Abstract. The spectral theory of higher-order symmetric tensors is an important tool to reveal some important properties of a hypergraph via its adjacency tensor, Laplacian tensor, and signless Laplacian tensor. Owing to the sparsity of these tensors, we propose an efficient approach to calculate products of these tensors and any vectors. Using the state-of-the-art L-BFGS approach, we develop a first-order optimization algorithm for computing H-and Z-eigenvalues of these large scale sparse tensors (CEST). With the aid of the Kurdyka-Lojasiewicz property, we prove that the sequence of iterates generated by CEST converges to an eigenvector of the tensor. When CEST is started from multiple randomly initial points, the resulting best eigenvalue could touch the extreme eigenvalue with a high probability. Finally, numerical experiments on small hypergraphs show that CEST is efficient and promising. Moreover, CEST is capable of computing eigenvalues of tensors corresponding to a hypergraph with millions of vertices. Recently, spectral hypergraph theory is proposed to explore connections between the geometry of a uniform hypergraph and H-and Z-eigenvalues of some related symmetric tensors. Cooper and Dutle [13] proposed in 2012 the concept of adjacency tensor for a uniform hypergraph. Two years later, Qi [49] gave definitions of Laplacian and signless Laplacian tensors associated with a hypergraph. When an even-uniform hypergraph is connected, the largest H-eigenvalues of the Laplacian and signless Laplacian tensors are equivalent if and only if the hypergraph is odd-bipartite [28]. This result gives a certification to check whether a connected even-uniform hypergraph is odd-bipartite or not.

Forty‐five subjects, including color normals, protanomalous, deuteranomalous, protanopes, and deuteranopes, judged dissimilarities of 26 Munsell color chips chosen to span the full color space (i.e., all three parameters—hue, saturation, and lightness—were varied). Each of the 325 pairs of colors was mounted on a standard grey background board. They were presented to subjects in different random orders and were viewed under a Macbeth daylight lamp. For each pair the subject circled a number varying from 0 (for “identical”) to 9 (for “maximally dissimilar”). Fifty data matrices obtained from 45 subjects (5 were from the same subject on different dates and 2 were from another subject) were analyzed by the INDSCAL method. The three‐dimensional solution yielded the “standard” three dimensions (lightness, red‐green, and yellow‐blue) with the classical “color circle” emerging, in a slightly distorted form, in the plane of the second and third dimensions. Seven dimensions seemed necessary to account fully for these data, however. In seven dimensions each of the “standard” dimensions is paired with a “folded” version. Accompanying lightness is a “folded” lightness dimensions, which we have called “lightness contrast.” The light and dark colors are at one end, contrasted with medium colors at the other. Similarly, “folded” red‐green roughly contrasts red and green with blue, yellow, and the greys, while “folded” yellow‐blue contrasts blue and yellow with red, green, and the greys. The seventh dimension, which may be artifactual, was called “split yellow.” It contrasts very brilliant (high Munsell value and chroma) yellow and orange colors with all the other colors. It is speculated that some of these extra dimensions may relate to anomalous receptor processes characteristic of deviant subjects. The INDSCAL subject space enables discrimination among all five subject types. Specifically, one of the “natural planes” (the red‐green versus “folded” yellow‐blue plane) of the seven‐dimensional solution can be divided into contiguous and fairly compact regions, with each subject type occupying a unique region.

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The p-spectral radius of a uniform hypergraph covers many important concepts, such as Lagrangian and spectral radius of the hypergraph, and is crucial for solving spectral extremal problems of hypergraphs. In this paper, we establish a spherically constrained maximization model and propose a first-order conjugate gradient algorithm to compute the p-spectral radius of a uniform hypergraph (CSRH). By the semialgebraic nature of the adjacency tensor of a uniform hypergraph, CSRH is globally convergent and obtains the global maximizer with a high probability. When computing the spectral radius of the adjacency tensor of a uniform hypergraph, CSRH stands out among existing approaches. Furthermore, CSRH is competent to calculate the p-spectral radius of a hypergraph with millions of vertices and to approximate the Lagrangian of a hypergraph. Finally, we show that the CSRH method is capable of ranking real-world data set based on solutions generated by the p-spectral radius model.