arXiv research

New method tackles over-parameterized matrix sensing with FGD, improving statistical and computational complexity.

Descending phase retrieval algorithms show a phase transition with increasing sample complexity.

New algorithms handle phase retrieval with rank d measurements, revealing phase transitions.

We review recent works on analyzing the dynamics of gradient-based algorithms in a prototypical statistical inference problem. Using methods and insights from the physics of glassy systems, these works showed how to understand quantitatively and qualitatively the performance of gradient-based algorithms. Here we review…

In this paper, we introduce a new type of relation between knots called the descendant relation. One knot $H$ is a descendant of another knot $K$ if $H$ can be obtained from a minimal crossing diagram of $K$ by some number of crossing changes. We explore properties of the descendant relation and study how certain knots…

The paper characterizes and examines nilpotent complex structures on stratified Lie algebras.

In the classical knot theory there is a well-known notion of descending diagram. From an arbitrary diagram one can easily obtain, by some crossing changes, a descending diagram which is a diagram of the unknot or unlink. In this paper the notion of descending diagram for knots and links in the real space is extended to…

The study explores $S^1$-invariant Laplacian flow on 6-manifolds.

K-means is a classical clustering algorithm with wide applications. However, soft K-means, or fuzzy c-means at m=1, remains unsolved since 1981. To address this challenging open problem, we propose a novel clustering model, i.e. Probabilistic K-Means (PKM), which is also a nonlinear programming model constrained on lin…

Given a Kaehler group $G$ and a primitive class $φ\in H^1(G;Z)$, we show that the rank gradient of $(G;φ)$ is zero if and only if Ker $φ$ is finitely generated. Using this approach, we give a quick proof of the fact (originally due to Napier and Ramachandran) that Kaehler groups are not properly ascending or descending…

Suppose $TM\setminus \{0\}$ and $T\widetilde M\setminus\{0\}$ are slashed tangent bundles of two smooth manifolds $M$ and $\widetilde M$, respectively. In this paper we characterize those diffeomorphisms $F\colon TM\setminus\{0\} \to T\widetilde M\setminus\{0\}$ that can be written as $F = (Dφ)|_{TM\setminus\{0\}}$ for…

The Regularized Nonlinear Acceleration (RNA) algorithm is an acceleration method capable of improving the rate of convergence of many optimization schemes such as gradient descend, SAGA or SVRG. Until now, its analysis is limited to convex problems, but empirical observations shows that RNA may be extended to wider set…

New algorithms improve Langevin Monte Carlo efficiency.

This is the second part in a series of two papers. The $k$-Dirac complex is a complex of differential operators which are natural to a particular $|2|$-graded parabolic geometry. In this paper we will consider the $k$-Dirac complex over a homogeneous space of the parabolic geometry and as a first result, we will prove …

Holomorphic quantum modular forms linked to knot volumes.

The paper studies unknotting operations and numbers for plus-welded knotoids.

The purpose of this note is introduce a new axiom (called the Descent Axiom) in the theory of $r$-spin cohomological field theories. This axiom explains the origin of gravitational descendants in this theory. Furthermore, the Descent Axiom immediately implies the Vanishing Axiom, explicating the latter (which has no a …

We propose a new algorithm that uses an auxiliary neural network to express the potential of the optimal transport map between two data distributions. In the sequel, we use the aforementioned map to train generative networks. Unlike WGANs, where the Euclidean distance is ${\it implicitly}$ used, this new method allows …

Decision trees perform well in complex interactions, even when interactions are not fully accounted for.

MSTGD optimizes gradient descent with stratified sampling for faster convergence.

We introduce a new numerical invariant of knots and links from the descending diagrams. It is considered to live between the unknotting number and the bridge number.

Gradient descent proves global convergence for 4-layer matrix factorization.

We study the existence of $S^1$-equivariant characteristic classes on certain natural infinite rank bundles over the loop space $LM$ of a manifold $M$. We discuss the different $S^1$-equivariant cohomology theories in the literature and clarify their relationships. We attempt to use $S^1$-equivariant Chern-Weil techniq…

Poly-free groups are constructed as iterated semidirect products of free groups. The class of poly-free groups includes the classical pure braid groups, fundamental groups of fiber-type hyperplane arrangements, and certain subgroups of the automorphism groups of free groups. The purpose of this article is to compute ce…

Study of 3d-3d correspondence involving $q$-Weyl algebra and 3d-index.

Phylogenetic tree inference using deep DNA sequencing is reshaping our understanding of rapidly evolving systems, such as the within-host battle between viruses and the immune system. Densely sampled phylogenetic trees can contain special features, including "sampled ancestors" in which we sequence a genotype along wit…

This paper investigates factors influencing SGD minima.

We study geodesic equations for a family of right-invariant Riemannian metrics on the group of diffeomorphisms of a compact manifold. The metrics descend to Fisher's information metric on the space of smooth probability densities. The right reduced geodesic equations are higher-dimensional generalisations of the $μ$--H…

Gradient descent in tensor factorization favors low-rank solutions.

The general perception is that kernel methods are not scalable, and neural nets are the methods of choice for nonlinear learning problems. Or have we simply not tried hard enough for kernel methods? Here we propose an approach that scales up kernel methods using a novel concept called "doubly stochastic functional grad…

We investigate Friedl-Lück's universal $L^2$-torsion for descending HNN extensions of finitely generated free groups, and so in particular for $F_n$-by-$\mathbb{Z}$ groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for $3$-manifold groups. We prove…

The concept of a C-class of differential equations goes back to E. Cartan with the upshot that generic equations in a C-class can be solved without integration. While Cartan's definition was in terms of differential invariants being first integrals, all results exhibiting C-classes that we are aware of are based on the…

FACMAC combines deep policy gradients with factored critic for multi-agent reinforcement learning.

Gradient descent with large steps leads to chaotic parameter space and unpredictable outcomes.

A new approach to maximum likelihood learning of discrete graphical models and RBM in particular is introduced. Our method, Perturb and Descend (PD) is inspired by two ideas (I) perturb and MAP method for sampling (II) learning by Contrastive Divergence minimization. In contrast to perturb and MAP, PD leverages trainin…

Robot untangles knots by walking and switching crossings.

We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix $X$ with gradient descent on a factorization of $X$. We conjecture and provide empirical and theoretical evidence that with small enough step sizes and initialization close enough to the origin, gradient descent on a f…

GD with large init shows incremental learning in matrix factorization.

A flow from hypersymplectic to hyperkähler structures is described.

PrecGD restores linear convergence in over-parameterized nonconvex matrix factorization.

AGD converges in polynomial iterations to optimal matrix factorization.

Gradient descent with noise converges to a unique optimum in nonconvex matrix factorization.

The paper examines how gradient descent stabilizes low-rank matrix factorization in noisy conditions.

Gradient flow with infinitesimal initialization converges to Greedy Low-Rank Learning for matrix factorization.

We study the projected gradient descent method on low-rank matrix problems with a strongly convex objective. We use the Burer-Monteiro factorization approach to implicitly enforce low-rankness; such factorization introduces non-convexity in the objective. We focus on constraint sets that include both positive semi-defi…

Study on descent properties of complex affine surfaces under proper morphisms.

The symplectic vortex equations admit a variational description as global minimum of the Yang-Mills-Higgs functional. We study its negative gradient flow on holomorphic pairs $(A,u)$ where $A$ is a connection on a principal $G$-bundle $P$ over a closed Riemann surface $Σ$ and $u: P \rightarrow X$ is an equivariant map …

New method for selective prediction under interventions learns causal structure from data.