arXiv research

Polynomial time algorithm matches correlated Gaussian matrices without vanishing correlation.

We study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot $K$ satisfies the Slope Conjecture then a $(p, q)$-cable of $K$ satisfies the conjecture, provided that $p/q$ is not a Jon…

The paper associates knots to numerical semigroups and shows their Alexander polynomials coincide with semigroups' Poincaré series.

Polynomial growth elements found in all subgroups of Out(F_n).

The paper develops AMP theory for sparse and robust regression with polynomial iterations.

Paper refutes conjecture on tensor power iteration convergence in overcomplete models.

We extend the construction of the DAHA-Jones polynomials for any reduced root systems and DAHA-superpolynomials in type A from the iterated torus knots (our previous paper) to links, including arbitrary algebraic links. Such a passage essentially corresponds to the usage of the products of Macdonald polynomials and is …

Abstract: Deltoid map connects complex dynamics and algebra.

Kernel methods can learn hierarchical polynomials efficiently.

PER-ETD improves ETD by reducing variance to polynomial complexity.

Simple gradient descent algorithm escapes saddle points efficiently.

Minimax optimal convergence rates for classes of stochastic convex optimization problems are well characterized, where the majority of results utilize iterate averaged stochastic gradient descent (SGD) with polynomially decaying step sizes. In contrast, SGD's final iterate behavior has received much less attention desp…

A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern--Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers.…

Polynomial-time RL algorithm for constant actions under linear Bellman completeness.

Study on periodic knots, proving limitations on their Alexander polynomials.

We calculate the twisted Reidemeister torsion of the complement of an iterated torus knot associated with a representation of its fundamental group to the complex special linear group of degree two. We also show that the twisted Reidemeister torsions associated with various representations appear in the asymptotic expa…

We say that a given knot $J\subset S^3$ is detected by its knot Floer homology and $A$-polynomial if whenever a knot $K\subset S^3$ has the same knot Floer homology and the same $A$-polynomial as $J$, then $K=J$. In this paper we show that every torus knot $T(p,q)$ is detected by its knot Floer homology and $A$-polynom…

Given a fibered link, consider the characteristic polynomial of the monodromy restricted to first homology. This generalizes the notion of the Alexander polynomial of a knot. We define a construction, called iterated plumbing, to create a sequence of fibered links from a given one. The resulting sequence of characteris…

In this paper, we study the online learning algorithm without explicit regularization terms. This algorithm is essentially a stochastic gradient descent scheme in a reproducing kernel Hilbert space (RKHS). The polynomially decaying step size in each iteration can play a role of regularization to ensure the generalizati…

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

New method improves DAG learning by using large coefficients for higher-order terms.

Polyak step size GD reaches final radius of convergence after log iterations.

We give a simple algorithm that determines whether a given post-critically finite topological polynomial is Thurston equivalent to a polynomial. If it is, the algorithm produces the Hubbard tree; otherwise, the algorithm produces the canonical obstruction. Our approach is rooted in geometric group theory, using iterati…

Researchers derived Kauffman bracket polynomial for Celtic link shadows using two methods.

We show how to efficiently project a vector onto the top principal components of a matrix, without explicitly computing these components. Specifically, we introduce an iterative algorithm that provably computes the projection using few calls to any black-box routine for ridge regression. By avoiding explicit principal …

Sharp analysis of power iteration for tensor PCA, improving convergence and stopping criteria.

Polynomial-time algorithm matches correlated random graphs with non-vanishing correlation.

Two new algorithms improve robust PCA and Schatten packing.

Cochran defined the nth-order integral Alexander module of a knot in the three sphere as the first homology group of the knot's (n+1)th-iterated abelian cover. The case n=0 gives the classical Alexander module (and polynomial). After a localization, one can get a finitely presented module over a principal ideal domain,…

A well-known issue of Batch Normalization is its significantly reduced effectiveness in the case of small mini-batch sizes. When a mini-batch contains few examples, the statistics upon which the normalization is defined cannot be reliably estimated from it during a training iteration. To address this problem, we presen…

Polynomial-time algorithm learns ReLU networks without assumptions.

In this paper we consider a problem of searching a space of predictive models for a given training data set. We propose an iterative procedure for deriving a sequence of improving models and a corresponding sequence of sets of non-linear features on the original input space. After a finite number of iterations N, the n…

Last SGD iterate bounds for overparameterized linear regression.

The AdaBoost algorithm was designed to combine many "weak" hypotheses that perform slightly better than random guessing into a "strong" hypothesis that has very low error. We study the rate at which AdaBoost iteratively converges to the minimum of the "exponential loss." Unlike previous work, our proofs do not require …

The paper generalizes polynomial functions on Lie groups and their properties.

Adversarial training is a technique for training robust machine learning models. To encourage robustness, it iteratively computes adversarial examples for the model, and then re-trains on these examples via some update rule. This work analyzes the performance of adversarial training on linearly separable data, and prov…

This paper proposes low-complexity algorithms for finding approximate second-order stationary points (SOSPs) of problems with smooth non-convex objective and linear constraints. While finding (approximate) SOSPs is computationally intractable, we first show that generic instances of the problem can be solved efficientl…

Paper provides tail bounds for stochastic mirror descent in heavy-tailed noise.

We describe an iterative construction of Lagrangian tori in the complex Grassmannian $\operatorname{Gr}(k,n)$, based on the cluster algebra structure of the coordinate ring of a mirror Landau-Ginzburg model proposed by Marsh-Rietsch. Each torus comes with a Laurent polynomial, and local systems controlled by the $k$-va…

Machine learning improves Buchberger's algorithm for polynomial systems.

The basin of infinity of a polynomial map $f : {\bf C} \arrow {\bf C}$ carries a natural foliation and a flat metric with singularities, making it into a metrized Riemann surface $X(f)$. As $f$ diverges in the moduli space of polynomials, the surface $X(f)$ collapses along its foliation to yield a metrized simplicial t…

New algorithm improves gradient-based ERM for smooth convex losses.

Bayesian method improves online NARMAX model identification.

Warm starts improve variational quantum algorithms by avoiding barren plateaus.

In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where 'rigid' means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown …

We solve principal component regression (PCR), up to a multiplicative accuracy $1+γ$, by reducing the problem to $\tilde{O}(γ^{-1})$ black-box calls of ridge regression. Therefore, our algorithm does not require any explicit construction of the top principal components, and is suitable for large-scale PCR instances. In…

Estimates hybrid dynamical systems with polynomial expansions and Markovian switching.

Fast algorithm recovers principal eigenvector from noisy matrices.