstration of the efficacy of using a massively parallel approach with cellular automata to tackle iterative problems like the Collatz Conjecture. | math | 1 |
Title: Sasakian structure associated with a second order ODE and Hamiltonian dynamical systems
Abstract: We define a contact metric structure on the manifold corresponding to a second order ordinary differential equation $d^2y/dx^2=f(x,y,y')$ and show that the contact metric structure is Sasakian if and only if the 1-form $\frac{1}{2}(dp-fdx)$ defines a Poisson structure. We consider a Hamiltonian dynamical system defined by this Poisson structure and show that the Hamiltonian vector field, which is a multiple of the Reeb vector field, admits a compatible bi-Hamiltonian structure for which $f$ can be regarded as a Hamiltonian function. As a particular case, we give a compatible bi-Hamiltonian structure of the Reeb vector field such that the structure equations correspond to the Maurer-Cartan equations of an invariant coframe on the Heisenberg group and the independent variable plays the role of a Hamiltonian function. We also show that the first Chern class of the normal bundle of an integral curve of a multiple of the Reeb vector field vanishes iff $f_x+ff_p = \Psi (x)$ for some $\Psi$. | math | 1 |
Title: Energy balance and damage for brittle fracture: nonlocal formulation
Abstract: A nonlocal model of peridynamic type for dynamic brittle damage is introduced consisting of two phases, one elastic and the other inelastic. Evolution from the elastic to the inelastic phase depends on material strength. Here inelasticity corresponds to material failure. Existence and uniqueness of the elastic displacement-failure set pair follow from the initial value problem. The displacement-failure pair satisfies energy balance. The length scale of nonlocality $\epsilon$ is taken to be small relative to the domain in $\mathbb{R}^d$, $d=2,3$. The new nonlocal model delivers a two point strain dynamics on a subset of $\mathbb{R}^d\times\mathbb{R}^d$. This dynamics provides an energy that interpolates between volume energy corresponding to recoverable strain and surface energy corresponding to non recoverable strain. The deformation energy resulting in material failure over a region $R$ is given by a $d-1$ dimensional integral that is uniformly bounded as $\epsilon\rightarrow 0$. For fixed $\epsilon$, this energy is nonzero for $d-1$ dimensional regions $R$ associated with flat crack surfaces. The failure energy is the Griffith fracture energy for a given crack $R$ in terms of its area for $d=3$ (or length for $d=2$). This energy follows directly from the nonlocal model without sending $\epsilon$ to zero. Simulations illustrate fracture evolution through generation of an internal traction free boundary as a wake left behind a moving strain concentration. Crack paths are seen to follow a maximal strain energy density criterion. | math | 0 |
Title: Mixed Poisson process with Min-U-Exp mixing variable
Abstract: This work continues the research done in Jordanova and Veleva (2023) where the history of the problem could be found. In order to obtain the structure distribution of the newly-defined Mixed Poisson process, here the operation "max" is replaced with "min". We start with the definition of Min-U-Exp distribution. Then, we compute its numerical characteristics and investigate some of its properties. The joint distribution of the inter-arrival times (which are dependent) is the Multivariate Exp-Min-U-Exp distribution of $II^{-nd}$ kind. Its univariate and multivariate versions are described, and the formulae for their numerical characteristics are obtained. The distribution of the moments of arrival of different events is called Erlang-Min-U-Exp. Different properties of these distributions are obtained, and their numerical characteristics are computed. Multivariate ordered Mixed Poisson-Min-U-Exp distribution describes the joint distribution of the time-intersection of a Mixed Poisson process with Min-U-Exp mixing variable. The corresponding distribution of the additive increments (which are also dependent) is the Mixed Poisson-Min-U-Exp one. The considered relations between these distributions simplify their understanding. | math | 0 |
Title: Xorbits: Automating Operator Tiling for Distributed Data Science
