arXiv research

Flow Matching models help generative models stay within the subspace of real data.

The paper studies ergodicity of flows on subspaces, generalizing earlier work.

We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restriction…

We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to a Perelman's modified backward Ricci type flow with some special restriction conditions. The restriction conditions are motivated by c…

A new DDR framework learns low-dimensional data representations using dynamical systems.

A new method for generative modeling of discrete data using geometric latent subspaces.

LOL method simplifies forming linear combinations of latent variables.

We prove a spectral flow formula for one-parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, we deduce sufficient conditions fo…

For an ancient solution of the mean curvature flow, we show that each time slice M_t is contained in an affine subspace with dimension bounded in terms of the density and the dimension of the evolving submanifold. Recall that an ancient solution is a family M_t that evolves under mean curvature flow for all negative ti…

We show that all closed $2$-dimensional singularities for higher codimension mean curvature flow that cannot be perturbed away have uniform entropy bounds and lie in a linear subspace of small dimension. The entropy and dimension of the subspace are both $\leq C\,(1+γ)$ for some universal constant $C$ and genus $γ$. Th…

It is shown that an equivariant Lagrangian sphere with a positivity condition on its Ricci curvature develops a type-II singularity under the Lagrangian mean curvature flow that rescales to the product of a grim reaper with a flat Lagrangian subspace. In particular this result applies to the Whitney spheres.

This paper addresses network anomography, that is, the problem of inferring network-level anomalies from indirect link measurements. This problem is cast as a low-rank subspace tracking problem for normal flows under incomplete observations, and an outlier detection problem for abnormal flows. Since traffic data is lar…

Dockless bike sharing systems need effective bike flow prediction models.

Study vector fields and flows on singular spaces like submanifolds.

Gradient flow solves multi-index regression for high-dimensional Gaussian data.

We consider a continuous curve of self-adjoint Fredholm extensions of a curve of closed symmetric operators with fixed minimal domain $D_m$ and fixed {\it intermediate} domain $D_W$. Our main example is a family of symmetric generalized operators of Dirac type on a compact manifold with boundary with varying well-posed…

We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities and present evidence in favor of a conjecture that parabolic rescalings at the singularities converge to singularity models that are shrinking Kahler-Ricci solitons. Specifically, the singularity model for these so…

We first bound the codimension of an ancient mean curvature flow by the entropy. As a consequence, all blowups lie in a Euclidean subspace whose dimension is bounded by the entropy and dimension of the evolving submanifolds. This drastically reduces the complexity of the system. Combined with \cite{CM12}, this gives th…

A new method splits surface flow discretizations into streamfunctions and harmonic fields.

Ancient solutions and translators identified for Lagrangian flow.

In this paper, we study Conley theory in Hilbert spaces and make some refinement of the construction of the stable Conley index developed by Gȩba, Izydorek, and Pruszko. For instance, we allow subspaces other than invariant subspaces in the the construction. As a main result, we show that the resulting stable Conley in…

Detect anomalies in complex networks using topological subspace detectors.

First, we prove a local spectral flow formula (Theorem 3.7) for a differentiable curve of selfadjoint Fredholm operators. This formula enables us to prove in a simple way a general spectral flow formula. Secondly, we prove a splitting formula (Theorem 4.12) for the spectral flow of a curve of selfadjoint elliptic opera…

We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying (weak) symplectic structures. Assuming vanishing index, we obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we defi…

New algorithms extract Koopman invariant subspaces from large-scale data.

We propose a framework for solving high-dimensional Bayesian inference problems using \emph{structure-exploiting} low-dimensional transport maps or flows. These maps are confined to a low-dimensional subspace (hence, lazy), and the subspace is identified by minimizing an upper bound on the Kullback--Leibler divergence …

The intent of this short note is to provide context for and an independent proof of the discovery of Klaus Kroencke that complex projective space with its canonical Fubini--Study metric is dynamically unstable under Ricci flow in all complex dimensions N>1. The unstable perturbation is not Kaehler. This provides a coun…

Analyzes word2vec-like models revealing linear subspaces learned during training.

I construct an algebraic model for a typical fiber on a 1+1 dimensional spacetime. The vector space comprising the fiber is composed of elements formed from the direct product of two copies of an element x in the D2=C2xC2 finite group algebra over the real numbers. The fiber contains subspaces whose elements are associ…

Let $M=Σ_1\times Σ_2$ be the product of two compact Riemannian manifolds of dimension $n\geq 2$ and two, respectively. Let $Σ$ be the graph of a smooth map $f:Σ_1\mapsto Σ_2$, then $Σ$ is an $n$-dimensional submanifold of $M$. Let ${\frak G}$ be the Grassmannian bundle over $M$ whose fiber at each point is the set of …

The paper studies mean curvature flow of spacelike-convex submanifolds in pseudo-Euclidean space.

Let $F:Σ^n \times [0,T)\to \R^{n+m}$ be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps $γ:(Σ^n, g_t)\to G(n, m)$ form a harmonic heat flow with respect to the time-dependent induced metric $g_t$. This provides a more systematic approach to investigating higher c…

In the paper we introduce new metric structures on $\mathfrak{g}$-foliations that are less rigid than the well-known structures: almost contact and 3-quasi-Sasakian structures as well as $f$-structures with parallelizable kernel and almost para-$φ$-structures with complemented frames. We discuss the properties of the n…

Constructs a Morse-Bott function on symplectic Grassmannians.

The group of real 4 by 4 upper triangular matrices with 1s on the diagonal has a left-invariant subRiemannian (or Carnot-Caratheodory) structure whose underlying distribution corresponds to the superdiagonal. We prove that the associated subRiemannian geodesic flow is not completely integrable. This provides the first …

The conservation laws of the third order quasilinear scalar evolution equations are considered via differential system and characteristic cohomology. We find a subspace of 2 forms in the infinite prolonged space in which every conservation law has a unique representative. The structure of this subspace naturally gives …

A new method uses Gaussian Processes to solve power flow problems with uncertain renewable and load inputs.

We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset H(q,n) of the variety of (q+n)-dimensional Lie algebras, parameterizing the space of all simply connected homogeneous spaces of dimension n with a q-dimensional isotropy,…

Study shows mixing of flows on specific geometric spaces.

A hypersymplectic structure on a 4-manifold $X$ is a triple $\underlineω$ of symplectic forms which at every point span a maximal positive-definite subspace of $Λ^2$ for the wedge product. This article is motivated by a conjecture of Donaldson: when $X$ is compact $\underlineω$ can be deformed through cohomologous hype…

Let Σbe a minimal submanifold of \R^{n+m} that can be represented as the graph of a smooth map f:\R^n-->\R^m. We apply a formula we derived in the study of mean curvature flow to obtain conditions under which Σmust be an affine subspace. Our result covers all known ones in the general case. The conditions are stated in…

Diffusion models' sampling paths lie in a low-dimensional subspace, resembling boomerangs.

Reinterprets DNF as a deep generative LDA model for complex data.

Classifies big mapping classes on infinite type surfaces.

We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic $n$-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve…

Study self-expanding solutions of mean curvature flow in various dimensions.

Paper bounds subspace estimator error from noisy projections.

PCA is one of the most widely used dimension reduction techniques. A related easier problem is "subspace learning" or "subspace estimation". Given relatively clean data, both are easily solved via singular value decomposition (SVD). The problem of subspace learning or PCA in the presence of outliers is called robust su…