Local Scale Covariance and Its Radical Implications for Cosmology

Local scale covariance posits that no privileged length scales should appear in the fundamental equations of local, Minkowskian physics—why should nature have scale, but not position preferences?—yet, they clearly do. A resolution is proposed wherein scale covariance is promoted to the status of Poincaré covariance, and privileged scales emerge as a result of `scale clustering’, similarly to the way privileged positions emerge in a translation covariant theory. The implied ability of particles to `move in scale’ has recently been shown by the author to offer a possible elegant solution to the missing matter problem. For cosmology, the implications are: (a) a novel component of the cosmological redshift, due to scale-motion over cosmological times; (b) a radically different scenario for the early universe, during which the conditions for such scale clustering are absent. The former is quantitatively analyzed, resulting in a unique cosmological model, empirically coinciding with standard Einstein–de-Sitter cosmology, only in some non-physical limit. The latter implication is qualitatively discussed as part of a critique of the conceptual foundations of ΛCDM which ignores scale covariance altogether.

Hart–Mas-Colell consistency and the core in convex games

This paper formally introduces Hart–Mas-Colell consistency for general (possibly multi-valued) solutions for cooperative games with transferable utility. This notion is used to axiomatically characterize the core on the domain of convex games. Moreover, we characterize all nonempty solutions satisfying individual rationality, anonymity, scale covariance, superadditivity, weak Hart–Mas-Colell consistency, and converse Hart–Mas-Colell consistency. This family consists of (a) the Shapley value, (b) all homothetic images of the core with the Shapley value as center of homothety and with positive ratios of homothety not larger than one, and (c) their relative interiors.

On the Scale Invariance in State of the Art CNNs Trained on ImageNet

The diffused practice of pre-training Convolutional Neural Networks (CNNs) on large natural image datasets such as ImageNet causes the automatic learning of invariance to object scale variations. This, however, can be detrimental in medical imaging, where pixel spacing has a known physical correspondence and size is crucial to the diagnosis, for example, the size of lesions, tumors or cell nuclei. In this paper, we use deep learning interpretability to identify at what intermediate layers such invariance is learned. We train and evaluate different regression models on the PASCAL-VOC (Pattern Analysis, Statistical modeling and ComputAtional Learning-Visual Object Classes) annotated data to (i) separate the effects of the closely related yet different notions of image size and object scale, (ii) quantify the presence of scale information in the CNN in terms of the layer-wise correlation between input scale and feature maps in InceptionV3 and ResNet50, and (iii) develop a pruning strategy that reduces the invariance to object scale of the learned features. Results indicate that scale information peaks at central CNN layers and drops close to the softmax, where the invariance is reached. Our pruning strategy uses this to obtain features that preserve scale information. We show that the pruning significantly improves the performance on medical tasks where scale is a relevant factor, for example for the regression of breast histology image magnification. These results show that the presence of scale information at intermediate layers legitimates transfer learning in applications that require scale covariance rather than invariance and that the performance on these tasks can be improved by pruning off the layers where the invariance is learned. All experiments are performed on publicly available data and the code is available on GitHub.

Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency

A solution on a set of transferable utility (TU) games satisfies strong aggregate monotonicity (SAM) if every player can improve when the grand coalition becomes richer. It satisfies equal surplus division (ESD) if the solution allows the players to improve equally. We show that the set of weight systems generating weighted prenucleoli that satisfy SAM is open, which implies that for weight systems close enough to any regular system, the weighted prenucleolus satisfies SAM. We also provide a necessary condition for SAM for symmetrically weighted nucleoli. Moreover, we show that the per capita nucleolus on balanced games is characterized by single-valuedness (SIVA), translation covariance (TCOV) and scale covariance (SCOV), and equal adjusted surplus division (EASD), a property that is comparable to but stronger than ESD. These properties together with ESD characterize the per capita prenucleolus on larger sets of TU games. EASD and ESD can be transformed to independence of (adjusted) proportional shifting, and these properties may be generalized for arbitrary weight systems p to I(A)Sp. We show that the p-weighted prenucleolus on the set of balanced TU games is characterized by SIVA, TCOV, SCOV, and IASp and on larger sets by additionally requiring ISp.