Abstract: Data science pipelines commonly utilize dataframe and array operations for tasks such as data preprocessing, analysis, and machine learning. The most popular tools for these tasks are pandas and NumPy. However, these tools are limited to executing on a single node, making them unsuitable for processing large-scale data. Several systems have attempted to distribute data science applications to clusters while maintaining interfaces similar to single-node libraries, enabling data scientists to scale their workloads without significant effort. However, existing systems often struggle with processing large datasets due to Out-of-Memory (OOM) problems caused by poor data partitioning. To overcome these challenges, we develop Xorbits, a high-performance, scalable data science framework specifically designed to distribute data science workloads across clusters while retaining familiar APIs. The key differentiator of Xorbits is its ability to dynamically switch between graph construction and graph execution. Xorbits has been successfully deployed in production environments with up to 5k CPU cores. Its applications span various domains, including user behavior analysis and recommendation systems in the e-commerce sector, as well as credit assessment and risk management in the finance industry. Users can easily scale their data science workloads by simply changing the import line of their pandas and NumPy code. Our experiments demonstrate that Xorbits can effectively process very large datasets without encountering OOM or data-skewing problems. Over the fastest state-of-the-art solutions, Xorbits achieves an impressive 2.66* speedup on average. In terms of API coverage, Xorbits attains a compatibility rate of 96.7%, surpassing the fastest framework by an impressive margin of 60 percentage points. Xorbits is available at https: //github.com/xorbitsai/xorbits. | cs | 0 |
Title: Energy-optimal Timetable Design for Sustainable Metro Railway Networks
Abstract: We present our collaboration with Thales Canada Inc, the largest provider of communication-based train control (CBTC) systems worldwide. We study the problem of designing energy-optimal timetables in metro railway networks to minimize the effective energy consumption of the network, which corresponds to simultaneously minimizing total energy consumed by all the trains and maximizing the transfer of regenerative braking energy from suitable braking trains to accelerating trains. We propose a novel data-driven linear programming model that minimizes the total effective energy consumption in a metro railway network, capable of computing the optimal timetable in real-time, even for some of the largest CBTC systems in the world. In contrast with existing works, which are either NP-hard or involve multiple stages requiring extensive simulation, our model is a single linear programming model capable of computing the energy-optimal timetable subject to the constraints present in the railway network. Furthermore, our model can predict the total energy consumption of the network without requiring time-consuming simulations, making it suitable for widespread use in managerial settings. We apply our model to Shanghai Railway Network's Metro Line 8 -- one of the largest and busiest railway services in the world -- and empirically demonstrate that our model computes energy-optimal timetables for thousands of active trains spanning an entire service period of one day in real-time (solution time less than one second on a standard desktop), achieving energy savings between approximately 20.93% and 28.68%. Given the compelling advantages, our model is in the process of being integrated into Thales Canada Inc's industrial timetable compiler. | math | 0 |
Title: Mapping and Validating a Point Neuron Model on Intel's Neuromorphic Hardware Loihi
Abstract: Neuromorphic hardware is based on emulating the natural biological structure of the brain. Since its computational model is similar to standard neural models, it could serve as a computational acceleration for research projects in the field of neuroscience and artificial intelligence, including biomedical applications. However, in order to exploit this new generation of computer chips, rigorous simulation and consequent validation of brain-based experimental data is imperative. In this work, we investigate the potential of Intel's fifth generation neuromorphic chip - `Loihi', which is based on the novel idea of Spiking Neural Networks (SNNs) emulating the neurons in the brain. The work is implemented in context of simulating the Leaky Integrate and Fire (LIF) models based on the mouse primary visual cortex matched to a rich data set of anatomical, physiological and behavioral constraints. Simulations on the classical hardware serve as the validation platform for the neuromorphic implementation. We find that Loihi replicates classical simulations very efficiently and scales notably well in terms of both time and energy performance as the networks get larger. | cs | 1 |
Title: From Function to Distribution Modeling: A PAC-Generative Approach to Offline Optimization
