Miracle-dz/UniFAR-FTUs huggingface dataset:
Innovation through intra and inter-regional interaction in economic geography

This paper analyses a two-region model with vertical innovations that enhance the quality of varieties of the horizontally differentiated manufactures produced in each of the two regions. We look at how the creation and diffusion of knowledge and increasing returns in manufacturing interact to shape the spatial economy. Innovations occur with a probability that depends on the inter-regional interaction between researchers (mobile workers). We find that, if the weight of interaction with foreign scientists is relatively more important for the success of innovation, the model accounts for re-dispersion of economic activities after an initial stage of progressive agglomeration as transport costs decrease from a high level.
Matrices With Gaussian Noise: Optimal Estimates for Singular Subspace Perturbation

The Davis–Kahan–Wedin <inline-formula> <tex-math notation="LaTeX">$\sin \Theta $ </tex-math></inline-formula> theorem describes how the singular subspaces of a matrix change when subjected to a small perturbation. This classic result is sharp in the worst case scenario. In this paper, we prove a stochastic version of the Davis–Kahan–Wedin <inline-formula> <tex-math notation="LaTeX">$\sin \Theta $ </tex-math></inline-formula> theorem when the perturbation is a Gaussian random matrix. Under certain structural assumptions, we obtain an optimal bound that significantly improves upon the classic Davis–Kahan–Wedin <inline-formula> <tex-math notation="LaTeX">$\sin \Theta $ </tex-math></inline-formula> theorem. One of our key tools is a new perturbation bound for the singular values, which may be of independent interest.
Geometry-induced dynamics of confined chiral active matter

Controlling the motion of active matter is a central issue that has recently garnered significant attention in fields ranging from non-equilibrium physics to chemical engineering and biology. Distinct methods for controlling active matter have been developed, and physical confinement to limited space and active matter with broken rotational symmetry (chirality) are two prominent mechanisms. However, the interplay between pattern formation due to physical constraints and the ordering by chiral motion needs to be better understood. In this study, we conduct numerical simulations of chiral self-propelled particles under circular boundary confinement. The collective motion of confined self-propelled particles can take drastically different forms depending on their chirality. The balance of orientation changes between particle interaction and the boundary wall is essential for generating ordered collective motion. Our results clarify the role of the steric boundary effect in controlling chiral active matter.
Exploring Depth Information for Face Manipulation Detection

Face manipulation detection has been receiving a lot of attention for the reliability and security of the face images. Recent studies focus on using auxiliary information or prior knowledge to capture robust manipulation traces, which are shown to be promising. As one of the important face features, the face depth map, which has shown to be effective in other areas such as the face recognition or face detection, is unfortunately paid little attention to in literature for detecting the manipulated face images. In this paper, we explore the possibility of incorporating the face depth map as auxiliary information to tackle the problem of face manipulation detection in real world applications. To this end, we first propose a Face Depth Map Transformer (FDMT) to estimate the face depth map patch by patch from a RGB face image, which is able to capture the local depth anomaly created due to manipulation. The estimated face depth map is then considered as auxiliary information to be integrated with the backbone features using a Multi-head Depth Attention (MDA) mechanism that is newly designed. Various experiments demonstrate the advantage of our proposed method for face manipulation detection.
An Analysis of Attention via the Lens of Exchangeability and Latent Variable Models

