arXiv research

Paper proves first non-trivial PTF testing lower bounds for NGCA.

New lower bound for knot genus using Links-Gould invariant.

The study optimizes polynomial regression for learning under Gaussian distributions.

New method uses almost orthonormal bases to prove low-degree lower bounds in complex statistical models.

Generalizes positivity conjecture to Roger--Yang skein algebras using polynomials.

In this paper, we investigate the Dirchlet eigenvalue problems of poly-Laplacian with any order and quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first $k$ eigenvalues which improve the previous results.

Paper conjectures Links-Gould invariant generalizes Alexander polynomial.

Recently twisted and higher order Alexander polynomials were used by Cochran, Harvey, Friedl--Kim and Turaev to give lower bounds on the Thurston norm. We first show how Reidemeister torsion relates to these Alexander polynomials. We then give lower bounds on the Thurston norm in terms of the Reidemeister torsion which…

In this paper we provide a lower bound for the long time on-diagonal heat kernel of minimal submanifolds in a Cartan-hadamard ambient manifold assuming that the submanifold is of polynomial volume growth. In particular cases, that lower bound is related with the number of ends of the submanifold.

Lower bounds on MALA and HMC for well-conditioned distributions.

We prove a new lower bound for the dilatation of an arbitrary pseudo-Anosov map on a surface of genus g with n punctures. Our bound improves the former super-exponential dependence on the genus by a polynomial dependence.

New SQ lower bound shows complexity nearly matches known upper bound for smoothed agnostic learning.

We compute lower bounds on the virtual crossing number and minimal surface genus of virtual knot diagrams from the arrow polynomial. In particular, we focus on several interesting examples.

Improved lower bound for knot coloring using quandles.

New computational lower bounds for clustering and related problems.

New bounds found for ribbon numbers of knots and links.

Every element in the first cohomology group of a 3--manifold is dual to embedded surfaces. The Thurston norm measures the minimal `complexity' of such surfaces. For instance the Thurston norm of a knot complement determines the genus of the knot in the 3--sphere. We show that the degrees of twisted Alexander polynomial…

This paper establishes strong lower bounds for learning in revealing POMDPs.

New bounds on virtual link genus using quantum supergroups.

The paper improves bounds on the complexity of computing link polynomials.

In this note we give a new lower bound on the virtual crossing number via the writhe polynomial, which refines a result of B. Mellor. The proof is based on a new interpretation of the writhe polynomial. The characterization of the writhe polynomial is also discussed.

We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus.

New algorithm for learning ReLU networks with Gaussian noise, improving previous results.

We give a lower bound on the number of non-simple closed curves on a hyperbolic surface, given upper bounds on both length and self-intersection number. In particular, we carefully show how to construct closed geodesics on pairs of pants, and give a lower bound on the number of curves in this case. The lower bound for …

This paper investigates symmetric ribbon numbers of low-complexity knots.

Algorithm distinguishes Gaussian mixtures from pure Gaussians in quasi-polynomial time.

We construct near-optimal coresets for kernel density estimates for points in $\mathbb{R}^d$ when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size $O(\sqrt{d}/\varepsilon\cdot \sqrt{\log 1/\varepsilon} )$, and we show a near-matching lower bound of size $Ω(\min\…

Factor graphs are important models for succinctly representing probability distributions in machine learning, coding theory, and statistical physics. Several computational problems, such as computing marginals and partition functions, arise naturally when working with factor graphs. Belief propagation is a widely deplo…

Gradient descent fails to learn simple neural networks efficiently.

Lower bound shows super-polynomial gap for estimating truncated Gaussian means.

Survey on using low-degree polynomials to assess statistical tasks complexity.

Twisted graph diagrams are virtual graph diagrams with bars on edges. A bijection between abstract graph diagrams and twisted graph diagrams is constructed. Then a polynomial invariant of Yamada-type is developed which provides a lower bound for the virtual crossing number of virtual graph diagrams.

New lower bounds show learning intersections of halfspaces is hard even for a few halfspaces.

New bounds show complex neural networks need many queries to learn.

GP-UCB performs suboptimally under certain conditions, as shown by a new regret lower bound.

We show that the if a sequence of normalized polynomials gives rise to a positive basis of the skein algebra of a surface, then it is sandwiched between the two types of Chebyshev polynomials. For the closed torus, we show that the normalized sequence of Chebyshev polynomials of type one $(\hat{T}_n)$ is the only one w…

Investigates polynomial time algorithms for computing Khovanov homology of braids.

We study the complexity of training neural network models with one hidden nonlinear activation layer and an output weighted sum layer. We analyze Gradient Descent applied to learning a bounded target function on $n$ real-valued inputs. We give an agnostic learning guarantee for GD: starting from a randomly initialized …

The paper improves bounds on skein tree depth and delta-crossing numbers for knots and links.

We study the question of whether parallelization in the exploration of the feasible set can be used to speed up convex optimization, in the local oracle model of computation. We show that the answer is negative for both deterministic and randomized algorithms applied to essentially any of the interesting geometries and…

New evidence shows computational barriers in graphon estimation using low-degree polynomials.

In this paper we study the approximate learnability of valuations commonly used throughout economics and game theory for the quantitative encoding of agent preferences. We provide upper and lower bounds regarding the learnability of important subclasses of valuation functions that express no-complementarities. Our main…

New lower bounds show challenges in clustering in moderate dimensions.

In the early 2000's Cochran and Harvey introduced non-commutative Alexander polynomials for 3-manifolds. Their degrees give strong lower bounds on the Thurston norm. In this paper we make the case that the vanishing of a certain Novikov-Sikorav homology module is the correct notion of a monic non-commutative Alexander …

New SQ lower bounds show learning mixtures of bounded covariance Gaussians is hard.

New formulas for knot polynomial evaluations from covering spaces.

New lower bounds for private covariance estimation of Gaussian distributions are proven.

Polynomial-time algorithm finds planted hypercube vectors in Gaussian mixtures.