Abstract: This paper considers the problem of offline optimization, where the objective function is unknown except for a collection of ``offline" data examples. While recent years have seen a flurry of work on applying various machine learning techniques to the offline optimization problem, the majority of these work focused on learning a surrogate of the unknown objective function and then applying existing optimization algorithms. While the idea of modeling the unknown objective function is intuitive and appealing, from the learning point of view it also makes it very difficult to tune the objective of the learner according to the objective of optimization. Instead of learning and then optimizing the unknown objective function, in this paper we take on a less intuitive but more direct view that optimization can be thought of as a process of sampling from a generative model. To learn an effective generative model from the offline data examples, we consider the standard technique of ``re-weighting", and our main technical contribution is a probably approximately correct (PAC) lower bound on the natural optimization objective, which allows us to jointly learn a weight function and a score-based generative model. The robustly competitive performance of the proposed approach is demonstrated via empirical studies using the standard offline optimization benchmarks. | cs | 0 |
Title: Dorfman connections and Courant algebroids
Abstract: We define Dorfman connections, which are to Courant algebroids what connections are to Lie algebroids. Several examples illustrate this analogy. A linear connection $\nabla\colon \mathfrak{X}(M)\times\Gamma(E)\to\Gamma(E)$ on a vector bundle $E$ over a smooth manifold $M$ is tantamount to a linear splitting $TE\simeq T^{q_E}E\oplus H_\nabla$, where $T^{q_E}E$ is the set of vectors tangent to the fibres of $E$. Furthermore, the curvature of the connection measures the failure of the horizontal space $H_\nabla$ to be integrable. We show that linear horizontal complements to $T^{q_E}E\oplus (T^{q_E}E)^\circ$ in the Pontryagin bundle over the vector bundle $E$ can be described in the same manner via a certain class of Dorfman connections $\Delta\colon \Gamma(TM\oplus E^*)\times\Gamma(E\oplus T^*M)\to\Gamma(E\oplus T^*M)$. Similarly to the tangent bundle case, we find that, after the choice of a linear splitting, the standard Courant algebroid structure of $TE\oplus T^*E\to E$ can be completely described by properties of the Dorfman connection. As an application, we study splittings of $TA\oplus T^*A$ over a Lie algebroid $A$ and, following Gracia-Saz and Mehta, we compute the representations up to homotopy defined by any linear splitting of $TA\oplus T^*A$ and the linear Lie algebroid $TA\oplus T^*A\to TM\oplus A^*$. Further, we characterise VB- and LA-Dirac structures in $TA\oplus T^*A$ via Dorfman connections. | math | 1 |
Title: Can AI Be as Creative as Humans?
Abstract: Creativity serves as a cornerstone for societal progress and innovation, but its assessment remains a complex and often subjective endeavor. With the rise of advanced generative AI models capable of tasks once reserved for human creativity, the study of AI's creative potential becomes imperative for its responsible development and application. This paper addresses the complexities in defining and evaluating creativity by introducing a new concept called Relative Creativity. Instead of trying to define creativity universally, we shift the focus to whether AI can match the creative abilities of a hypothetical human. This perspective draws inspiration from the Turing Test, expanding upon it to address the challenges and subjectivities inherent in evaluating creativity. This methodological shift facilitates a statistically quantifiable evaluation of AI's creativity, which we term Statistical Creativity. This approach allows for direct comparisons of AI's creative abilities with those of specific human groups. Building on this foundation, we discuss the application of statistical creativity in contemporary prompt-conditioned autoregressive models. In addition to defining and analyzing a measure of creativity, we introduce an actionable training guideline, effectively bridging the gap between theoretical quantification of creativity and practical model training. Through these multifaceted contributions, the paper establishes a cohesive, continuously evolving, and transformative framework for assessing and fostering statistical creativity in AI models. | cs | 0 |
Title: Nonconforming virtual element method for an incompressible miscible displacement problem in porous media
Abstract: This article presents a priori error estimates of the miscible displacement of one incompressible fluid by another through a porous medium characterized by a coupled system of nonlinear elliptic and parabolic equations. The study utilizes the $H(\rm{div})$ conforming virtual element method (VEM) for the approximation of the velocity, while a non-conforming virtual element approach is employed for the concentration. The pressure is discretised using the standard piecewise discontinuous polynomial functions. These spatial discretization techniques are combined with a backward Euler difference scheme for time discretization. The article also includes numerical results that validate the theoretical estimates presented. | math | 0 |
Title: Relative Fourier transforms and expectations on coideal subalgebras