With the attention mechanism, transformers achieve significant empirical successes. Despite the intuitive understanding that transformers perform relational inference over long sequences to produce desirable representations, we lack a rigorous theory on how the attention mechanism achieves it. In particular, several intriguing questions remain open: (a) What makes a desirable representation? (b) How does the attention mechanism infer the desirable representation within the forward pass? (c) How does a pretraining procedure learn to infer the desirable representation through the backward pass? We observe that, as is the case in BERT and ViT, input tokens are often exchangeable since they already include positional encodings. The notion of exchangeability induces a latent variable model that is invariant to input sizes, which enables our theoretical analysis. - To answer (a) on representation, we establish the existence of a sufficient and minimal representation of input tokens. In particular, such a representation instantiates the posterior distribution of the latent variable given input tokens, which plays a central role in predicting output labels and solving downstream tasks. - To answer (b) on inference, we prove that attention with the desired parameter infers the latent posterior up to an approximation error, which is decreasing in input sizes. In detail, we quantify how attention approximates the conditional mean of the value given the key, which characterizes how it performs relational inference over long sequences. - To answer (c) on learning, we prove that both supervised and self-supervised objectives allow empirical risk minimization to learn the desired parameter up to a generalization error, which is independent of input sizes. Particularly, in the self-supervised setting, we identify a condition number that is pivotal to solving downstream tasks.
Quantizing Heavy-Tailed Data in Statistical Estimation: (Near) Minimax Rates, Covariate Quantization, and Uniform Recovery

Modern datasets often exhibit heavy-tailed behavior, while quantization is inevitable in digital signal processing and many machine learning problems. This paper studies the quantization of heavy-tailed data in several fundamental statistical estimation problems where the underlying distributions have bounded moments of some order (no greater than 4). We propose to truncate and properly dither the data prior to a uniform quantization. Our major standpoint is that (near) minimax rates of estimation error could be achieved by computationally tractable estimators based on the quantized data produced by the proposed scheme. In particular, concrete results are worked out for covariance estimation, compressed sensing (also interpreted as sparse linear regression), and matrix completion, all agreeing that the quantization only slightly worsens the multiplicative factor. Additionally, while prior results focused on the quantization of responses (i.e., measurements), we study compressed sensing where the covariates (i.e., sensing vectors) are also quantized; in this case, though our recovery program is non-convex (since our covariance matrix estimator lacks positive semi-definiteness), we prove that all local minimizers enjoy near-optimal estimation error. Moreover, by the concentration inequality of the product process and a covering argument, we establish a near minimax uniform recovery guarantee for quantized compressed sensing with heavy-tailed noise. Finally, numerical simulations are provided to corroborate our theoretical results.
Lattice operads and operad filtrations

We elaborate on the notion of a filtration of an operad defined in terms of a lattice-valued operad serving as an indexing object. That covers ordinary integer-indexed filtrations of associative algebras and operads as a special case, yet the notion appears to be natural enough to encompass examples of other kind as well. The characteristic property of lattice operads is that of a certain distributivity of partial compositions with respect to meets and joins. We observe that some well-known families of lattices of combinatorial origin, such as Tamari lattices, assemble to operads subject to this particular property. Other examples include an operad of integer paritions supported on Young's lattice, operads of integer compositions of types $A, B$ and $D$, which we relate to operads of regular polytopes.
A Unified Object Counting Network With Object Occupation Prior

The counting task, which plays a fundamental role in numerous applications (e.g., crowd counting, traffic statistics), aims to predict the number of objects with various densities. Existing object counting tasks are designed for a single object class. However, it is inevitable to encounter newly coming data with new classes in our real world. We name this scenario as evolving object counting. In this paper, we build the first evolving object counting dataset and propose a unified object counting network as the first attempt to address this task. The proposed network consists of two key components: a class-agnostic mask module and a class-incremental module. The class-agnostic mask module learns generic object occupation prior by predicting a class-agnostic binary mask (e.g., 1 denotes there exists an object at the considering position in an image and 0 otherwise). The class-incremental module is used to handle new classes and provides discriminative class guidance for density map prediction. The combined outputs of the class-agnostic mask module and image feature extractor are used to predict the final density map. When new classes arrive, we first add new neural nodes to the last regression and classification layers of the class-incremental module. Then, instead of retraining the model from scratch, we utilize knowledge distillation to help the model retain and consolidate what it has previously learned. We also employ a support sample bank to store a small number of typical training samples for each class, which are used to prevent the model from forgetting key information from old data. With this design, our model can efficiently and effectively adapt to new classes while maintaining good performance on already-seen data without large-scale retraining. Extensive experiments on the collected dataset demonstrate favorable performance. The dataset and code will be available at: https://github.com/Tanyjiang/EOCO.
Linear programming word problems formulation using EnsembleCRF NER labeler and T5 text generator with data augmentations

