arXiv research

Mirror flows converge to a limiting flow with a convex potential.

Derives Mirror Descent from gradient flow on a Riemannian manifold.

Mirror flow optimizes separable data problems, converging to a maximum margin classifier.

Mirror flow in shallow neural networks shows similar implicit bias to gradient flow, with key differences in curvature penalties.

New method improves generative modeling on convex domains using regularized mirror maps and Student-t priors.

The Sinkhorn flow converges to a Wasserstein mirror gradient flow from the Sinkhorn algorithm.

In this article we construct Lagrangian torus fibrations for general quintic \cy hypersurfaces near the large complex limit and their mirror manifolds using gradient flow method. Then we prove the Strominger-Yau-Zaslow mirror conjecture for this class of \cy manifolds in symplectic category.

Continuous-time Sinkhorn flow generalizes and unifies existing dynamics.

In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in t…

Mirror descent algorithm recovers low-rank matrices in matrix sensing.

Continuous-time mirror descent solves sparse phase retrieval efficiently.

Defines a volume functional for Hermitian connections on manifolds, proving its properties and connections.

Stochastic mirror descent improves performance on ensemble models.

Mirror Langevin Algorithm converges with zero bias.

Proposes a new algorithm for robust learning in Schrödinger bridge problems.

The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. The hypersurface case has been much studied since the eighties. Recently, several theorems on regularity, global existence and convergence of the flow in various ambient spaces and codimensio…

Gradient methods work well on overparameterized diagonal linear networks.

In this paper we consider a Ricci de Turck flow of spaces with isolated conical singularities, which preserves the conical structure along the flow. We establish that a given initial regularity of Ricci curvature is preserved along the flow. Moreover under additional assumptions, positivity of scalar curvature is prese…

We consider the following problem: given two parallel and identically oriented bundles of light rays in n-dimensional Euclidean space and given a diffeomorphism between the rays of the former bundle and the rays of the latter one, is it possible to realize this diffeomorphism by means of several mirror reflections? We …

Symmetric graphs flow without singularities on their axis.

New mirror maps improve PMD performance in reinforcement learning.

We consider the so-called inverse $F$-curvature flow (IFCF) $\dot x = -F^{-1}ν$ in ARW spaces, i.e. in Lorentzian manifolds with a special future singularity. Here, $F$ denotes a curvature function of class $(K^*)$, which is homogenous of degree one, e.g. the $n$-th root of the Gaussian curvature, and $ν$ the past dire…

Develops a gradient flow for Muon optimizer, a method for optimization.

The paper examines properties of deformed Donaldson-Thomas connections on G2-manifolds.

We present a unified method, based on convex optimization, for managing the power produced and consumed by a network of devices over time. We start with the simple setting of optimizing power flows in a static network, and then proceed to the case of optimizing dynamic power flows, i.e., power flows that change with ti…

We describe mirror symmetry on higher dimensional tori, paying special attention to the behaviour of D-branes under mirror symmetry. To find the mirror D-branes the description of mirror symmetry on D-branes due to Ooguri, Oz en Yin is used. This method allows us to deal with the coisotropic D-branes recently introduce…

Motivated by Strominger-Yau-Zaslow's mirror symmetry proposal and Kontsevich's homological mirror symmetry conjecture, we study mirror phenomena (in A-model) of certain results from Donaldson-Thomas theory for Calabi-Yau 4-folds.

Study homological mirror symmetry for Hirzebruch surfaces using Morse homotopy.

This paper deforms complex tori and their mirrors using gerbes.

Constructs mirror pairs for solvmanifolds using Lie groups.

We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups $G_2$ and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to twisted connected sum $G_2$ manifolds, mirrors of such Spin(7) manifolds can be fou…

Homological mirror symmetry for toric Fano surfaces using Morse homotopy.

New analysis shows GMD can converge linearly under PL-like conditions.

In this article we explore some finer properties of equi-areal mirrors and introduce techniques for developing new mirror surfaces that simultaneously minimize angular and areal distortion.

Study connects mirror symmetry invariants to K-stability for toric manifolds.

This paper focuses on a topological version on the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, the SYZ conjecture suggests that mirror pairs of Calabi-Yau manifolds are related by the existence of dual special Lagrangian torus fibrations. We explore this conjecture without reference to the special La…

We explore the tan-concavity of the Lagrangian phase operator for the study of the deformed Hermitian Yang-Mills (dHYM) metrics. This new property compensates for the lack of concavity of the Lagrangian phase operator as long as the metric is almost calibrated. As an application, we introduce the tangent Lagrangian pha…

NGMs create mirrored features to assess neural network feature importance.

We revisit our construction of mirror symmetries for compactifications of Type II superstrings on twisted connected sum $G_2$ manifolds. For a given $G_2$ manifold, we discuss evidence for the existence of mirror symmetries of two kinds: one is an autoequivalence for a given Type II superstring on a mirror pair of $G_2…

Find first (0,2) mirror symmetry examples on Hopf surfaces.

The paper discusses a solution to homological mirror symmetry for complex tori, especially when the matrix is singular.

Inspired by the paper on quantum knots and knot mosaics [23] and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology [2,3,7,24], we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. Tame knot theory is equivalent to knot…

New algorithm reduces optimization complexity in adaptive mirror descent.

New algorithm improves sampling from constrained spaces.

Develops parameter-free online mirror descent for optimal dynamic regret.

We discuss mirror symmetry in generalized Calabi-Yau compactifications of type II string theories with background NS fluxes. Starting from type IIB compactified on Calabi-Yau threefolds with NS three-form flux we show that the mirror type IIA theory arises from a purely geometrical compactification on a different class…

In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect the geometry of the Calabi-Yau itself like a mirror. Strominger, Yau and Zaslow c…

Researchers match complex affine structures in mirror constructions.