相关论文: Constrained PSLQ Search for Machin-like Identities…
Previously we have proposed a new method of transforming quotients into integer reciprocals in the Machin-like formulas for $\pi$. As a further development, here we show how to generate a multi-term Machin-like formula for $\pi$ with a…
We present a simple recurrent formula to generate the Machin-like expression for calculating $\pi/4$. The method works for any denominator in the starting term and always provides a finite decomposition. We show that the terms in the…
The PSLQ algorithm computes integer relations for real numbers and Gaussian integer relations for complex numbers. We endeavour to extend PSLQ to find integer relations consisting of algebraic integers from some quadratic extension fields…
We present a new form of the Machin-like formula for $\pi$ that can be generated by using iteration. This form of the Machin-like formula may be promising for computation of the constant $\pi$ due to rapidly increasing integers at each step…
Lehmer defined a measure $$ \mu=\sum\limits_{j=1}^J\frac{1}{\log_{10}\left(\left|\beta_j\right|\right)}, $$ where the $\beta_j$ may be either integers or rational numbers in a Machin-like formula for $\pi$. When the $\beta_j$ are integers,…
The PSLQ algorithm is one of the most popular algorithm for finding nontrivial integer relations for several real numbers. In the present work, we present an incremental version of PSLQ. For some applications needing to call PSLQ many…
Most of metric learning approaches are dedicated to be applied on data described by feature vectors, with some notable exceptions such as times series, trees or graphs. The objective of this paper is to propose a metric learning algorithm…
Identifying low-dimensional structure in high-dimensional probability measures is an essential pre-processing step for efficient sampling. We introduce a method for identifying and approximating a target measure $\pi$ as a perturbation of a…
Researchers currently use a number of approaches to predict and substantiate information-computation gaps in high-dimensional statistical estimation problems. A prominent approach is to characterize the limits of restricted models of…
A new method for analyzing high-dimensional categorical data, Linear Latent Structure (LLS) analysis, is presented. LLS models belong to the family of latent structure models, which are mixture distribution models constrained to satisfy the…
This paper introduces Latent Relational Analysis (LRA), a method for measuring relational similarity. LRA has potential applications in many areas, including information extraction, word sense disambiguation, machine translation, and…
The celebrated integer relation finding algorithm PSLQ has been successfully used in many applications. PSLQ was only analyzed theoretically for exact input data, however, when the input data are irrational numbers, they must be approximate…
Mutual information (MI) is a general measure of statistical dependence with widespread application across the sciences. However, estimating MI between multi-dimensional variables is challenging because the number of samples necessary to…
Over the past few years, table interpretation tasks have made significant progress due to their importance and the introduction of new technologies and benchmarks in the field. This work experiments with a hybrid approach for detecting…
We establish theoretical recovery guarantees of a family of Riemannian optimization algorithms for low rank matrix recovery, which is about recovering an $m\times n$ rank $r$ matrix from $p < mn$ number of linear measurements. The…
Many AI researchers and cognitive scientists have argued that analogy is the core of cognition. The most influential work on computational modeling of analogy-making is Structure Mapping Theory (SMT) and its implementation in the Structure…
In this paper we present a two-term Machin-like formula for pi \[\frac{\pi}{4} = 2^{k - 1}\arctan\left(\frac{1}{u_1}\right) + \arctan\left(\frac{1}{u_2}\right)\] with small Lehmer's measure $e \approx 0.245319$ and describe iteration…
In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to…
Relations such as "is influenced by", "is known for" or "is a competitor of" are inherently graded: we can rank entity pairs based on how well they satisfy these relations, but it is hard to draw a line between those pairs that satisfy them…
Many constructs that characterize language, like its complexity or emotionality, have a naturally continuous semantic structure; a public speech is not just "simple" or "complex," but exists on a continuum between extremes. Although large…