Abstract: For an algebraic compact quantum group $H$ we establish a bijection between the set of right coideal $*$-subalgebras $A\to H$ and that of left module quotient $*$-coalgebras $H\to C$. It turns out that the inclusion $A\to H$ always splits as a map of right $A$-modules and right $H$-comodules, and the resulting expectation $E:H\to A$ is positive (and lifts to a positive map on the full $C^*$ completion on $H$) if and only if $A$ is invariant under the squared antipode of $H$. The proof proceeds by Tannaka-reconstructing the coalgebra $C$ corresponding to $A\to H$ by means of a fiber functor from $H$-equivariant $A$-modules to Hilbert spaces, while the characterization of those $A\to H$ which admit positive expectations makes use of a Fourier transform turning elements of $H$ into functions on $C$. | math | 1 |
Title: Stochastic Broadcast Control of Multi-Agent Swarms
Abstract: We present a model for controlling swarms of mobile agents via broadcast control, assumed to be detected by a random set of agents in the swarm. The agents that detect the control signal become ad-hoc leaders of the swarm. The agents are assumed to be velocity controlled, identical, anonymous, memory-less units with limited capabilities of sensing their neighborhood. Each agent is programmed to behave according to a linear local gathering process, based on the relative position of all its neighbors. The detected exogenous control, which is a desired velocity vector, is added by the leaders to the local gathering control. The graph induced by the agents adjacency is referred to as the visibility graph. We show that for piece-wise constant system parameters and a connected visibility graph, the swarm asymptotically aligns in each time-interval on a line in the direction of the exogenous control signal, and all the agents move with identical speed. These results hold for two models of pairwise influence in the gathering process, uniform and scaled. The impact of the influence model is mostly evident when the visibility graph is incomplete. These results are conditioned by the preservation of the connectedness of the visibility graph. In the second part of the report we analyze sufficient conditions for preserving the connectedness of the visibility graph. We show that if the visibility graph is complete then certain bounds on the control signal suffice to preserve the completeness of the graph. However, when the graph is incomplete, general conditions, independent of the leaders topology, could not be found. | cs | 1 |
Title: Almost global well-posedness for quasilinear strongly coupled wave-Klein-Gordon systems in two space dimensions
Abstract: We prove almost global well-posedness for quasilinear strongly coupled wave-Klein-Gordon systems with small and localized data in two space dimensions. We assume only mild decay on the data at infinity as well as minimal regularity. We systematically investigate all the possible quadratic null form type quasilinear strong coupling nonlinearities, and provide a new, robust approach for the proof. In a second paper we will complete the present results to full global well-posedness. | math | 1 |
Title: Asymptotics for the Taylor coefficients of certain infinite products
Abstract: Let $(m_1,\ldots,m_J)$ and $(r_1,\ldots,r_J)$ be two sequences of $J$ positive integers satisfying $1\le r_j< m_j$ for all $j=1,\ldots,J$. Let $(\delta_1,\ldots,\delta_J)$ be a sequence of $J$ nonzero integers. In this paper, we study the asymptotic behavior of the Taylor coefficients of the infinite product $$\prod_{j=1}^J\Bigg(\prod_{k\ge 1}\big(1-q^{r_j+m_j(k-1)}\big)\big(1-q^{-r_j+m_jk}\big)\Bigg)^{\delta_j}.$$ | math | 1 |
Title: Task and Explanation Network
Abstract: Explainability in deep networks has gained increased importance in recent years. We argue herein that an AI must be tasked not just with a task but also with an explanation of why said task was accomplished as such. We present a basic framework -- Task and Explanation Network (TENet) -- which fully integrates task completion and its explanation. We believe that the field of AI as a whole should insist -- quite emphatically -- on explainability. | cs | 0 |
Title: Braid variety cluster structures, II: general type
Abstract: We show that braid varieties for any complex simple algebraic group $G$ are cluster varieties. This includes open Richardson varieties inside the flag variety $G/B$. | math | 1 |
Title: Orlov spectra: bounds and gaps
Abstract: The Orlov spectrum is a new invariant of a triangulated category. It was introduced by D. Orlov building on work of A. Bondal-M. van den Bergh and R. Rouquier. The supremum of the Orlov spectrum of a triangulated category is called the ultimate dimension. In this work, we study Orlov spectra of triangulated categories arising in mirror symmetry. We introduce the notion of gaps and outline their geometric significance. We provide the first large class of examples where the ultimate dimension is finite: categories of singularities associated to isolated hypersurface singularities. Similarly, given any nonzero object in the bounded derived category of coherent sheaves on a smooth Calabi-Yau hypersurface, we produce a new generator by closing the object under a certain monodromy action and uniformly bound this new generator's generation time. In addition, we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived Fukaya category of a symplectic surface of genus greater than one. | math | 1 |
