Quadratic Space Research Articles

Isometric rigidity of Wasserstein spaces over Euclidean spheres

We study the structure of isometries of the quadratic Wasserstein space W2(Sn,‖⋅‖) over the sphere endowed with the distance inherited from the norm of Rn+1. We prove that W2(Sn,‖⋅‖) is isometrically rigid, meaning that its isometry group is isomorphic to that of (Sn,‖⋅‖). This is in striking contrast to the non-rigidity of its ambient space W2(Rn+1,‖⋅‖) but in line with the rigidity of the geodesic space W2(Sn,∢). One of the key steps of the proof is the use of mean squared error functions to mimic displacement interpolation in W2(Sn,‖⋅‖). A major difficulty in proving rigidity for quadratic Wasserstein spaces is that one cannot use the Wasserstein potential technique. To illustrate its general power, we use it to prove isometric rigidity of Wp(S1,‖⋅‖) for 1≤p<2.

Should I make it round? Suitability of circular and linear layouts for comparative tasks with matrix and connective data

AbstractVisual representations based on circular shapes are frequently used in visualization applications. One example are circos plots within bioinformatics, which bend graphs into a wheel of information with connective lines running through the center like spokes. The results are aesthetically appealing and impressive visualizations that fit long data sequences into a small quadratic space. However, the authors' experiences are that when asked, a visualization researcher would generally advise against making visualizations with radial layouts. Upon reviewing the literature we found that there is evidence that circular layouts are preferable in some cases, but we found no clear evidence for what layout is preferable for matrices and connective data in particular, which both are common data types in circos plots. In this work, we thus performed a user study to compare circular and linear layouts. The tasks are inspired by genomics data, but our results generalize to many other application areas, involving comparison and connective data. To build the prototype we utilized Gosling, a grammar for visualizing genomics data. We contribute empirical evidence on the suitedness of linear versus circular layouts, adding to the specific and general knowledge concerning perception of circular graphs. In addition, we contribute a case study evaluation of the grammar Gosling as a rapid prototyping language, confirming its utility and providing guidance on suitable areas for future development.

Fixed point groups of involutions of type O(q,k) over a field of characteristic two

ABSTRACT For G = O ⁡ ( q , k ) , the orthogonal group over a field k of characteristic 2 with respect to a quadratic form q , we discuss the G-conjugacy classes of fixed points of involutions. When the quadratic space is either totally singular or nonsingular, a full classification of the isomorphism classes is given. We also give some implications of these results for a general quadratic space over a field of characteristic 2.

Nonlinearity-Adaptive Data-Driven Power Flow Constraint for Distribution Network Optimization

Data-driven power flow (PF) analysis methods have higher practical value to deal with the inaccurate branch parameters in medium and low-voltage distribution networks, which can be easily integrated into the classic optimization framework as a constraint. However, the PF mapping relationship of the measurement data shows stronger nonlinearity, with the high penetration of distributed generation (DG), leading to the low accuracy of the linear PF constraint. To this end, we propose an incomplete dimension lifting (IDL) data-driven PF method for distribution network optimization. The state variables are divided into optimization variables and uncontrollable variables, among them, the optimization variables keep the original linear state space or converted into quadratic space to simplify the optimization solution; and the uncontrollable variables is dimension lifted to fit the nonlinearity of PF. In addition, in order to deal with the problem of gradient direction error caused by nonlinear model overfitting, the first-order sensitivity constraint is involved in parameter regression process. The standard IEEE cases with high penetration of DG demonstrate that the proposed method can achieve higher accurate PF constraint and better optimization results, compared to the existing data-driven PF constrain.

Parameter-Free and Scalable Incomplete Multiview Clustering With Prototype Graph.

Multiview clustering (MVC) seamlessly combines homogeneous information and allocates data samples into different communities, which has shown significant effectiveness for unsupervised tasks in recent years. However, some views of samples may be incomplete due to unfinished data collection or storage failure in reality, which refers to the so-called incomplete multiview clustering (IMVC). Despite many IMVC pioneer frameworks have been introduced, the majority of their approaches are limited by the cubic time complexity and quadratic space complexity which heavily prevent them from being employed in large-scale IMVC tasks. Moreover, the massively introduced hyper-parameters in existing methods are not practical in real applications. Inspired by recent unsupervised multiview prototype progress, we propose a novel parameter-free and scalable incomplete multiview clustering framework with the prototype graph termed PSIMVC-PG to solve the aforementioned issues. Different from existing full pair-wise graph studying, we construct an incomplete prototype graph to flexibly capture the relations between existing instances and discriminate prototypes. Moreover, PSIMVC-PG can directly obtain the prototype graph without pre-process of searching hyper-parameters. We conduct massive experiments on various incomplete multiview tasks, and the performances show clear advantages over existing methods. The code of PSIMVC-PG can be publicly downloaded at https://github.com/wangsiwei2010/PSIMVC-PG.