We propose an ensemble approach to predict the labels in linear programming word problems. The entity identification and the meaning representation are two types of tasks to be solved in the NL4Opt competition. We propose the ensembleCRF method to identify the named entities for the first task. We found that single models didn't improve for the given task in our analysis. A set of prediction models predict the entities. The generated results are combined to form a consensus result in the ensembleCRF method. We present an ensemble text generator to produce the representation sentences for the second task. We thought of dividing the problem into multiple small tasks due to the overflow in the output. A single model generates different representations based on the prompt. All the generated text is combined to form an ensemble and produce a mathematical meaning of a linear programming problem.
DRG-Net: Interactive Joint Learning of Multi-lesion Segmentation and Classification for Diabetic Retinopathy Grading

—Diabetic Retinopathy (DR) is a leading cause of vision loss in the world, and early DR detection is necessary to prevent vision loss and support an appropriate treatment. In this work, we leverage interactive machine learning and introduce a joint learning framework, termed DRG-Net, to effectively learn both disease grading and multi-lesion segmentation. Our DRG-Net consists of two modules: (i) DRG-AI-System to classify DR Grading, localize lesion areas and provide visual explanation and (ii) DRG-Expert-Interaction to receive feedback from user-expert and improve the DRG-AI-System. To deal with sparse data, we utilize transfer learning mechanisms to extract invariant feature representations by using Wasserstein distance and adversarial learning-based entropy minimization. Besides, we propose a novel attention strategy at both low- and high-level features to automatically select the most signiﬁcant lesion information and provide explainable properties. In terms of human interaction, we further develop DRG-Net as a tool that enables expert users to correct the system’s predictions, which may then be used to update the system as a whole. Moreover, thanks to the attention mechanism and loss functions constraint between lesion features and classiﬁcation features, our approach can be robust given a certain level of noise in the feedback of users. We have benchmarked DRG-Net on the two largest DR datasets, i.e., IDRID and FGADR, and compared it to various state-of-the-art deep learning networks. In addition to outperforming other SOTA approaches, DRG-Net is effectively updated using user feedback, even in a weakly-supervised manner.
Polarization phase relationships via multiple Morse wavelets. II. Data analysis

Three examples are provided of the use of multiple generalized Morse wavelets for polarization analysis of three–component recordings where phase relationships between components of the signal are transient. The example three–component signals are (i) synthetic, (ii) the radial, transverse and vertical components of an earthquake, and (iii) the radial, transverse and normal components of the solar magnetic field measured by the Ulysses spacecraft upstream of an interplanetary shock. Working in the scale–time plane (a,b), the multiple–wavelet approach detects coherent motion with elliptical polarization in all three cases. Our graphical presentations fully take into account restrictions imposed by two factors: the minimum usable scale, and wraparound effects caused by Fourier transform computations, and consequently a maximum usable scale.
Normalizing flow neural networks by JKO scheme

Normalizing flow is a class of deep generative models for efficient sampling and likelihood estimation, which achieves attractive performance, particularly in high dimensions. The flow is often implemented using a sequence of invertible residual blocks. Existing works adopt special network architectures and regularization of flow trajectories. In this paper, we develop a neural ODE flow network called JKO-iFlow, inspired by the Jordan-Kinderleherer-Otto (JKO) scheme, which unfolds the discrete-time dynamic of the Wasserstein gradient flow. The proposed method stacks residual blocks one after another, allowing efficient block-wise training of the residual blocks, avoiding sampling SDE trajectories and score matching or variational learning, thus reducing the memory load and difficulty in end-to-end training. We also develop adaptive time reparameterization of the flow network with a progressive refinement of the induced trajectory in probability space to improve the model accuracy further. Experiments with synthetic and real data show that the proposed JKO-iFlow network achieves competitive performance compared with existing flow and diffusion models at a significantly reduced computational and memory cost.
Topology of $3$-manifolds with uniformly positive scalar curvature