Title: Characterization of circular D0L systems
Abstract: We prove that every non-circular D0L system contains arbitrarily long repetitions. This result was already published in 1993 by Mignosi and S\'e\'ebold, however their proof is only a sketch. We give here a complete proof. Further, employing our previous result, we give a simple algorithm to test circularity of an injective D0L system. | math | 1 |
Title: Polynomial-time Approximation Scheme for Equilibriums of Games
Abstract: Whether a PTAS (polynomial-time approximation scheme) exists for equilibriums of games has been an open question, which relates to the practicality of methods in algorithmic game theory and the problem of non-stationarity in training and curse of dimensionality in multi-agent reinforcement learning. This paper introduces our theory that implies a method that is sufficient and necessary to be the PTAS for perfect equilibriums of dynamic games. The theory consists of cone interior dynamic programming and primal-dual unbiased regret minimization. The former enables the dynamic programming operator to iteratively converge to a perfect equilibrium based on a concept called policy cone. The latter enables the line search method to approximate a Nash equilibrium based on two concepts called primal-dual bias and unbiased central path, solving a subproblem of the former. Validity of our discovery is cross-corroborated by a combination of theorem proofs, graphs of the three core concepts, and experimental results. | cs | 0 |
Title: Time Protection: the Missing OS Abstraction
Abstract: Timing channels enable data leakage that threatens the security of computer systems, from cloud platforms to smartphones and browsers executing untrusted third-party code. Preventing unauthorised information flow is a core duty of the operating system, however, present OSes are unable to prevent timing channels. We argue that OSes must provide time protection in addition to the established memory protection. We examine the requirements of time protection, present a design and its implementation in the seL4 microkernel, and evaluate its efficacy as well as performance overhead on Arm and x86 processors. | cs | 1 |
Title: Beyond Information Exchange: An Approach to Deploy Network Properties for Information Diffusion
Abstract: Information diffusion in Online Social Networks is a new and crucial problem in social network analysis field and requires significant research attention. Efficient diffusion of information are of critical importance in diverse situations such as; pandemic prevention, advertising, marketing etc. Although several mathematical models have been developed till date, but previous works lacked systematic analysis and exploration of the influence of neighborhood for information diffusion. In this paper, we have proposed Common Neighborhood Strategy (CNS) algorithm for information diffusion that demonstrates the role of common neighborhood in information propagation throughout the network. The performance of CNS algorithm is evaluated on several real-world datasets in terms of diffusion speed and diffusion outspread and compared with several widely used information diffusion models. Empirical results show CNS algorithm enables better information diffusion both in terms of diffusion speed and diffusion outspread. | cs | 1 |
Title: Big Ramsey Degrees in Ultraproducts of Finite Structures
Abstract: We develop a transfer principle of structural Ramsey theory from finite structures to ultraproducts. We show that under certain mild conditions, when a class of finite structures has finite small Ramsey degrees, under the (Generalized) Continuum Hypothesis the ultraproduct has finite big Ramsey degrees for internal colorings. The necessity of restricting to internal colorings is demonstrated by the example of the ultraproduct of finite linear orders. Under CH, this ultraproduct $\fLL^*$ has, as a spine, $\eta_1$, an uncountable analogue of the order type of rationals $\eta$. Finite big Ramsey degrees for $\eta$ were exactly calculated by Devlin in \cite{Devlin}. It is immediate from \cite{Tod87} that $\eta_1$ fails to have finite big Ramsey degrees. Moreover, we extend Devlin's coloring to $\eta_1$ to show that it witnesses big Ramsey degrees of finite tuples in $\eta$ on every copy of $\eta$ in $\eta_1,$ and consequently in $\fLL^*$. This work gives additional confirmation that ultraproducts are a suitable environment for studying Ramsey properties of finite and infinite structures. | math | 1 |
Title: Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients
Abstract: We prove heat kernel bounds for the operator (1 + |x|^{\alpha})\Delta in R^N, through Nash inequalities and weighted Hardy inequalities. | math | 1 |
Title: Symbolic Dynamics of Magnetic Bumps
Abstract: For n convex magnetic bumps in the plane, whose boundary has a curvature somewhat smaller than the absolute value of the constant magnetic field inside the bump, we construct a complete symbolic dynamics of a classical particle moving with speed one. | math | 1 |
