Social media platforms enable users to share diverse types of information, including geolocation data that captures their movement patterns. Such geolocation data can be leveraged to reconstruct the trajectory of a user's visited Points of Interest (POIs). A key requirement in numerous applications is the ability to measure the similarity between such trajectories, as this facilitates the retrieval of trajectories that are similar to a given reference trajectory. This is the main focus of our work. Existing methods predominantly rely on applying a similarity function to each candidate trajectory to identify those that are sufficiently similar. However, this approach becomes computationally expensive when dealing with large-scale datasets. To mitigate this challenge, we propose TISIS, an efficient method that uses trajectory indexing to quickly find similar trajectories that share common POIs in the same order. Furthermore, to account for scenarios where POIs in trajectories may not exactly match but are contextually similar, we introduce TISIS*, a variant of TISIS that incorporates POI embeddings. This extension allows for more comprehensive retrieval of similar trajectories by con
Measuring visual similarity is critical for image understanding. But what makes two images similar? Most existing work on visual similarity assumes that images are similar because they contain the same object instance or category. However, the reason why images are similar is much more complex. For example, from the perspective of category, a black dog image is similar to a white dog image. However, in terms of color, a black dog image is more similar to a black horse image than the white dog image. This example serves to illustrate that visual similarity is ambiguous but can be made precise when given an explicit contextual perspective. Based on this observation, we propose the concept of contextual visual similarity. To be concrete, we examine the concept of contextual visual similarity in the application domain of image search. Instead of providing only a single image for image similarity search (\eg, Google image search), we require three images. Given a query image, a second positive image and a third negative image, dissimilar to the first two images, we define a contextualized similarity search criteria. In particular, we learn feature weights over all the feature dimensions
In this paper we investigate $p$-adic self-similar sets and $p$-adic self-similar measures. We show that $p$-adic self-similar sets are $p$-adic path set fractals, and that the converse is not necessarily true. For $p$-adic self-similar sets and $p$-adic self-similar measures, we show the existence of a unique essential class. We show that, under mild assumptions, the decimation of $p$-adic self-similar sets is maximal. For $p$-adic self-similar measures, we show that many results involving local dimension are similar to those of their real counterparts, with fewer complications. Most of these results use the additional structure of self-similarity, and are not true in general for $p$-adic path set fractals.
We construct examples and provide a classification of self-similar solutions to the two-dimensional incompressible Euler equations whose pseudo-velocity fields possess more than one stagnation point. These solutions are also homogeneous steady states of the Euler equations. In contrast, we prove that any homogeneous self-similar solution with bounded vorticity away from the origin necessarily admits only a single stagnation point, located at the origin. The solutions we construct develop velocity cusps along rays from the origin, and this allows for additional stagnation points of the pseudo-velocity field.
Let $K\subset\mathbb R^d$ be a compact subset equipped with a $δ$-Ahlfors regular measure $μ$. For any $τ>1/d$ and any ``inhomogeneous'' vector $\boldsymbolθ\in\mathbb R^d$, let $W_d(ψ_τ,\boldsymbolθ)$ denote the set of $(ψ_τ,\boldsymbolθ)$-well approximable numbers, where $ψ_τ(q)=q^{-τ}$. Assuming a local estimate for the $μ$-measure of the intersections of $K$ with the neighborhoods of ``rational'' vectors $(\mathbf p+\boldsymbolθ)/q$, we establish a sharp upper bound for the Hausdorff dimension of $K\cap W_d(ψ_τ,\boldsymbolθ)$, together with some nontrivial lower bounds when $τ$ is below a certain threshold. One of the lower bounds becomes sharp in the one-dimensional homogeneous case ($d=1$, $θ=0$) for a class of sufficiently thick self-similar sets $K$, and moreover $K\cap W_1(ψ_τ,0)$ has full $(δ+\frac{2}{1+τ}-1)$-Hausdorff measure. These results have several applications: (1) the set of homogeneous very well approximable numbers has full Hausdorff dimension within strongly irreducible self-similar sets in $\mathbb R^d$, extending a recent result of Chen [arXiv:2510.17096]; (2) the set of inhomogeneous very well approximable numbers has full Hausdorff dimension within suff
