Approximate Vector Research Articles

Winning of inhomogeneous bad for curves

AbstractWe prove the absolute winning property of weighted inhomogeneous badly approximable vectors on nondegenerate analytic curves. This answers a question by Beresnevich, Nesharim, and Yang. In particular, our result is an inhomogeneous version of the main result in (Duke Math J 171(14), 2022) by Beresnevich, Nesharim, and Yang. Also, the generality of the inhomogeneous part that we considered extends the previous result in An et al. (Adv Math 324:148–202, 2018). Moreover, our results even contribute to classical results, namely establishing the inhomogeneous Schmidt’s conjecture in arbitrary dimensions.

On perturbations for spectrum and singular value decompositions followed by deflation techniques

The calculation of the dominant eigenvalues of a symmetric matrix A together with its eigenvectors, followed by the calculation of the deflation of A1=A−ρUkUkT corresponds to one step of the Wielandt deflation technique, where ρ is a shift and Uk are eigenvectors of A. In this paper, we investigate how the eigenspace of A1 changes when A1 is perturbed to A˜1=A−ρU˜kU˜kT, where U˜k are approximate eigenvectors of A. We establish the bounds for the angle of eigenspaces of A1 and A˜1 based on the Davis-Kahan theorem. Moreover, in the practical implementation for singular value decomposition, once one or several singular triplets converge to a preset accuracy, they should be deflated by B1=B−γWkVkH with γ being a shift, Wk and Vk are singular vectors of B, so that they will not be re-computed. We investigate how the singular subspaces of B1=B−γWkVkH change when B1 is perturbed to B˜1=B−γW˜kV˜kH, W˜k and V˜k are approximate singular vectors of B. We also establish the bounds for the angle of singular subspaces of B1 and B˜1 based on the Wedin theorem.

Momentum recursive DARTS

DARTS has emerged as a popular method for neural architecture search (NAS) owing to its efficiency and simplicity. It employs gradient-based bi-level optimization to iteratively optimize the upper-level architecture parameters and lower-level super-network weights. The key challenge in DARTS is the accurate estimation of gradients for two-level object functions, leading to significant errors in gradient approximation. To address this issue, we propose a new approach, MR-DARTS, that incorporates a momentum term and a recursive scheme to improve gradient estimation. Specifically, we leverage historical information by using a running average of past observed gradients to enhance the quality of current gradient estimation in both upper-level and lower-level functions. Our theoretical analysis shows that the variance of our estimated gradient decreases with each iteration. By utilizing momentum and a recursive scheme, MR-DARTS effectively controls the error in stochastic gradient updates that result from inaccurate gradient estimation. Furthermore, we utilize the Neumann series approximation and Hessian Vector Product scheme to reduce computational requirements and memory usage. We evaluate our proposed method on several benchmarks and demonstrate its effectiveness through comprehensive experiments.

A remark on the set of exactly approximable vectors in the simultaneous case

We compute the Hausdorff dimension of the set of ψ \psi -exactly approximable vectors, in the simultaneous case, in dimension strictly larger than 2 2 and for approximating functions ψ \psi with order at infinity less than or equal to − 2 -2 . Our method relies on the analogous result in dimension 1 1 , proved by Yann Bugeaud and Carlos Moreira, and a version of Jarník’s theorem on fibres.

Hierarchical Vector-Quantized Variational Autoencoder and Vector Credibility Mechanism for High-Quality Image Inpainting

Image inpainting infers the missing areas of a corrupted image according to the information of the undamaged part. Many existing image inpainting methods can generate plausible inpainted results from damaged images with the fast-developed deep-learning technology. However, they still suffer from over-smoothed textures or textural distortion in the cases of complex textural details or large damaged areas. To restore textures at a fine-grained level, we propose an image inpainting method based on a hierarchical VQ-VAE with a vector credibility mechanism. It first trains the hierarchical VQ-VAE with ground truth images to update two codebooks and to obtain two corresponding vector collections containing information on ground truth images. The two vector collections are fed to a decoder to generate the corresponding high-fidelity outputs. An encoder then is trained with the corresponding damaged image. It generates vector collections approximating the ground truth by the help of the prior knowledge provided by the codebooks. After that, the two vector collections pass through the decoder from the hierarchical VQ-VAE to produce the inpainted results. In addition, we apply a vector credibility mechanism to promote vector collections from damaged images and approximate vector collections from ground truth images. To further improve the inpainting result, we apply a refinement network, which uses residual blocks with different dilation rates to acquire both global information and local textural details. Extensive experiments conducted on several datasets demonstrate that our method outperforms the state-of-the-art ones.

Memory and Communication Efficient Federated Kernel k-Means.

