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Daniel Krefl

Publications and source records attributed to Daniel Krefl.

At least 19 recordsLinked to original sources

Zero- and Few-Shots Knowledge Graph Triplet Extraction with Large Language Models

In this work, we tested the Triplet Extraction (TE) capabilities of a variety of Large Language Models (LLMs) of different sizes in the Zero- and Few-Shots settings. In detail, we proposed a pipeline that dynamically gathers contextual information from a Knowledge Base (KB), both in the form of context triplets and of (sentence, triplets) pairs as examples, and provides it to the LLM through a prompt. The additional context allowed the LLMs to be competitive with all the older fully trained baselines based on the Bidirectional Long Short-Term Memory (BiLSTM) Network architecture. We further conducted a detailed analysis of the quality of the gathered KB context, finding it to be strongly correlated with the final TE performance of the model. In contrast, the size of the model appeared to only logarithmically improve the TE capabilities of the LLMs.

cs.CL

Product Jacobi-Theta Boltzmann machines with score matching

The estimation of probability density functions is a non trivial task that over the last years has been tackled with machine learning techniques. Successful applications can be obtained using models inspired by the Boltzmann machine (BM) architecture. In this manuscript, the product Jacobi-Theta Boltzmann machine (pJTBM) is introduced as a restricted version of the Riemann-Theta Boltzmann machine (RTBM) with diagonal hidden sector connection matrix. We show that score matching, based on the Fisher divergence, can be used to fit probability densities with the pJTBM more efficiently than with the original RTBM.

stat.ML

Sampling the Riemann-Theta Boltzmann Machine

We show that the visible sector probability density function of the Riemann-Theta Boltzmann machine corresponds to a gaussian mixture model consisting of an infinite number of component multi-variate gaussians. The weights of the mixture are given by a discrete multi-variate gaussian over the hidden state space. This allows us to sample the visible sector density function in a straight-forward manner. Furthermore, we show that the visible sector probability density function possesses an affine transform property, similar to the multi-variate gaussian density.

stat.ML

Riemann-Theta Boltzmann Machine

A general Boltzmann machine with continuous visible and discrete integer valued hidden states is introduced. Under mild assumptions about the connection matrices, the probability density function of the visible units can be solved for analytically, yielding a novel parametric density function involving a ratio of Riemann-Theta functions. The conditional expectation of a hidden state for given visible states can also be calculated analytically, yielding a derivative of the logarithmic Riemann-Theta function. The conditional expectation can be used as activation function in a feedforward neural network, thereby increasing the modelling capacity of the network. Both the Boltzmann machine and the derived feedforward neural network can be successfully trained via standard gradient- and non-gradient-based optimization techniques.

stat.ML

Modelling conditional probabilities with Riemann-Theta Boltzmann Machines

The probability density function for the visible sector of a Riemann-Theta Boltzmann machine can be taken conditional on a subset of the visible units. We derive that the corresponding conditional density function is given by a reparameterization of the Riemann-Theta Boltzmann machine modelling the original probability density function. Therefore the conditional densities can be directly inferred from the Riemann-Theta Boltzmann machine.

stat.ML

Machine Learning of Calabi-Yau Volumes

We employ machine learning techniques to investigate the volume minimum of Sasaki-Einstein base manifolds of non-compact toric Calabi-Yau 3-folds. We find that the minimum volume can be approximated via a second order multiple linear regression on standard topological quantities obtained from the corresponding toric diagram. The approximation improves further after invoking a convolutional neural network with the full toric diagram of the Calabi-Yau 3-folds as the input. We are thereby able to circumvent any minimization procedure that was previously necessary and find an explicit mapping between the minimum volume and the topological quantities of the toric diagram. Under the AdS/CFT correspondence, the minimum volumes of Sasaki-Einstein manifolds correspond to central charges of a class of 4d N=1 superconformal field theories. We therefore find empirical evidence for a function that gives values of central charges without the usual extremization procedure.

hep-th

Non-Perturbative Quantum Geometry III

The Nekrasov-Shatashvili limit of the refined topological string on toric Calabi-Yau manifolds and the resulting quantum geometry is studied from a non-perturbative perspective. The quantum differential and thus the quantum periods exhibit Stokes phenomena over the combined string coupling and quantized Kaehler moduli space. We outline that the underlying formalism of exact quantization is generally applicable to points in moduli space featuring massless hypermultiplets, leading to non-perturbative band splitting. Our prime example is local P1xP1 near a conifold point in moduli space. In particular, we will present numerical evidence that in a Stokes chamber of interest the string based quantum geometry reproduces the non-perturbative corrections for the Nekrasov-Shatashvili limit of 4d supersymmetric SU(2) gauge theory at strong coupling found in the previous part of this series. A preliminary discussion of local P2 near the conifold point in moduli space is also provided.

hep-th

Mellin-Barnes Representation of the Topological String

We invoke integrals of Mellin-Barnes type to analytically continue the Gopakumar-Vafa resummation of the topological string free energy in the string coupling constant, leading to additional non-perturbative terms. We also discuss in a similar manner the refined and Nekrasov-Shatashvili limit version thereof. The derivation is straight-forward and essentially boils down to taking residue. This allows us to confirm some related conjectures in the literature at tree-level.

