Triangle Inequality Research Articles

Freeprotmap: waiting-free prediction method for protein distance map

BackgroundProtein residue–residue distance maps are used for remote homology detection, protein information estimation, and protein structure research. However, existing prediction approaches are time-consuming, and hundreds of millions of proteins are discovered each year, necessitating the development of a rapid and reliable prediction method for protein residue–residue distances. Moreover, because many proteins lack known homologous sequences, a waiting-free and alignment-free deep learning method is needed.ResultIn this study, we propose a learning framework named FreeProtMap. In terms of protein representation processing, the proposed group pooling in FreeProtMap effectively mitigates issues arising from high-dimensional sparseness in protein representation. In terms of model structure, we have made several careful designs. Firstly, it is designed based on the locality of protein structures and triangular inequality distance constraints to improve prediction accuracy. Secondly, inference speed is improved by using additive attention and lightweight design. Besides, the generalization ability is improved by using bottlenecks and a neural network block named local microformer. As a result, FreeProtMap can predict protein residue–residue distances in tens of milliseconds and has higher precision than the best structure prediction method.ConclusionSeveral groups of comparative experiments and ablation experiments verify the effectiveness of the designs. The results demonstrate that FreeProtMap significantly outperforms other state-of-the-art methods in accurate protein residue–residue distance prediction, which is beneficial for lots of protein research works. It is worth mentioning that we could scan all proteins discovered each year based on FreeProtMap to find structurally similar proteins in a short time because the fact that the structure similarity calculation method based on distance maps is much less time-consuming than algorithms based on 3D structures.

Krylov complexity is not a measure of distance between states or operators

We ask whether Krylov complexity is mutually compatible with the circuit and Nielsen definitions of complexity. We show that the Krylov complexities between three states fail to satisfy the triangle inequality and so cannot be a measure of distance: there is no possible metric for which Krylov complexity is the length of the shortest path to the target state or operator. We show this explicitly in the simplest example, a single qubit, and in general. Published by the American Physical Society 2024

Novel picture fuzzy distance measure [formula omitted] and its application in medical diagnosis

In this research work, we introduced novel distance measure in picture fuzzy sets and prove all the triangle inequality associated with it. There is an introduction to the use of the distance measure in medical diagnostics. In addition to this, we have shown a comparison between two distance functions ρ and ρ∗ in this research work. Our observation indicates that the novel fuzzy distance measure ρ∗ exhibits greater efficacy in the context of disease diagnosis compared to the conventional measure ρ.

A Rényi-type quasimetric with random interference detection

This paper introduces a new dissimilarity measure between two discrete and finite probability distributions. The followed approach is grounded jointly on mixtures of probability distributions and an optimization procedure. We discuss the clear interpretation of the constitutive elements of the measure under an information-theoretical perspective by also highlighting its connections with the Rényi divergence of infinite order. Moreover, we show how the measure describes the inefficiency in assuming that a given probability distribution coincides with a benchmark one by giving formal writing of the random interference between the considered probability distributions. We explore the properties of the considered tool, which are in line with those defining the concept of quasimetric—i.e. a divergence for which the triangular inequality is satisfied. As a possible usage of the introduced device, an application to rare events is illustrated. This application shows that our measure may be suitable in cases where the accuracy of the small probabilities is a relevant matter.

Interpolation of generalized Ky Fan inequality for normal map on Eaton triple: Using gradients of G-increasing functions

In this paper, by using gradients of Gateaux differentiable G-increasing functions, we establish some interpolation results for the following G-majorization triangle inequality: (x+y)↓≺Gx↓+y↓ for vectors x,y of an Eaton triple with normal map (⋅)↓. In particular, we show refinements for the majorization Ky Fan's eigenvalue inequality λ(x+y)≺mλ(x)+λ(y) for x,y∈Hn, where Hn is the space of n×n Hermitian matrices and λ(⋅) is the eigenvalue operator on Hn. We derive analogous results for an inner product space with its standard norm, and for real skew-symmetric matrices with their singular values.

Quantum projections on conceptual subspaces: A deeper dive into methodological challenges and opportunities.