A federated kernel k -means (FedKKM) algorithm is developed in this article to conduct distributed clustering with low memory consumption on user devices. In FedKKM, a federated eigenvector approximation (FEA) algorithm is designed to iteratively determine the low-dimensional approximate vectors of the transformed feature vectors, using only low-dimensional random feature vectors. To maintain high communication efficiency in each iteration of FEA, a communication-efficient Lanczos algorithm (CELA) is further designed in FEA to reduce the communication cost. Based on the low-dimensional approximate vectors, the clustering result is obtained by leveraging a distributed linear k -means algorithm. A theoretical analysis shows that: 1) FEA has a convergence rate of O(1/T) , where T is the number of iterations; 2) the scalability of FedKKM is not affected by the dataset size since the communication cost of FedKKM is independent of the number of users' data; and 3) FedKKM is a (1+ϵ) approximation algorithm. The experimental results show that FedKKM achieves the comparable clustering quality to that of a centralized kernel k -means. Compared with state-of-the-art schemes, FedKKM reduces the memory consumption on user devices by up to 94% and also reduces the communication cost by more than 40%.

The pair correlation function of multi-dimensional low-discrepancy sequences with small stochastic error terms

In any dimension d≥2, there is no known example of a low-discrepancy sequence which possesses Poisssonian pair correlations. This is in some sense rather surprising, because low-discrepancy sequences always have β-Poissonian pair correlations for all 0<β<1d and are therefore arbitrarily close to having Poissonian pair correlations (which corresponds to the case β=1d). In this paper, we further elaborate on the closeness of the two notions. We show that d-dimensional Kronecker sequences for badly approximable vectors α→ with an arbitrary small uniformly distributed stochastic error term generically have β=1d-Poissonian pair correlations.

Recommendations for Velocity Adjustment in Surface Hopping.

This study investigates velocity adjustment directions after hopping in surface hopping dynamics. Using fulvene and a protonated Schiff base (PSB4) as case studies, we investigate the population decay and reaction yields of different sets of dynamics with the velocity adjusted in either the nonadiabatic coupling, gradient difference, or momentum directions. For the latter, in addition to the conventional algorithm, we investigated the performance of a reduced kinetic energy reservoir approach recently proposed. Our evaluation also considered velocity adjustment in the directions of approximate nonadiabatic coupling vectors. While results for fulvene are susceptible to the adjustment approach, PSB4 is not. We correlated this dependence to the topography near the conical intersections. When nonadiabatic coupling vectors are unavailable, the gradient difference direction is the best adjustment option. If the gradient difference is also unavailable, a semiempirical vector direction or the momentum direction with a reduced kinetic energy reservoir becomes an excellent option to prevent an artificial excess of back hoppings. The precise velocity adjustment direction is less crucial for describing the nonadiabatic dynamics than the kinetic energy reservoir's size.

Concrete analysis of approximate ideal-SIVP to decision ring-LWE reduction

&lt;p style='text-indent:20px;'&gt;A seminal 2013 paper by Lyubashevsky, Peikert, and Regev proposed basing post-quantum cryptography on ideal lattices and supported this proposal by giving a polynomial-time security reduction from the approximate Shortest Independent Vectors Problem (SIVP) to the Decision Learning With Errors (DLWE) problem in ideal lattices. We give a concrete analysis of this multi-step reduction. We find that the tightness gap in the reduction is so great as to vitiate any meaningful security guarantee, and we find reasons to doubt the feasibility in the foreseeable future of the quantum part of the reduction. In addition, when we make the reduction concrete it appears that the approximation factor in the SIVP problem is far larger than expected, a circumstance that causes the corresponding approximate-SIVP problem most likely not to be hard for proposed cryptosystem parameters. We also discuss implications for systems such as Kyber and SABER that are based on module-DLWE.&lt;/p&gt;

Chunk2vec: A novel resemblance detection scheme based on Sentence‐BERT for post‐deduplication delta compression in network transmission

AbstractDelta compression, as a complementary technique for data deduplication, has gained widespread attention in network storage systems. It can eliminate redundant data between non‐duplicate but similar chunks that cannot be identified by data deduplication. The network transmission overhead between servers and clients can be greatly reduced by using data deduplication and delta compression techniques. Resemblance detection is a technique that identifies similar chunks for post‐deduplication delta compression in network storage systems. The existing resemblance detection approaches fail to detect similar chunks with arbitrary similarity by setting a similarity threshold, which can be suboptimal. In this paper, the authors propose Chunk2vec, a resemblance detection scheme for delta compression that utilizes deep learning techniques and Approximate Nearest Neighbour Search technique to detect similar chunks with any given similarity range. Chunk2vec uses a deep neural network, Sentence‐BERT, to extract an approximate feature vector for each chunk while preserving its similarity with other chunks. The experimental results on five real‐world datasets indicate that Chunk2vec improves the accuracy of resemblance detection for delta compression and achieves higher compression ratio than the state‐of‐the‐art resemblance detection technique.