Title: LLaMA Pro: Progressive LLaMA with Block Expansion
Abstract: Humans generally acquire new skills without compromising the old; however, the opposite holds for Large Language Models (LLMs), e.g., from LLaMA to CodeLLaMA. To this end, we propose a new post-pretraining method for LLMs with an expansion of Transformer blocks. We tune the expanded blocks using only new corpus, efficiently and effectively improving the model's knowledge without catastrophic forgetting. In this paper, we experiment on the corpus of code and math, yielding LLaMA Pro-8.3B, a versatile foundation model initialized from LLaMA2-7B, excelling in general tasks, programming, and mathematics. LLaMA Pro and its instruction-following counterpart (LLaMA Pro-Instruct) achieve advanced performance among various benchmarks, demonstrating superiority over existing open models in the LLaMA family and the immense potential of reasoning and addressing diverse tasks as an intelligent agent. Our findings provide valuable insights into integrating natural and programming languages, laying a solid foundation for developing advanced language agents that operate effectively in various environments. | cs | 0 |
Title: Lambda Module structure on higher $K$-groups
Abstract: In this article, we show that for a quasicompact scheme $X$ and $n>0,$ the $n$-th $K$-group $K_{n}(X)$ is a $\lambda$-module over a $\lambda$-ring $K_{0}(X)$ in the sense of Hesselholt. | math | 0 |
Title: An analysis of the logic of Riesz Spaces with strong unit
Abstract: We study \L ukasiewicz logic enriched with a scalar multiplication with scalars taken in $[0,1]$. Its algebraic models, called {\em Riesz MV-algebras}, are, up to isomorphism, unit intervals of Riesz spaces with a strong unit endowed with an appropriate structure. When only rational scalars are considered, one gets the class of {\em DMV-algebras} and a corresponding logical system. Our research follows two objectives. The first one is to deepen the connections between functional analysis and the logic of Riesz MV-algebras. The second one is to study the finitely presented MV-algebras, DMV-algebras and Riesz MV-algebras, connecting them from logical, algebraic and geometric perspective. | math | 1 |
Title: Hochschild cohomology of the second kind: Koszul duality and Morita invariance
Abstract: We define Hochschild cohomology of the second kind for differential graded (dg) or curved algebras as a derived functor in a compactly generated derived category of the second kind, and show that it is invariant under Morita equivalence of the second kind. A bimodule version of Koszul duality is constructed and used to show that Hochschild cohomology of the second kind is preserved under (nonconilpotent) Koszul duality. Hochschild cohomology of the second kind of an algebra often computes the ordinary Hochschild cohomology of geometrically meaningful dg categories. Examples include the category of infinity local systems on a topological space, the bounded derived category of a complex algebraic manifold and the category of matrix factorizations. | math | 0 |
Title: Recursive Monte Carlo and Variational Inference with Auxiliary Variables
Abstract: A key design constraint when implementing Monte Carlo and variational inference algorithms is that it must be possible to cheaply and exactly evaluate the marginal densities of proposal distributions and variational families. This takes many interesting proposals off the table, such as those based on involved simulations or stochastic optimization. This paper broadens the design space, by presenting a framework for applying Monte Carlo and variational inference algorithms when proposal densities cannot be exactly evaluated. Our framework, recursive auxiliary-variable inference (RAVI), instead approximates the necessary densities using meta-inference: an additional layer of Monte Carlo or variational inference, that targets the proposal, rather than the model. RAVI generalizes and unifies several existing methods for inference with expressive approximating families, which we show correspond to specific choices of meta-inference algorithm, and provides new theory for analyzing their bias and variance. We illustrate RAVI's design framework and theorems by using them to analyze and improve upon Salimans et al.'s Markov Chain Variational Inference, and to design a novel sampler for Dirichlet process mixtures, achieving state-of-the-art results on a standard benchmark dataset from astronomy and on a challenging datacleaning task with Medicare hospital data. | cs | 1 |
Title: Non-local games and quantum symmetries of quantum metric spaces