Knowledge graph embedding models (KGEMs) developed for link prediction learn vector representations for entities in a knowledge graph, known as embeddings. A common tacit assumption is the KGE entity similarity assumption, which states that these KGEMs retain the graph's structure within their embedding space, \textit{i.e.}, position similar entities within the graph close to one another. This desirable property make KGEMs widely used in downstream tasks such as recommender systems or drug repurposing. Yet, the relation of entity similarity and similarity in the embedding space has rarely been formally evaluated. Typically, KGEMs are assessed based on their sole link prediction capabilities, using ranked-based metrics such as Hits@K or Mean Rank. This paper challenges the prevailing assumption that entity similarity in the graph is inherently mirrored in the embedding space. Therefore, we conduct extensive experiments to measure the capability of KGEMs to cluster similar entities together, and investigate the nature of the underlying factors. Moreover, we study if different KGEMs expose a different notion of similarity. Datasets, pre-trained embeddings and code are available at: ht
Similar subtrajectory search is a finer-grained operator that can better capture the similarities between one query trajectory and a portion of a data trajectory than the traditional similar trajectory search, which requires the two checked trajectories are similar to each other in whole. Many real applications (e.g., trajectory clustering and trajectory join) utilize similar subtrajectory search as a basic operator. It is considered that the time complexity is O(mn^2) for exact algorithms to solve the similar subtrajectory search problem under most trajectory distance functions in the existing studies, where m is the length of the query trajectory and n is the length of the data trajectory. In this paper, to the best of our knowledge, we are the first to propose an exact algorithm to solve the similar subtrajectory search problem in O(mn) time for most of widely used trajectory distance functions (e.g., WED, DTW, ERP, EDR and Frechet distance). Through extensive experiments on three real datasets, we demonstrate the efficiency and effectiveness of our proposed algorithms.
This paper investigates a system of nonlinear reaction-diffusion equations modeling the industrial synthesis of ammonia. By applying Lie group analysis, we construct self-similar solutions and derive a reduced system of ordinary differential equations. Using comparison principles and barrier techniques, we establish sufficient conditions for the existence of global-in-time solutions in both slow-diffusion ($γ_i > 0$) and fast-diffusion ($γ_i < 0$) regimes. Detailed asymptotic analysis near the diffusion front reveals power-law behavior of concentration profiles, with explicit expressions for the decay exponents. The theoretical results are illustrated by numerical simulations, demonstrating the spatio-temporal evolution of reactant concentrations under realistic parameter values. The study provides rigorous mathematical foundations for predicting and optimizing ammonia production in catalytic reactors, with potential extensions to other chemically reacting systems.
When students make a mistake in an exercise, they can consolidate it by ``similar exercises'' which have the same concepts, purposes and methods. Commonly, for a certain subject and study stage, the size of the exercise bank is in the range of millions to even tens of millions, how to find similar exercises for a given exercise becomes a crucial technical problem. Generally, we can assign a variety of explicit labels to the exercise, and then query through the labels, but the label annotation is time-consuming, laborious and costly, with limited precision and granularity, so it is not feasible. In practice, we define ``similar exercises'' as a retrieval process of finding a set of similar exercises based on recall, ranking and re-rank procedures, called the \textbf{FSE} problem (Finding similar exercises). Furthermore, comprehensive representation of the semantic information of exercises was obtained through representation learning. In addition to the reasonable architecture, we also explore what kind of tasks are more conducive to the learning of exercise semantic information from pre-training and supervised learning. It is difficult to annotate similar exercises and the annotatio
Finding players with similar profiles is an important problem in sports such as football. Scouting for new players requires a wealth of information about the available players so that similar profiles to that of a target player can be identified. However, information about the position of the players in the field is seldom used. For this reason, a novel approach based on spatial data analysis is introduced to produce a spatial similarity index that can help to identify similar players. The use of this new spatial similarity index is illustrated to identify similar players using spatial data from the Spanish competition "La Liga", season 2019-2020.