In alignment with the distributional hypothesis of language, the work "Quantum Projections on Conceptual Subspaces" (Martínez-Mingo A, Jorge-Botana G, Martinez-Huertas JÁ, et al. Quantum projections on conceptual subspaces. Cogn Syst Res 2023; 82: 101154) proposed a methodology for generating conceptual subspaces from textual information based on previous work (Martinez-Mingo A, Jorge-Botana G and Olmos R. Quantum approach for similarity evaluation in LSA vector space models. 2020). These subspaces enable the utilization of the quantum model of similarity put forth by Pothos and Busemeyer (Pothos E, Busemeyer J. A quantum probability explanation for violations of symmetry in similarity judgments. In Proceedings of the annual meeting of the cognitive science society, 2011, Vol. 33, No. 33), allowing for the empirical examination of the violations of assumptions concerning symmetry and triangular inequality (Tversky A. Features of similarity. Psychol Rev 1977; 84: 327-352; Yearsley JM, Barque-Duran A, Scerrati E, et al. The triangle inequality constraint in similarity judgments. Prog Biophys Mol Biol 2017; 130: 26-32), as well as the diagnosticity effect (Tversky A. Features of similarity. Psychol Rev 1977; 84: 327-352; Yearsley JM, Pothos EM, Barque-Duran A, et al. Context effects in similarity judgments. J Exp Psychol Gen 2022; 151: 711-717), within a data-driven environment. These psychological biases, deeply studied by authors such as Tversky and Kahneman, inform us about the limitations of modeling psychological similarity measures using tools from classical geometry. This commentary aims to offer methodological clarifications, discuss theoretical and practical implications, and speculate on future directions in this field of research. Concretely, it aims to propose the use of different contours (conceptual or contextual) to generate the subspaces, which lead to subspaces of terms or contexts. Once these contours are defined, a differentiation is proposed between Aggregated Terms Subspaces (ATSs), Aggregated Contexts Subspaces (ACSs), and Aggregated Features Subspaces (AFSs) depending on whether we define the subspaces by grouping the terms or contexts within the contour, or from the latent dimensions of the semantic space obtained in the contour window. Finally, new data is provided on the violation of the triangular inequality assumption through the application of the quantum similarity model to ATSs.

Wasserstein Differential Privacy

Differential privacy (DP) has achieved remarkable results in the field of privacy-preserving machine learning. However, existing DP frameworks do not satisfy all the conditions for becoming metrics, which prevents them from deriving better basic private properties and leads to exaggerated values on privacy budgets. We propose Wasserstein differential privacy (WDP), an alternative DP framework to measure the risk of privacy leakage, which satisfies the properties of symmetry and triangle inequality. We show and prove that WDP has 13 excellent properties, which can be theoretical supports for the better performance of WDP than other DP frameworks. In addition, we derive a general privacy accounting method called Wasserstein accountant, which enables WDP to be applied in stochastic gradient descent (SGD) scenarios containing subsampling. Experiments on basic mechanisms, compositions and deep learning show that the privacy budgets obtained by Wasserstein accountant are relatively stable and less influenced by order. Moreover, the overestimation on privacy budgets can be effectively alleviated. The code is available at https://github.com/Hifipsysta/WDP.

More Quickly-RRT*: Improved Quick Rapidly-exploring Random Tree Star algorithm based on optimized sampling point with better initial solution and convergence rate

RRT* (Rapidly-exploring Random Tree Star), as a variant of RRT (Rapidly-exploring Random Tree), is widely used to solve path planning problems because of its asymptotic optimality. However, the algorithm is inefficient due to the high initial path cost and the slow convergence rate. In this paper, we propose a More Quickly-RRT* (MQ-RRT*) path planning algorithm based on optimized sampling points to solve the problems. A sparse sampling mechanism is proposed in MQ-RRT* to improve the global search efficiency by reducing repetitive sampling. To make the random tree oriented when expanding, a dynamic goal-biased strategy is proposed, which can reduce the sampling time. Like Q-RRT* (Quick-RRT*), MQ-RRT* expands the set of possible parent nodes in the ChooseParent and Rewire phases, which reduces the path cost. On this basis, a method for creating a new parent node close to the obstacle is proposed. The creation process can be divided into two steps: Remove-tips and CreateNodes, which further reduces the cost of path generation and makes the path smoother by using the triangle inequality principle. Finally, numerical simulations are used to compare the proposed algorithm with RRT*, Q-RRT*, GuILD (Guided Incremental Local Densification), and F-RRT* (Fast-RRT*), which verifies that the proposed algorithm has certain advantages in path cost, convergence rate, and path smoothness.

Quantum mechanics on a p-adic Hilbert space: Foundations and prospects

We review some recent results on the mathematical foundations of a quantum theory over a scalar field that is a quadratic extension of the non-Archimedean field of [Formula: see text]-adic numbers. In our approach, we are inspired by the idea — first postulated in [I. V. Volovich, [Formula: see text]-adic string, Class. Quantum Grav. 4 (1987) L83–L87] — that space, below a suitably small scale, does not behave as a continuum and, accordingly, should be modeled as a totally disconnected metrizable topological space, ruled by a metric satisfying the strong triangle inequality. The first step of our construction is a suitable definition of a [Formula: see text]-adic Hilbert space. Next, after introducing all necessary mathematical tools — in particular, various classes of linear operators in a [Formula: see text]-adic Hilbert space — we consider an algebraic definition of physical states in [Formula: see text]-adic quantum mechanics. The corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a [Formula: see text]-adic counterpart of the POVMs associated with a standard quantum system over the complex numbers. Interestingly, it turns out that the typical convex geometry of the space of states of a standard quantum system is replaced, in the [Formula: see text]-adic setting, with an affine geometry; therefore, a symmetry transformation of a [Formula: see text]-adic quantum system may be defined as a map preserving this affine geometry. We argue that, as a consequence, the group of all symmetry transformations of a [Formula: see text]-adic quantum system has a richer structure with respect to the case of standard quantum mechanics over the complex numbers.