hep-th

Exact Chern-Simons / Topological String duality

We invoke universal Chern-Simons theory to analytically calculate the exact free energy of the refined topological string on the resolved conifold. In the unrefined limit we reproduce non-perturbative corrections for the resolved conifold found elsewhere in the literature, thereby providing strong evidence that the Chern-Simons / topological string duality is exact, and in particular holds at arbitrary N as well. In the refined case, the non-perturbative corrections we find are novel and appear to be non-trivial. We show that non-perturbatively special treatment is needed for rational valued deformation parameter. Above results are also extend to refined Chern-Simons with orthogonal groups.

hep-th

Non-Perturbative Quantum Geometry II

The Nekrasov-Shatashvili limit of beta-ensembles with polynomial potential and N=2 supersymmetric gauge theories in the Omega-background is intimately related to complex one-dimensional quantum mechanics. Multi-instanton corrections in quantum mechanics, inferable from exact quantization conditions, imply additional non-perturbative corrections to the Nekrasov-Shatashvili free energies. Besides filling some of the gaps in previous derivations, we present analytic expressions for such additional non-perturbative corrections in the case of SU(2) gauge theory expanded at strong coupling. In contrast, at weak coupling these additional non-perturbative corrections appear to be negligible.

hep-th

B-Model Approaches to Instanton Counting

This is the 13th article in the collection of reviews "Exact results in N=2 supersymmetric gauge theories", ed. J. Teschner. It discusses the relation between the instanton partition functions and the partition function of the topological string from the perspective of the B-model. The instanton partition functions provide solutions to the holomorphic anomaly equations characterising the partition functions of the topological string.

hep-th

Non-Perturbative Quantum Geometry

The beta-ensemble with cubic potential can be used to study a quantum particle in a double-well potential with symmetry breaking term. The quantum mechanical perturbative energy arises from the ensemble free energy in a novel large N limit. A relation between the generating functions of the exact non-perturbative energy, similar in spirit to the one of Dunne-Unsal, is found. The exact quantization condition of Zinn-Justin and Jentschura is equivalent to the Nekrasov-Shatashvili quantization condition on the level of the ensemble. Refined topological string theory in the Nekrasov-Shatashvili limit arises as a large N limit of quantum mechanics.

hep-th

Refined Chern-Simons versus Vogel universality

We study the relation between the partition function of refined SU(N) and SO(2N) Chern-Simons on the 3-sphere and the universal Chern-Simons partition function in the sense of Mkrtchyan and Veselov. We find a four-parameter generalization of the integral representation of universal Chern-Simons that includes refined SU(N) and SO(2N) Chern-Simons for special values of parameters. The large N expansion of the integral representation of refined SU(N) Chern-Simons explicitly shows the replacement of the virtual Euler characteristic of the moduli space of complex curves with a refined Euler characteristic related to the radius deformed c=1 string free energy.

hep-th

Penner Type Ensemble for Gauge Theories Revisited

The Penner type beta-ensemble for Omega-deformed N=2 SU(2) gauge theory with two massless flavors arising as a limiting case from the AGT conjecture is considered. The partition function can be calculated perturbatively in a saddle-point approximation. A large N limit reproduces the gauge theory partition function expanded in a strong coupling regime, for any beta and beyond tree-level, confirming previous results obtained via special geometry and the holomorphic anomaly equation. The leading terms and gap of the gauge theory free energy at the monopole/dyon point follow as a corollary.

hep-th

ABCD of Beta Ensembles and Topological Strings

We study beta-ensembles with Bn, Cn, and Dn eigenvalue measure and their relation with refined topological strings. Our results generalize the familiar connections between local topological strings and matrix models leading to An measure, and illustrate that all those classical eigenvalue ensembles, and their topological string counterparts, are related one to another via various deformations and specializations, quantum shifts and discrete quotients. We review the solution of the Gaussian models via Macdonald identities, and interpret them as conifold theories. The interpolation between the various models is plainly apparent in this case. For general polynomial potential, we calculate the partition function in the multi-cut phase in a perturbative fashion, beyond tree-level in the large-N limit. The relation to refined topological string orientifolds on the corresponding local geometry is discussed along the way.

hep-th

Holomorphic Anomaly in Gauge Theory on ALE space

We consider four-dimensional Omega-deformed N=2 supersymmetric SU(2) gauge theory on A1 space and its lift to five dimensions. We find that the partition functions can be reproduced via special geometry and the holomorphic anomaly equation. Schwinger type integral expressions for the boundary conditions at the monopole/dyon point in moduli space are inferred. The interpretation of the five-dimensional partition function as the partition function of a refined topological string on A1x(local P1xP1) is suggested.

hep-th

Quantum Geometry of Refined Topological Strings

We consider branes in refined topological strings. We argue that their wave-functions satisfy a Schrödinger equation depending on multiple times and prove this in the case where the topological string has a dual matrix model description. Furthermore, in the limit where one of the equivariant rotations approaches zero, the brane partition function satisfies a time-independent Schroedinger equation. We use this observation, as well as the back reaction of the brane on the closed string geometry, to offer an explanation of the connection between integrable systems and N=2 gauge systems in four dimensions observed by Nekrasov and Shatashvili.

hep-th

Shift versus Extension in Refined Partition Functions

We have recently shown that the global behavior of the partition function of N=2 gauge theory in the general Omega-background is captured by special geometry in the guise of the (extended) holomorphic anomaly equation. We here analyze the fate of our results under the shift of the mass parameters of the gauge theory. The preferred value of the shift, noted previously in other contexts, restores the Z_2 symmetry of the instanton partition function under inversion of the Omega-background, and removes the extension. We comment on various connections.

hep-th