Abstract: We generalize Banica's construction of the quantum isometry group of a metric space to the class of quantum metric spaces in the sense of Kuperberg and Weaver. We also introduce quantum isometries between two quantum metric spaces, and we show that if a pair of quantum metric spaces are algebraically quantum isometric, then their quantum isometry groups are monoidally equivalent. Motivated by the recent work on the graph isomorphism game, we introduce a new two-player non-local game called the metric isometry game, where players can win classically if and only if the metric spaces are isometric. Winning quantum strategies of this game align with quantum isometries of the metric spaces. | math | 1 |
Title: Various Covering Spectra for Complete Metric Spaces
Abstract: We study various covering spectra for complete noncompact length spaces with universal covers (including Riemannian manifolds and the pointed Gromov Hausdorff limits of Riemannian manifolds with lower bounds on their Ricci curvature). We relate the covering spectrum to the (marked) shift spectrum of such a space. We define the slipping group generated by elements of the fundamental group whose translative lengths are 0. We introduce a rescaled length, the rescaled covering spectrum and the rescaled slipping group. Applying these notions we prove that certain complete noncompact Riemannian manifolds with nonnegative or positive Ricci curvature have finite fundamental groups. Throughout we suggest further problems both for those interested in Riemannian geometry and those interested in metric space theory. | math | 1 |
Title: Understanding the Effects of RLHF on LLM Generalisation and Diversity
Abstract: Large language models (LLMs) fine-tuned with reinforcement learning from human feedback (RLHF) have been used in some of the most widely deployed AI models to date, such as OpenAI's ChatGPT or Anthropic's Claude. % , or Meta's LLaMA-2. While there has been significant work developing these methods, our understanding of the benefits and downsides of each stage in RLHF is still limited. To fill this gap, we present an extensive analysis of how each stage of the process (i.e.~supervised fine-tuning (SFT), reward modelling, and RLHF) affects two key properties: out-of-distribution (OOD) generalisation and output diversity. OOD generalisation is crucial given the wide range of real-world scenarios in which these models are being used, while output diversity refers to the model's ability to generate varied outputs and is important for a variety of use cases. We perform our analysis across two base models on both summarisation and instruction following tasks, the latter being highly relevant for current LLM use cases. We find that RLHF generalises better than SFT to new inputs, particularly as the distribution shift between train and test becomes larger. However, RLHF significantly reduces output diversity compared to SFT across a variety of measures, implying a tradeoff in current LLM fine-tuning methods between generalisation and diversity. Our results provide guidance on which fine-tuning method should be used depending on the application, and show that more research is needed to improve the tradeoff between generalisation and diversity. | cs | 0 |
Title: Domination structure for number three
Abstract: From a research of several recent papers, in the first part, we are concerned with domination number in cubic graphs and give a sufficient condition of Reed's conjecture. In the second part, from a perspective, we study the structure of a minimum dominating set in 3-connected graphs. It is derived from a collection of cycles with length 0 mod 3. | math | 1 |
Title: Functional Dimensionality of Koopman Eigenfunction Space
Abstract: This work presents the general form solution of Koopman Partial Differential Equation and shows that its functional dimensionality is finite. The dimensionality is as the dimensionality of the dynamics. Thus, the representation of nonlinear dynamics as a linear one with a finite set of Koopman eigenfunctions without error is possible. This formulation justifies the flowbox statement and provides a simple numerical method to find such representation. | math | 0 |
Title: Assistive Tele-op: Leveraging Transformers to Collect Robotic Task Demonstrations
Abstract: Sharing autonomy between robots and human operators could facilitate data collection of robotic task demonstrations to continuously improve learned models. Yet, the means to communicate intent and reason about the future are disparate between humans and robots. We present Assistive Tele-op, a virtual reality (VR) system for collecting robot task demonstrations that displays an autonomous trajectory forecast to communicate the robot's intent. As the robot moves, the user can switch between autonomous and manual control when desired. This allows users to collect task demonstrations with both a high success rate and with greater ease than manual teleoperation systems. Our system is powered by transformers, which can provide a window of potential states and actions far into the future -- with almost no added computation time. A key insight is that human intent can be injected at any location within the transformer sequence if the user decides that the model-predicted actions are inappropriate. At every time step, the user can (1) do nothing and allow autonomous operation to continue while observing the robot's future plan sequence, or (2) take over and momentarily prescribe a different set of actions to nudge the model back on track. We host the videos and other supplementary material at https://sites.google.com/view/assistive-teleop. | cs | 1 |