Inexact sequential injective algorithm for weakly univalent vector equation and its application to regularized smoothing Newton algorithm for mixed second-order cone complementarity problems

&lt;p style='text-indent:20px;'&gt;It is known that the complementarity problems and the variational inequality problems are reformulated equivalently as a vector equation by using the natural residual or Fischer-Burmeister function. In this paper, we first propose an inexact sequential injective algorithm (ISIA) for a vector equation, and show the global convergence under weak univalence assumption. Roughly speaking, the ISIA generates the sequence of inexact solutions of approximate vector equations, which consist of the injectives converging to the original vector-valued function. Although the ISIA is simple and conceptual, it can be a prototype to many other algorithms such as a smoothing Newton algorithm, semismooth Newton algorithm, etc. Next, we apply the ISIA prototype to the regularized smoothing Newton algorithm (ReSNA) for mixed second-order cone complementarity problems (MSOCCPs). Exploiting the ISIA convergence scheme, we prove that the ReSNA is globally convergent under Cartesian &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ P_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; assumption.&lt;/p&gt;

On a stable calculation of the normal to a surface given approximately

The paper proposes a stable method for constructing a normal to a surface given approximately. The normal is calculated as the gradient of the function in the surface equation. As is known, the problem of calculating the derivative is ill-posed. In the paper, an approach is adopted to solving this problem as to the problem of calculating the values of an unbounded operator. To construct its stable solution, the principle of minimum of the smoothing functional in Morozov’s formulation is used. The normal is obtained in the form of a Fourier series in the expansion in terms of eigenfunctions of the Laplace operator in a rectangle with boundary conditions of the second kind. The functional stabilizer uses the Laplacian, which makes it possible to obtain a normal in the form of a Fourier series that converges uniformly to the exact normal vector as the error in the surface definition tends to zero. The resulting approximate normal vector can be used to solve various problems of mathematical physics using surface integrals, normal derivatives, simple and double layer potentials.

LoNe Sampler: Graph Node Embeddings by Coordinated Local Neighborhood Sampling

Local graph neighborhood sampling is a fundamental computational problem that is at the heart of algorithms for node representation learning. Several works have presented algorithms for learning discrete node embeddings where graph nodes are represented by discrete features such as attributes of neighborhood nodes. Discrete embeddings offer several advantages compared to continuous word2vec-like node embeddings: ease of computation, scalability, and interpretability. We present LoNe Sampler, a suite of algorithms for generating discrete node embeddings by Local Neighborhood Sampling, and address two shortcomings of previous work. First, our algorithms have rigorously understood theoretical properties. Second, we show how to generate approximate explicit vector maps that avoid the expensive computation of a Gram matrix for the training of a kernel model. Experiments on benchmark datasets confirm the theoretical findings and demonstrate the advantages of the proposed methods.

Quasi-Synchronous Random Access for Massive MIMO-Based LEO Satellite Constellations

Low earth orbit (LEO) satellite constellation-enabled communication networks are expected to be an important part of many Internet of Things (IoT) deployments due to their unique advantage of providing seamless global coverage. In this paper, we investigate the random access problem in massive multiple-input multiple-output-based LEO satellite systems, where the multi-satellite cooperative processing mechanism is considered. Specifically, at edge satellite nodes, we conceive a training sequence padded multi-carrier system to overcome the issue of imperfect synchronization, where the training sequence is utilized to detect the devices’ activity and estimate their channels. Considering the inherent sparsity of terrestrial-satellite links and the sporadic traffic feature of IoT terminals, we utilize the orthogonal approximate message passing-multiple measurement vector algorithm to estimate the delay coefficients and user terminal activity. To further utilize the structure of the receive array, a two-dimensional estimation of signal parameters via rotational invariance technique is performed for enhancing channel estimation. Finally, at the central server node, we propose a majority voting scheme to enhance activity detection by aggregating backhaul information from multiple satellites. Moreover, multi-satellite cooperative linear data detection and multi-satellite cooperative Bayesian dequantization data detection are proposed to cope with perfect and quantized backhaul, respectively. Simulation results verify the effectiveness of our proposed schemes in terms of channel estimation, activity detection, and data detection for quasi-synchronous random access in satellite systems.

