arXiv research

Develops techniques to estimate thresholds for logarithmic surfaces, proving existence of Kahler-Einstein metrics.

Paper proposes a new estimator for real log canonical thresholds.

Counterexample disproves conjectures about log canonical thresholds.

We study global log canonical thresholds of del Pezzo surfaces.

We compute the log canonical thresholds of non-negatively curved singular hermitian metrics on ample linearized line bundles on bi-equivariant group compactifications of complex reductive groups. To this end, we associate to any such metric a convex function whose asymptotic behavior determines the log canonical thresh…

We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an application, we show that they have a Kaehler-Einstein metric if they are general.

Sharp inequalities for weighted log canonical thresholds derived.

Paper bounds tensor decomposition's RLCT, aiding Bayesian inference.

Survey on singularity theory and its relation to minimal model program.

We relate the global log canonical threshold of a variety with torus action to the global log canonical threshold of its quotient. We apply this to certain Fano varieties and use Tian's criterion to prove the existence of Kahler-Einstein metrics on them. In particular, we obtain simple examples of Fano threefolds being…

We study global log canonical thresholds of cubic surfaces with canonical singularities, and we prove the existence of a Kahler-Einstein metric on two singular cubic surfaces.

We prove a conjecture about log canonical thresholds and volumes of klt singularities.

This paper studies a specific blow-up algorithm for sop polynomials and their RLCT.

Study shows boundedness of klt singularities in 3D or with bounded Kollár components.

The global log canonical threshold (or Tian's alpha-invariant) plays an important role in the geometry of Fano varieties. Tian showed that Fano manifolds with big alpha-invariant can be equipped with a Kahler-Einstein metric. In recent years Donaldson drafted a programme to determine when a smooth Fano variety X admits…

The paper proves the openness of K-semistability for Fano varieties.

Study on discrete Okounkov bodies and their applications.

Solves Tian's stabilization problem for toric Fano manifolds.

Paper answers Jin and Rubinstein's question about Fano manifolds.

Extends probabilistic approach for Kahler-Einstein metrics on Fano manifolds.

Reverse Hölder inequalities on Fano metrics with applications to geodesics and singularities.

We provide a sufficient condition for polarisations of Fano varieties to be K-stable in terms of Tian's alpha invariant, which uses the log canonical threshold to measure singularities of divisors in the linear system associated to the polarisation. This generalises a result of Odaka-Sano in the anti-canonically polari…

Gaussian latent tree models, or more generally, Gaussian latent forest models have Fisher-information matrices that become singular along interesting submodels, namely, models that correspond to subforests. For these singularities, we compute the real log-canonical thresholds (also known as stochastic complexities or l…

New approach finds analytic interpretation of algebraic invariants for balanced metrics.

The paper analyzes the error in variational Bayesian NMF compared to Bayesian NMF.

New method corrects Laplace/BIC errors in singular models, revealing effective dimension.

Let X be a normal complex projective variety with at worst klt singularities, and L a big line bundle on X. We use valuations to study the log canonical threshold of L, as well as another invariant, the stability threshold. The latter generalizes a notion by Fujita and Odaka, and can be used to characterize when a Q-Fa…

In this paper we study the problem of existence of orbifold Kaehler-Einstein metrics on del Pezzo surfaces of degree 1 with Du Val singular points. Moreover we compute global log canonical thresholds of del Pezzo surfaces of degree 1 with Du Val singularities and of del Pezzo surfaces of Picard rank 1 with Du Val singu…

Study the geometry of matrix multiplication in deep neural networks.

We prove the existence of Kahler-Einstein metrics on a nonsingular section of the Grassmannian $\mathrm{Gr}(2, 5)\subset\mathbb{P}^9$ by a linear subspace of codimension 3, and the Fermat hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$. We also show that a global log canonical threshold of the Mukai--Umemura variet…

Paper shows LDA and SMF have similar generalization errors.

We compute global log canonical thresholds of a large class of quasismooth well-formed del Pezzo weighted hypersurfaces in $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$. As a corollary we obtain the existence of orbifold Kähler--Einstein metrics on many of them, and classify exceptional and weakly exceptional quasismooth well-…

Paper proves integrability and entropy compactness for Kähler potentials with uniform log-log threshold.

In this article we construct a canonical Kähler-Einstein current on a LC (log canonical) pairs of log general type as the limit of a sequence of canonical Kähler-Einstein currents on KLT(Kawamata log terminal) pairs of log general type. We call the volume form associated with the canonical Kähler-Einstein current the c…

Motivated by the study of Fano type varieties we define a new class of log pairs that we call asymptotically log Fano varieties and strongly asymptotically log Fano varieties. We study their properties in dimension two under an additional assumption of log smoothness, and give a complete classification of two dimension…

Sparse reconstruction approaches using the re-weighted l1-penalty have been shown, both empirically and theoretically, to provide a significant improvement in recovering sparse signals in comparison to the l1-relaxation. However, numerical optimization of such penalties involves solving problems with l1-norms in the ob…

We consider degenerations of complex projective Calabi--Yau varieties and study the singularities of $L^2$, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibers are shown to be related to well-known topological invarian…

Counterexample disproves log canonical Beauville--Bogomolov decomposition.

Establishes K"ahler-Ricci flow on log canonical varieties.

Constructs Kahler-Einstein metrics near isolated log canonical singularities.

Proves finitely generated associated graded rings for valuations on log Fano pairs.

Thresholded Lasso bandit minimizes regret in sparse linear bandits.

Critical volatility triggers log-normal to power-law transitions in interconnected systems.

Introduces new stability concept for Fano fibrations.

Non-negative matrix factorization (NMF) is a new knowledge discovery method that is used for text mining, signal processing, bioinformatics, and consumer analysis. However, its basic property as a learning machine is not yet clarified, as it is not a regular statistical model, resulting that theoretical optimization me…

Paper develops DLTF to learn optimized dictionaries for efficient thresholded feature recovery.

Algorithm aligns correlated Erdős-Rényi graphs efficiently.

The paper studies fundamental groups of compact Kahler varieties with nef anti-canonical bundles.