Looking at the posterior: accuracy and uncertainty of neural-network predictions

Bayesian inference can quantify uncertainty in the predictions of neural networks using posterior distributions for model parameters and network output. By looking at these posterior distributions, one can separate the origin of uncertainty into aleatoric and epistemic contributions. One goal of uncertainty quantification is to inform on prediction accuracy. Here we show that prediction accuracy depends on both epistemic and aleatoric uncertainty in an intricate fashion that cannot be understood in terms of marginalized uncertainty distributions alone. How the accuracy relates to epistemic and aleatoric uncertainties depends not only on the model architecture, but also on the properties of the dataset. We discuss the significance of these results for active learning and introduce a novel acquisition function that outperforms common uncertainty-based methods. To arrive at our results, we approximated the posteriors using deep ensembles, for fully-connected, convolutional and attention-based neural networks.Comment: 26 pages, 10 figures, 5 table

Finite-time Lyapunov exponents of deep neural networks

We compute how small input perturbations affect the output of deep neural networks, exploring an analogy between deep networks and dynamical systems, where the growth or decay of local perturbations is characterised by finite-time Lyapunov exponents. We show that the maximal exponent forms geometrical structures in input space, akin to coherent structures in dynamical systems. Ridges of large positive exponents divide input space into different regions that the network associates with different classes. These ridges visualise the geometry that deep networks construct in input space, shedding light on the fundamental mechanisms underlying their learning capabilities.Comment: 6 pages, 4 figure

(2,0) theory on circle fibrations

We consider (2,0) theory on a manifold M_6 that is a fibration of a spatial S^1 over some five-dimensional base manifold M_5. Initially, we study the free (2,0) tensor multiplet which can be described in terms of classical equations of motion in six dimensions. Given a metric on M_6 the low energy effective theory obtained through dimensional reduction on the circle is a Maxwell theory on M_5. The parameters describing the local geometry of the fibration are interpreted respectively as the metric on M_5, a non-dynamical U(1) gauge field and the coupling strength of the resulting low energy Maxwell theory. We derive the general form of the action of the Maxwell theory by integrating the reduced equations of motion, and consider the symmetries of this theory originating from the superconformal symmetry in six dimensions. Subsequently, we consider a non-abelian generalization of the Maxwell theory on M_5. Completing the theory with Yukawa and phi^4 terms, and suitably modifying the supersymmetry transformations, we obtain a supersymmetric Yang-Mills theory which includes terms related to the geometry of the fibration.Comment: 24 pages, v2 References added, typos correcte

Deconstructing graviphoton from mass-deformed ABJM

Mass-deformed ABJM theory has a maximally supersymmetric fuzzy two-sphere vacuum solution where the scalar fields are proportional to the TGRVV matrices. We construct these matrices using Schwinger oscillators. This shows that the ABJM gauge group that corresponds to the fuzzy two-sphere geometry is $U(N)\times U(N-1)$. We deconstruct the graviphoton term in the D4 brane theory. The normalization of this term is fixed by topological reasons. This gives us the correct normalization of the deconstructed U(1) gauge field and fixes the Yang -Mills coupling constant to the value which corresponds to M5 brane compactified on \mb{R}^ {1,2} \times S^3/{\mb{Z}_k}. The graviphoton term also enable us to show that the zero mode contributions to the partition functions for the D4 and the M5 brane agree.Comment: 26 page

Partition Functions for Maxwell Theory on the Five-torus and for the Fivebrane on S1XT5

We compute the partition function of five-dimensional abelian gauge theory on a five-torus T5 with a general flat metric using the Dirac method of quantizing with constraints. We compare this with the partition function of a single fivebrane compactified on S1 times T5, which is obtained from the six-torus calculation of Dolan and Nappi. The radius R1 of the circle S1 is set to the dimensionful gauge coupling constant g^2= 4\pi^2 R1. We find the two partition functions are equal only in the limit where R1 is small relative to T5, a limit which removes the Kaluza-Klein modes from the 6d sum. This suggests the 6d N=(2,0) tensor theory on a circle is an ultraviolet completion of the 5d gauge theory, rather than an exact quantum equivalence.Comment: v4, 37 pages, published versio

The Conformal Anomaly of M5-Branes

We show that the conformal anomaly for N M5-branes grows like $N^3$. The method we employ relates Coulomb branch interactions in six dimensions to interactions in four dimensions using supersymmetry. This leads to a relation between the six-dimensional conformal anomaly and the conformal anomaly of N=4 Yang-Mills. Along the way, we determine the structure of the four derivative interactions for the toroidally compactified (2,0) theory, while encountering interesting novelties in the structure of the six derivative interactions.Comment: 38 pages, LaTeX; references adde

Contact Manifolds, Contact Instantons, and Twistor Geometry

Recently, Kallen and Zabzine computed the partition function of a twisted supersymmetric Yang-Mills theory on the five-dimensional sphere using localisation techniques. Key to their construction is a five-dimensional generalisation of the instanton equation to which they refer as the contact instanton equation. Subject of this article is the twistor construction of this equation when formulated on K-contact manifolds and the discussion of its integrability properties. We also present certain extensions to higher dimensions and supersymmetric generalisations.Comment: v3: 28 pages, clarifications and references added, version to appear in JHE

One-loop effective actions and higher spins. Part II

In this paper we continue and improve the analysis of the effective actions obtained by integrating out a scalar and a fermion field coupled to external symmetric sources, started in the previous paper. The first subject we study is the geometrization of the results obtained there, that is we express them in terms of covariant Jacobi tensors. The second subject concerns the treatment of tadpoles and seagull terms in order to implement off-shell covariance in the initial model. The last and by far largest part of the paper is a repository of results concerning all two point correlators (including mixed ones) of symmetric currents of any spin up to 5 and in any dimensions between 3 and 6. In the massless case we also provide formulas for any spin in any dimension