The degree matrix of a graph is the diagonal matrix with diagonal entries equal to the degrees of the vertices of $X$. If $X_1$ and $X_2$ are graphs with respective adjacency matrices $A_1$ and $A_2$ and degree matrices $D_1$ and $D_2$, we say that $X_1$ and $X_2$ are degree similar if there is an invertible real matrix $M$ such that $M^{-1}A_1M=A_2$ and $M^{-1}D_1M=D_2$. If graphs $X_1$ and $X_2$ are degree similar, then their adjacency matrices, Laplacian matrices, unsigned Laplacian matrices and normalized Laplacian matrices are similar. We first show that the converse is not true. Then, we provide a number of constructions of degree-similar graphs. Finally, we show that the matrices $A_1-μD_1$ and $A_2-μD_2$ are similar over the field of rational functions $\mathbb{Q}(μ)$ if and only if the Smith normal forms of the matrices $tI-(A_1-μD_1)$ and $tI-(A_2-μD_2)$ are equal.
Non-autonomous self-similar sets are a family of compact sets which are, in some sense, highly homogeneous in space but highly inhomogeneous in scale. The main purpose of this note is to clarify various regularity properties and separation conditions relevant for the fine local scaling properties of these sets. A simple application of our results is a precise formula for the Assouad dimension of non-autonomous self-similar sets in $\mathbb{R}^d$ satisfying a certain ``bounded neighbourhood'' condition, which generalizes earlier work of Li--Li--Miao--Xi and Olson--Robinson--Sharples. We also see that the bounded neighbourhood assumption is, in few different senses, as general as possible.
Knowledge distillation is a widely applicable technique for training a student neural network under the guidance of a trained teacher network. For example, in neural network compression, a high-capacity teacher is distilled to train a compact student; in privileged learning, a teacher trained with privileged data is distilled to train a student without access to that data. The distillation loss determines how a teacher's knowledge is captured and transferred to the student. In this paper, we propose a new form of knowledge distillation loss that is inspired by the observation that semantically similar inputs tend to elicit similar activation patterns in a trained network. Similarity-preserving knowledge distillation guides the training of a student network such that input pairs that produce similar (dissimilar) activations in the teacher network produce similar (dissimilar) activations in the student network. In contrast to previous distillation methods, the student is not required to mimic the representation space of the teacher, but rather to preserve the pairwise similarities in its own representation space. Experiments on three public datasets demonstrate the potential of our a
We consider connections between similar sublattices and coincidence site lattices (CSLs), and more generally between similar submodules and coincidence site modules of general (free) $\mathbb{Z}$-modules in $\mathbb{R}^d$. In particular, we generalise results obtained by S. Glied and M. Baake [1,2] on similarity and coincidence isometries of lattices and certain lattice-like modules called $\mathcal{S}$-modules. An important result is that the factor group $\mathrm{OS}(M)/\mathrm{OC}(M)$ is Abelian for arbitrary $\mathbb{Z}$-modules $M$, where $\mathrm{OS}(M)$ and $\mathrm{OC}(M)$ are the groups of similar and coincidence isometries, respectively. In addition, we derive various relations between the indices of CSLs and their corresponding similar sublattices. [1] S. Glied, M. Baake, Similarity versus coincidence rotations of lattices, Z. Krist. 223, 770--772 (2008). DOI: 10.1524/zkri.2008.1054 [2] S. Glied, Similarity and coincidence isometries for modules, Can. Math. Bull. 55, 98--107 (2011). DOI: 10.4153/CMB-2011-076-x
Equal pay laws increasingly require that workers doing "similar" work are paid equal wages within firm. We study such "equal pay for similar work" (EPSW) policies theoretically and test our model's predictions empirically using evidence from a 2009 Chilean EPSW. When EPSW only binds across protected class (e.g., no woman can be paid less than any similar man, and vice versa), firms segregate their workforce by gender. When there are more men than women in a labor market, EPSW increases the gender wage gap. By contrast, EPSW that is not based on protected class can decrease the gender wage gap.