In this article, we classify (non-compact) $3$-manifolds with uniformly positive scalar curvature. Precisely, we show that an oriented $3$-manifold has a complete metric with uniformly positive scalar curvature if and only if it is homeomorphic to an (possibly) infinite connected sum of spherical $3$-manifolds and some copies of $\mathbb{S}^1\times \mathbb{S}^2$. Further, we study an oriented $3$-manifold with mean convex boundary and with uniformly positive scalar curvature. If the boundary is a disjoint union of closed surfaces, then the manifold is an (possibly) infinite conned sum of spherical $3$-manifolds, some handlebodies and some copies of $\mathbb{S}^1\times \mathbb{S}^2$.
Delving into Semantic Scale Imbalance

Model bias triggered by long-tailed data has been widely studied. However, measure based on the number of samples cannot explicate three phenomena simultaneously: (1) Given enough data, the classification performance gain is marginal with additional samples. (2) Classification performance decays precipitously as the number of training samples decreases when there is insufficient data. (3) Model trained on sample-balanced datasets still has different biases for different classes. In this work, we define and quantify the semantic scale of classes, which is used to measure the feature diversity of classes. It is exciting to find experimentally that there is a marginal effect of semantic scale, which perfectly describes the first two phenomena. Further, the quantitative measurement of semantic scale imbalance is proposed, which can accurately reflect model bias on multiple datasets, even on sample-balanced data, revealing a novel perspective for the study of class imbalance. Due to the prevalence of semantic scale imbalance, we propose semantic-scale-balanced learning, including a general loss improvement scheme and a dynamic re-weighting training framework that overcomes the challenge of calculating semantic scales in real-time during iterations. Comprehensive experiments show that dynamic semantic-scale-balanced learning consistently enables the model to perform superiorly on large-scale long-tailed and non-long-tailed natural and medical datasets, which is a good starting point for mitigating the prevalent but unnoticed model bias.
Hidden Symmetries, Rapid Turns and Cosmic Acceleration

Hidden symmetries provide a powerful tool for finding exact solutions in multifield cosmological models. We review how, using such symmetries, one can find inflationary solutions in two-field models, which lead to the generation of primordial black holes. We also discuss an exact solution in a two-field cosmological model, which describes dark energy. This solution is obtained with the use of a hidden symmetry, although the latter is broken by a constant term in the scalar potential. All of the above solutions are characterized by field-space trajectories with rapid turns.
Supermagic labeling of $C_n\Box C_m$

A supermagic labeling (often also called supermagic labeling) of a graph $G(V,E)$ with $|E|=k$ is a bijection from $E$ to the set of first $k$ positive integers such that the sum of labels of all incident edges of every vertex $x\in V$ is equal to the same integer $c$. An existence of a supermagic labeling of Cartesian product of two cycles, $C_{n}\Box C_m$ for $n,m\geq4$ and both $n,m$ even and for any $C_n\Box C_n$ with $n\geq3$ was proved by Ivan\v{c}o. Ivan\v{c}o also conjectured that such labeling is possible for any $C_n\Box C_m$ with $n,m\geq3$. We prove his conjecture for all $n,m$ odd that are not relatively prime.
Recurrent Problems in the LOCAL model

