In this paper, we reconsider the merits of unfolding solutions based on loss functions involving a normalization on the variance per subject. In the literature, solutions based on Stress-2 are often diagnosed to be degenerate in the majority of cases. Here, the focus lies on two frequently occurring types of degeneracies. The first type typically locates some subject points far away from a compact cluster of the other points. In the second type of solution, the object points lie on a circle. In this paper, we argue that these degenerate solutions are well fitting and informative. To reveal the information, we introduce mixtures of plots based on the ideal point model of unfolding, the vector model, and on the signed distance model. In addition to a different representation, we provide a new iterative majorization algorithm to optimize the average squared correlation between the distances in the configuration and the transformed data per individual. It is shown that this approach is equivalent to minimizing Kruskal’s Stress-2.

Multidimensional unfolding methods suffer from the degeneracy problem in almost all circumstances. Most degeneracies are easily recognized: the solutions are perfect but trivial, characterized by approximately equal distances between points from different sets. A definition of an absolutely degenerate solution is proposed, which makes clear that these solutions only occur when an intercept is present in the transformation function. Many solutions for the degeneracy problem have been proposed and tested, but with little success so far. In this paper, we offer a substantial modification of an approach initiated bythat introduced a normalization factor based on thevariance in the usual least squares loss function. Heiser unpublishedthesis, (1981) and showed that the normalization factor proposed by Kruskal and Carroll was not strong enough to avoid degeneracies. The factor proposed in the present paper, based on the coefficient of variation, discourages or penalizes nonmetric transformations of the proximities with small variation, so that the procedure steers away from solutions with small variation in the interpoint distances. An algorithm is described for minimizing the re-adjusted loss function, based on iterative majorization. The results of a simulation study are discussed, in which the optimal range of the penalty parameters is determined. Two empirical data sets are analyzed by our method, clearly showing the benefits of the proposed loss function.

The Candecomp/Parafac (CP) model is a well-known tool for summarizing a three-way array by extracting a limited number of components. Unfortunately, in some cases, the model suffers from the so-called degeneracy, that is a solution with diverging and uninterpretable components. To avoid degeneracy, orthogonality constraints are usually applied to one of the component matrices. This solves the problem only from a technical point of view because the existence of orthogonal components underlying the data is not guaranteed. For this purpose, we consider some variants of the CP model where the orthogonality constraints are relaxed either by constraining only a pair, or a subset, of components or by stimulating the CP solution to be possibly orthogonal. We theoretically clarify that only the latter approach, based on the least absolute shrinkage and selection operator and named the CP-Lasso, is helpful in solving the degeneracy problem. The results of the application of CP-Lasso on simulated and real life data show its effectiveness.

In this paper, we investigate Kolmogorov-type theorems for small perturbations of degenerate Hamiltonian systems. These systems are index by a parameter ξ as $ H(y,x,\xi) = \langle\omega(\xi),y\rangle {+ \bar h(y,\xi)}+\varepsilon P(y,x,\xi,\varepsilon) $, where ɛ > 0. We assume that the frequency mapping $\omega(\cdot)$, $\bar h(y,\cdot)=O(|y|^2)$ and the perturbation $\varepsilon P(y,x,\cdot, \varepsilon)$ maintain Hölder continuity about ξ. We prove that the persistent invariant tori retain the same frequency as those of the unperturbed tori, under a certain topological degree condition and a weak convexity condition for the frequency mapping. Notably, this paper presents, to our understanding, pioneering results on the KAM theorem under such conditions with only assumption of Hölder continuous dependence of frequency mapping ω on the parameter.

This article interrogates how Franz Schreker's Die Gezeichneten and its early reception reflected an uneasiness about the confines of manhood. As an opera with a complex genesis and a difficult reception history, Die Gezeichneten's allure comes from its resistance to being reduced to only one thing. I nevertheless seek to locate this opera around the time of its premiere towards the end of the First World War. I contend that Die Gezeichneten and its immediate reception charted a key transition in Austro-German masculinity. Specifically, the opera's early performances marked a move away from the period's normative models of bourgeois masculinity (and their corresponding ideas about appearance, health and nationhood) and towards an alternative masculinity preoccupied with degeneracy. I focus on the opera's masks, arguing that, through acts of concealment and disclosure, the opera's two male protagonists struggle to negotiate expectations of an emotionally controlled modern manhood, calling attention to wartime anxieties about what it meant to be a man. Such anxieties resulted in a hardening of attitudes towards the masculine gender, which influenced contemporary music criticism too. Die Gezeichneten's highly sensationalist early reviews relied on a language of degeneracy. Yet I suggest that the opera's initial reception captured a critical moment in this language's history before it was subsumed under Nazi ideology.