Greedy optimization of resistance-based graph robustness with global and local edge insertions

The total effective resistance, also called the Kirchhoff index, provides a robustness measure for a graph G. We consider two optimization problems of adding k new edges to G such that the resulting graph has minimal total effective resistance (i.e., is most robust)—one where the new edges can be anywhere in the graph and one where the new edges need to be incident to a specified focus node. The total effective resistance and effective resistances between nodes can be computed using the pseudoinverse of the graph Laplacian. The pseudoinverse may be computed explicitly via pseudoinversion, yet this takes cubic time in practice and quadratic space. We instead exploit combinatorial and algebraic connections to speed up gain computations in an established generic greedy heuristic. Moreover, we leverage existing randomized techniques to boost the performance of our approaches by introducing a sub-sampling step. Our different graph- and matrix-based approaches are indeed significantly faster than the state-of-the-art greedy algorithm, while their quality remains reasonably high and is often quite close. Our experiments show that we can now process larger graphs for which the application of the state-of-the-art greedy approach was impractical before.

Discovery of interesting frequent item sets in an uncertain database using ant colony optimization

ABSTRACT Applications like business basket analysis, digital service analytics, bio-informatics, and mobile commerce have greatly benefited from the information retrieval of significant features from massive databases for improved decision-making. Item set mining is used to find intriguing patterns in databases. Discovering item sets in an uncertain database is a tedious task. Only mathematical correlations between the elements in an item set are the exclusive subject of recurring item set mining research. The finding is direct to optimal. This article introduces an ant colony that maps the viable solution space to a directed graph with quadratic space complexity. The proposed model evaluates an uncertain transaction database's item set. Compared to the current methods, the findings demonstrate the importance of the proposed model.

A scalable parallel algorithm for global sequence alignment with customizable scoring scheme

SummarySequence alignment is a critical computational problem in various domains, including genomics, proteomics, and natural language processing. The Needleman‐Wunsch (NW) algorithm is a classical dynamic programming approach for finding the optimal global alignment between two sequences. However, its quadratic time and space complexity make it impractical for aligning large‐scale sequences, which are increasingly common in modern applications. In this article, we propose a parallel variation of the NW algorithm that enables scalable global sequence alignment with customizable scoring schemes. Our approach re‐formulates the dependencies in the NW algorithm to enable parallel execution, thereby leveraging the computational power of modern parallel architectures, such as graphics processing unit (GPU). Furthermore, our algorithm supports arbitrary linear scoring schemes, which allows us to use domain‐specific knowledge to improve alignment accuracy. We establish the correctness of our algorithm and evaluate its performance using real DNA and user trajectory sequences on GPUs. Our parallel algorithm has shown impressive results in our experiments, with a peak performance of 27.99 GCUPS (giga cell updates per second) and a maximum speedup of 48.18 times compared to the traditional sequential implementation. Additionally, our algorithm demonstrates remarkable scalability, enabling the alignment of sequences of any length while ensuring balanced work distribution and optimal utilization of resources. Our primary objective is to harness the computational capabilities of a single GPU and fully utilize the processing power of multi‐core CPUs.

Diagonal restriction of Eisenstein series and Kudla–Millson theta lift

Abstract We consider the Kudla–Millson theta series associated to a quadratic space of signature ( N , N ) {(N,N)} . By combining a “see-saw” argument with the Siegel–Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree N is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon, Pozzi and Vonk to totally real fields.