Weighted badly approximable complex vectors and bounded orbits of certain diagonalizable flows

We show an analog of a theorem of [J. An, A. Ghosh, L. Guan and T. Ly, Bounded orbits of diagonalizable flows on finite volume quotients of products of [Formula: see text], Adv. Math. 354 (2019) 106743] on weighted badly approximable vectors for totally imaginary number fields. We show that for [Formula: see text] and [Formula: see text] a lattice subgroup, the points of [Formula: see text] with bounded orbits under a one-parameter Ad-semisimple subgroup of [Formula: see text] form a hyperplane-absolute-winning set. As an application, we also provide a generalization of a result of [R. Esdahl-Schou and S. Kristensen, On badly approximable complex numbers, Glasgow Math. J. 52 (2010) 349–355] about the set of badly approximable complex numbers.

Dain’s invariant for black hole initial data

Dynamical black holes in the non-perturbative regime are not mathematically well understood. Studying approximate symmetries of spacetimes describing dynamical black holes gives an insight into their structure. Utilising the property that approximate symmetries coincide with actual symmetries when they are present allows one to construct geometric invariants characterising the symmetry. In this paper, we extend Dain’s construction of geometric invariants characterising stationarity to the case of initial data sets for the Einstein equations corresponding to black hole spacetimes. We prove the existence and uniqueness of solutions to a boundary value problem showing that one can always find approximate Killing vectors in black hole spacetimes and these coincide with actual Killing vectors when they are present. In the time-symmetric setting we make use of a 2 + 1 decomposition to construct a geometric invariant on a marginally outer trapped surface that vanishes if and only if the Killing initial data equations are locally satisfied.

Investigating Path Integration Cues in Sand Scorpion Homing Behavior

Many sand scorpions are faithful to the burrows they dig; however, it is unknown how these animals get back home after hunting excursions. Of the many mechanisms of homing that exist, path integration (PI) is one of the more common tools used by arachnids. In PI, an animal integrates its distance and direction while leaving its home, enabling it to compute an approximate homebound vector for the return trip. The objective of our study was to test whether scorpions use PI to return home under absolute darkness in the lab. We first allowed the animals to establish burrows in homing arenas. Then, after they left their burrow, we recorded the scorpion’s location in the homing arena before we transferred it to the center of the testing arena. We used overhead IR cameras to record its movements in the testing arena. If scorpions exhibited PI, we predicted they would follow a vector in the test arena that approximated the same angle and distance from the capture point to their burrow in their home arena. However, under the conditions of this experiment, we found no evidence that scorpions moved along such homebound vectors. We speculated that scorpions may need a reliable reference cue to accommodate path integration.

Variational quantum approximate support vector machine with inference transfer

A kernel-based quantum classifier is the most practical and influential quantum machine learning technique for the hyper-linear classification of complex data. We propose a Variational Quantum Approximate Support Vector Machine (VQASVM) algorithm that demonstrates empirical sub-quadratic run-time complexity with quantum operations feasible even in NISQ computers. We experimented our algorithm with toy example dataset on cloud-based NISQ machines as a proof of concept. We also numerically investigated its performance on the standard Iris flower and MNIST datasets to confirm the practicality and scalability.

Approximate and discrete Euclidean vector bundles

Abstract We introduce $\varepsilon $ -approximate versions of the notion of a Euclidean vector bundle for $\varepsilon \geq 0$ , which recover the classical notion of a Euclidean vector bundle when $\varepsilon = 0$ . In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that $\varepsilon $ -approximate vector bundles can be used to represent classical vector bundles when $\varepsilon&gt; 0$ is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples.

Novel approach to design matched digital filter with Abelian group and fuzzy particle swarm optimization vector quantization

This paper presents a new method for designing matched digital filters with discrete valued coefficients. The fuzzy particle swarm optimization vector quantization (FPSOVQ) has been applied to obtain the optimum codebook in design of matched wavelet function. Abelian group has been used to extract the similarity present in the input voiced signal. Fuzzy particle swarm optimization (FPSO) process is used to find approximate ideal vector quantization (VQ) codebook to be carried out for compression of data. FPSOVQ scheme utilises features of fuzzy inference method (FIM) and expert particle swarm optimization (PSO). The generated codebook consists of set of highly ideal features, which are considered as the filter coefficients. These coefficients are used in the designing of the filter. All of the phonemes in the American English language were included in the 30 sentences that were chosen from the IEEE database. The sentences were originally down-sampled from 25 kHz to 8 kHz. The magnitude responses of each filter have been drawn, which indicates the characteristics of the filter. A comparison has been provided using energy compaction ratio as a parameter to judge the performance of matched designed filter with db4 filter. Experimental results show the advantage of the developed algorithm as the average value of energy compaction ratio for sampled voice signals is 2.8046 times lower for matched designed filter.