Exponential random graph models, or ERGMs, are a flexible and general class of models for modeling dependent data. While the early literature has shown them to be powerful in capturing many network features of interest, recent work highlights difficulties related to the models’ ill behavior, such as most of the probability mass being concentrated on a very small subset of the parameter space. This behavior limits both the applicability of an ERGM as a model for real data and inference and parameter estimation via the usual Markov chain Monte Carlo algorithms. To address this problem, we propose a new exponential family of models for random graphs that build on the standard ERGM framework. Specifically, we solve the problem of computational intractability and “degenerate” model behavior by an interpretable support restriction. We introduce a new parameter based on the graph-theoretic notion of degeneracy, a measure of sparsity whose value is commonly low in real-world networks. The new model family is supported on the sample space of graphs with bounded degeneracy and is called degeneracy-restricted ERGMs, or DERGMs for short. Since DERGMs generalize ERGMs—the latter is obtained from the former by setting the degeneracy parameter to be maximal—they inherit good theoretical properties, while at the same time place their mass more uniformly over realistic graphs. The support restriction allows the use of new (and fast) Monte Carlo methods for inference, thus making the models scalable and computationally tractable. We study various theoretical properties of DERGMs and illustrate how the support restriction improves the model behavior. We also present a fast Monte Carlo algorithm for parameter estimation that avoids many issues faced by Markov Chain Monte Carlo algorithms used for inference in ERGMs.

The fossil record is scarce and incomplete by nature. Animals and ecological processes devour soft tissue and important bony details over time and, when the dust settles, we are faced with a patchy record full of variation. Fossil taxa are usually defined by craniodental characteristics, so unless postcranial bones are found associated with a skull, assignment to taxon is unstable. Naming a locomotor category based on fossil bone morphology by analogy to living hominoids is not uncommon, and when no single locomotor label fits, postcrania are often described as exhibiting a ‘mosaic’ of traits. Here, we contend that the unavoidable variation that characterises the fossil record can be described far more rigorously based on extensive work in human neurobiology and neuroanatomy, movement sciences and motor control and biomechanics research. In neurobiology, degeneracy is a natural mechanism of adaptation allowing system elements that are structurally different to perform the same function. This concept differs from redundancy as understood in engineering, where the same function is performed by identical elements. Assuming degeneracy, structurally different elements are able to produce different outputs in a range of environmental contexts, favouring ecological robusticity by enabling adaptations. Furthermore, as degeneracy extends to genome level, genetic variation is sustained, so that genes which might benefit an organism in a different environment remain part of the genome, favouring species’ evolvability.

Finite difference computations that involve spatial adaptation commonly employ an equidistribution principle. In these cases, a new mesh is constructed such that a given monitor function is equidistributed in some sense. Typical choices of the monitor function involve the solution or one of its many derivatives. This straightforward concept has proven to be extremely effective and practical. However, selections of core monitoring functions are often challenging and crucial to the computational success. This paper concerns six different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities. While the first four monitoring strategies are within the so-called primitive regime, the rest belong to a later category of the modified type, which requires the priori knowledge of certain important quenching solution characteristics. Simulated examples are given to illustrate our study and conclusions.

This article discusses key characteristics of a semi-adaptive finite difference method for solving singular degenerate reaction-diffusion equations. Numerical stability, monotonicity, and convergence are investigated. Numerical experiments illustrate the discussion. The study reconfirms and improves several of our earlier results.

This paper is devoted to the study of systems of entities that are capable of generating other entities of the same kind and, possibly, self-reproducing. The main technical issue addressed is to quantify the requirements that such entities must meet to be able to produce a progeny that is not degenerative, i.e., that has the same reproductive capability as the progenitor. A novel theory that allows an explicit quantification of these requirements is presented. The notion of generation rank of an entity is introduced, and it is proved that the generation process, in most cases, is degenerative in that it strictly and irreversibly decreases the generation rank from parent to descendent. It is also proved that there exists a threshold of rank such that this degeneracy can be avoided if and only if the entity has a generation rank that meets that threshold – this is the von Neumann rank threshold. On the basis of this threshold, an information threshold is derived, which quantifies the minimum amount of information that must be provided to specify an entity such that its descendents are not degenerative. Furthermore, a complexity threshold is obtained, which quantifies the minimum length of the description of that entity in a given language. As an application, self-assembly for a 2 Degrees of Freedom planar robot is considered, and simulation results are presented. A robot arm capable of picking up and placing the components of another arm, in the presence of errors, is considered to have successfully reproduced if these are placed within an allowable tolerance. The example shows that, due to the kinematics of the robot, errors can grow from one generation to the next, until the reproduction process fails eventually. However, error correction (via error sensing and feedback control) can then be used to prevent such degeneracy. The von Neumann generation rank and information thresholds are computed for this example, and are consistent with the simulation results in predicting degeneracy in the case without error correction, and predicting successful self-reproduction in the case with error correction.

We challenge the classical belief that products of degenerate quadratic forms must remain degenerate and we show that this fails in general, e.g. over tensor triangulated categories with duality. This opens new ways of constructing non-degenerate quadratic forms and hence classes in Witt groups. In addition, we encapsulate in a Leibniz-type formula the behavior of the product with respect to the symmetric cone construction. We illustrate these ideas by computing the total Witt group of regular projective spaces.