Asymptotically Better Query Optimization Using Indexed Algebra

Query optimization is essential for the efficient execution of queries. The necessary analysis, if we can and should apply optimizations and transform the query plan, is already challenging. Traditional techniques focus on the availability of columns at individual operators, which does not scale for analysis of data flow through the query. Tracking available columns per operator takes quadratic space, which can result in multi-second optimization time for deep algebra trees. Instead, we need to re-think the naïve algebra representation to efficiently support data flow analysis. In this paper, we introduce Indexed Algebra , a novel representation of relational algebra that makes common optimization tasks efficient. Indexed Algebra enables efficient reasoning with an auxiliary index structure based on link/cut trees that support dynamic updates and queries in O (log n ). This approach not only improves the asymptotic complexity, but also allows elegant and concise formulations for the data flow questions needed for query optimization. While large queries see theoretically unbounded improvements, Indexed Algebra also improves optimization time of the relatively harmless queries of TPC-H and TPC-DS by more than 1.8×.

Boundary value problems for a second‐order elliptic partial differential equation system in Euclidean space

Let be a bounded regular domain, let be the standard Dirac operator in , and let be the Clifford algebra constructed over the quadratic space . For fixed, denotes the space of ‐vectors in . In the framework of Clifford analysis, we consider two boundary value problems for a second‐order elliptic system of partial differential equations of the form in , where is a smooth ‐vector valued function. The boundary conditions of the problems contain the inner and outer products of the ‐vector solution with both the Dirac operator and the normal vector to , ensuring the well‐posedness for the problems. Investigation of the spectral properties of the sandwich operator is considered by using the Fredholm theory. Finally, it is shown that satisfactory problem‐solving properties, in general, fail when we replace the standard Dirac operator by those, obtained via unusual orthogonal bases of .

On the exotic isometry flow of the quadratic Wasserstein space over the real line

Kloeckner discovered that the quadratic Wasserstein space over the real line (denoted by W2(R)) is quite peculiar, as its isometry group contains an exotic isometry flow. His result implies that it can happen that an isometry Φ fixes all Dirac measures, but still, Φ is not the identity of W2(R). This is the only known example of this surprising and counterintuitive phenomenon. Kloeckner also proved that the image of each finitely supported measure under these isometries (and thus under all isometry) is a finitely supported measure. Recently we showed that the exotic isometry flow can be represented as a unitary group on L2((0,1)). In this paper, we calculate the generator of this group, and we show that every exotic isometry (and thus every isometry) maps the set of all absolutely continuous measures belonging to W2(R) onto itself.

Memory-efficient Transformer-based network model for Traveling Salesman Problem

Combinatorial optimization problems such as Traveling Salesman Problem (TSP) have a wide range of real-world applications in transportation, logistics, manufacturing. It has always been a difficult problem to solve large-scale TSP problems quickly because of memory usage limitations. Recent research shows that the Transformer model is a promising approach. However, the Transformer has several severe problems that prevent it from quickly solving TSP combinatorial optimization problems, such as quadratic time complexity, especially quadratic space complexity, and the inherent limitations of the encoder and decoder itself. To address these issues, we developed a memory-efficient Transformer-based network model for TSP combinatorial optimization problems, termed Tspformer, with two distinctive characteristics: (1) a sampled scaled dot-product attention mechanism with O(Llog(L)) (L is the length of input sequences) time and space complexity, which is the most different between our work and other works. (2) due to the reduced space complexity, GPU/CPU memory usage is significantly reduced. Extensive experiments demonstrate that Tspformer significantly outperforms existing methods and provides a new solution to the TSP combinatorial optimization problems. Our Pytorch code will be publicly available on GitHub https://github.com/yhnju/tspFormer.

An explicit lifting construction of CAP forms on O(1,5)

In this paper, we explicitly construct nontempered cusp forms on the orthogonal group O(1,5) of signature [Formula: see text]. Given a definite quaternion algebra [Formula: see text] over [Formula: see text], the orthogonal group is attached to the indefinite quadratic space of rank 6 with the anisotropic part defined by the reduced norm of [Formula: see text]. Our construction can be viewed as a generalization of the previous work by the first two authors joint with Masanori Muto to the case of any definite quaternion algebras, for which we note that the work just mentioned takes up the case where the discriminant of [Formula: see text] is two. Unlike the previous work the method of the construction is to consider the theta lifting from Maass cusp forms to O(1,5), following the formulation by Borcherds. The cuspidal representations generated by our cusp forms are studied in detail. We determine all local components of the cuspidal representations and show that our cusp forms are CAP forms.