Provably Scale-Covariant Continuous Hierarchical Networks Based on Scale-Normalized Differential Expressions Coupled in Cascade

This article presents a theory for constructing hierarchical networks in such a way that the networks are guaranteed to be provably scale covariant. We first present a general sufficiency argument for obtaining scale covariance, which holds for a wide class of networks defined from linear and nonlinear differential expressions expressed in terms of scale-normalized scale-space derivatives. Then, we present a more detailed development of one example of such a network constructed from a combination of mathematically derived models of receptive fields and biologically inspired computations. Based on a functional model of complex cells in terms of an oriented quasi quadrature combination of first- and second-order directional Gaussian derivatives, we couple such primitive computations in cascade over combinatorial expansions over image orientations. Scale-space properties of the computational primitives are analysed, and we give explicit proofs of how the resulting representation allows for scale and rotation covariance. A prototype application to texture analysis is developed, and it is demonstrated that a simplified mean-reduced representation of the resulting QuasiQuadNet leads to promising experimental results on three texture datasets.

No Need for Dark-Matter, Dark-Energy or Inflation, Once Ordinary Matter Is Properly Represented?

In a recent Foundations of Physics paper [5] by the current author it was shown that, when the self-force problem of classical electrodynamics is properly solved, it becomes a plausible ontology underlying the statistical description of quantum mechanics. In the current paper we extend this result, showing that ordinary matter, thus represented, possibly suffices in explaining the outstanding observations currently requiring for this task the contrived notions of dark-matter, dark-energy and inflation. The single mandatory 'fix' to classical electrodynamics, demystifying both very small and very large scale physics, should be contrasted with other adhoc solutions to either problems. Instrumental to our cosmological model is scale covariance (and 'spontaneous breaking' thereof), a formal symmetry of classical electrodynamics treated on equal footing with its Poincare covariance, which is incompatible with the (absolute) metrical attributes of the GR metric tensor.

No Need for Dark-Matter, Dark-Energy or Inflation, Once Ordinary Matter is Properly Represented?

In a recent Foundations of Physics paper [5] by the current author it was shown that, when the self force problem of classical electrodynamics is properly solved, it becomes a plausible ontology underlying the statistical description of quantum mechanics. In the current paper we extend this result, showing that ordinary matter, thus represented, possibly suffices in explaining the outstanding observations currently requiring for this task the contrived notions of dark-matter, dark-energy and inflation. The single mandatory `fix' to classical electrodynamics, demystifying both very small and very large scale physics, should be contrasted with other ad hoc solutions to either problems. Instrumental to our cosmological model is scale covariance (and `spontaneous breaking' thereof), a formal symmetry of classical electrodynamics treated on equal footing with its Poincare covariance, which is incompatible with the (absolute) metrical attributes of the GR metric tensor.

No Need for Dark-Matter, Dark-Energy or Inflation, Once Ordinary Matter is Properly Represented?

In a recent Foundations of Physics paper [5] by the current author it was shown that, when the self-force problem of classical electrodynamics is properly solved, the representation of matter which results becomes a plausible ontology underlying QM's statistical description. In the current paper we extend this result, showing that ordinary matter, thus represented, possibly suffices in explaining the outstanding observations currently requiring for this task the contrived notions of dark-matter, dark-energy and inflation. The single `fix' to classical electrodynamics, demystifying both very small and very large scale physics, should be contrasted with other ad hoc solutions to either problems. Instrumental to our cosmological model is scale covariance (and `spontaneous breaking' thereof), a formal symmetry of CE which we consider to be just as important as its Poincare covariance.