Title: Federated Class-Incremental Learning with Prototype Guided Transformer
Abstract: Existing federated learning methods have effectively addressed decentralized learning in scenarios involving data privacy and non-IID data. However, in real-world situations, each client dynamically learns new classes, requiring the global model to maintain discriminative capabilities for both new and old classes. To effectively mitigate the effects of catastrophic forgetting and data heterogeneity under low communication costs, we designed a simple and effective method named PLoRA. On the one hand, we adopt prototype learning to learn better feature representations and leverage the heuristic information between prototypes and class features to design a prototype re-weight module to solve the classifier bias caused by data heterogeneity without retraining the classification layer. On the other hand, our approach utilizes a pre-trained model as the backbone and utilizes LoRA to fine-tune with a tiny amount of parameters when learning new classes. Moreover, PLoRA does not rely on similarity-based module selection strategies, thereby further reducing communication overhead. Experimental results on standard datasets indicate that our method outperforms the state-of-the-art approaches significantly. More importantly, our method exhibits strong robustness and superiority in various scenarios and degrees of data heterogeneity. Our code will be publicly available. | cs | 0 |
Title: Ahlfors' contribution to the theory of meromorphic functions
Abstract: This is an expanded version of one of the Lectures in memory of Lars Ahlfors in Haifa in 1996. Some mistakes are corrected and references added. It contains a survey of his work on meromorphic functions and related topics written in 1929-1941. | math | 1 |
Title: The Principal Ideal Theorem in Spectral Synthesis
Abstract: In an earlier paper we solved a long-standing problem which goes back to Laurent Schwartz's work on mean-periodic functions. Namely, we completely characterised those locally compact Abelian groups having spectral synthesis. The method is based on the localisation concept. In this paper we show that localisation can be used to prove another basic result in spectral synthesis: the principal ideal theorem. | math | 0 |
Title: Towards Open Set Deep Networks
Abstract: Deep networks have produced significant gains for various visual recognition problems, leading to high impact academic and commercial applications. Recent work in deep networks highlighted that it is easy to generate images that humans would never classify as a particular object class, yet networks classify such images high confidence as that given class - deep network are easily fooled with images humans do not consider meaningful. The closed set nature of deep networks forces them to choose from one of the known classes leading to such artifacts. Recognition in the real world is open set, i.e. the recognition system should reject unknown/unseen classes at test time. We present a methodology to adapt deep networks for open set recognition, by introducing a new model layer, OpenMax, which estimates the probability of an input being from an unknown class. A key element of estimating the unknown probability is adapting Meta-Recognition concepts to the activation patterns in the penultimate layer of the network. OpenMax allows rejection of "fooling" and unrelated open set images presented to the system; OpenMax greatly reduces the number of obvious errors made by a deep network. We prove that the OpenMax concept provides bounded open space risk, thereby formally providing an open set recognition solution. We evaluate the resulting open set deep networks using pre-trained networks from the Caffe Model-zoo on ImageNet 2012 validation data, and thousands of fooling and open set images. The proposed OpenMax model significantly outperforms open set recognition accuracy of basic deep networks as well as deep networks with thresholding of SoftMax probabilities. | cs | 1 |
Title: Weight functions on Berkovich curves
Abstract: Let $C$ be a curve over a complete discretely valued field $K$. We give tropical descriptions of the weight function attached to a pluricanonical form on $C$ and the essential skeleton of $C$. We show that the Laplacian of the weight function equals the pluricanonical divisor on Berkovich skeleta, and we describe the essential skeleton of $C$ as a combinatorial skeleton of the Berkovich skeleton of the minimal $snc$-model. In particular, if $C$ has semi-stable reduction, then the essential skeleton coincides with the minimal skeleton. As an intermediate step, we describe the base loci of logarithmic pluricanonical line bundles on minimal $snc$-models. | math | 1 |