A POISSON ALGEBRA STRUCTURE OVER THE EXTERIOR ALGEBRA OF A QUADRATIC SPACE

We construct a Poisson algebra structure of degree $-2$ over the exterior algebra of a quadratic space. Here we do not use Clifford algebra as in [4].

Higher Order Finite Element Methods for Some One-dimensional Boundary Value Problems

In this paper, third-order compact and fourth-order finite element methods (FEMs) based on simple modifications of traditional FEMs are proposed for solving one-dimensional Sturm-Liouville boundary value problems (BVPs). The key idea is based on interpolation error estimates. A simple posterior error analysis of the original piecewise linear finite element space leads to a third-order accurate solution in the L2 norm, second-order in the H1, and the energy norm. The novel idea is also applied to obtain a fourth-order FEM based on the quadratic finite element space. The basis functions in the new fourth-order FEM are more compact compared with that of the classic cubic basis functions. Numerical examples presented in this paper have confirmed the convergence order and analysis. A generalization to a class of nonlinear two-point BVPs is also discussed and tested.

Explicit class number formulas for Siegel–Weil averages of ternary quadratic forms

In this paper, we investigate the interplay between positive-definite integral ternary quadratic forms and class numbers. We generalize a result of Jones relating the theta function for the genus of a quadratic form to the Hurwitz class numbers, obtaining an asymptotic formula (with a main term and error term away from finitely many bad square classes t j Z 2 t_j\mathbb {Z}^2 ) relating the number of lattice points in a quadratic space of a given norm with a sum of class numbers related to that norm and the squarefree part of the discriminant of the quadratic form on this lattice.

Cusp Density and Commensurability of Non-arithmetic Hyperbolic Coxeter Orbifolds

For three distinct infinite families (R_m), (S_m), and (T_m) of non-arithmetic 1-cusped hyperbolic Coxeter 3-orbifolds, we prove incommensurability for a pair of elements X_k and Y_l belonging to the same sequence and for most pairs belonging two different ones. We investigate this problem first by means of the Vinberg space and the Vinberg form, a quadratic space associated to each of the corresponding fundamental Coxeter prism groups, which allows us to deduce some partial results. The complete proof is based on the analytic behavior of another commensurability invariant. It is given by the cusp density, and we prove and exploit its strict monotonicity.

Indexing the extended Dyck-CFL reachability for context-sensitive program analysis

Many context-sensitive dataflow analyses can be formulated as an extended Dyck-CFL reachability problem, where function calls and returns are modeled as partially matched parentheses. Unfortunately, despite many works on the standard Dyck-CFL reachability problem, solving the extended version is still of quadratic space complexity and nearly cubic time complexity, significantly limiting the scalability of program analyses. This paper, for the first time to the best of our knowledge, presents a cheap approach to transforming the extended Dyck-CFL reachability problem to conventional graph reachability, a much easier and well-studied problem. This transformation allows us to benefit from recent advances in reachability indexing schemes, making it possible to answer any reachability query in a context-sensitive dataflow analysis within almost constant time plus only a few extra spaces. We have implemented our approach in two common context-sensitive dataflow analyses, one determines pointer alias relations and the other tracks information flows. Experimental results demonstrate that, compared to their original analyses, we can achieve orders of magnitude (10 2 × to 10 5 ×) speedup at the cost of only a moderate space overhead. Our implementation is publicly available.

Toward Efficient Similarity Search under Edit Distance on Hybrid Architectures

Edit distance is the most widely used method to quantify similarity between two strings. We investigate the problem of similarity search under edit distance. Given a collection of sequences, the goal of similarity search under edit distance is to find sequences in the collection that are similar to a given query sequence where the similarity score is computed using edit distance. The canonical method of computing edit distance between two strings uses a dynamic programming-based approach that runs in quadratic time and space, which may not provide results in a reasonable amount of time for large sequences. It advocates for parallel algorithms to reduce the time taken by edit distance computation. To this end, we present scalable parallel algorithms to support efficient similarity search under edit distance. The efficiency and scalability of the proposed algorithms is demonstrated through an extensive set of experiments on real datasets. Moreover, to address the problem of uneven workload across different processing units, which is mainly caused due to the significant variance in the size of the sequences, different data distribution schemes are discussed and empirically analyzed. Experimental results have shown that the speedup achieved by the hybrid approach over inter-task and intra-task parallelism is 18 and 13, respectively.