While there has been substantial progress in learning suitable distance metrics, these techniques in general lack transparency and decision reasoning, i.e., explaining why the input set of images is similar or dissimilar. In this work, we solve this key problem by proposing the first method to generate generic visual similarity explanations with gradient-based attention. We demonstrate that our technique is agnostic to the specific similarity model type, e.g., we show applicability to Siamese, triplet, and quadruplet models. Furthermore, we make our proposed similarity attention a principled part of the learning process, resulting in a new paradigm for learning similarity functions. We demonstrate that our learning mechanism results in more generalizable, as well as explainable, similarity models. Finally, we demonstrate the generality of our framework by means of experiments on a variety of tasks, including image retrieval, person re-identification, and low-shot semantic segmentation.
This papers explores the self similar solutions of the Vlasov-Poisson system and their relation to the gravitational collapse of dynamically cold systems. Analytic solutions are derived for power law potential in one dimension, and extensions of these solutions in three dimensions are proposed. Next the self similarity of the collapse of cold dynamical systems is investigated numerically. The fold system in phase space is consistent with analytic self similar solutions, the solutions present all the proper self-similar scalings. An additional point is the appearance of an $x^{-(1/2)}$ law at the center of the system for initial conditions with power law index larger than $-(1/2)$. It is found that the first appearance of the $x^{-(1/2)}$ law corresponds to the formation of a singularity very close to the center. Finally the general properties of self similar multi dimensional solutions near equilibrium are investigated. Smooth and continuous self similar solutions have power law behavior at equilibrium. However cold initial conditions result in discontinuous phase space solutions, and the smoothed phase space density looses its auto similar properties. This problem is easily solved
A linear isometry $R$ of $\mathbb{R}^d$ is called a similarity isometry of a lattice $Γ\subseteq \mathbb{R}^d$ if there exists a positive real number $β$ such that $βRΓ$ is a sublattice of (finite index in) $Γ$. The set $βRΓ$ is referred to as a similar sublattice of $Γ$. A (crystallographic) point packing generated by a lattice $Γ$ is a union of $Γ$ with finitely many shifted copies of $Γ$. In this study, the notion of similarity isometries is extended to point packings. We provide a characterization for the similarity isometries of point packings and identify the corresponding similar subpackings. Planar examples will be discussed, namely, the $1 \times 2$ rectangular lattice and the hexagonal packing (or honeycomb lattice). Finally, we also consider similarity isometries of point packings about points different from the origin by studying similarity isometries of shifted point packings. In particular, similarity isometries of a certain shifted hexagonal packing will be computed and compared with that of the hexagonal packing.
The concept of degree of similarity (η) is proposed to quantitatively describe the similarity of a parameter (e.g. the maximum amplitude Rmax) of a solar cycle relative to a referenced one, and the prediction method of similar cycles is further developed. For two parameters, the solar minimum (Rmin) and rising rate (βa), which can be directly measured a few months after the minimum, a synthesis degree of similarity (ηs) is defined as the weighted-average of the η values around Rmin and βa with the weights given by the coefficients of determination of Rmax with Rmin and βa, respectively. The monthly values of the whole referenced cycle can be predicted by averaging the corresponding values in the most similar cycles with the weights given by the ηs values. Cycles 14 and 10 are found to be the two most similar cycles of Cycle 24. As an application, Cycle 24 is predicted to peak around January 2013{\pm}8 (months) with a size of about Rmax =83.0{\pm}16.7 and to end around September 2019.
We consider methods for quantifying the similarity of vertices in networks. We propose a measure of similarity based on the concept that two vertices are similar if their immediate neighbors in the network are themselves similar. This leads to a self-consistent matrix formulation of similarity that can be evaluated iteratively using only a knowledge of the adjacency matrix of the network. We test our similarity measure on computer-generated networks for which the expected results are known, and on a number of real-world networks.