The paper considers the SUPPORTED model of distributed computing introduced by Schmid and Suomela [HotSDN'13], generalizing the LOCAL and CONGEST models. In this framework, multiple instances of the same problem, differing from each other by the subnetwork to which they apply, recur over time, and need to be solved efficiently online. To do that, one may rely on an initial preprocessing phase for computing some useful information. This preprocessing phase makes it possible, in some cases, to overcome locality-based time lower bounds. A first contribution of the current paper is expanding the spectrum of problem types to which the SUPPORTED model applies. In addition to subnetwork-defined recurrent problems, we introduce also recurrent problems of two additional types: (i) instances defined by partial client sets, and (ii) instances defined by partially fixed outputs. Our second contribution is illustrating the versatility of the SUPPORTED framework by examining recurrent variants of three classical graph problems. The first problem is Minimum Client Dominating Set (CDS), a recurrent version of the classical dominating set problem with each recurrent instance requiring us to dominate a partial client set. We provide a constant time approximation scheme for CDS on trees and planar graphs. The second problem is Color Completion (CC), a recurrent version of the coloring problem in which each recurrent instance comes with a partially fixed coloring (of some of the vertices) that must be completed. We study the minimum number of new colors and the minimum total number of colors necessary for completing this task. The third problem we study is a recurrent version of Locally Checkable Labellings (LCL) on paths of length $n$. We show that such problems have complexities that are either $\Theta(1)$ or $\Theta(n)$, extending the results of Foerster et al. [INFOCOM'19].
On Korobov bound concerning Zaremba's conjecture

We prove in particular that for any suﬃciently large prime p there is 1 6 a < p such that all partial quotients of a/p are bounded by O (log p/ log log p ) . For composite denominators a similar result is obtained. This improves the well–known Korobov bound concerning Zaremba’s conjecture from the theory of continued fractions.
Critically fixed Thurston maps: classification, recognition, and twisting

An orientation-preserving branched covering map f : S → S is called a critically fixed Thurston map if f fixes each of its critical points. It was recently shown that there is an explicit one-to-one correspondence between Möbius conjugacy classes of critically fixed rational maps and isomorphism classes of planar embedded connected graphs. In the paper, we generalize this result to the whole family of critically fixed Thurston maps. Namely, we show that each critically fixed Thurston map f is obtained by applying the blow-up operation, introduced by Kevin Pilgrim and Tan Lei, to a pair (G,φ), where G is a planar embedded graph in S without isolated vertices and φ is an orientation-preserving homeomorphism of S that fixes each vertex of G. This result allows us to provide a classification of combinatorial equivalence classes of critically fixed Thurston maps. We also develop an algorithm that reconstructs (up to isotopy) the pair (G,φ) associated with a critically fixed Thurston map f . Finally, we solve some special instances of the twisting problem for the family of critically fixed Thurston maps obtained by blowing up pairs (G, idS2).
Noncrossing partitions of an annulus

The noncrossing partition poset associated to a Coxeter group $W$ and Coxeter element $c$ is the interval $[1,c]_T$ in the absolute order on $W$. We construct a new model of noncrossing partititions for $W$ of classical affine type, using planar diagrams (affine types $\tilde A$ and $\tilde C$ in this paper and affine types $\tilde D$ and $\tilde B$ in the sequel). The model in type $\tilde A$ consists of noncrossing partitions of an annulus. In type $\tilde C$, the model consists of symmetric noncrossing partitions of an annulus or noncrossing partitions of a disk with two orbifold points. Following the lead of McCammond and Sulway, we complete $[1,c]_T$ to a lattice by factoring the translations in $[1,c]_T$, but the combinatorics of the planar diagrams leads us to make different choices about how to factor.
Wormhole MAML: Meta-Learning in Glued Parameter Space

In this paper, we introduce a novel variation of model-agnostic meta-learning, where an extra multiplicative parameter is introduced in the inner-loop adaptation. Our variation creates a shortcut in the parameter space for the inner-loop adaptation and increases model expressivity in a highly controllable manner. We show both theoretically and numerically that our variation alleviates the problem of conflicting gradients and improves training dynamics. We conduct experiments on 3 distinctive problems, including a toy classification problem for threshold comparison, a regression problem for wavelet transform, and a classification problem on MNIST. We also discuss ways to generalize our method to a broader class of problems.