Development and Validation of an Interactive E-Module in Triangle Inequality and Its Theorems for Grade 8 Learners

Development and Validation of an Interactive E-Module in Triangle Inequality and Its Theorems for Grade 8 Learners

DBVS-APF-RRT*: A global path planning algorithm with ultra-high speed generation of initial paths and high optimal path quality

During the last decade, RRT* algorithm has been widely concerned by researchers because of its asymptotic optimality. However, the slow convergence rate of the RRT* algorithm leads to poor quality of the optimal paths at small number of iterations. Meanwhile, searching the initial path slowly limits its application scenarios. To overcome these problems, a directionally biased variable step APF-RRT* (DBVS-APF-RRT*) algorithm is proposed in this paper. Firstly, a novel directionally biased variable step sampling strategy is used in RRT* algorithm to quickly generate the initial path. Then, global random sampling and key region sampling strategies are added to improve global search ability and optimal path quality. At the same time, an artificial potential field (APF) method is introduced to improve the ability of obstacle avoidance. In addition, we propose a pruning strategy based on triangular inequality with direct connection of goal points to reduce the number of redundant nodes and shorten the path length. Finally, DBVS-APF-RRT* algorithm is compared with RRT*, Bias-RRT*, Informed-RRT* and Bias-P-RRT* algorithms to verify its superiority of optimal path quality, stability of the algorithm and its rapidity of finding the initial path.

A Duality-Based Proof of the Triangle Inequality for the Wasserstein Distances

A Duality-Based Proof of the Triangle Inequality for the Wasserstein Distances

Proof -Based Learning Design Using APOS Theory on Triangle Inequality Theorem Material

This research aims to produce a series of activities that can support evidence-based learning on the material of the Triangle Inequality Theorem. The research method used was design research type validation study and involved 22 4th semester students of the Mathematics Education study program at PGRI University Palembang. The research was conducted through 3 stages, namely preliminary design, design experiment and restropective analysis. Data collection was carried out using observation techniques, documentation and activity sheets. The research data were analyzed qualitatively, namely comparing the results of observations during the learning process with the HLT that had been designed. The results showed that the designed activities can support evidence-based learning for Triangle Inequality Theorem material.

Bayesian Consistency with the Supremum Metric

We present simple conditions for Bayesian consistency in the supremum metric. The key to the technique is a triangle inequality which allows us to explicitly use weak convergence, a consequence of the standard Kullback--Leibler support condition for the prior. A further condition is to ensure that smoothed versions of densities are not too far from the original density, thus dealing with densities which could track the data too closely. A key result of the paper is that we demonstrate supremum consistency using weaker conditions compared to those currently used to secure $\mathbb{L}_1$ consistency.

The Least Core of Routing Game Without Triangle Inequality

The Least Core of Routing Game Without Triangle Inequality

Operator inequalities via the triangle inequality

Operator inequalities via the triangle inequality

UBETI: A DV-Hop Localization Algorithm for IoT With Distance Upper Bound Estimation by Triangle Inequality

UBETI: A DV-Hop Localization Algorithm for IoT With Distance Upper Bound Estimation by Triangle Inequality

A Study of Triangle Inequality Violations in Social Network Clustering

A Study of Triangle Inequality Violations in Social Network Clustering

The Schatten $ p $-quasinorm on Euclidean Jordan algebras

<abstract><p>In this article, we proved that a Schatten $ p $-(quasi)norm for $ 0 < p < 1 $, defined on Euclidean Jordan algebras, satisfied a relaxed triangle inequality with an optimal constant $ 2^{\frac{1}{p} - 1} $; hence, it indeed induced a quasinorm. This confirmed the validity of a conjecture raised by Huang, Chen, and Hu.</p></abstract>

Accelerating Image Classification based on a Model for Estimating Descriptor-to-Class Distance

The article describes a method of image classification based on the estimation of the distance to the etalon class. The implementation of estimates gives a significant gain in classification speed compared to linear search while maintaining a decent level of accuracy. The methodology is based on the use of the triangle inequality for images given by a set of binary vectors as descriptors of the image key points. The evaluation is applied to the "object descriptor – etalon" classification method, which is based on the descriptor voting procedure. An analysis of evaluation options is carried out using the parameters of the etalon sets in the form of a medoid and the closest or farthest points from it. The gain in classification time compared to the traditional method proportionally depends on the number of descriptors in the etalon description. Software simulation of classifiers with the implementation of evaluation shows a gain in speed of 350-450 times for the description of 500 descriptors while maintaining one hundred percent classification accuracy on the training set of similar NFT images. A control sample experiment shows that the classifier with estimation can respond better to image details compared to the traditional method.